Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Existence of Solutions of Antiperiodic Boundary Value Problems for a Non Linear Third Order Impulsive Integro Dynamic System on Time Scales

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 629--643 | DOI:10.5890/DNC.2022.12.005

D. Arunkumar, M. Sivabalan, K. Sathiyanathan

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore - 641020, India

Download Full Text PDF

 

Abstract

This paper explores the existence of solutions for the system of impulsive integrodifferential equation with anti-periodic boundary value problems on time scales. Some sufficient conditions for the existence of solutions are established by using Schauder's fixed point theorem and Schaefer fixed point theorem.

References

  1. [1]  Cabaca, A. and Liz, E. (1998), Boundary value problems for higher order ordinary differential equations with impulses, Nonlinear Analysis, 32, 775-786.
  2. [2]  Franco, D. and Nieto, J. (2000), First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Analysis, 42(2), 163-173.
  3. [3]  Cui, Y., Choi, S.K., and Koo, N. (2014), Impulsive stabilization of dynamic equations on time scales, Abstract and Applied Analysis, Article ID 609808, 6 pages.
  4. [4]  Guo, D. and Liu, X. (1995), Multiple positive solutions of boundary value problems for impulsive differential equations, Nonlinear Analysis, 25, 327-337.
  5. [5]  Jankowski, T. (2004), Ordinary differential equations with nonlinear boundary conditions of antiperiodic type, Computer and Mathematics with Applications, 47, 1419-1428.
  6. [6]  Liu, Y. (2007), Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 327, 435-452.
  7. [7]  Moghaddam, B.P. and Mostaghim, Z.S. (2014), A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations, Ain Shams Engineering Journal, 5, 585-594.
  8. [8]  Liz, (1995), Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonlinear Analysis, 25, 1191-1198.
  9. [9]  Arunkumar, D., Sivabalan, M., and Sathiyanathan, K. (2019), Existence of impulsive integro differential equations with integral boundary conditions, American International Journal of Research in Science, Technology, Engineering and Mathematics, ISSN: 2328-3491, 5-13.
  10. [10]  Wang, Y. and Shi, Y.M. (2005), Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, Journal of Mathematical Analysis and Applications, 309, 56-69.
  11. [11]  Zhai, H., Yu, C., and Wang, J. (2018), Existence of solutions for a class of fractional ordinary differential equations with integral and antiperiodic boundary value conditions, International Journal of Electrical and Electronics, 883-888.
  12. [12]  Benchohra, M., Hamidi, N., and Henderson, J. (2013), Fractional differential equations with anti-periodic boundary conditions, Journal of Numerical Functional Analysis and Optimization, 34, 404-414.
  13. [13]  Henderson, H. and Luca, R. (2017), Positive solutions for an impulsive second-order nonlinear boundary value problem, Mediterranean Journal of Mathematics, 14(93).
  14. [14]  Lupulescu, V. and Zada, A. (2010), Linear impulsive dynamic system on time scales, Electronic Journal of Qualitative Theory of Differential equations, 11, 1-30.
  15. [15]  Shah, S.O. and Zada, A. (2019), Connection between Ulam-Hyers stability and uniform exponential stability of time varing linear dynamic system on time scales, Sohag Journal of Mathematics, 1, 1-4.
  16. [16]  Liu, X.Z. and Zhang, K.X. (2016), Existence uniqueness and stability results for functional differential equations on time scales, Dynamic System and Applications, 25.
  17. [17]  Krishnaveni, V. Sathiyanathan, K., and Arunkumar, D. (2016), Impulsive integro differential equation with anti periodic boundary conditions on time scales, International Journal of Control Theory and Applications, 9(41), 1136-1150.
  18. [18]  Baxter, L.H., Lyons, J.W., and Neugebauer, J.T. (2016), Differentiating solutions of a boundary value problem on a time scale, Bulletin of the Australian Mathematical Society, 94, 101-109.
  19. [19]  Zada, A., Shah, S.O., and Li, Y. (2017), Hyers-Ulam stability of nonlinear impulsive volterra integro delay dynamic system on time scales, Journal of Nonlinear Science and Applications, 10, 5701-5711.
  20. [20]  Syed, O.S. and Zada, A. (2019), On the stability analysis of non linear Hammerstein impulsive integro dynamic system on time scales, Journal of Mathematics, 51(7), 89-98.
  21. [21]  Guo, D.J. and Liu, X.Z. (1993), Extremal solutions of nonlinear impulsive integro-differential equations in Banach spaces, Journal of Mathematical Analysis and Applications, 177, 538-553.
  22. [22]  Lu, H.O. (1999), Extremal solutions of nonlinear first order impulsive integro-differential equations in Banach spaces, Indian Journal of Pure and Applied Mathematics, 30, 1181-1197.
  23. [23]  Hao, X.N., Liu, L.S., and Wu, Y.H. (2011), Positive solutions for second order impulsive differential equations with integral boundary conditions, Communications in Nonlinear Science and Numerical Simulation, 16, 101-111.
  24. [24]  Fen, F.T. and Karaca, I.Y. (2016), Positive solutions of nth-order impulsive differential equations with integral boundary conditions, Analele Stiintifice ale Universitatii Ovidius Constanta, 24(1), 243-261.
  25. [25]  Kumlin, P. (2003/2004), TMA 401/MAN 670 Functional Analysis, Chalmers $\&$ GU.