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A total of 88 sentences were analysed. As a result 71 sentences (80,68%) were found in other documents.

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The following list contains titles and addresses of documents in which similar sentences were found. With a click on the number of found sentences („x Sentences”) the corresponding sentences are highlighted in the document as well as in the navigation bar by a colored border Another click on „x Sentences” resets the highlighting. By clicking on ✓ each source document can be hidden in the report so that they are not counted as similarities. The same applies to found sentences which may be marked as valid citations. A second click on the symbold reverts this process again.

46 Sentences were found in a text with the title: „PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS”, located at:✓
https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf

20 Sentences were found in a text with the title: „Periodic boundary value problems for first-order impulsive difference equations with time delay”, located at:✓
https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc

18 Sentences were found in a text with the title: „Periodic boundary value problems for first-order impulsive ...”, located at:✓
https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-018-1539-5
https://link.springer.com/article/10.1186/s13662-018-1539-5
https://www.semanticscholar.org/paper/Periodic-boundary-value-problems-for-first-order-Tian-Wang/90f52c39b391149bd4d047e3711d6fe2e25e51e8
https://www.researchgate.net/journal/Advances-in-Difference-Equations-1687-1847/publication/323559736_Periodic_boundary_value_problems_for_first-order_impulsive_difference_equations_with_time_delay/links/5fc46bc5299bf104cf941a9d/Periodic-boundary-value-problems-for-first-order-impulsive-difference-equations-with-time-delay.pdf

8 Sentences were found in a text with the title: „Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf”, located at:✓
http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf

8 Sentences were found in a text with the title: „excerpt.pdf”, located at:✓
https://downloads.hindawi.com/books/978977594550/excerpt.pdf

5 Sentences were found in a text with the title: „Download PDF - Advances in Difference Equations”, located at:✓
https://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/s13662-018-1539-5.pdf

4 Sentences were found in a text with the title: „Top PDF Periodic boundary value problems for nonlinear first ...”, located at:✓
https://1library.net/title/periodic-boundary-problems-nonlinear-impulsive-dynamic-equations-scales

4 Sentences were found in a text with the title: „Top PDF Some properties of algebraic difference equations of ...”, located at:✓
https://1library.net/title/properties-algebraic-difference-equations-order

4 Sentences were found in a text with the title: „Periodic boundary value problems for first-order ... - X-MOL”, located at:✓
https://www.x-mol.com/paper/1335612593800704128?recommendPaper=1277746314451505152

4 Sentences were found in a text with the title: „Top PDF first-order difference operators - 1Library”, located at:✓
https://1library.net/subject/first-order-difference-operators

3 Sentences were found in a text with the title: „Periodic boundary value problem for the first order functional ...”, located at:✓
https://www.researchgate.net/publication/225911046_Periodic_boundary_value_problem_for_the_first_order_functional_differential_equations_with_impulses

3 Sentences were found in a text with the title: „Impulsive Differential Equations and Inclusions”, located at:✓
https://books.google.com/books?id=JlFHOT8dkucC

3 Sentences were found in a text with the title: „Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times”, located at:✓
https://core.ac.uk/download/pdf/82658699.pdf
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.563.6840&rep=rep1&type=pdf
http://www.dma.uvigo.es/~eliz/pdf/jmaa96i.pdf

2 Sentences were found in a text with the title: „10.1.1.4.161.pdf - download”, located at:✓
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.4.161&rep=rep1&type=pdf

2 Sentences were found in a text with the title: „Search results ADVANCES IN TIME DELAY SYSTEMS 1ST ...”, located at:✓
https://3i.antikvar-24.ru/1843.html

2 Sentences were found in a text with the title: „Impulsive Differential Equations and Inclusions - StudyLib”, located at:✓
https://studylib.net/doc/18821535/impulsive-differential-equations-and-inclusions

2 Sentences were found in a text with the title: „correlation extremal principle - 专否”, located at:✓
https://zhuanfou.com/data/search?q=correlation extremal principle

2 Sentences were found in a text with the title: „1410.4829”, located at:✓
http://arxiv.org/abs/1410.4829

2 Sentences were found in a text with the title: „effrandom.pdf”, located at:✓
http://www.math.uni-heidelberg.de/logic/merkle/ps/effrandom.pdf

2 Sentences were found in a text with the title: „On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings”, located at:✓
http://www.emis.de/journals/HOA/FPTA/Volume2010/414232.pdf

2 Sentences were found in a text with the title: „414232”, located at:✓
https://fixedpointtheoryandapplications.springeropen.com/track/pdf/10.1155/2010/414232?site=fixedpointtheoryandapplications.springeropen.com

2 Sentences were found in a text with the title: „Habilitationsschrift eingereicht bei der Fakult at f ur Mathematik und ...”, located at:✓
http://math.uni-heidelberg.de/logic/merkle/ps/effrandom.ps

2 Sentences were found in a text with the title: „Algebra & Number Theory vol. 12 (2018), no. 8 - MSP”, located at:✓
https://msp.org/ant/2018/12-8/ant-v12-n8-s.pdf

2 Sentences were found in a text with the title: „PII: 0047-259X(91)90100-G - 0047-259x(91)90100-g”, located at:✓
http://www.sciencedirect.com/science/article/pii/0047-259X(91)90100-G

2 Sentences were found in a text with the title: „aa_lrm_final_version.pdf”, located at:✓
https://web.math.ucsb.edu/~agboola/papers/aa_lrm_final_version.pdf

2 Sentences were found in a text with the title: „2aa_lrm_paper.pdf”, located at:✓
https://web.math.ucsb.edu/~agboola/papers/2aa_lrm_paper.pdf

2 Sentences were found in a text with the title: „414232.pdf - 414232”, located at:✓
https://fixedpointtheoryandapplications.springeropen.com/articles/10.1155/2010/414232

2 Sentences were found in a text with the title: „Introductory Econometrics: Intuition, Proof, and Practice”, located at:✓
https://books.google.com/books?id=niaFEZOkA1IC

2 Sentences were found in a text with the title: „10.1.1.67.8984.pdf - download”, located at:✓
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.67.8984&rep=rep1&type=pdf

2 Sentences were found in a text with the title: „Symmetries in Algebra and Number Theory (SANT): Proceedings ...”, located at:✓
https://books.google.com/books?id=8V6dH0slpnMC

2 Sentences were found in a text with the title: „Multiparameter Processes: An Introduction to Random Fields”, located at:✓
https://books.google.com/books?id=XADpBwAAQBAJ

2 Sentences were found in a text with the title: „Advanced Topics in Relation Algebras: Relation Algebras, ...”, located at:✓
https://books.google.com/books?id=CXMzDwAAQBAJ

2 Sentences were found in a text with the title: „Analysis and Mathematical Physics - Page 137 - Google Books Result”, located at:✓
https://books.google.com/books?id=87nZccF3vMwC

2 Sentences were found in a text with the title: „Periodic boundary value problem for first-order impulsive ...”, located at:✓
https://www.researchgate.net/publication/222728564_Periodic_boundary_value_problem_for_first-order_impulsive_functional_differential_equations
https://www.sciencedirect.com/science/article/pii/S0022247X02001336

2 Sentences were found in a text with the title: „Monotone iterative technique for initial-value problems of ...”, located at:✓
https://www.sciencedirect.com/science/article/pii/S0377042707005328
https://www.sciencedirect.com/science/article/abs/pii/S0377042707005328
https://dl.acm.org/doi/abs/10.1016/j.cam.2007.10.002
https://www.researchgate.net/publication/242981032_Monotone_iterative_technique_for_initial-value_problems_of_nonlinear_singular_discrete_systems

2 Sentences were found in a text with the title: „Impulsive Boundary Value Problems for First-order Ordinary ...”, located at:✓
https://link.springer.com/article/10.1007/s10255-007-0381-4

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17 matches:

Monotone iterative technique for causal differential equations with upper and lower solutions in the reversed order » P, Tian, S, Wu, Y: Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions. Appl. Math. Comput. 203, 266-272 (2008) 8. Wang, P, Zhang, J: Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. J. Comput. Appl. Math. 221, 158-164 (2008) 9. Wang, P, Wang, W: Boundary value problems for first order impulsive difference equations. Int. J. Difference Equ. 1, 249-259 (2006) 10.« - Wenli Wang
^{https://cyberleninka.org/article/n/714256}

81065042.pdf » value conditions. Appl. Math. Comput. 203, 266-272 (2008) http://dx.doi.org/10.1155/2014/165129 Wang and Wang Advances in Difference Equations (2015) 2015:93 Page 13 of 13 17. Wang, P, Zhang, J: Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. J. Comput. Appl. Math. 221, 158-164 (2008) 18. Wang, P, Wang, W: Boundary value problems for first order impulsive difference equations. Int. J. Difference Equ. 1, 249-259 (2006) 19.«
^{https://core.ac.uk/download/pdf/81065042.pdf}

Vol8_Iss5_600--609_Generalized_monotone_iterative_meth.pdf » Lett., 20 (2007), 637–644.1 [15] P. Wang, M. Wu, Y. Wu, Practical stability in terms of two measures for discrete hybrid systems, Nonlinear Anal. Hybrid Syst., 2 (2008) 58–64.1 [16] P. Wang, J. Zhang, Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, J. Comput. Appl. Math., 221 (2008), 158–164.1 [17] P. Wang, W. Wang, Anti-periodic boundary value problem for first order impulsive delay difference equations, Adv. Difference Equ., 2015 (2015), 13 pag«
^{http://www.kurims.kyoto-u.ac.jp/EMIS/journals/TJNSA/includes/files/articles/Vol8_Iss5_600--609_Generalized_monotone_iterative_meth.pdf}

10.22436/jnsa.008.05.12.ris » stability in terms of two measures for discrete hybrid systems AU - P. Wang AU - M. Wu AU - Y. Wu JO - Nonlinear Anal. Hybrid Syst. PY - 2008 DA - 2008// VL - 2 ID - Wang2008 ER - TY - JOUR TI - Monotone iterative technique for initial-value problems of nonlinear singular discrete systems AU - P. Wang AU - J. Zhang JO - J. Comput. Appl. Math. PY - 2008 DA - 2008// VL - 221 ID - Wang2008 ER - TY - JOUR TI - Anti-periodic boundary value problem for first order impulsive delay difference eq«
^{https://www.isr-publications.com/jnsa/articles/1840/references?format=ris&flavour=full}

Monotone iterative technique for initial-value problems of ... »by P Wang · 2008 · Cited by 12 — ... iterative technique for initial-value problems of nonlinear singular discrete systems☆. Author links open overlay panelPeiguangWang Jing ...«
^{https://www.sciencedirect.com/science/article/pii/S0377042707005328}

Monotone iterative technique for initial-value problems of ... »by P Wang · 2008 · Cited by 12 — Monotone iterative technique for initial-value problems of nonlinear singular discrete systems☆. Author links open overlay panelPeiguangWang JingZhang.«
^{https://www.sciencedirect.com/science/article/abs/pii/S0377042707005328}

Monotone iterative technique for initial-value problems of ... »by P Wang · 2008 · Cited by 12 — Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. Share on. Authors: Peiguang Wang. College of Electronic and ...«
^{https://dl.acm.org/doi/abs/10.1016/j.cam.2007.10.002}

Monotone iterative technique for initial-value problems of ... »Oct 27, 2020 — Download Citation | Monotone iterative technique for initial-value problems of nonlinear singular discrete systems | This paper studies a class ...«
^{https://www.researchgate.net/publication/242981032_Monotone_iterative_technique_for_initial-value_problems_of_nonlinear_singular_discrete_systems}

[PDF] Generalized monotone iterative method for integral boundary ... »Monotone iterative technique for initial-value problems of nonlinear singular discrete systems · P. Wang, J. Zhang. Mathematics. 2008. 10. Alert. Research Feed.«
^{https://www.semanticscholar.org/paper/Generalized-monotone-iterative-method-for-integral-Wang-Tian/1dcd275c40e7007f43e27f5ac7976e9babf4a6c6}

Monotone iterative technique for causal differential equations ... »Monotone iterative technique for initial-value problems of nonlinear singular discrete systems · P. Wang, J. Zhang. Mathematics. 2008. 10. Alert. Research Feed.«
^{https://www.semanticscholar.org/paper/Monotone-iterative-technique-for-causal-equations-Wang/76872d4365183e498f7a8c12ebef80b523a45a45}

Download PDF - Boundary Value Problems »by W Wang · 2016 · Cited by 2 — Wang, P, Zhang, J: Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. J. Comput. Appl. Math. 221, 158-164 (2008). 9.«
^{https://boundaryvalueproblems.springeropen.com/track/pdf/10.1186/s13661-016-0649-9.pdf}

1h1xfu7481h1r1.pdf » Iteration Method for Singular Perturbation Initial Value Problems with Delays, Mathematical Problems in Engineering, vol. 2014, Article ID 850343, 8 pages (2014). [14] P. Wanga and J. Zhangb, Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, Journal of Computational and Applied Mathematics, 221, 158–164 (2008). [15] A. Wazwaz, A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems, Applied Mathemat« - ()
^{http://www.naturalspublishing.com/files/published/1h1xfu7481h1r1.pdf}

Periodic boundary value problems for first-order impulsive difference equations with time delay » (2008) 20. Wang, P., Wang, W.: Anti-periodic boundary value problem for first order impulsive delay difference equations. Adv. Differ. Equ. 2015(1), Article ID 93 (2015) 21. Wang, P., Zhang, J.: Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. J. Comput. Appl. Math. 221(1), 158–164 (2008) 22. Zhu, W.: Invariant and attracting sets of impulsive delay difference equations with continuous variables. Comput. Math.« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

192947199.pdf » value conditions. Appl. Math. Comput. 203, 266-272 (2008) http://dx.doi.org/10.1155/2014/165129 Wang and Wang Advances in Difference Equations (2015) 2015:93 Page 13 of 13 17. Wang, P, Zhang, J: Monotone iterative technique for initial-value problems of nonlinear singular discrete systems. J. Comput. Appl. Math. 221, 158-164 (2008) 18. Wang, P, Wang, W: Boundary value problems for first order impulsive difference equations. Int. J. Difference Equ. 1, 249-259 (2006) 19.«
^{https://core.ac.uk/download/pdf/192947199.pdf}

6a10067f9e179790589bb3f285b7bc1b7696.pdf » (2009), 525–529. 1 [13] P. G. Wang, T. T. Kong, Rapid convergence of solution of nonlinear singular difference system, J. Nat. Sci. Heilongjiang Univ., 27 (2010), 561–564.1, 2.3 [14] P. Wang, J. Zhang, Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, J. Comput. Appl. Math., 221 (2008), 158–164. 1 [15] X. Yang, X. Liao, C. Li, On a difference equation wtih maximum, Appl. Math. Comput., 181 (2006), 1–5. 1 [16] L. Zhang, J.«
^{https://www.semanticscholar.org/paper/The-quadratic-convergence-of-approximate-solutions-Wang-Liu/9d8d6a10067f9e179790589bb3f285b7bc1b7696}

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14 matches:

IRJMETS960448.pdf » 36(1990),119-130. [9] LIUX, Non-linear boundary value problem for first order impulsive differential equations, Application Analysis 36(1990),119-130. [10] Pierson.C.-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential equations Dynamic Systems,1(1993),185-196. http://www.irjmets.com/« - Rampyari
^{https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf}

impulsive-periodic-boundary-value-problems-for-dynamic-equations-on-time-scale.pdf » nonresonance impulsive periodic problems of first order,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 423–433, 1997. 15 C. Pierson-Gorez, “Impulsive differential equations of first order with periodic boundary conditions,” Differential Equations and Dynamical Systems, vol. 1, no. 3, pp. 185–196, 1993. 16 Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 199«
^{https://cyberleninka.org/article/n/1372332}

Periodic Boundary Value Problems for Higher Order Impulsive Functional Differential Equations » 489-493. NIETO, J. J.; 2002, Periodic boundary value problems for first order impulsive ordinary differential equations, Nonlinear Analysis, 51, 1223-1232. PIERSON-GOREZ, C.; 1993, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations and Dynamical Systems, 185-196. VATSALA, A. S.; SUN, Y.; 1992, Periodic boundary value problems of impulsive differential equations, Nonlinear Analysis, 44, 145-158. YANG, X.; SHEN, J.; 2007, Periodic boundary va« - Çağrı
^{https://dergipark.org.tr/en/pub/sdufeffd/issue/11266/134631}

Impulsive Boundary Value Problems for First-order Ordinary ... »Pierson-Gorez, C. Impulsive differential equations of first order with periodic boundary conditions. Differential Equations Dynam. Systems, 1:185–196 ...«
^{https://link.springer.com/article/10.1007/s10255-007-0381-4}

Impulsive Periodic Boundary Value Problems for Dynamic ... »by ER Kaufmann · 2009 · Cited by 6 — Pierson-Gorez C: Impulsive differential equations of first order with periodic boundary conditions. Differential Equations and Dynamical ...«
^{https://link.springer.com/article/10.1155/2009/603271}

Topics in Functional Differential and Difference Equations »[ 11 ] Pierson - Gorez , C. , Impulsive differential equations of first order with periodic boundary conditions , differential Equations Dynam . Systems , 1 , 1993 ...«
^{https://books.google.com/books?id=Yz6GFA-jtgwC}

Read online Higher-Order Nonlinear Ordinary Differential ... »1993, impulsive differential equations of first order with periodic boundary conditions, differential equations and dynamical systems, 185-196. Recall that an ...«
^{https://if.artexsamara.ru/408}

188787955.pdf » nonresonance impulsive periodic problems of first order,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 423–433, 1997. 15 C. Pierson-Gorez, “Impulsive differential equations of first order with periodic boundary conditions,” Differential Equations and Dynamical Systems, vol. 1, no. 3, pp. 185–196, 1993. 16 Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 199«
^{https://core.ac.uk/download/pdf/188787955.pdf}

2119.pdf » Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonlinear Anal. 25 (1995), no. 11, 1191–1198. [19] C. Pierson-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations Dynam. Systems 1 (1993), no. 3, 185–196. [20] J. Qi and K. Wang, Upper and lower solutions for impulsive differential equations with application to ODE, Northeast. Math. J. 18 (2002), no. 3, 189–196. [21] A. M«
^{http://emis.maths.adelaide.edu.au/journals/HOA/ADE/2005/2119.pdf}

First- and second-order dynamic equations with impulse » Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonlinear Anal. 25 (1995), no. 11, 1191–1198. [19] C. Pierson-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations Dynam. Systems 1 (1993), no. 3, 185–196. [20] J. Qi and K. Wang, Upper and lower solutions for impulsive differential equations with application to ODE, Northeast. Math. J. 18 (2002), no. 3, 189–196. [21] A. M« - F. M. Atici and D. C. Biles
^{http://www.emis.de/journals/HOA/ADE/Volume2005_2/193525.pdf}

art:10.1155/ADE.2005.119.pdf » Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonlinear Anal. 25 (1995), no. 11, 1191–1198. [19] C. Pierson-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations Dynam. Systems 1 (1993), no. 3, 185–196. [20] J. Qi and K.Wang,Upper and lower solutions for impulsive differential equations with application to ODE, Northeast. Math. J. 18 (2002), no. 3, 189–196. [21] A. M. «
^{http://advancesindifferenceequations.springeropen.com/track/pdf/10.1155/ADE.2005.119?site=advancesindifferenceequations.springeropen.com}

10.1.1.586.2068.pdf » Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonlinear Anal. 25 (1995), no. 11, 1191–1198. [19] C. Pierson-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations Dynam. Systems 1 (1993), no. 3, 185–196. [20] J. Qi and K. Wang, Upper and lower solutions for impulsive differential equations with application to ODE, Northeast. Math. J. 18 (2002), no. 3, 189–196. [21] A. M«
^{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.586.2068&rep=rep1&type=pdf}

art:10.1155/2009/603271.pdf » nonresonance impulsive periodic problems of first order,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 423–433, 1997. 15 C. Pierson-Gorez, “Impulsive differential equations of first order with periodic boundary conditions,” Differential Equations and Dynamical Systems, vol. 1, no. 3, pp. 185–196, 1993. 16 Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 199«
^{http://advancesindifferenceequations.springeropen.com/track/pdf/10.1155/2009/603271?site=advancesindifferenceequations.springeropen.com}

Basic Theory for Nonresonance Impulsive Periodic Problems of First Order » Appl. Anal. 36 1990 , 119]130. 10. E. Liz, Boundary Value Problems for New Types of Differential Equations,'' Ph.D. w xThesis, Univ. Vigo-Spain, 1994. In Spanish 11. C. Pierson-Gorez, Impulsive differential equations of first order with periodic boundary .conditions, Differential Equations Dynam. Systems 1 1993 , 185]196. 12. A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect,'' w xViska Skoda, Kiev, 1987. In Russian 13. D. R. Smart, Fixed Point Theorems,'' Cambrid« - Nieto, J. J.
^{https://www.researchgate.net/profile/Juan_Nieto9/publication/223150681_Basic_Theory_for_Nonresonance_Impulsive_Periodic_Problems_of_First_Order/links/547ca5410cf2cfe203c1f98e.pdf}

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1 match:

IRJMETS960448.pdf » Lakshmikantham.V, The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments, Application analysis, 36(1990),119-130. [9] LIUX, Non-linear boundary value problem for first order impulsive differential equations, Application Analysis 36(1990),119-130. [10] Pierson.C.-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential equations Dynamic Systems,1(1993),185-196. http://www.irjmets.com/« - Rampyari
^{https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf}

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2 matches:

IRJMETS960448.pdf » 26(1996), 1913-1922. [4] Jain.P.K., Ahuja.O.P,and Khalil Ahmed,Functional analysis, New age International(P) Limited Publishers,1995. [5] Lakshmi kantham. V, Bainov. D. D and Simeonov. P. S, Theory of impulsive differential equation, World Scientific Publishing company private limited, Singapore,1989. [6] Lakshmikantham.V, Ledu.S Differential and Integral Inequalities, 1 (1969). [7] Ladde. G. S, Lakshmikantham. V, and Vatsala. A. S Monotone iterative techniques for Non-Linear Differential « - Rampyari
^{https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf}

Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems »V. Lakshmikantham, D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equation, World Scientific Publishing Company 1989. A.M. Samoilenko, N.A. ...«
^{https://books.google.com/books?id=5zlUCwAAQBAJ}

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12 matches:

Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — He, Z., Ge, W.: Monotone iterative technique and periodic boundary value problem for first order impulsive functional differential equations.«
^{https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-018-1539-5}

Periodic boundary value problem for first-order impulsive ... »Dec 2, 2020 — The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations. Article.«
^{https://www.researchgate.net/publication/222728564_Periodic_boundary_value_problem_for_first-order_impulsive_functional_differential_equations}

Periodic boundary value problems for first order functional ... »Sep 2, 2020 — The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations. Article.«
^{https://www.researchgate.net/publication/257194540_Periodic_boundary_value_problems_for_first_order_functional_differential_equations_with_impulse}

Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — He, Z., Ge, W.: Monotone iterative technique and periodic boundary value problem for first order impulsive functional differential equations.«
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The Monotone Iterative Technique and Periodic Boundary ... »by ZM He · 2002 · Cited by 7 — The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations. Authors; Authors and ...«
^{https://link.springer.com/article/10.1007/s101140200155}

Periodic boundary value problems for first-order impulsive ... »The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations · Z. He, Wei Gao Ge.«
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Nonlinear boundary problem of first order impulsive integro ... »by L Chen · 2007 · Cited by 20 — And periodic boundary value problem for first-order and second-order integro-differential equations with or without impulse have been extensively discussed, ...«
^{https://www.sciencedirect.com/science/article/pii/S0377042706001476}

The Monotone Iterative Technique and Periodic Boundary ... »by ZM He · 2002 · Cited by 7 — The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations. Abstract. This paper uses ...«
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The Monotone Iterative Technique and Periodic Boundary ... »The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations. 推荐CAJ下载 · PDF下载 ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » institutional affiliations. Received: 12 December 2017 Accepted: 26 February 2018 Tian et al. Advances in Difference Equations (2018) 2018:79 Page 14 of 14 References 1. He, Z., Ge, W.: Monotone iterative technique and periodic boundary value problem for first order impulsive functional differential equations. Acta Math. Sin. 18(2), 253–262 (2002) 2. Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985) 3.« - Wenli Wang
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cheng.dvi » 467—473. [24] Hale J. K. and Cruz M. A., Existence, uniqueness, and continuous dependence for hereditary systems, Ann. Mat. Pura Appl., 85 (1970), 63—81. [25] He Z. M. and Ge W. G., The monotone iterative technique and periodic boundary value problem for first order impulsive functional differential equations, Acta Math. Sin. (Engl. Ser.), 18 (2002), No. 2, 253—262. [26] Hsing D. P. K., Existence and uniqueness theorem for the one dimensional backwards two-body problem of electrodynamics, Phy.« - valery
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excerpt.pdf » using set-valued maps, Applicable Analysis 58 (1995), no. 3-4, 325–333. [151] , Existence results for first-order impulsive differential equations, Journal of Mathematical Analysis and Applications 193 (1995), no. 1, 96–113. [152] , Impulsive differential equations with variable times, Nonlinear Analysis 26 (1996), no. 12, 1913–1922. [153] , First order impulsive initial and periodic problems with variable moments, Journal of Mathematical Analysis and Applications 233 (1999), no. 2, 730–739. [«
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » using set-valued maps, Applicable Analysis 58 (1995), no. 3-4, 325–333. [151] , Existence results for first-order impulsive differential equations, Journal of Mathematical Analysis and Applications 193 (1995), no. 1, 96–113. [152] , Impulsive differential equations with variable times, Nonlinear Analysis 26 (1996), no. 12, 1913–1922. [153] , First order impulsive initial and periodic problems with variable moments, Journal of Mathematical Analysis and Applications 233 (1999), no. 2, 730–739. [«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

IRJMETS960448.pdf » 4 (1996), 65-73. [2] Frigon.M and Regand, O, Existence results for first order impulsive differential Equations, Journal of mathematical, Analysis and Application, 193 (1995), 96-113. [3] Frigon.M and Regand, O, Impulsive differential Equations, with variable times, Non-linear Analysis, 26(1996), 1913-1922. [4] Jain.P.K., Ahuja.O.P,and Khalil Ahmed,Functional analysis, New age International(P) Limited Publishers,1995. [5] Lakshmi kantham. V, Bainov. D. D and Simeonov.« - Rampyari
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index.pdf » solutions in discontinuous systems. Arch. Math. (Brno) 34:73–82, 1998. 66 [11] A. F. Filippov. Differential equations with discontinuous righthand sides, Kluwer, Dordrecht, 1988. [12] M. Frigon and D. O’Regan. Impulsive differential equations with variable times. Nonlinear Analysis, 26:1913–1922, 1996. [13] J. Guckenheimer and P. Holmes. Nonlinear oscillations, Dynamical systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [14] A. Halanay and D.«
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10.1.1.684.6227.pdf » and Appl. 14(2005), no. 1-10. Franco D., Liz E. & Nieto J. J. A contribution to the study of functional differential equations with impulses, Math. Nachr. 28(2000), 49-60. Frigon M. & O'Regan D., Impulsive Differential Equations with Variable Times. Nonlinear Analysis: Theory, Methods & Applications, 26(12):1913-1922, 1996. Granas and J. Dugundji, Fixed Point Theory. SpringerMonographs inMathematics, Springer, New York, 2003. Lakshmikantham V., Bainov D.« - PC 2
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10.1.1.241.4961.pdf »Anal. Appl., 288 182-198 (2003). [33] Devi, J. Vasundara and A. S. Vatsala, Generalized quasilinearization for an impulsive differential equation with variable moments of impulse, Dynam. Systems Appl., 12 369–382 (2003). [34] M. Frigon and D. O’Regan, Impulsive differential equations with variable times,Nonlinear Analysis, 26 1913–1922 (1996). [35] V. Laks«
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AIIC2014.Vol.3.pdf » and Appl. 14(2005), no. 1-10. Franco D., Liz E. & Nieto J. J. A contribution to the study of functional differential equations with impulses, Math. Nachr. 28(2000), 49-60. Frigon M. & O'Regan D., Impulsive Differential Equations with Variable Times. Nonlinear Analysis: Theory, Methods & Applications, 26(12):1913-1922, 1996. Granas and J. Dugundji, Fixed Point Theory. SpringerMonographs inMathematics, Springer, New York, 2003. Lakshmikantham V., Bainov D.« - PC 2
^{http://eujournal.org/files/journals/1/books/AIIC2014.Vol.3.pdf}

10.1.1.505.7852.pdf »Anal. Appl., 288, 2003, 182-198. [29] Devi, J. Vasundara and A. S. Vatsala, Generalized quasilinearization for an impulsive differential equation with variable moments of impulse, Dynam. Systems Appl., 12, 2003, 369-382. [30] M. Frigon and D. O’Regan, Impulsive differential equations with variable times, Nonlinear Analysis, 26, 1996, 1913-1922. [31] V. Lak«
^{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.505.7852&rep=rep1&type=pdf}

akalin.pdf »Anal. Appl., 288, 2003, 182-198. [29] Devi, J. Vasundara and A. S. Vatsala, Generalized quasilinearization for an impulsive differential equation with variable moments of impulse, Dynam. Systems Appl., 12, 2003, 369-382. [30] M. Frigon and D. O’Regan, Impulsive differential equations with variable times, Nonlinear Analysis, 26, 1996, 1913-1922. [31] V. Lak«
^{http://www.cs.ubbcluj.ro/~studia-m/2004-4/akalin.pdf}

1102.4462 »Vatsala, Generalized quasilinearization for an impulsive differential equation with variable moments of impulse, Dy-nam. Systems Appl., 12 (2003) 369-382. [20] Feckan, M., Bifurcation of periodic and chaotic solutions in discontinuous systems. Arch. Math. (Brno), 34 (1998) 73–82. [21] M. Frigon, D. O’Regan, Impulsive «
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IRJMETS960448.pdf » boundary value problem for first order differential equations with impulses at variable times, Journal of mathematical, Analysis and Application, 4 (1996), 65-73. [2] Frigon.M and Regand, O, Existence results for first order impulsive differential Equations, Journal of mathematical, Analysis and Application, 193 (1995), 96-113. [3] Frigon.M and Regand, O, Impulsive differential Equations, with variable times, Non-linear Analysis, 26(1996), 1913-1922. [4] Jain.P.K., Ahuja.O.P,and Khalil Ahmed,« - Rampyari
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27 matches:

IRJMETS960448.pdf » periodic boundary value problems, initial and periodic problems of first order impulsive differential equations at variable time. III. REFERENCES [1] Bajo.I and Liz.E, Periodic boundary value problem for first order differential equations with impulses at variable times, Journal of mathematical, Analysis and Application, 4 (1996), 65-73. [2] Frigon.M and Regand, O, Existence results for first order impulsive differential Equations, Journal of mathematical, Analysis and Application, 193 (199« - Rampyari
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excerpt.pdf » Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York, 1993. [31] I. Bajo and E. Liz, Periodic boundary value problem for first order differential equations with impulses at variable times, Journal of Mathematical Analysis and Applications 204 (1996), no. 1, 65–73. [32] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekk«
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Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York, 1993. [31] I. Bajo and E. Liz, Periodic boundary value problem for first order differential equations with impulses at variable times, Journal of Mathematical Analysis and Applications 204 (1996), no. 1, 65–73. [32] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekk«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

a965903-065.pdf » systems. Frunze: Ilim. p.352.−1974. [15] Trenogin V.A. Development and application of the asymptotic method of Lyusternik-Vishik. // UMN, 1970. T.25. №4 (154).− S.123-156. [16] Tupchiev V.A. The asymptotic solution of boundary value problem for first order differential equations with a small parameter in the derivatives .// Akad. 1962 T.143. №6.− S.1296-1299. [17] Tupchiev V.A. The existence, uniqueness and asymptotic behavior of the solution of the boundary problem for a system of« - MT
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Positive Solutions to Periodic Boundary Value Problems for Four-Order Differential Equations » A Project Supported by the NNSF of China 10871063 and ScientificResearch Fund of human Provincial Education Department 10C0258 . References 1 I. Bajo and E. Liz, “Periodic boundary value problem for first order differential equations with impulses at variable times,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 65–73, 1996. 2 J. Chu, X. Lin, D. Jiang, D. O’Regan, and R. P. Agarwal, “Multiplicity of positive periodic solutions to second order differential equations,”« - H. Zhu and Z. Luo
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1687-2770-2013-195.pdf » PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 3. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) 4. Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204, 65-73 (1996) 5. Belley, J, Virgilio, M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl. 306, 646-662 (2005) 6. Belley, J, V«
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Periodic Boundary Value Problem for First Order Differential ... »by I Bajo · 1996 · Cited by 77 — Regular Article. Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times.«
^{https://www.sciencedirect.com/science/article/pii/S0022247X96904242}

Monotone Iterative Techniques and a Periodic Boundary ... »Request PDF | Monotone Iterative Techniques and a Periodic Boundary Value Problem for First Order Differential Equations with a Functional Argument | This ...«
^{https://www.researchgate.net/publication/265634256_Monotone_Iterative_Techniques_and_a_Periodic_Boundary_Value_Problem_for_First_Order_Differential_Equations_with_a_Functional_Argument}

Functional boundary value problem for first order impulsive ... »Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times. Article. Nov 1996; J MATH ANAL APPL · Ignacio Bajo ...«
^{https://www.researchgate.net/publication/268868024_Functional_boundary_value_problem_for_first_order_impulsive_differential_equations_at_variable_times}

[PDF] Periodic Boundary Value Problem for First Order ... »Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times · I. Bajo, Eduardo Liz · Published 1996 · Mathematics ...«
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Impulsive Boundary Value Problems for First-order Ordinary ... »... value problem for first order differential equations with impulses at variable times. J. Math. Appl. Anal., 204:65–73 (1996)zbMATHCrossRefMathSciNetGoogle ...«
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Monotone Iterative Techniques and a Periodic ... - De Gruyter »by A Qi · 2000 · Cited by 1 — Monotone Iterative Techniques and a Periodic Boundary Value Problem for First Order Differential Equations with a Functional Argument ...«
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Dirichlet boundary value problem for differential equation with ... »by J Tomeček · 2017 · Cited by 10 — [6] Bajo, I.—Liz, E.: Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math. Anal.«
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First-order nonlinear differential equations with state ... - CORE »by L Rachunek · 2013 · Cited by 5 — Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204, 65-73 (1996). 5.«
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Second order BVPs with state dependent impulses ... - EUDML »by I Rachůnková · 2014 · Cited by 18 — ... Harlow, 1993 Zbl0815.34001; [5] Bajo I., Liz E., Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math.«
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Solution of a periodic boundary-value problem by direct application ... »Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times. DOI: 10.1006/jmaa.1996.0424 · Journal of Mathematical ...«
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ÝÉÔÉÒÄÁÉÓ ÉÍÃÄØÓÉ »A. Qi and Y. Liu, Monotone iterative techniques and a periodic boundary value problem for first order differential equations with a functional argument. Georgian ...«
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Impulsive functional differential equations with variable times ... »by M Benchohra · 2004 · Cited by 56 — Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. v204. 65-73.Google Scholar Google ...«
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Cumulatice Index - Mathematical Institute Home Page »Liu Y. & A. Qi: Monotone Iterative Techniques and a Periodic Boundary Value Problem for First Order Differential Equations with a Functional Argument, ...«
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‪Eduardo Liz‬ - ‪Google Scholar‬ »Periodic boundary value problem for first order differential equations with impulses at variable times. I Bajo, E Liz. Journal of mathematical analysis and ...«
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0010_20131203_dw_isaac_9th_congress_abstracts.pdf » the Carathéodory conditions, the impulse functions Ji, Mi, and the barriers γi, i = 1, . . . , p, are continuous, c ∈ R2, and ℓ is a linear and bounded operator. [1] Bajo, I. and Liz, E., Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204 (1996), 65–73. [2] [2] Rach˚unková, I. and Tomecˇek, J., New approach to BVPs with state-dependent impulses. Boundary Value Problems 2013, 2013:22. [3] [3] Rach˚unková, I. and To«
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A new approach to BVPs with state-dependent impulses » 360, 105-108 (2006) 10. Qi, J, Fu, X: Existence of limit cycles of impulsive differential equations with impulses at variable times. Nonlinear Anal. 44, 345-353 (2001) 11. Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204, 65-73 (1996) 12. Belley, J, Virgilio, M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl. 306, 646-662 (2005) 13. Belley, J, Virgilio, M: Perio« - Irena Rachunková, Jan Tomecek
^{http://www.boundaryvalueproblems.com/content/pdf/1687-2770-2013-22.pdf}

First-order nonlinear differential equations with state-dependent impulses » PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 3. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) 4. Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204, 65-73 (1996) 5. Belley, J, Virgilio, M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl. 306, 646-662 (2005) 6. Belley, J, V« - Luká Rachunek, Irena Rachunková
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bib & pub.pdf » Nonlinear Dynam. 29 (2002), 281-296. [198] C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl., 310(2005), 26-29. [199] I. Bajo, E. Liz, Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math. Anal. Appl., 204 (1996), 65-73. [200] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and v« - http://www.verypdf.com
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10.1.1.360.675.pdf » Anal. Appl., 236 (1999), 312–326. [3] D. D. Bainov and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Applications,” Longman, Harlow, 1993. [4] I. Bajo and E. Liz, Periodic Boundary Value Problem for First Order Differential Equations with Impulses at variable times, J. Math. Anal. Appl., 204 (1996), 65–73. [5] G. Ballinger and X. Liu, “Practical Stability of Impulsive Delay Differential Equations and Applications to Control Problems, Optimization methods and applicat«
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Positive Solutions to Periodic Boundary Value Problems for Four-Order Differential Equations » A Project Supported by the NNSF of China 10871063 and ScientificResearch Fund of human Provincial Education Department 10C0258 . References 1 I. Bajo and E. Liz, “Periodic boundary value problem for first order differential equations with impulses at variable times,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 65–73, 1996. 2 J. Chu, X. Lin, D. Jiang, D. O’Regan, and R. P. Agarwal, “Multiplicity of positive periodic solutions to second order differential equations,”«
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Nonlocal fractional differential inclusions with impulses at variable times » https://zbmath.org/?q=an:1084.34002 http://www.utgjiu.ro/math/sma/v14/v14.html http://www.utgjiu.ro/math/sma 322 A. Ouahab and S. Seghiri [8] I. Bajo and E. Liz, Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math. Anal. Appl. 204 (1996), no. 1, 65–73. MR1418522. Zbl 0876.34020. [9] J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60. Dekker, New York, 1« - Abdelghani Ouahab , Sarah Seghiri
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5 matches:

JNFA201620.pdf »J. Nonlinear Funct. Anal. 2016 (2016), Article ID 20 Copyright c© 2016 Mathematical Research Press. EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF NONLINEAR NEUTRAL DIFFERENCE EQUATIONS WITH VARIABLE DELAY ABDELOUAHEB ARDJOUNI1,∗, AHCENE DJOUDI2 1Department of Mathematics and Informatics, Faculty of Sciences and Technology, Univ Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria 2Department of Mathematics, Faculty of Sciences, Univ Annaba, P.O.«
^{http://jnfa.mathres.org/issues/JNFA201620.pdf}

JNFA201729.pdf » of periodic solutions for nonlinear neutral dynamic equations with wariable delay on a time scale, Commun. Nonlinaer Sci. Numer. Simulation 17 (2012), 3061-3069. [6] A. Ardjouni, A. Djoudi, Existence of positive periodic solutions for a system of nonlinear neutral difference equations with variable delay, J. Nonlinear Funt. Anal. 2016 (2016), Article ID 20. [7] B. Bacova, B. Dorociakova, R. Olach, Existence of positive solutions of nonlinear neutral differential equations asymptotic to zero, R«
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for a system of nonlinear neutral difference equations with variable delay.«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Abdelouaheb Ardjouni, Ahcene Djoudi, Existence of positive ... »Abdelouaheb Ardjouni, Ahcene Djoudi, Existence of positive periodic solutions for a system of nonlinear neutral difference equations with variable delay, 2016 ( ...«
^{http://jnfa.mathres.org/archives/886}

Periodic boundary value problems for first-order impulsive difference equations with time delay » in Mathematics. Spinger, New York (2005) 14. Keney, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, Tokyo (2001) 15. Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for a system of nonlinear neutral difference equations with variable delay. J. Nonlinear Funct. Anal. 2016, Article ID 20 (2016) 16. He, Z., Zhang, X.: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Appl« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — It is reckoned that these results may play an important role in the theory of difference equations, and are useful in many practical problems in ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Periodic boundary value problems for first-order impulsive difference equations with time delay » equation and established the existence of extremal solutions by utilizing the monotone iterative technique. An example was given to illustrate the results obtained. It is reckoned that these results may play an important role in the theory of difference equations, and are useful in many practical problems in the aforesaid fields. Acknowledgements The authors would like to thank the reviewers and the editors for their valuable suggestions and comments.« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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13 matches:

Dissipativity of Pseudorational Behaviors » we have derived a new necessary and sufficient condition for entire functions in the Paley-Wiener class to be symmetrically factorizable. We also has given some conditions for pseudorational behaviors to be dissipative. An example was given to illustrate the results. References [1] R. P. Boas Jr., Entire Functions. Academic Press, 1954. [2] F. M. Callier, “On polynomial spectral factorization by symmetric factor extraction,” IEEE Trans. Automat. Control, vol. 30, pp. 453– 464, 1985. [3] N« - Masaki Ogura, Yutaka Yamamoto
^{https://www.conferences.hu/mtns2010/proceedings/Papers/147_097.pdf}

Solution of an inverse problem for “fixed-fixed” and “fixed-free ... »by X Wu · 2007 — ... and the spring constants from the natural frequencies of the “fixed-fixed” and “fixed-free” spring-mass systems. An example was given to illustrate the results.«
^{https://link.springer.com/article/10.1007/s11741-007-0104-3}

Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — An example was given to illustrate the results obtained. It is reckoned that these results may play an important role in the theory of difference ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Existence and uniqueness of solutions of multi-point boundary ... »An example was given to illustrate the results. ResearchGate Logo. Discover the world's research. 20+ million members; 135+ million publications; 700k+ ...«
^{https://www.researchgate.net/publication/283367376_Existence_and_uniqueness_of_solutions_of_multi-point_boundary_value_problems_for_integro-differential_equations_of_fractional_order}

Three-point boundary value problems of fractional functional ... »Oct 11, 2020 — [Show full abstract] applying, the fixed point theorem. An example was given to illustrate the results. Read more. Last Updated: 11 Oct 2020 ...«
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Asymptotic solution for a class of impulsive differential equations »Finally, an example was given to illustrate the results. Keywords: singular perturbation,; asymptotic solution,; impulsive differential equation. FullText(HTML).«
^{http://hdsfdxzkb.xml-journal.net/en/article/id/24626}

Existence of Periodic Solutions for Nonlinear Neutral ... - EBSCO »by H Ming-hui · 2014 — An example was given to illustrate the results. [ABSTRACT FROM AUTHOR]. Copyright of Journal of Guangdong University of Technology is the property of ...«
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Airiti Library華藝線上圖書館_具有分布偏差变元的二阶中立型时 ... »by 李鹏松 · 2010 — An example was given to illustrate the results. 並列摘要〈TOP〉. The oscillation of second-order neutral delay differential equations with ...«
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中立型多变时滞随机微分方程的稳定性 »by 王春生 · 2015 · Cited by 2 — Also an example was given to illustrate the results in the paper. Key words: Banach fixed points, neutral stochastic differential equations,some ...«
^{http://lxbwk.njournal.sdu.edu.cn/CN/10.6040/j.issn.1671-9352.0.2014.237}

Dissipativity of Pseudorational Behaviors » we have derived a new necessary and sufficient condition for entire functions in the Paley-Wiener class to be symmetrically factorizable. We also has given some conditions for pseudorational behaviors to be dissipative. An example was given to illustrate the results. References [1] R. P. Boas Jr., Entire Functions. Academic Press, 1954. [2] F. M. Callier, “On polynomial spectral factorization by symmetric factor extraction,” IEEE Trans. Automat. Control, vol. 30, pp. 453– 464, 1985. [3] N« - Masaki Ogura, Yutaka Yamamoto
^{http://fwn06.housing.rug.nl/mtns2010/Papers/147_097.pdf}

Periodic boundary value problems for first-order impulsive difference equations with time delay » two new comparison principles. With these, we constructed monotone sequences from a corresponding linear equation and established the existence of extremal solutions by utilizing the monotone iterative technique. An example was given to illustrate the results obtained. It is reckoned that these results may play an important role in the theory of difference equations, and are useful in many practical problems in the aforesaid fields. Acknowledgements The authors would like to thank the r« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

CDC2013.pdf » the number of switches a low-pass filter was added to the last rule. A sufficient condition on the matrices Gi and Λi has been derived to get the exponential stability of the system with these three switching rules. Finally, an example was given to illustrate the results. This work leaves many questions open. In particular, question of existence of solutions has to be investigated when closing the loop with a switching rule. In our analysis we have supposed that y lies in Cpw([0, 1]) but this « - ()
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — With these, we constructed monotone sequences from a corresponding linear equation and established the existence of extremal solutions by ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructed monotone sequences from a corresponding linear equation and established the existence of extremal solutions by utilizing the monotone iterative technique. An example was given to illustrate the results obtained. It is reckoned that these results may play an important role in the theory of difference equations, a« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructed monotone ...«
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Top PDF Periodic boundary value problems for first order impulsive ... »In this paper, we studied impulsive delay diﬀerence equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we ...«
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Download PDF - Advances in Difference Equations »by J Tian · 2018 · Cited by 9 — In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we ...«
^{https://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/s13662-018-1539-5.pdf}

Periodic boundary value problems for first-order impulsive difference equations with time delay » equations and impulsive delay difference equations. These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructed monotone sequences from a corresponding linear equation and established the existence of extremal solutions by utilizing the monotone iterative technique. An example was g« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

in this paper we studied | English examples in context | Ludwig »In this paper, we studied impulsive delay difference equations with periodic boundary conditions. 5. Advances in Difference Equations. In this paper, we studied ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » analysis, biology, etc. However, there are not many related results for impulsive difference equations and impulsive delay difference equations. These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructed monotone sequences from a corresponding linear equation and established« - Wenli Wang
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motivated us to | English examples in context | Ludwig »These motivated us to work on the present topic. 6.«
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she worked on the topic | English examples in context | Ludwig »These motivated us to work on the present topic. 6.«
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions.«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Download PDF - Advances in Difference Equations »by J Tian · 2018 · Cited by 9 — These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the ...«
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Top PDF first-order difference operators - 1Library »... are not many related results for impulsive diﬀerence equations and impulsive delay diﬀerence equations. These motivated us to work on the present topic.«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » many fields such as numerous settings and forms, computing, electrical circuit analysis, biology, etc. However, there are not many related results for impulsive difference equations and impulsive delay difference equations. These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructe« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — Research in difference equations has been active in recent years and has played an important role in numeral fields, including biology, computing, electrical circuit analysis, etc. ... However, there are not many related results for impulsive difference equations and impulsive delay difference equations.«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Download PDF - Advances in Difference Equations »by J Tian · 2018 · Cited by 9 — However, there are not many related results for impulsive difference equations and impulsive delay difference equations. These motivated us to work on the ...«
^{https://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/s13662-018-1539-5.pdf}

Global exponential stability of impulsive delay difference ... »Oct 11, 2020 — However, there are not many related results for impulsive difference equations and impulsive delay difference equations. ... Periodic boundary ...«
^{https://www.researchgate.net/publication/222809957_Global_exponential_stability_of_impulsive_delay_difference_equation}

Asymptotic behaviors of non-autonomous impulsive difference ... »Nov 30, 2020 — However, there are not many related results for impulsive difference equations and impulsive delay difference equations. ... Periodic boundary ...«
^{https://www.researchgate.net/publication/233730742_Asymptotic_behaviors_of_non-autonomous_impulsive_difference_equation_with_delays}

Periodic boundary value problems for first-order impulsive difference equations with time delay » value problems. On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology, etc. However, there are not many related results for impulsive difference equations and impulsive delay difference equations. These motivated us to work on the present topic. In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upp« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit ...«
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Ludwig • Find your English sentence - Ludwig Guru »On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology ...«
^{https://ludwig.guru/s/in+numerous+settings}

Download PDF - Advances in Difference Equations »by J Tian · 2018 · Cited by 9 — On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology ...«
^{https://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/s13662-018-1539-5.pdf}

Download Recent Advances In Differential Equations By H H Dai »On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology ...«
^{http://zuaobiyoji56.inoxdvr.com/217.html}

Periodic boundary value problems for first-order impulsive difference equations with time delay » particular, in recent years, there has been an increasing interest in extending impulsive differential equations to time-delay systems and boundary value problems. On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology, etc. However, there are not many related results for impulsive difference equations and impulsive delay difference equations.« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other ...«
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control medicine | English examples in context | Ludwig »Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other fields. 3.«
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Download PDF - Advances in Difference Equations »by J Tian · 2018 · Cited by 9 — plications in economics, dynamic systems, optimal control, medicine, population dynam- ics, and many other fields. In particular, in recent years, there has been ...«
^{https://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/s13662-018-1539-5.pdf}

Top PDF first-order difference operators - 1Library »Such equations have extensive ap- plications in economics, dynamic systems, optimal control, medicine, population dynam- ics, and many other ﬁelds.«
^{https://1library.net/subject/first-order-difference-operators}

Top PDF first order difference - 1Library »Such equations have extensive ap- plications in economics, dynamic systems, optimal control, medicine, population dynam- ics, and many other ﬁelds.«
^{https://1library.net/subject/first-order-difference}

Impulsive Differential Equations Periodic Solutions and Applications »Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other fields. In particular ...«
^{https://ci.atomexpo2010.ru/432}

Periodic boundary value problems for first-order impulsive difference equations with time delay » recognized that the theory of impulsive equations offers a general framework for the mathematical modeling of many real-world phenomena where the states undergo abrupt changes. Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other fields. In particular, in recent years, there has been an increasing interest in extending impulsive differential equations to time-delay systems and boundary value problems. On the oth« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — It is well recognized that the theory of impulsive equations offers a general framework for the mathematical modeling of many real-world ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Top PDF Boundary value problems for the diffusion equation with ... »It is well recognized that the theory of impulsive equations oﬀers a general framework for the mathematical modeling of many real-world phenomena where the ...«
^{https://1library.net/title/boundary-value-problems-diffusion-equation-piecewise-continuous-delay}

Top PDF first-order difference operators - 1Library »It is well recognized that the theory of impulsive equations oﬀers a general framework for the mathematical modeling of many real-world phenomena where the ...«
^{https://1library.net/subject/first-order-difference-operators}

Periodic boundary value problems for first-order impulsive difference equations with time delay » (3.10) in [v, w]. 4 Conclusion The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized that the theory of impulsive equations offers a general framework for the mathematical modeling of many real-world phenomena where the states undergo abrupt changes. Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other fields.« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized ...«
^{https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-018-1539-5}

Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Periodic boundary value problems for first-order impulsive ... »The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized that the theory of ...«
^{https://www.researchgate.net/journal/Advances-in-Difference-Equations-1687-1847/publication/323559736_Periodic_boundary_value_problems_for_first-order_impulsive_difference_equations_with_time_delay/links/5fc46bc5299bf104cf941a9d/Periodic-boundary-value-problems-for-first-order-impulsive-difference-equations-with-time-delay.pdf}

Top PDF first-order difference operators - 1Library »The theme of the paper is a study on periodic boundary value problems for ﬁrst-order impulsive delay diﬀerence equations. It is well recognized that the theory ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » 13 of 14 which shows that all of the conditions of (2.1) are satisfied. Therefore, from Theorem 3.5, we arrive at the existence of extremal solutions of (3.10) in [v, w]. 4 Conclusion The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized that the theory of impulsive equations offers a general framework for the mathematical modeling of many real-world phenomena where the states undergo abrupt changes.« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »+)) ≥ I* (β (t3)) ≥ u (t3). Suppose that I* (β (t4)) ≥ u (t4), using the condition (H3) and proceeding as in step ...«
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IRJMETS960448.pdf » the condition (H0) instead of (H1). (b) β (t3+) ≥ β (t3). If β (t) does not hit σ for t > t3, the proof is over. Suppose that there exists one point t4 ϵ (t3, T) such that t4 = γ (β (t4)). In this case, which implies that β (t) ≥ u (t) for all t ϵ (t3, t4) and it remains to prove that I* (β (t4)) ≥ u (t4). For it, since I* is increasing, I* (β (t3+)) ≥ I* (β (t3)) ≥ u (t3). Suppose that I* (β (t4)) < u (t4), using the condition (H3) and proceeding as in step one, we get a contradiction.« - Rampyari
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »suppose that there exists one-point t4 ϵ (t3, T) such that t4 = γ (β ( t4)). In this case it follows that β (t) ≥ u (t) for all t ϵ (t3, t4) and it remains to show that. I* (β (t4)) ...«
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aa_lrm_final_version.pdf » = ϕs(g) ω. On the other hand, we also have ϕs((s i)ω) = ϕs((s i)σ mφn) = ϕs(s m · tn · si · t−n) = ϕs(s m+iqn) = σm+iq n (βs) = (σm · φn) · σi(βs) = ϕs(s i)ω. Hence ϕs ∈ Lπr(πnr), as claimed. (b) The proof of this assertion is very similar to that of [7, Lemma 6.6], which is in turn an analogue of [21, 5.4]. Set H = 〈s〉. Then Ωt acts transitively on πrH(πnr) ⊆πr G(πnr), and so the algebra Lπr(πnr) H := MapΩt( πrH(πnr), L t) may be identified with a subfield of Lt via identifying b ∈ Lπr(πnr)H w«
^{https://web.math.ucsb.edu/~agboola/papers/aa_lrm_final_version.pdf}

2aa_lrm_paper.pdf » also have ϕs((s i)ω) = ϕs((s i)σ mφn) = ϕs(s m · tn · si · t−n) = ϕs(s m+iqn) = σm+iq n (βs) = (σm · φn) · σi(βs) = ϕs(s i)ω. Hence ϕs ∈ Lπr(πnr), as claimed. 26 A. AGBOOLA AND L. R. MCCULLOH (b) The proof of this assertion is very similar to that of [2, Lemma 6.6], which is in turn an analogue of [9, 5.4]. Set H = 〈s〉, viewed as a subset of πrG(πnr). Then Ωt acts transitively on H, and so the algebra Lπr(πnr) H := MapΩt(H,L c) may be identified with a subfield of Lt via identifying b ∈ LH wi«
^{https://web.math.ucsb.edu/~agboola/papers/2aa_lrm_paper.pdf}

IRJMETS960448.pdf » (a) β (t3+) < β (t). In this case, it is possible to prove that β (t) does not meet σ for t > t3, and thus β (t3+) ≥ u (t3), implies that β (t) ≥ u (t) for every t ϵ (t3, T]. The proof of this assertion is very similar to the corresponding one in the first step, using the condition (H0) instead of (H1). (b) β (t3+) ≥ β (t3). If β (t) does not hit σ for t > t3, the proof is over. Suppose that there exists one point t4 ϵ (t3, T) such that t4 = γ (β (t4)).« - Rampyari
^{https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf}

414232.pdf » that x ∈ ⋂∞ i 1 F Ti , that is, F Ŝ ⊂⋂∞ i 1 F Ti . Now F Ŝ ⋂∞ i 1 F Ti / ∅. Again, using the convexity of ‖ · ‖2, we can show that Ŝ satisfies property G2 , and hence it is generalized nonexpansive, as desired. ii Since the proof of this assertion is very similar to that of Theorem 3.1, it is omitted. iii and iv follow directly from ii . Remark 4.3. Theorem 4.2 i generalizes 16, Theorem 3.3 from a finite family to a countable one. Following Reich 6 , we introduced the following concept.«
^{https://fixedpointtheoryandapplications.springeropen.com/articles/10.1155/2010/414232}

Introductory Econometrics: Intuition, Proof, and Practice »The proof of this assertion is very similar to the proof in exercise 4.12b. We attempt it in exercise 10.4. This procedure is sometimes abbreviated as TSLS rather ...«
^{https://books.google.com/books?id=niaFEZOkA1IC}

Advanced Topics in Relation Algebras: Relation Algebras, ... »The proof of this assertion is very similar to the proof of the corresponding result for canonical extensions (Theorem 14.36), and is left as an exercise. Theorem ...«
^{https://books.google.com/books?id=CXMzDwAAQBAJ}

Symmetries in Algebra and Number Theory (SANT): Proceedings ... »(The proof of this assertion is very similar to the proof of the fact that every cuspidal function is compactly supported). It follows that each Yakov Varshavsky: ...«
^{https://books.google.com/books?id=8V6dH0slpnMC}

Analysis and Mathematical Physics - Page 137 - Google Books Result »We are going now to prove the assertion of Part 2. The proof of this assertion is very similar to the proof of [2, Theorem 1]. As previously we assume, that the ...«
^{https://books.google.com/books?id=87nZccF3vMwC}

Multiparameter Processes: An Introduction to Random Fields »Then, for all t > 0, the following holds Pr—a.s., for all a e S, Er |f(X,+t, • . . ; Xtr-t) 3, — Ex, |f(X, 5 * * * * Xı.) - (1) The proof of this assertion is very similar to that of ...«
^{https://books.google.com/books?id=XADpBwAAQBAJ}

Algebra & Number Theory vol. 12 (2018), no. 8 - MSP »Adebisi Agboola and Leon R. McCulloh. Proof. (a) The proof of this assertion is very similar to that of [McCulloh 1987, Proposition 5.4]. It suffices to show that the ...«
^{https://msp.org/ant/2018/12-8/ant-v12-n8-s.pdf}

Habilitationsschrift eingereicht bei der Fakult at f ur Mathematik und ... »The proof of this assertion is very similar to the corresponding parts of the. proof of Theorem 41 and we omit the details. Again the sequence X is constructed.«
^{http://math.uni-heidelberg.de/logic/merkle/ps/effrandom.ps}

10.1.1.67.8984.pdf » l0 + : : :+ ls.Then any set X is wtt-redu ible to a set R in C, where the redu tion an be hosen su h that for any s, the pre x of X of length m0 + : : : + ms an be omputed from the pre x of R of length l0 + : : :+ ls.The proof of this assertion is very similar to the orresponding parts of theproof of Theorem 14 and we omit the details. Again the set X is onstru ted instages, where during stage s we extend an admissible pre x of length ls 1 to anadmissible pre x of length ls, hen e by (i) the o«
^{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.67.8984&rep=rep1&type=pdf}

PII: 0047-259X(91)90100-G » Finally, since 6, - (log n/n)1’(2p + ‘), a suitable choice of K: giving an appropriate value of c7 depending on p and d ensures (5.4). 1 Proof of Theorem 3.3. We begin by proving assertion (ii). The proof of this assertion is very similar to the proof of Theorem 3.2. Assume again, as in the proof of Theorem 3.2, that n 2 No (note that this N, may be different from that in Theorem 3.2 depending on the choice of J,,). Also, recall that 6, =J;i, where J, is as defined in the statement of the the«
^{http://www.sciencedirect.com/science/article/pii/0047-259X(91)90100-G}

Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times » Ž . Ž q. Ž . Ž .a b t - b t . In this case, we may prove that b t does not3 3 Ž q. Ž . Ž . Ž .meet s for t ) t and thus b t G u t implies that b t G u t for3 3 3 Ž xevery t g t , T . The proof of this assertion is very similar to the corre-3 Ž . Ž .sponding one in the first step, using the condition H instead of H .0 1 Ž . Ž q. Ž . Ž .b b t G b t . If b t does not hit s for t ) t , the proof is3 3 3 Ž .complete. Then, we can suppose that there exists one point t g t , T4 3 Ž Ž ..« - Bajo, I. et al.
^{https://core.ac.uk/download/pdf/82658699.pdf}

10.1.1.4.161.pdf » : :+ ls.Then any sequen e X is wtt-redu ible to a sequen e R in C, where the redu tion an be hosen su h that for any s, the pre x of X of length m0 + : : :+ms anbe omputed from the pre x of R of length l0 + : : :+ ls.The proof of this assertion is very similar to the orresponding parts of theproof of Theorem 41 and we omit the details. Again the sequen e X is onstru tedin stages, where during stage s we extend an admissible pre x of length ls 1 toan admissible pre x of length ls, hen e by (i) «
^{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.4.161&rep=rep1&type=pdf}

414232 » that x ∈ ⋂∞ i 1 F Ti , that is, F Ŝ ⊂⋂∞ i 1 F Ti . Now F Ŝ ⋂∞ i 1 F Ti / ∅. Again, using the convexity of ‖ · ‖2, we can show that Ŝ satisfies property G2 , and hence it is generalized nonexpansive, as desired. ii Since the proof of this assertion is very similar to that of Theorem 3.1, it is omitted. iii and iv follow directly from ii . Remark 4.3. Theorem 4.2 i generalizes 16, Theorem 3.3 from a finite family to a countable one. Following Reich 6 , we introduced the following concept.«
^{https://fixedpointtheoryandapplications.springeropen.com/track/pdf/10.1155/2010/414232?site=fixedpointtheoryandapplications.springeropen.com}

On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings » that x ∈ ⋂∞ i 1 F Ti , that is, F Ŝ ⊂⋂∞ i 1 F Ti . Now F Ŝ ⋂∞ i 1 F Ti / ∅. Again, using the convexity of ‖ · ‖2, we can show that Ŝ satisfies property G2 , and hence it is generalized nonexpansive, as desired. ii Since the proof of this assertion is very similar to that of Theorem 3.1, it is omitted. iii and iv follow directly from ii . Remark 4.3. Theorem 4.2 i generalizes 16, Theorem 3.3 from a finite family to a countable one. Following Reich 6 , we introduced the following concept.« - W. Nilsrakoo and S. Saejung
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effrandom.pdf » X is wtt-reducible to a sequence R in C, where the reduction can be chosen such that for any s, the prefix of X of length m0 + . . . + ms can be computed from the prefix of R of length l0 + . . . + ls. The proof of this assertion is very similar to the corresponding parts of the proof of Theorem 41 and we omit the details. Again the sequence X is constructed in stages, where during stage s we extend an admissible prefix of length ls−1 to an admissible prefix of length ls, hence by (i) the c«
^{http://www.math.uni-heidelberg.de/logic/merkle/ps/effrandom.pdf}

1410.4829 » = ϕs(g) ω. On the other hand, we also have ϕs((s i)ω) = ϕs((s i)σ mφn) = ϕs(s m · tn · si · t−n) = ϕs(s m+iqn) = σm+iq n (βs) = (σm · φn) · σi(βs) = ϕs(s i)ω. Hence ϕs ∈ Lπr(πnr), as claimed. (b) The proof of this assertion is very similar to that of [7, Lemma 6.6], which is in turn an analogue of [21, 5.4]. Set H = 〈s〉. Then Ωt acts transitively on πrH(πnr) ⊆πr G(πnr), and so the algebra Lπr(πnr) H := MapΩt( πrH(πnr), L t) may be identified with a subfield of Lt via identifying b ∈ Lπr(πnr) H «
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IRJMETS960448.pdf » www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [96] Therefore, there are two possibilities: (i) β ( t3+) < β (t3). In this case, it is possible to prove that β (t) does not meet σ for t > t3 and thus β (t3+) ≥ u (t3) implies that β (t) ≥ u (t) for every t ϵ ( t3, T]. The proof of this assertion is very similar to the corresponding one in the first step, using the condition (H0) instead of (H1).« - Rampyari
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IRJMETS960448.pdf » in Engineering, Technology and Science [98] Using (4.1.12), (4.1.13), condition (H3) and the same arguments employed in step one it can be proved that β (t3+) ≥ I*( β (t3)) ≥ u (t3). This implies that u (t) does not meet σ for t > t3, but the condition (H1) does not assure the same β (t). Therefore, there are two possibilities: (a) β (t3+) < β (t). In this case, it is possible to prove that β (t) does not meet σ for t > t3, and thus β (t3+) ≥ u (t3), implies that β (t) ≥ u (t) for « - Rampyari
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Using (4.1.12), (4.1.13), condition (H3) and the same arguments employed in step one it can be proved that β (t3. +) ≥ I*( β (t3)) ≥ u (t3). This implies that u (t) ...«
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Now assume that there exists one-point t3 ϵ (t2, T) such that t3 = γ (β (t3)). Since u (t2. +) ≤ β (t2), u (t) ≤ β (t), for all t ϵ [t2, t3]. Using equations (4.1.6), (4.1.7), ...«
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IRJMETS960448.pdf » To prove β (0+) ≥ u0 implies β (t) ≥ u (t), for all t ϵ J. The proof of this step will follow the same procedure as in step one. The main difference is that the graph of the function β can meet the curve σ several times. It is easy to see that u (t) hits σ before than β (t). Let t2 = γ (u (t2)). It follows that u (0+) ≤ β (0+) implies u (t) ≤ β (t), for all t ϵ [0, t2]. But, u (t2+) = u (t2) + I (u (t2)) ≤ u (t2) ≤ β (t2). If β (t) does not hit σ, then u (t2+) ≤ β (t2) ≤ β (t), for all t ϵ (t2« - Rampyari
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IRJMETS960448.pdf » α (t) ≤ u (t) for all t ϵ ( t2 , T). Hence the proof of step one. Step 2: To prove β (0+) ≥ u0 implies β (t) ≥ u (t), for all t ϵ J. The proof of this step will follow the same procedure as in step one. The main difference is that the graph of the function β can meet the curve σ several times. It is easy to see that u (t) hits σ before than β (t). Let t2 = γ (u (t2)). u (0+) ≤ β (0+) implies u (t) ≤ β (t), for all t ϵ [ 0, t2]. But u (t2+) = u (t2) + I (u (t2)) (using (1.1.9)) ≤ u (t2) ≤ β (t2« - Rampyari
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IRJMETS960448.pdf » (1.1.9) follows. Hence u (t) hits σ only for t = t2 and then α (t2+) implies α (t) ≤ u (t) for all t ϵ ( t2 , T). Hence the proof of step one. Step 2: To prove β (0+) ≥ u0 implies β (t) ≥ u (t), for all t ϵ J. The proof of this step will follow the same procedure as in step one. The main difference is that the graph of the function β can meet the curve σ several times. It is easy to see that u (t) hits σ before than β (t). Let t2 = γ (u (t2)). u (0+) ≤ β (0+) implies u (t) ≤ β (t), for all« - Rampyari
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Hence u (t) hits σ only for t = t2 and then α (t2. +) implies α (t) ≤ u (t) for all t ϵ ( t2 , T). Hence the proof of step one. Step 2: To prove β (0+) ≥ u0 implies β (t) ≥ u ...«
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Then there exists one point s ϵ (t1, t2) such that α(s) = I*(u (s)) and α (t) > I* (u (t)) for s < t ≤ t2. Differentiating with respect to s αˈ(s). ≥ (I*) ˈ(u (s)) uˈ(s).«
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »for ordinary differential equations α (t) ≤ u (t), for all t ϵ J. Now assume that there exists t2 ϵ (t1, T) such that u (t2) = γ (u (t2)).«
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IRJMETS960448.pdf » with respect to t pˈ(t) ≥ 1 – γˈ(α (t)) f ( t, α (t)) > 0. (using condition H1) As a consequence, p (t) > 0, for all t > t1. Therefore α (t) does not hit σ for t > t1. If u (t) ≠ γ (u (t)) for t ϵ J, using α (t1+) ≤ u (t1), and the comparison principle for ordinary differential equations α (t) ≤ u (t), for all t ϵ J. Now assume that there exists t2 ϵ [t1, T] such that u (t2) = γ (u (t2)). Since u (t1) ≥ α (t1+), it follows that α (t) ≤ u (t) for t ϵ (t1, t2). To show that u (t2+) = I*(u (« - Rampyari
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IRJMETS960448.pdf » and γ-1 (0) > α (0+), α (t) hits the curve σ first. Therefore, there exists t1 ϵ (0, T) such that t1 = γ (α (t1)). Using α (t) ≤ u (t) on (0, t1) and I (x) ≤ 0, α (t1+) ≤ α (t1) + I (α (t1)) ≤ α (t1) ≤ u (t1). To prove α (t) hits σ exactly once, let p (t) = t – γ (α (t)). If α (t1+) ≤ α (t1), then γ (α (t1+) = γ (α (t1)) = t1. (1.1.14) Using the mean value theorem, there exists a point ξ ϵ (α (t1+), α (t1)) Such that γ (α (t1)) – γ (α (t1+)) = γˈ ( ϵ ) [α (t1) – α (t1+) ] < 0. (1.1.15) « - Rampyari
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IRJMETS960448.pdf » www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [97] Let α (t) and u (t) do not hit the curve σ : t = γ (x) for t ϵ ( 0, T). Then α (t) ≤ for t ϵ J. Thus, we can consider at least one of them hits σ. Since γ is decreasing and γ-1 (0) > α (0+), α (t) hits the curve σ first. Therefore, there exists t1 ϵ (0, T) such that t1 = γ (α (t1)). Using α (t) ≤ u (t) on (0, t1) and I (x) ≤ 0, α (t1+) ≤ α (t1) + I (α (t1)) ≤ α (t1) ≤ « - Rampyari
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »3. Although condition (H0) may seem very restrictive to find the function β, if β can be defined as β (t) = C, where C is a constant such that f (t, c) ≤ 0 on J, then we have (a) βˈ(t) = 0 ≥ f (t, β (t)). (b) β (t+) = β (t) ≥ β (t) + I (β (t)), whenever I ϵ C1 ( R, Rˈ), (c ) γˈ (β (t)) βˈ(t) = 0 < 1.«
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IRJMETS960448.pdf »For it, since I* is increasing, I* (β (t3+)) ≥ I* (β (t3)) ≥ u (t3). Suppose that I* (β (t4)) < u (t4), using the condition (H3) and proceeding as in step one, we get a contradiction. Therefore, β (t4+) ≥ I* (β (t4)) ≥ u (t4). Employing the same procedures successively, u (t) ≤ β (t) for all t ϵ J. Hence the proof. II. CONCLUSION In this paper an attempt is made to study about the existence of solutions of functional boundary value problems, periodic boundary value problems, initial and« - Rampyari
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IRJMETS960448.pdf » condition (H0) instead of (H1). (ii) β (t3+) ≥ β (t3). If β (t) does not hit σ for t > t3, then the proof is over. suppose that there exists one-point t4 ϵ (t3, T) such that t4 = γ (β ( t4)). In this case it follows that β (t) ≥ u (t) for all t ϵ (t3, t4) and it remains to show that I* (β (t4)) ≥ u (t4). For it, since I* is increasing, I* (β (t3+)) ≥ I* (β (t3)) ≥ u (t3). Suppose that I* (β (t4)) ≥ u (t4), using the condition (H3) and proceeding as in step one, we get a contradiction.« - Rampyari
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IRJMETS960448.pdf » ≤ β (t2), u (t) ≤ β (t), for all t ϵ [t2, t3]. Using equations (4.1.6), (4.1.7), condition (H3) and the same arguments employed in step one it can be proved that β (t3+) ≥ I* (β (t3)) ≥ u (t3). This implies, u (t) does not meet σ for t > t3, but the condition (H1) does not assure the same for β (t). http://www.irjmets.com/ e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:04/April-2021 Impact Factor- 5.354 www.irjmets.com w« - Rampyari
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Using equations (4.1.6), (4.1.7), condition (H3) and the same arguments employed in step one it can be proved that β (t3. +) ≥ I* (β (t3)) ≥ u (t3). This implies, u ...«
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PERIODIC BOUNDARY VALUE PROBLEM FOR ... - IRJMETS »Then there exists one point s ϵ (t1, t2) such that α(s) = I*(u (s)) and α (t) > I* (u (t)) for s < t ≤ t2. Differentiating with respect to s αˈ(s). ≥ (I*) ˈ(u (s)) uˈ(s).«
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IRJMETS960448.pdf » and Science Volume:03/Issue:04/April-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [95] Then α (t) ≤ u (t), for t ϵ J. Thus, we consider at least one of them meets σ. Since γ is increasing and γ-1(0) < α (0+), α (t) hits the curve σ first. Therefore there exists t1 ϵ (0, T) such that t1 = γ (α (t1)). Using α (t) ≤ u (t) on (0, t1) and I (x) ≤ 0, α (t1+) ≤ α (t1) + I (α (t1)) ≤ α (t1) ≤ u (« - Rampyari
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IRJMETS960448.pdf » aim consists in applying these principles and introducing some new results in this direction to develop the method of upper and lower solutions in order to find an existence result for the following periodic boundary value problem with impulses at variable times, Consider the periodic boundary value problem with impulses at variable times, 𝑢′ (t) = f (t, u (t)), t ϵ J, t ≠ γ (u (t)) (1) u (t+) = u (t) + I (u (t)), t = γ (u (t)), (2) u (0) = u (T), (3) where J = [0, T], f ϵ C (J x R, R), I ϵ C1 (R,R), and γ ϵ C1 (R, R). 1Our aim consists in applying these principles and introducing some new results in this direction to develop the method of upper and lower solutions in order to find an existence result for the following periodic boundary value problem with impulses at variable times, Consider the periodic boundary value problem with impulses at variable times, 𝑢′ (t) = f (t, u (t)), t ϵ J, t ≠ γ (u (t)) (1) u (t+) = u (t) + I (u (t)), t = γ (u (t)), (2) u (0) = u (T), (3) where J = [0, T], f ϵ C (J x R, R), I ϵ C1 (R,R), and γ ϵ C1 (R, R).« - Rampyari
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Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times » some comparison principles have appeared see 4, 8 for impulsive differential equations at variable times. Our aim consists in applying these principles and introducing some new results in this direction to develop the method of upper and lower solutions in order to find an existence result for the following periodic boundary value problem with impulses at variable times, u9 t s f t , u t , t g J , t / g u t 1.1Ž .« - Bajo, I. et al.
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IRJMETS960448.pdf » value problem for first order impulsive differential equations at variable time. Keywords: Differential equation, Periodic boundary value problem, Impulsive Differential equation. I. INTRODUCTION There exist several papers about boundary value problems with impulsive effects at fixed points. Our aim consists in applying these principles and introducing some new results in this direction to develop the method of upper and lower solutions in order to find an existence result for the followin« - Rampyari
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excerpt.pdf » differential equations and inclusions, including scenarios of neutral equations, as well as semilinear models. The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer’s fixed point theorem, a Martelli fixed point theorem for multivalued condensing maps, and a Covitz-Nadler fixed point theorem for multivalued maps. 3.2. Impulsive functional differential equations In this section, we will establish existence theory for first- and «
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » differential equations and inclusions, including scenarios of neutral equations, as well as semilinear models. The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer’s fixed point theorem, a Martelli fixed point theorem for multivalued condensing maps, and a Covitz-Nadler fixed point theorem for multivalued maps. 3.2. Impulsive functional differential equations In this section, we will establish existence theory for first- and «
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

Impulsive Differential Equations and Inclusions - StudyLib »The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer's fixed point theorem, a Martelli fixed ...«
^{https://studylib.net/doc/18821535/impulsive-differential-equations-and-inclusions}

Impulsive Differential Equations and Inclusions »The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer's fixed point theorem, a Martelli fixed ...«
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excerpt.pdf » occurs in the derivative of the state variable, which are sometimes modelled by neutral differential equations or neutral differential inclusions. This chapter presents a theory for the existence of solutions of impulsive functional differential equations and inclusions, including scenarios of neutral equations, as well as semilinear models. The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer’s fixed point theorem, a Martelli«
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » occurs in the derivative of the state variable, which are sometimes modelled by neutral differential equations or neutral differential inclusions. This chapter presents a theory for the existence of solutions of impulsive functional differential equations and inclusions, including scenarios of neutral equations, as well as semilinear models. The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer’s fixed point theorem, a Martelli«
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JPMS-004-004-CORRECTED-PRABAKARAN-EXISTENCE-SPACES.pdf » state of the system. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable as well as in the independent variable, as in the so called neutral differential difference equations. The literature for neutral functional differential equations is the book by Hale and Lun« - user
^{https://www.cibtech.org/J-PHYSICS-MATHEMATICAL-SCIENCES/PUBLICATIONS/2014/Vol-4-No-2/JPMS-004-004-CORRECTED-PRABAKARAN-EXISTENCE-SPACES.pdf}

excerpt.pdf » played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral differential equations or neutral differential inclusions. This chapter presents a theory for the existence of solutions of impulsive functional differential equations and inclusions, including scenar«
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Discrete Approximations and Optimization of Evolution Inclusions and Equations » played an important role in areas involving hereditary phenomena for which a delay argument arises in the modeling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modeled by neutral differential equations or neutral differential inclusions. Indeed, the states of many evolutionary processes are often subject to instantaneous perturbations and experience abrupt changes at certain m« - Qamar Din
^{http://prr.hec.gov.pk/jspui/bitstream/123456789/1679/2/1505S.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral differential equations or neutral differential inclusions. This chapter presents a theory for the existence of solutions of impulsive functional differential equations and inclusions, including scenar«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

JNFA201523.pdf » Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades [13, 14, 15]. There are also a number of applications in which the delayed argument occurs in the derivative of the state NONLINEAR NEUTRAL FUZZY INTEGRODIFFERENTIAL SYSTEMS IN FUZZY SEMIGROUPS 3 variable as well as in the independent variable, as in the so called neutral differential difference equations.«
^{http://jnfa.mathres.org/issues/JNFA201523.pdf}

Structural, Syntactic, and Statistical Pattern Recognition: ... »... 1 (two modes: dark and bright), but there are also a number of applications in which the images present more than two models in a natural way, like medical ...«
^{https://books.google.com/books?id=I4BrCQAAQBAJ}

Two-Phase Flow Heat Exchangers: Thermal-Hydraulic ... »There are also a number of applications in which a compact geometry is needed to facilitate a highly efficient heating or cooling process involving boiling or ...«
^{https://books.google.com/books?id=2yr0CAAAQBAJ}

Theory of Functional Differential Equations »There are also a number of applications in which the delayed argument occurs in the derivative of the state variable as well as in the independent variable, the ...«
^{https://books.google.com/books?id=8_r2BwAAQBAJ}

Introduction to Functional Differential Equations »—r — r There are also a number of applications in which the delayed argument occurs in the derivative of the state variable as well as in the independent ...«
^{https://books.google.com/books?id=KZTaBwAAQBAJ}

Impulsive Differential Equations and Inclusions »There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral ...«
^{https://books.google.com/books?id=JlFHOT8dkucC}

Reliability of Computer Systems and Networks: Fault ... »In many problems, this is the primary measure of interest; however, there are also a number of applications in which another approach is important. In a digital ...«
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US20080130876A1 - Method for Private-Key Encryption of ... »However, there are also a number of applications in which security, although essential, has a lesser significance due to, for example, the low cost of the products ...«
^{http://www.google.co.in/patents/US20080130876}

IMPULSE® 2-2 with Integrated ... - Kurt J. Lesker Company »However, there are also a number of applications in which is it not. When smooth, dense film qualities are desired or required, HIPIMS can be an excellent ...«
^{https://www.lesker.com/newweb/process_instruments/kjlc-impulse-hipims-powersupply.cfm}

Thesis » and not the derivative of the state variables, then the functional differential equation is called retarded functional differential equations or retarded differential equations [Hal77]. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable as well as in the independent variable. Such functional differential equations are called neutral differential equations [Hal77]. Neutral differential equations are less common compared to re« - Vahid Samadi Bokharaie
^{http://149.157.192.253/publications/Vahid_Thesis.pdf}

S0025-5718-05-01814-4.pdf » simplest type of past dependence is that it is carried through the state variable but not through its derivative. Then the equation can be expressed as delay differential equations (DDEs). There are also a number of applications in which the delayed argument occurs in the derivative of the state variables as well as in the state variable itself. Therefore, their models can be formulated with linear neutral delay differential equations (NDDEs)(see [9]). Since analytical solutions of the above «
^{http://www.ams.org/mcom/2006-75-253/S0025-5718-05-01814-4/S0025-5718-05-01814-4.pdf}

S0025-5718-05-01814-4.pdf » simplest type of past dependence is that it is carried through the state variable but not through its derivative. Then the equation can be expressed as delay differential equations (DDEs). There are also a number of applications in which the delayed argument occurs in the derivative of the state variables as well as in the state variable itself. Therefore, their models can be formulated with linear neutral delay differential equations (NDDEs)(see [9]). Since analytical solutions of the above «
^{http://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01814-4/S0025-5718-05-01814-4.pdf}

Template » beyond 1Km while VCSEL based optical links can only currently achieve up to 300m. The extended reach offered by the POLYSYS approach is of particular relevance as the size of the data centers increases. There are also a number of applications in which it is a plus to limit the number of optical fibers used for a particular link and in all these scenarios high speed serial links like those being developed in POLYSYS are particularly attractive for the simplicity and reduced size.« - PCRL
^{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.675.5053&rep=rep1&type=pdf}

Controllability of nonlinear stochastic neutral fractional dynamical systems » often only a first approximation to the true situation and that a more realistic model would include some of the past states of the system. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable as well as in the independent variable, the so-called neutral differential difference equations. Such problems are more difficult to motivate but often arise in the study of two or more simple oscillatory systems with some interconnections« - Rajendran Mabel Lizzy, Krishnan Balachandran, Juan J. Trujillo
^{http://www.zurnalai.vu.lt/nonlinear-analysis/article/view/13374}

10.1.1.799.8031.pdf » ENSDF is basically a nuclear structure data file representing the evaluated status of measurements, and specific parameters such as u and the associated energy release from spontaneous fission may not be included. There are also a number of applications in which the measured data, alone (and hence ENSDP) are insufficient to solve specific problems. In these two instances the ENSDP data, need to be supplemented with data from other sources and xrith estimates of those asyet unmeasured d«
^{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.799.8031&rep=rep1&type=pdf}

Reconstruction of two-dimensional signals from level crossings - Proceedings of the IEEE » @ 1990 I E E E PROCEEDINGS OF THE IEEE, VOL. 78, NO. 1 , JANUARY 1990 more, one theory of human vision relies primarily on edge detection as the mechanism by which humans process visual information. There are also a number of applications in which it i s desired to recover a signal from its threshold crossings. For instance, an image which has been corrupted by a memoryless nonlinear distortion containing at least one monotonic region, can be recovered from threshold c« - IEEE
^{https://core.ac.uk/download/pdf/188665572.pdf}

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excerpt.pdf » and inclusions involving impulses, our attention in this chapter is turned to functional differential equations and inclusions each undergoing impulse effects. These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral diff«
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » and inclusions involving impulses, our attention in this chapter is turned to functional differential equations and inclusions each undergoing impulse effects. These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral diff«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

Impulsive Differential Equations and Inclusions - StudyLib »These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling ...«
^{https://studylib.net/doc/18821535/impulsive-differential-equations-and-inclusions}

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excerpt.pdf » differential inequality such as x′(t) ≤ f (t, x(t)), 3 Impulsive functional differential equations & inclusions 3.1. Introduction While the previous chapter was devoted to ordinary differential equations and inclusions involving impulses, our attention in this chapter is turned to functional differential equations and inclusions each undergoing impulse effects. These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument aris«
^{https://downloads.hindawi.com/books/978977594550/excerpt.pdf}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » point theorem for contraction multivalued operators. 3 Impulsive functional differential equations & inclusions 3.1. Introduction While the previous chapter was devoted to ordinary differential equations and inclusions involving impulses, our attention in this chapter is turned to functional differential equations and inclusions each undergoing impulse effects. These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument aris«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

Impulsive Differential Equations and Inclusions »Introduction While the previous chapter was devoted to ordinary differential equations and inclusions involving impulses, our attention in this chapter is turned to ...«
^{https://books.google.com/books?id=JlFHOT8dkucC}

Top PDF Practical Stability of Impulsive Differential Equations ... »While the previous chapter was devoted to ordinary diﬀerential equations and in- clusions involving impulses, our attention in this chapter is turned to functional ...«
^{https://1library.net/title/practical-stability-impulsive-differential-equations-supremum-integral-inequalities}

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IRJMETS960448.pdf » *1Department Of Mathematics, Sri Shakthi Institute Of Engineering And Technology, Coimbatore-641062, Tamilnadu, India. Email- 1082usha@gmail.com ABSTRACT This paper deals with existence results for a periodic boundary value problem and the corresponding initial value problem for first order impulsive differential equations at variable time. Keywords: Differential equation, Periodic boundary value problem, Impulsive Differential equation.« - Rampyari
^{https://irjmets.com/rootaccess/forms/uploads/IRJMETS960448.pdf}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Periodic boundary value problems for first-order ... - X-MOL »... equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.«
^{https://www.x-mol.com/paper/1335612593800704128?recommendPaper=1277746314451505152}

(PDF) Boundary value problem for first order impulsive ... »Dec 4, 2020 — An interesting example is also given to verify the results obtained. View. Show abstract ... Recently, the existence of solutions for such problems ...«
^{https://www.researchgate.net/publication/265891240_Boundary_value_problem_for_first_order_impulsive_difference_equations}

Two-Point Boundary Value Problems for First Order Causal ... »Apr 10, 2021 — An interesting example is also given to verify the results obtained. View. Show abstract. Difference equations involving causal operators with ...«
^{https://www.researchgate.net/publication/348243962_Two-Point_Boundary_Value_Problems_for_First_Order_Causal_Difference_Equations}

Top PDF Some properties of algebraic difference equations of ... »... equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.«
^{https://1library.net/title/properties-algebraic-difference-equations-order}

Top PDF Periodic boundary value problems for nonlinear first ... »... equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.«
^{https://1library.net/title/periodic-boundary-problems-nonlinear-impulsive-dynamic-equations-scales}

Periodic boundary value problems for first-order impulsive difference equations with time delay » methods of lower and upper solutions, an existence theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained. MSC: 39A10; 34B37 Keywords: Impulsive difference equations; Time delay; Comparison principle; Periodic boundary value problem; Extremal solutions 1 Introduction The mathematical model of many real-world phenomena ca« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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exportPdf » property holds for the tactical continuous, and strategic discrete and continuous subtree location problems in a tree network with the ordered median objective, where the λ-weights take at most two different values. These results extend some existing results in the literature. With these nestedness results, we solve the problems in polynomial time. Finally we pose an open problem on identifying the nestedness property for the (k1, k2)-trimmed problem. 1. Introduction. Network location proble«
^{https://www.aimsciences.org/article/doi/10.3934/jimo.2012.8.41}

Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — ... equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

existing results in the | English examples in context | Ludwig »These results extend some existing results in the literature. 6. Advances in Difference Equations. Our theorems extend several existing results in the current ...«
^{https://ludwig.guru/s/existing+results+in+the}

Multiplicity of solutions for Schrödinger-Poisson system with ... »by X Peng · 2021 — ... main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.«
^{https://www.aimspress.com/article/doi/10.3934/math.2021126?viewType=HTML}

Periodic boundary value problems for first-order ... - X-MOL »... theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature.«
^{https://www.x-mol.com/paper/1335612593800704128?recommendPaper=1277746314451505152}

(PDF) A note on the subtree ordered median problem in ... »Oct 6, 2020 — These results extend some existing results in the literature. With these nestedness results, we solve the problems in polynomial time. Finally we ...«
^{https://www.researchgate.net/publication/277688299_A_note_on_the_subtree_ordered_median_problem_in_networks_based_on_nestedness_property}

(PDF) Multiplicity of solutions for Schrödinger-Poisson system ... »Feb 1, 2021 — These results extend some existing results in the literature. ResearchGate Logo. Discover the world's research. 20+ million members; 135+ ...«
^{https://www.researchgate.net/publication/348127908_Multiplicity_of_solutions_for_Schrodinger-Poisson_system_with_critical_exponent_in_mathbbR3}

A note on the subtree ordered median problem in networks ... »by H Tang · 2012 · Cited by 9 — These results extend some existing results in the literature. With these nestedness results, we solve the problems in polynomial time. Finally we ...«
^{https://research.polyu.edu.hk/en/publications/a-note-on-the-subtree-ordered-median-problem-in-networks-based-on}

American Institute of Mathematical Sciences »Aug 1, 2012 — These results extend some existing results in the literature. With these nestedness results, we solve... Huajun Tang, T. C. Edwin Cheng, Chi To ...«
^{https://www.aimsciences.org/journal/1547-5816/2012/8/1}

NAVER Academic > 具正负系数的二阶非线性中立型时滞差分 ... »... positive solutions are obtained. These results extend some existing results in the literature. The examples are presented to illustrate the effects of our theorems ...«
^{https://academic.naver.com/article.naver?doc_id=296039866}

Top PDF Some properties of algebraic difference equations of ... »... theorem of extremal solutions to ﬁrst-order impulsive diﬀerence equations with delay is obtained. These results extend some existing results in the literature.«
^{https://1library.net/title/properties-algebraic-difference-equations-order}

Top PDF Periodic boundary value problems for nonlinear first ... »... theorem of extremal solutions to ﬁrst-order impulsive diﬀerence equations with delay is obtained. These results extend some existing results in the literature.«
^{https://1library.net/title/periodic-boundary-problems-nonlinear-impulsive-dynamic-equations-scales}

correlation extremal principle - 专否 »These results extend some existing results in the literature. An interesting example is also given to verify the results obtained. Impulsive difference equations ...«
^{https://zhuanfou.com/data/search?q=correlation extremal principle}

Periodic boundary value problems for first-order impulsive difference equations with time delay » derived. By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained. MSC: 39A10; 34B37 Keywords: Impulsive difference equations; Time delay; Comparison principle; Periodic boundary value problem; Extremal solutions 1 I« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Periodic boundary value problems for first-order ... - X-MOL »By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order ...«
^{https://www.x-mol.com/paper/1335612593800704128?recommendPaper=1277746314451505152}

Monotone iterative technique for initial-value problems of ... »Oct 27, 2020 — By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to ...«
^{https://www.researchgate.net/publication/242981032_Monotone_iterative_technique_for_initial-value_problems_of_nonlinear_singular_discrete_systems}

Periodic boundary value problem for the first order functional ... »By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order ...«
^{https://www.researchgate.net/publication/225911046_Periodic_boundary_value_problem_for_the_first_order_functional_differential_equations_with_impulses}

Top PDF Periodic boundary value problems for nonlinear first ... »By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to ﬁrst-order ...«
^{https://1library.net/title/periodic-boundary-problems-nonlinear-impulsive-dynamic-equations-scales}

Top PDF Some properties of algebraic difference equations of ... »By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to ﬁrst-order ...«
^{https://1library.net/title/properties-algebraic-difference-equations-order}

Periodic boundary value problems for first-order impulsive difference equations with time delay » conditions for the existence and uniqueness of solutions to the corresponding linear problem of the boundary value problem are derived. By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

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Positive solutions of singular multi-point discrete boundary ... »by M Mohamed · 2018 — By using the Krasnoselskii's fixed point theorem, sufficient conditions for the existence of positive solutions are established. Under new conditions when f is ...«
^{https://ui.adsabs.harvard.edu/abs/2018AIPC.1974c0001M/abstract}

Existence of multiple positive solutions of a nonlinear ... - GWDG »by G Wanga · Cited by 12 — fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established. Keywords: Positive ...«
^{http://ftp.gwdg.de/pub/EMIS/journals/EJQTDE/p1252.pdf}

Existence of solutions of boundary value problems for singular ... »Feb 3, 2016 — Guo-Krasnoselskii fixed point theorem, sufficient conditions for the existence and uniqueness results of the problem were established. In [26] ...«
^{https://link.springer.com/content/pdf/10.1007/s12190-016-0994-y.pdf}

Periodicity and stability in neutral nonlinear differential ... »Mar 31, 2021 — By applications of Green's function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic ...«
^{https://www.researchgate.net/publication/326002866_Periodicity_and_stability_in_neutral_nonlinear_differential_equations_by_Krasnoselskii's_fixed_point_theorem}

Existence nonexistence and multiplicity of periodic solutions ... »Nov 30, 2020 — By applications of Green's function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic ...«
^{https://www.researchgate.net/publication/238860658_Existence_nonexistence_and_multiplicity_of_periodic_solutions_for_a_kind_of_functional_differential_equation_with_parameter}

MULTIPLICITY RESULTS OF FOURTH-ORDER SINGULAR ... »By applications of Green's function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.«
^{http://www.jaac-online.com/article/doi/10.11948/2017029?viewType=HTML}

Positive solutions for a second order multi-point boundary ... »by A Lakmeche · 2018 — ... fixed point theorem, sufficient conditions for the existence of nontrivial solution are obtained, which improve the results of J. Chen et al. [3].«
^{https://arxiv.org/abs/1801.02528}

Academic Journals - Tubitak »By using the Avery-Peterson fixed point theorem, sufficient conditions for the existence of at least three positive solutions to the boundary value problem ...«
^{https://journals.tubitak.gov.tr/math/abstract.htm;jsessionid=8E54A24FFAAE27EBBBEB154F29C1113F?id=10081}

order differential equation with singularit - Wiley Online Library »by Z Cheng · 2014 · Cited by 21 — of Green's functions and Schauder's fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.«
^{https://onlinelibrary.wiley.com/doi/pdf/10.1002/mma.2975}

PROCEEDINGS_REPORT.pdf » existence of positive solutions for a second order multi-point boundary value problem with delay. Under certain growth conditions on the nonlinearity, and by the mean of Leray-Schauder fixed point theorem, sufficient conditions for the existence of nontrivial solution are obtained, which improve the results of literature J. Chen and al. [2] References [1] D. Bai, and Y. Xu, Existence of positive solutions for boundary-value problems of second order delay differential equations, Appl.«
^{https://icramcs2019.sciencesconf.org/page/customizable_page_2}

Abstract and Applied Analysis Articles (Project Euclid) » singular third-order differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green’s function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established. New Explicit Bounds on Gamidov Type Integral Inequalities for Functions in Two Variables and Their Applications Kelong Cheng, Chunxiang Guo.«
^{http://projecteuclid.org/feeds/euclid.aaa_article_rss.xml}

207045967.pdf » i= αiu(ξi) = , u′() + m–∑ i= αiu(ηi) = , where  < ξ < ξ < · · · < ξm– < ,  < η < η < · · · < ηm– < , ξi < ηi, αi >  for i = , , . . . ,m – . By using fixed point index theory [] and the Legget-Williams fixed point theorem [], sufficient conditions for the existence of countably many positive solutions are established. © 2013 Tokmak and Karaca; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (ht«
^{https://core.ac.uk/download/pdf/207045967.pdf}

/mnt/raid/home/ejqtde/temp/WaSorFin.dvi » fractional derivative of order α, f : [0, ∞) → [0, ∞), a : [0, 1] → (0, ∞) and θ : (0, 1) → (0, 1] are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established. Keywords: Positive solution; advanced arguments; fractional differential equations; fixed point index theory; Leggett-Williams fixed point theorem.«
^{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.449.2735&rep=rep1&type=pdf}

0354-51801912627C.pdf » Chengdu, Sichuan 610064, PR China Abstract. In this paper, we consider a kind of second-order neutral differential equation with timedependent deviating arguments. By applications of Krasnoselskii’s fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established. 1. Introduction In this paper, we consider the following second-order neutral differential equation with time-dependent deviating arguments (x(t) − cx(t −τ))′′ + a(t)x(t) = f (t, x(t −δ«
^{https://doi.org/10.1186/s13661-017-0790-0}

Periodic boundary value problems for first-order impulsive difference equations with time delay » impulsive difference equations with time delay. Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer’s fixed point theorem, sufficient conditions for the existence and uniqueness of solutions to the corresponding linear problem of the boundary value problem are derived. By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order imp« - Wenli Wang
^{https://doaj.org/article/44b36bdcf33443f193e93510ff509cbc}

LAKMECHE.dvi » solutions for a multi-point boundary value problem for a second order delay differential equation. Under certain growth conditions on the nonlinearity, and by the mean of Leray-Schauder fixed point theorem, sufficient conditions for the existence of nontrivial solution are obtained, which improve the results of J. Chen et al. [3]. MOTS-CLÉS : Solution positive, Equation différentielle à retard, Problème aux limites en plusieurs points, Théorème de point fixe de Leray-Schauder.« - Abdelkader Lakmeche, Horiya Habbaze, Ahmed Lakmeche
^{https://hal.archives-ouvertes.fr/hal-01677056/file/LAKMECHE.pdf}

Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity » have monotonicity of the nonlinearity f (t, u) since we derive the properties of the corresponding integral kernel function and get a more accurate inequality than the literature [12]. By a fixed point theorem, sufficient conditions for the existence of positive solutions of boundary value problem (1.1) are obtained. It is worth mentioning that the nonlinearity f (t, u) does not need to be nonnegative and lower bounded. We also focus on studying the impact of the parameter λ on the existence o« - Mei Jia
^{https://doaj.org/article/9847c679dc4d4c2cbd1b545386123882}

art:10.1186/1687-1847-2014-96.pdf » )] , i = , , . . . ,n, (.) where all the coefficients of system (.) are positive continuous almost periodic functions. In [], with the help of a variable substitution and by applying Schauder’s fixed point theorem, sufficient conditions for the existence of positive almost periodic solutions are obtained. ©2014 Liao and Xu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0)«
^{http://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/1687-1847-2014-96?site=advancesindifferenceequations.springeropen.com}

art:10.1186/1687-1847-2011-2.pdf »China Full list of author information is available at the end of the article Abstract In this article, we consider the existence of at least one positive solution to the three-point boundary value problem for nonlinear fractional-order differential equation with an advanced argument{ CDαu(t) + a(t)f (u(θ(t))) = 0, 0 < t < 1, u(0) = u′′(0) = 0, βu(η) = u(1), where 2«
^{http://advancesindifferenceequations.springeropen.com/track/pdf/10.1186/1687-1847-2011-2?site=advancesindifferenceequations.springeropen.com}

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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's fixed point ...«
^{https://link.springer.com/article/10.1186/s13662-018-1539-5}

Periodic boundary value problem for the first order functional ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's fixed point theorem, sufficient ...«
^{https://www.researchgate.net/publication/225911046_Periodic_boundary_value_problem_for_the_first_order_functional_differential_equations_with_impulses}

(PDF) Causal difference equations with upper and lower solutions in ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's fixed point theorem, sufficient ...«
^{https://www.researchgate.net/publication/332351976_Causal_difference_equations_with_upper_and_lower_solutions_in_the_reverse_order}

Search - SpringerOpen »Notions of lower and upper solutions are introduced, with which two new ... Authors: Jingfeng Tian, Wenli Wang and Wing-Sum Cheung. Citation: Advances in ...«
^{https://www.springeropen.com/Search?searchType=publisherSearch&sort=Relevance&page=4482}

Top PDF Some properties of algebraic difference equations of ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's ﬁxed point theorem, suﬃcient ...«
^{https://1library.net/title/properties-algebraic-difference-equations-order}

Top PDF Periodic boundary value problems for nonlinear first ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's ﬁxed point theorem, suﬃcient ...«
^{https://1library.net/title/periodic-boundary-problems-nonlinear-impulsive-dynamic-equations-scales}

Download Link AN INTRODUCTION TO DIFFERENCE EQUATIONS ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's fixed point theorem, sufficient ...«
^{https://ep.uvart.ru/13.html}

Search results ADVANCES IN TIME DELAY SYSTEMS 1ST ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefers fixed point theorem, sufficient ...«
^{https://3i.antikvar-24.ru/1843.html}

TWO POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Fredholm boundary value problems for first ...«
^{https://db.mimitoy.ru/376}

Anti-Periodic Boundary Value Problems for Nonlinear Langevin ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Periodic boundary value for impulsive ...«
^{https://uo.photo-bk.ru/588}

Differential equations and boundary value problems computing and ... »... impulsive difference equations with time delay Notions of lower and upper solutions are introduced, with which two new comparison theorems are established ...«
^{https://1.litrf.site/Gdk}

Download differential equations and boundary value problems ... »... impulsive difference equations with time delay Notions of lower and upper solutions are introduced, with which two new comparison theorems are established ...«
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Download : Differential Equations And Boundary Value Problems ... »... impulsive difference equations with time delay Notions of lower and upper solutions are introduced, with which two new comparison theorems are established ...«
^{https://ir.fileg.site/BL7}

Download : An Introduction To Difference Equations Undergraduate ... »... impulsive difference equations with time delay Notions of lower and upper solutions are introduced, with which two new comparison theorems are established ...«
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Impulsive Boundary Value Problems for Dynamical Inclusions on ... »Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. 21 nov 2019 in this paper, we study the ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » is available at the end of the article Abstract This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer’s fixed point theorem, sufficient conditions for the existence and uniqueness of solutions to the corresponding linear problem of the boundary value problem are derived. By utilizi« - Wenli Wang
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions ...«
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Periodic boundary value problems for first-order impulsive ... »by J Tian · 2018 · Cited by 9 — This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions ...«
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Periodic boundary value problems for first-order impulsive ... »This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions of lower and ...«
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Periodic boundary value problems for first-order impulsive difference equations with time delay » wscheung@hku.hk 3Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong Full list of author information is available at the end of the article Abstract This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer’s fixed point theorem, sufficient conditions for the exis« - Wenli Wang
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IRJMETS960448.pdf » [8] Lakshmikantham.V, The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments, Application analysis, 36(1990),119-130. [9] LIUX, Non-linear boundary value problem for first order impulsive differential equations, Application Analysis 36(1990),119-130. [10] Pierson.C.-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential equations Dynamic Systems,1(1993),185-196. http://www.i« - Rampyari
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81058935.pdf » Mi, J, Han, M: Periodic boundary value problems for the first order impulsive functional differential equations. Appl Math Comput. 165, 433–446 (2005). doi:10.1016/j.amc.2004.06.022 5. Zhang, F, Li, M, Yan, J: Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay. Comput Math Appl. 51, 927–936 (2006). doi:10.1016/j.camwa.2005.11.028 6. Chen, L, Sun, J: Nonlinear boundary problem of first order impulsive integro-differential equations. J Comput Appl «
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excerpt.pdf » Banach spaces, Journal of Mathematical Analysis and Applications 263 (2001), 763–780. [49] , Impulsive neutral functional differential equations in Banach spaces, Applicable Analysis 80 (2001), 353–365. [50] , On a periodic boundary value problem for first order impulsive differential inclusions, Dynamic Systems and Applications 10 (2001), no. 4, 477–488. [51] , On nonresonance impulsive functional differential inclusions with periodic boundary conditions, International Journal of Applied Mat«
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cai.dvi »ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 37, Number 1, 2007 PERIODIC BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE DIFFERENTIAL EQUATION AT RESONANCE GUOLAN CAI, ZENGJI DU AND WEIGAO GE ABSTRACT. We develop a general theorem concerning the existence of solutions to the periodic boundary value problem for the first-order impulsive differential equation, { x′(t) = f(t, x(t)) t ∈ J \ {t1, t2, . . . , tk} x(ti) = Ii(x(ti)) i = 1, 2 . . . , k x(0) = x(T ).«
^{https://projecteuclid.org/euclid.rmjm/1181069320}

Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf » Banach spaces, Journal of Mathematical Analysis and Applications 263 (2001), 763–780. [49] , Impulsive neutral functional differential equations in Banach spaces, Applicable Analysis 80 (2001), 353–365. [50] , On a periodic boundary value problem for first order impulsive differential inclusions, Dynamic Systems and Applications 10 (2001), no. 4, 477–488. [51] , On nonresonance impulsive functional differential inclusions with periodic boundary conditions, International Journal of Applied Mat«
^{http://213.230.96.51:8090/files/ebooks/Matematika/Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions (Hindawi, 2006)(ISBN 977594550X)(O)(381s) MCde .pdf}

Periodic boundary value problem for first-order impulsive ... »by Z He · 2002 · Cited by 46 — This paper investigates the existence of minimal and maximal solutions of the periodic boundary value problem for first-order impulsive differential equations by ...«
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Nonhomogeneous boundary value problem for first-order ... »by F Zhang · 2006 · Cited by 11 — Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay*. Author links open overlay panelFengqinZhang ...«
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Antiperiodic Boundary Value Problems for First-Order ... »by Y Luo · 2020 — This paper discusses the antiperiodic boundary value problem for first-order impulsive differential inclusions. By using Martelli's fixed point ...«
^{https://link.springer.com/article/10.1007/s13226-020-0471-z}

(PDF) Periodic Boundary Value Problem for First Order ... »Periodic Boundary Value Problem for First Order Impulsive Differential Equation at Resonance. February 2007; Rocky Mountain Journal of Mathematics 37(1).«
^{https://www.researchgate.net/publication/38371020_Periodic_Boundary_Value_Problem_for_First_Order_Impulsive_Differential_Equation_at_Resonance}

Boundary value problem for first order impulsive differential ... »Request PDF | On Jan 1, 2006, Binbin Du and others published Boundary value problem for first order impulsive differential equation with a parameter and ...«
^{https://www.researchgate.net/publication/265507495_Boundary_value_problem_for_first_order_impulsive_differential_equation_with_a_parameter_and_infinite_delay}

PERIODIC BOUNDARY VALUE PROBLEM FOR FIRST ... »maximal and minimal solutions of the periodic boundary value problem for first order impulsive differential equations with supremum is considered. Key Words ...«
^{http://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/2000827c_133.pdf}

Mathematics in the Context of Fuzzy Sets: Basic Ideas, Concepts, and Some Remarks on the History and Recent Trends of Development » Differential Equations. Springer-Verlag, New York, Heidelberg, Berlin, 1977. Math. Model. Anal., 16(2):304–314, 2011. http://dx.doi.org/10.1007/BF02256795 314 I. Stamova [4] Z. He, P. Wang and W. Ge. Periodic boundary value problem for first order impulsive differential equations with supremum. Indian J. Pure Appl. Math., 34(1):133–143, 2003. [5] J. Kato. On Liapunov–Razumikhin type theorems for functional differential equations. Funkcial. Ekvac., 16:225–239, 1973. [6] V. Lakshmikantha« - A. Šostaks
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xie.dvi »Analysis, Vol. 3, 2009, no. 12, 581 - 590 Integral Boundary Value Problems for First Order Impulsive Differential Inclusions1 Jingli Xie 2, Guoping Chen and Xiaofei He Department of Mathematics, Jishou University Jishou, Hunan 416000, P.R. China Abstract In this paper, we study the integral boundary value problem for first order i«
^{http://www.m-hikari.com/ijma/ijma-password-2009/ijma-password9-12-2009/xieIJMA9-12-2009.pdf}

2652.pdf »See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/38371020 Periodic Boundary Value Problem for First Order Impulsive Differential Equation at Resonance ARTICLE in ROCKY MOUNTAIN JOURNAL OF MATHEMATICS · FEBRUARY 2007 Impact Factor: 0.4 · DOI: 10.1216/rmjm/1181069320 · Source: OAI CITATIONS 5 READS 20 3 AUTHORS, INCLUDING: Zengji Du 45 PUBLICATIONS «
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10.1.1.504.3980.pdf » Nonlinear Anal. (2006), doi: 10.1016/ j.na.2006.06.033. [9] A. M. Samoilenko and N. A. Perestyuk; Impulsive Differential Equations, World Scientific, Singapore, (1995). [10] F. Q. Zhang, M. L. Li, J. R. Yan; Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay, Comput. Math. Appl., 51(2006), 927-936. Yu Tian School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China E-mail address: tianyu2992@163.com 1. Introduc«
^{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.504.3980&rep=rep1&type=pdf}

doi:10.1016/j.jmaa.2006.01.027 » ∣∣Ik(x)∣∣ ak + bk|x|αk , k = 1, . . . , p, hold for all t ∈ J and x ∈ R. The sublinear case of the impulsive functions has also been considered recently in [46]. In [1], Nieto considered the following periodic boundary value problem for first order impulsive differential equation:⎧⎨ ⎩ x′(t) + F (t, x(t)) = 0, a.e. t ∈ [0, T ] \ {t1, . . . , tp}, x(t + k ) − x(tk) = Ik(x(tk)), k = 1, 2, . . . , p, x(0) = x(T ), (2) where 0 = t0 < t1 < · · · < tp < tp+1 = 1, F is an impulsive Caratheodory f«
^{https://core.ac.uk/download/pdf/82543090.pdf}

10.1.1.456.8441.pdf » · · ,ur) −g(y,u1, · · · ,ur) ≤ L(x−y) for all α(0) ≤ y ≤ x ≤ β(0) and α(ti) ≤ ui ≤ β(ti), i = 1, · · · ,r. Using fixed point theorems and the lower and upper solution methods, in [30], a pioneer paper concerning the solvability of periodic boundary value problem for first order impulsive differential equation ( IBVP for short ), Nieto studied the solvability of  x′(t) + λx(t) = F(t, x(t)), t ∈ [0,T] \{t1, · · · , tp}, x(t+k ) −x(tk) = Ik(x(tk)), k = 1, · · · ,p x(0 = x(T), (1.4) where λ 6= «
^{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.456.8441&rep=rep1&type=pdf}

1687-2770-2012-38 » Mi, J, Han, M: Periodic boundary value problems for the first order impulsive functional differential equations. Appl Math Comput. 165, 433–446 (2005). doi:10.1016/j.amc.2004.06.022 5. Zhang, F, Li, M, Yan, J: Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay. Comput Math Appl. 51, 927–936 (2006). doi:10.1016/j.camwa.2005.11.028 6. Chen, L, Sun, J: Nonlinear boundary problem of first order impulsive integro-differential equations. J Comput Appl «
^{http://boundaryvalueproblems.springeropen.com/track/pdf/10.1186/1687-2770-2012-38?site=boundaryvalueproblems.springeropen.com}

s13661-014-0186-3 »Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 69, 83-96 (2005) 6. Qian, D, Li, X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 303, 288-303 (2005) 7. Cai, G, Du, Z, «
^{http://boundaryvalueproblems.springeropen.com/track/pdf/10.1186/s13661-014-0186-3?site=boundaryvalueproblems.springeropen.com}

A STUDY ON PERIODIC BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE DIFFERENTIAL EQUATIONS AT VARIABLE TIME

Dr. R. SEENIVASAN.

Asst. Professor, Dept of Mathematical Economics, School of Economics, M.K.University, Madurai – 625 021.

see666@rediffmail.com

ABSTRACT:

This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer’s fixed point theorem, sufficient conditions for the existence and uniqueness of solutions to the corresponding linear problem of the boundary value problem are derived. By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained. This paper deals with existence results for a periodic boundary value problem and the corresponding initial value problem for first order impulsive differential equations at variable time.

Keywords: Difference equations , Time delay Periodic boundary value problem, Impulsive Differential equation.

INTRODUCTION:

While the previous chapter was devoted to ordinary differential equations and inclusions involving impulses, our attention in this chapter is turned to functional differential equations and inclusions each undergoing impulse effects. These equations and inclusions have played an important role in areas involving hereditary phenomena for which a delay argument arises in the modelling equation or inclusion. There are also a number of applications in which the delayed argument occurs in the derivative of the state variable, which are sometimes modelled by neutral differential equations or neutral differential inclusions. This chapter presents a theory for the existence of solutions of impulsive functional differential equations and inclusions, including scenarios of neutral equations, as well as semilinear models. The methods used throughout the chapter range over applications of the Leray-Schauder nonlinear alternative, Schaefer’s fixed point theorem, a Martelli fixed point theorem for multivalued condensing maps, and a Covitz-Nadler fixed point theorem for multivalued maps. There exist several papers about boundary value problems with impulsive effects at fixed points. Our aim consists in applying these principles and introducing some new results in this direction to develop the method of upper and lower solutions in order to find an existence result for the following periodic boundary value problem with impulses at variable times, Consider the periodic boundary value problem with impulses at variable times,

𝑢 ′ (t) = f (t, u (t)), t ϵ J, t ≠ γ (u (t)) --------------------(1)

u (t+) = u (t) + I (u (t)), t = γ (u (t)), ---------------------(2)

u (0) =u (T),--------------------------------------------------- (3)

where J = [0, T], f ϵ C (J x R, R), I ϵ C1 (R,R),

and γ ϵ C1 (R, R).

Periodic boundary value problems for first order functional differential equations with impulse

Mathematics of computing

Mathematical analysis

Numerical analysis

Numerical differentiation

Existence results for the initial value problem corresponding to Periodic boundary value problem The initial value problem corresponding to the periodic boundary value problem (1.1) – (1.3) is of the form

uˈ (t) = f (t, u (t)), t ≥ t0, t ≠ γ (u (t)) ------------------(1.1.1)

u (t+) = u (t) + I (u (t)), t = γ (u (t)), -------------------(1.1.2)

u (t0 +) = u0,---------------------------------------------- (1.1.3)

with (t0, u0) ϵ J x R.

Theorem 1.1.1.

Let α, β ϵ PC (J, R)

such that αˈ (t) ≤ f (t, α (t)), t ϵ J, t ≠ γ (α (t))----------------------- (1.1.4)

α (t+) ≤ α (t) + I (α (t)), t = γ (α (t))------------------------ (1.1.5)

and

βˈ (t) ≥ f (t, β (t)), t ϵ J, t ≠ γ (β (t))---------------------- (1.1.6)

β (t+) ≥ β (t) + I (β (t)), t = γ (β (t)).----------------------- (1.1.7)

Assume that the following conditions are satisfied:

(H0) γˈ (β (t)) βˈ(t) < 1, for all t ϵ J.

(H1) γˈ (x) f (t, x) < 1 for all t ϵ J. for all x ϵ R.

(H2) γ (x) is increasing and γ (α (0+)) > 0.

(H3) I* (x) = x + I (x) is increasing and (I*) ˈ (x) f (t, x) > f (t, I* (x)).

Then the solution u (t) of equation (4.1), (4.2) with u (0+) = u0 satisfies α (0+) ≤ u0 ≤ β (0+) implies

α (t) ≤ u (t) ≤ β (t), for all t ϵ J

Then α (t) ≤ u (t), for t ϵ J.

Proof:

The proof of the theorem is divided into two steps.

Step 1:

To prove α (0+) ≤ u0 implies α (t) ≤ u (t), for all t ϵ J.

Let α (t) and u (t) do not hit the curve σ: t = γ (x) for t ϵ (0, T).

Thus, we consider at least one of them meets σ.

Since γ is increasing and γ-1(0) < α (0+), α (t) hits the curve σ first.

Therefore there exists t1 ϵ (0, T) such that t1 = γ (α (t1)).

Using α (t) ≤ u (t) on (0, t1) and I (x) ≤ 0,

α (t1 +) ≤ α (t1) + I (α (t1))

≤ α (t1) ≤ u (t1).

To prove α (t) hits σ exactly once, let p (t) = t – γ (α (t)). If α (t1 +) = α(t1),

then γ (α (t1 +) = γ (α (t1)) = t1. (1.1.8)

Using the mean value theorem, there exists a point ζ ϵ (α (t1 +), α (t1)) such that γ (α (t1)) – γ (α (t1 + ) = γˈ (ζ ) [ α (t1) – α (t1 +) ] > 0.

This implies that p (t1) = t1 – γ (α (t1 +)) ≥ t1 – γ (α (t1 +)) ≥ t1 – t1 = 0 (using 1.1.8) Differentiating with respect to t pˈ(t) ≥ 1 – γˈ( α(t)) f (t, α(t)) > 0. (using H1) As a consequence, p (t) > 0, for all t > t1 Therefore, α (t) does not hit σ for t > t1. If u (t) ≠ γ (u (t)) for t ϵ J, using α (t1 + ) ≤ u (t1) and the comparison principle for ordinary differential equations α (t) ≤ u (t), for all t ϵ J.

Now assume that there exists t2 ϵ (t1, T) such that u (t2) = γ (u (t2)).

Since u (t1) ≥ α (t1 + ), it follows that α (t) ≤ u (t) for t ϵ (t1 t2).

To show that u (t2 +) = I*(u (t2)) ≥ α (t2), (1.1.9)

suppose that (1.1.9) is not true.

Since α (t1) ≤ u ( t1) and I* is increasing, α (t1 1) ≤ I* (α (t1)) ≤ I*(u (t1)).

Then there exists one point s ϵ (t1, t2) such that α(s) = I*(u (s))

and α (t) > I* (u (t)) for s < t ≤ t2.

Differentiating with respect to s

αˈ(s) ≥ (I*) ˈ(u (s)) uˈ(s)

= (I*)ˈ(u (s)) f (s, u (s))

> f (s, I*(u (s))) (using condition (H3))

= f (s, α (s)),

since α (s) = I*(u (s)).

But this is contradiction to αˈ (t) ≤ f (t, α (t)) and hence (1.1.9) follows.

Hence u (t) hits σ only for t = t2 and then α (t2 +) implies α (t) ≤ u (t)

for all t ϵ ( t2 , T).

Hence the proof of step one.

Step 2:

To prove β (0+) ≥ u0 implies β (t) ≥ u (t), for all t ϵ J. The proof of this step will follow the same procedure as in step one. The main difference is that the graph of the function β can meet the curve σ several times. It is easy to see that u (t) hits σ before than β (t).

Let t2 = γ (u (t2)).

u (0+) ≤ β (0+) implies u (t) ≤ β (t), for all t ϵ [ 0, t2].

But

u (t2 +) = u (t2) + I (u (t2)) (using (1.1.9))

≤ u (t2) ≤ β (t2). If β (t) does not hit σ,

then u (t2 +) ≤ β (t2) → u (t) ≤ β (t), for all t ϵ (t2, T].

Now assume that there exists one-point t3 ϵ (t2, T) such that t3 = γ (β (t3)).

Since u (t2 +) ≤ β (t2), u (t) ≤ β (t), for all t ϵ [t2, t3].

Using equations (4.1.6), (4.1.7), condition (H3) and the same arguments employed in step one it can be proved that β (t3 +) ≥ I* (β (t3)) ≥ u (t3).

This implies, u (t) does not meet σ for t > t3, but the condition (H1) does not assure the same for β (t).

Therefore, there are two possibilities:

β ( t3 +) < β (t3).

In this case, it is possible to prove that β (t) does not meet σ for t > t3 and

thus β (t3 +) ≥ u (t3) implies that β (t) ≥ u (t) for every t ϵ ( t3, T].

The proof of this assertion is very similar to the corresponding one in the first step, using

the condition (H0) instead of (H1).

β (t3 +) ≥ β (t3).

If β (t) does not hit σ for t > t3, then the proof is over.

suppose that there exists one-point t4 ϵ (t3, T) such that t4 = γ (β ( t4)).

In this case it follows that β (t) ≥ u (t) for all t ϵ (t3, t4) and it remains to show that I* (β (t4)) ≥ u (t4). For it, since I* is increasing, I* (β (t3 +)) ≥ I* (β (t3)) ≥ u (t3).

Suppose that I* (β (t4)) ≥ u (t4), using the condition (H3) and proceeding as in step one,

we get a contradiction.

Therefore, (β (t4 +)) ≥ I* (β (t4)) ≥ u (t4),

Employing the same procedures successively,

u (t) ≤ β (t) for all t ϵ J.

The proof is complete.

Result 1.1.1.

The conclusions of theorem (4.1.1) remain valid, for the function

γ ϵ C1 (R, (0, + ∞)), instead of assuming the condition γ (α ( 0+)) > 0.

Result 1.1.2.

The same arguments employed in the proof of theorem (1.1.1) permit us to show its validity when we consider several curves σk : t = γk (x), k = 1,2,…p and the corresponding impulses Ik satisfying the same conditions that γ and I satisfy respectively, together with the following assumptions,

γ k (x) < γ k+1 (x), k = 1,2,., p –1.

(ii) γ k+1 (I* (x)) > γ k (x), k = 1, 2,…, p –1.

Result 1.1.3.

Although condition (H0) may seem very restrictive to find the function β,

if β can be defined as β (t) = C, where C is a constant such that f (t, c) ≤ 0 on J,

then we have

(a) βˈ(t) = 0 ≥ f (t, β (t)).

(b) β (t+) = β (t) ≥ β (t) + I (β (t)), whenever I ϵ C1 ( R, Rˈ),

(c ) γˈ (β (t)) βˈ(t) = 0 < 1.

Theorem 1.1.2.

Let α, β ϵ PC (J, R) such that

αˈ(t) ≤ f ( t, α (t)), t ϵ J, t ≠ γ (α (t)) -----------------------(1.1.10)

αˈ(t+) ≤ α (t) + I (α(t)), t = γ (α (t))--------------------------- (1.1.11)

and

βˈ(t) ≥ f (t, β (t)), t ϵ J, t ≠ γ (β (t))---------------------------- (1.1.12)

βˈ(t+) ≥ β (t) + I (β (t)), t = γ (β (t)) ---------------------------(1.1.13)

Assume that the following conditions are satisfied:

(H0) γˈ (α (t)) αˈ(t) < 1, for all t ϵ J.

(H1) γˈ(x) f (t, x) < 1, for all t ϵ J, for all x ϵ R.

(H2) γ (x) is decreasing and γ (β ( 0+)) < 0.

(H3) I* (x) = x + I (x) is increasing and ( I*)ˈ(x) f (t, x) < f (t, I* (x)).

Then the solutions u (t) of (4.1), (4.2) with u (0+) = u0 satisfies

α (0+) ≤ u (0) ≤ β (0+) implies α (t) ≤ u(t) ≤ β(t), for all t ϵ J.

Proof:

Step 1:

To prove α (0+) ≤ u0 implies α (t) ≤ u (t), for all t ϵ J.

Let α (t) and u (t) do not hit the curve σ : t = γ (x) for t ϵ ( 0, T).

Then α (t) ≤ for t ϵ J.

Thus, we can consider at least one of them hits σ.

Since γ is decreasing and γ-1 (0) > α (0+), α (t) hits the curve σ first.

Therefore, there exists t1 ϵ (0, T) such that t1 = γ (α (t1)).

Using α (t) ≤ u (t) on (0, t1) and I (x) ≤ 0,

α (t1 +) ≤ α (t1) + I (α (t1)) ≤

α (t1) ≤ u (t1).

To prove α (t) hits σ exactly once, let p (t) = t – γ (α (t)).

If α (t1 +) ≤ α (t1), then γ (α (t1 +) = γ (α (t1)) = t1. (1.1.14)

Using the mean value theorem, there exists a point ξ ϵ (α (t1 +), α (t1)) Such that γ (α (t1)) – γ (α (t1 +)) = γˈ ( ϵ ) [α (t1) – α (t1 +) ] < 0. (1.1.15)

This implies that

p (t1) = t1 – γ (α (t1 +)) ≤ t1 – γ (α (t1))

≤ t1 – t1 = 0. (using (1.1.14))

Differentiating with respect to

t pˈ(t) ≥ 1 – γˈ(α (t)) f ( t, α (t)) > 0. (using condition H1)

As a consequence, p (t) > 0, for all t > t1.

Therefore α (t) does not hit σ for t > t1.

If u (t) ≠ γ (u (t)) for t ϵ J,

using α (t1 +) ≤ u (t1), and the comparison principle for ordinary differential equations α (t) ≤ u (t), for all t ϵ J.

Now assume that there exists t2 ϵ [t1, T] such that u (t2) = γ (u (t2)). Since u (t1) ≥ α (t1 +), it follows that α (t) ≤ u (t) for t ϵ (t1, t2). To show that u (t2 +) = I*(u (t2)) ≥ α (t2), (1.1.16)

suppose that (1.1.16) is not true.

Since α (t1) ≤ u (t1) and I* is decreasing

α (t1 +) ≤ I* (α (t1)) ≤ I* ( u (t1)).

Then there exists one point s ϵ (t1, t2) such that α (s) = I* (u (s)) and α (t) > I* (u (t)) for s < t ≤ t2. αˈ (s) ≥ (I*)ˈ ( u (s)) uˈ (s) = (I*)ˈ (u (s)) f (s, u (s)) > f (s, I* (u (s))) (using condition (H3)) = f (s, α (s)),

since α (s) =I* (u (s)).

But this is a contraction to αˈ(t) ≤ f (t, α (t)) and hence (1.1.16) follows.

Hence u (t) hits σ only for t = t2 and then α (t2) ≤ u (t2 +) implies α (t) ≤ u (t)

for all t ϵ [ t2, T].

Hence the proof of step one.

Step 2:

To prove β (0+) ≥ u0 implies β (t) ≥ u (t), for all t ϵ J.

The proof of this step will follow the same procedure as in step one.

The main difference is that the graph of the function β can meet the curve σ several times.

It is easy to see that u (t) hits σ before than β (t).

Let t2 = γ (u (t2)).

It follows that u (0+) ≤ β (0+) implies u (t) ≤ β (t), for all t ϵ [0, t2].

But, u (t2 +) = u (t2) + I (u (t2))

≤ u (t2) ≤ β (t2).

If β (t) does not hit σ, then u (t2 +) ≤ β (t2) ≤ β (t), for all t ϵ (t2, T].

Now assume that there exists one-point t3 ϵ [t2, T] such that t3 = γ (β (t3)).

Since u (t2 +) ≤ β (t2), then u (t) ≤ β (t), for all t ϵ [t2, t3].

Using (4.1.12), (4.1.13), condition (H3) and the same arguments employed in step one it can be proved that β (t3 +) ≥ I*( β (t3)) ≥ u (t3).

This implies that u (t) does not meet σ for t > t3, but the condition (H1) does not assure the same β (t).

Therefore, there are two possibilities:

β (t3 +) < β (t).

In this case, it is possible to prove that β (t) does not meet σ for t > t3, and thus β (t3 +) ≥ u (t3), implies that β (t) ≥ u (t) for every t ϵ (t3, T].

The proof of this assertion is very similar to the corresponding one in the first

step, using the condition (H0) instead of (H1). (b) β (t3 +) ≥ β (t3). If β (t) does not hit σ for t > t3, the proof is over. Suppose that there exists one point t4 ϵ (t3, T) such that t4 = γ (β (t4)). In this case, which implies that β (t) ≥ u (t) for all t ϵ (t3, t4) and it remains to prove that

I* (β (t4)) ≥ u (t4). For it, since I* is increasing,

I* (β (t3 +)) ≥ I* (β (t3)) ≥ u (t3).

Suppose that I* (β (t4)) < u (t4), using the condition (H3) and proceeding as in

step one, we get a contradiction.

Therefore, β (t4 +) ≥ I* (β (t4)) ≥ u (t4).

Employing the same procedures successively, u (t) ≤ β (t) for all t ϵ J.

Hence the proof.

CONCLUSION :

The theme of the paper is a study on periodic boundary value problems for first-order impulsive delay difference equations. It is well recognized that the theory of impulsive equations offers a general framework for the mathematical modeling of many real-world phenomena where the states undergo abrupt changes. Such equations have extensive applications in economics, dynamic systems, optimal control, medicine, population dynamics, and many other fields. In particular, in recent years, there has been an increasing interest in extending impulsive differential equations to time-delay systems and boundary value problems. On the other hand, difference equations play an important role in many fields such as numerous settings and forms, computing, electrical circuit analysis, biology, etc. However, there are not many related results for impulsive difference equations and impulsive delay difference equations. These motivated us to work on the present topic.

In this paper, we studied impulsive delay difference equations with periodic boundary conditions. Based on the new concepts of lower and upper solutions, we established two new comparison principles. With these, we constructed monotone sequences from a corresponding linear equation and established the existence of extremal solutions by utilizing the monotone iterative technique. An example was given to illustrate the results obtained. It is reckoned that these results may play an important role in the theory of difference equations, and are useful in many practical problems in the aforesaid fields.

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