Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Existence and Stability Results on Nonlinear Delay Integro-Differential Equations with Random Impulses

  • Received : 2014.07.09
  • Accepted : 2015.02.21
  • Published : 2016.06.23
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Abstract

In this paper, the existence, uniqueness, stability via continuous dependence and Ulam stabilities of nonlinear integro-differential equations with random impulses are studied under sufficient condition. The results are obtained by using Leray-Schauder alternative fixed point theorem and Banach contraction principle.

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References

  1. B. Ahmad and B. S. Alghamdi, Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions, Computer Physics Communications, 179(6)(2008), 409-416. https://doi.org/10.1016/j.cpc.2008.04.008
  2. B. Ahmad, On the existence of T-periodic solutions for Duffing type integro-differential equations with p-Laplacian, Lobachevskii Journal of Mathematics, 29(1)(2008), 1-4. https://doi.org/10.1134/S1995080208010010
  3. A. Anguraj, M. Mallika Arjunan and E. Hernandez, Existence results for an impulsive partial neutral functional differential equations with state- dependent delay, Appl. Anal., 86(7)(2007), 861-872. https://doi.org/10.1080/00036810701354995
  4. A. Anguraj, S. Wu and A. Vinodkumar, Existence and Exponential Stability of Semilinear Functional Differential Equations with Random Impulses under Non-uniqueness, Nonlinear Anal. TMA, 74(2011), 331-342. https://doi.org/10.1016/j.na.2010.07.007
  5. A. Anguraj, A. Vinodkumar, Existence, Uniqueness and Stability Results of Random Impulsive Semilinear Differential Systems, Nonlinear Anal. Hybrid Syst. 4(3)(2010), 475-483. https://doi.org/10.1016/j.nahs.2009.11.004
  6. A. Anguraj, A.Vinodkumar, Existence and Uniqueness of Neutral Functional Differential Equations with Random Impulses, I. J. Nonlinear Sci., 8:4(2009), 412-418.
  7. T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006.
  8. M. Gowrisankar, P.Mohankumar, and A. Vinodkumar, Stability Results of Random Impulsive Semilinear Differential Equations, Acta Mathematica Scientia, 34B(4)(2014), 1055-1071.
  9. R. Haloi, Dwijendra N. Pandey, and D. Bahuguna, Existence and Uniqueness of a Solution for a Non-Autonomous Semilinear Integro-Differential Equation With Deviated Argument, Differ. Equ. Dyn. Syst., 20(1)(2012), 1-16. https://doi.org/10.1007/s12591-011-0099-x
  10. E. Hernandez, M. Rabello, and H.R.Henriquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl., 331(2007), 1135-1158. https://doi.org/10.1016/j.jmaa.2006.09.043
  11. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  12. R. Iwankievicz and S. R. K. Nielsen, Dynamic response of non-linear systems to Poisson distributed random impulses, Journal of Sound Vibration, 156(1992), 407-423. https://doi.org/10.1016/0022-460X(92)90736-H
  13. V. Lakshmikantham, D. D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  14. X. Li, J. Wang, Ulam-Hyers-Rassias stability of semilinear differential equations with impulses, Electron. J. Diff. Equ., 2013(172)(2013), 1-8. https://doi.org/10.1186/1687-1847-2013-1
  15. A. M. Samoilenko, N.A Perestyuk., Impulsive Differential Equations, World Scientific, Singapore, 1995.
  16. J. M. Sanz-Serna and A. M. Stuart, Ergodicity of dissipative differential equations subject to random impulses, J. Diff. Eqns. 155(1999), 262-284. https://doi.org/10.1006/jdeq.1998.3594
  17. K. Tatsuyuki, K. Takashi and S. Satoshi, Drift motion of granules in chara cells induced by random impulses due to the myosinctininteraction, Physica A 248(1998), 21-27. https://doi.org/10.1016/S0378-4371(97)00455-X
  18. S. M. Ulam, A collection of mathematical problems, New York, Interscience Publishers, 1968.
  19. A. Vinodkumar, M. Gowrisankar, and P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial differential equations, J. The Egyptian Mathematical Society, 23(2015), 31-36. https://doi.org/10.1016/j.joems.2014.01.005
  20. A. Vinodkumar, A. Anguraj, Existence of random impulsive abstract neutral non-autonomous differential inclusions with delays, Nonlinear Anal. Hybrid Syst. 5(2011), 413-426. https://doi.org/10.1016/j.nahs.2011.04.002
  21. A.Vinodkumar, Existence results on random impulsive semilinear functional differential inclusions with delays, Ann. Funct. Anal., 3(2)(2012), 89-106. https://doi.org/10.15352/afa/1399899934
  22. J. Wang, L. Lv, and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63(2011), 1-10.
  23. J. Wang, L. Lv, and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 2530-2538. https://doi.org/10.1016/j.cnsns.2011.09.030
  24. J. Wang, Y. Zhou, and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64(2012), 3389-3405. https://doi.org/10.1016/j.camwa.2012.02.021
  25. J. Wang, M. Feckan, and Y. Zhou, Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395(2012), 258-264. https://doi.org/10.1016/j.jmaa.2012.05.040
  26. Wei Wei, Xuezhu Li, and Xia Li, New stability results for fractional integral equation,Comput. Math. Appl., 64(2012), 3468-3476. https://doi.org/10.1016/j.camwa.2012.02.057
  27. S. J. Wu, X. Z. Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin., 20(1)(2004), 147-154. https://doi.org/10.1007/s10255-004-0157-z
  28. S. J. Wu, Y. R. Duan, Oscillation, stability, and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49(9-10)(2005), 1375-1386. https://doi.org/10.1016/j.camwa.2004.12.009
  29. S. J. Wu, X. L. Guo and S. Q. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Math. Appl. Sin., 22(4)(2006), 595-600. https://doi.org/10.1007/s10114-005-0689-z
  30. S. J. Wu, X. L. Guo and Y. Zhou, p-moment stability of functional differential equations with random impulses, Comput. Mathe. Appl., 52(2006), 1683-1694. https://doi.org/10.1016/j.camwa.2006.04.026
  31. S. J. Wu, X. L. Guo and R.H. Zhai,Almost sure stability of functional differential equations with random impulses, DCDIS, Series A: Math. Anal., 15(2008), 403-415.
  32. Z. Yan, Nonlocal problems for delay integrodifferential equations in Banach spaces, Differ. Eqn. Appl., 2(1)(2010), 15-24.
  33. S. Zhang, J. Sun, Stability analysis of second-order differential systems with Erlang distribution random impulses, Advances in Difference Equations, 2013(4)(2013), 403-415.