In this article, we study the existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Banach fixed point theorem. In the later case we explore the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval.
[1] A. Anguraj, A. Vinodkumar, and K. Malar, Existence and stability results for random impulsive fractional pan- tograph equations, Filomat, 30(14) (2016), 3839–3854.
[2] A. Anguraj, K. Ravikumar, and J. J. Nieto, On stability of stochastic differential equations with random impulses driven by Poisson jumps, Stochastics An International Journal of Probability and Stochastic Processes, (2020), 1–15.
[3] S. Dragomir, Some Gronwall type inequalities and applications, RGMIA Monographs, Nova Biomedical, Mel- bourne, (2002).
[4] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Mathematicae, 50(1-2) (1995), 143–190.
[5] M. Gowrisankar, P. Mohankumar, and A. Vinodkumar, Stability results of random impulsive semilinear differential equations, Acta Math. Sci, 34(4) (2014), 1055–1071.
[6] N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equations, Journal of Mathematical Analysis and Applications, 385 (2012), 86–91.
[7] K. Malar, Existence and uniqueness results for random impulsive integro-differential equation, Global Journal of Pure and Applied Mathematics. 6 (2018), 809-817.
[8] X. Mao, Stochastic differential equations and applications, M. Horwood, Chichester, (1997).
[9] N. Ngoc, Ulam-Hyers-Rassias stability of a nonlinear stochastic integral equation of Volterra type, Differ. Equ. Appl, 9(2) (2017), 183–193.
[10] T. M. Rassias, On the stability of fractional equations and a problem of Ulam, Acta Appl. Math, 62(1) (2000), 23–130.
[11] J. M. Sanz-Serna and A. M. Stuart, Ergodicity of dissipative differential equations subject to random impulses, Journal of differential equations, 155(2) (1999), 262–284.
[12] R. Shah and A. Zada, A fixed point approach to the stability of a nonlinear volterra integrodifferential equation with delay, Hacet. J. Math. Stat, 47(3) (2018), 615–623.
[13] A. Vinodkumar, M. Gowrisankar, and P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial differential equations, J. Egyptian Math. Soc, 23(1) (2015), 31–36.
[14] A. Vinodkumar, K. Malar, M. Gowrisankar, and P. Mohankumar, Existence, uniqueness and stability of random impulsive fractional differential equations, Acta Math. Sci, 36B(2) (2016), 428–442.
[15] T. Wang and S. Wu, Random impulsive model for stock prices and its application for insurers, Master thesis (in Chinese), Shanghai, East China Normal University, (2008).
[16] S. Wu and X. Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin, 20(1) (2004), 147–154.
[17] S. Wu and B. Zhou, Existence and uniqueness of stochastic differential equations with random impulses and markovian switching under Non-Lipschitz conditions, J. Acta Math. Sin, 27(3) (2011), 519–536.
[18] X. Zhao, Mean square Hyers-Ulam stability of stochastic differential equations driven by Brownian motion, Adv. Difference. Equ, 2016(1) (2016), 1–12.
[19] Y. Zhou and S. Wu, Existence and uniqueness of solutions to stochastic differential equations with random impulses under Lipschitz conditions, Chinese J. Appl. Proba. Statist, 26(4) (2010), 347–356.
Anguraj, A., Ramkumar, K., & Ravikumar, K. (2022). Existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Computational Methods for Differential Equations, 10(1), 191-199. doi: 10.22034/cmde.2020.32591.1512
MLA
Annamalai Anguraj; Kasinathan Ramkumar; Kasinathan Ravikumar. "Existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays". Computational Methods for Differential Equations, 10, 1, 2022, 191-199. doi: 10.22034/cmde.2020.32591.1512
HARVARD
Anguraj, A., Ramkumar, K., Ravikumar, K. (2022). 'Existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays', Computational Methods for Differential Equations, 10(1), pp. 191-199. doi: 10.22034/cmde.2020.32591.1512
VANCOUVER
Anguraj, A., Ramkumar, K., Ravikumar, K. Existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Computational Methods for Differential Equations, 2022; 10(1): 191-199. doi: 10.22034/cmde.2020.32591.1512
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