On stability of nonlocal neutral stochastic integro differential equations with random impulses and Poisson jumps

Sahar M. A. Maqbol; R. S. Jain; B. S. Reddy

doi:10.56754/0719-0646.2502.211

DOI:

https://doi.org/10.56754/0719-0646.2502.211

Abstract

This article aims to examine the existence and Hyers-Ulam stability of non-local random impulsive neutral stochastic integrodifferential delayed equations with Poisson jumps. Initially, we prove the existence of mild solutions to the equations by using the Banach fixed point theorem. Then, we investigate stability via the continuous dependence of solutions on the initial value. Next, we study the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, we give an illustrative example of our main result.

Keywords

Existence of mild solutions , Hyers-Ulam (HU) stability , random impulsive , stochastic integro differential equations , time delays

Mathematics Subject Classification:

93B05 , 34K45 , 45J05

Sahar M. A. Maqbol School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India. https://orcid.org/0000-0002-5329-2583
R. S. Jain School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India. https://orcid.org/0000-0003-0481-4944
B. S. Reddy School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India. https://orcid.org/0000-0001-7578-3685

Pages: 211–229
Date Published: 2023-08-04
Vol. 25 No. 2 (2023)

A. Anguraj and K. Ravikumar, “Existence and stability of impulsive stochastic partial neutral functional differential equations with infinite delays and Poisson jumps”, J. Appl. Nonlinear Dyn., vol. 9, no. 2, pp. 245–255, 2020.

A. Anguraj and K. Ravikumar, “Existence, uniqueness and stability of impulsive stochastic partial neutral functional differential equations with infinite delays driven by a fractional Brownian motion”, J. Appl. Nonlinear Dyn., vol. 9, no. 2, pp. 327–337, 2020.

A. Anguraj , K. Ravikumar and J. J. Nieto, “On stability of stochastic differential equations with random impulses driven by Poisson jumps”, Stochastics, vol. 93, no. 5, pp. 682–696, 2021.

A. Anguraj, K. Ramkumar and K. Ravikumar, “Existence and Hyers-Ulam stability of ran- dom impulsive stochastic functional integrodifferential equations with finite delays”, Comput. Methods Differ. Equ., vol. 10, no. 1, pp. 191–199, 2022.

D. Baleanu, R. Kasinathan, R. Kasinathan and V. Sandrasekaran, “Existence, uniqueness and Hyers-Ulam stability of random impulsive stochastic integro-differential equations with nonlocal conditions”, AIMS Math., vol. 8, no. 2, pp. 2556–2575, 2023.

D. Chalishajar, R. Kasinathan, R. Kasinathan and G. Cox, “Existence Uniqueness and Stability of Nonlocal Neutral Stochastic Differential Equations with Random Impulses and Poisson Jumps”, RNA, vol. 5, no. 3, pp. 250–62, 2022.

S. Deng, X. Shu and J. Mao, “Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with non compact semigroup via Mönch fixed point”, J. Math. Anal. Appl., vol. 467, no. 1, pp. 398–420, 2018.

S. Dragomir, Some Gronwall type inequalities and applications, Melbourne, Australia: RGMIA Monographs, 2002. Available: https://rgmia.org/papers/monographs/standard.pdf.

M. Gowrisankar, P. Mohankumar and A. Vinodkumar, “Stability results of random impulsive semilinear differential equations”, Acta Math. Sci. (English Ed.), vol. 34, no. 4, pp. 1055–1071, 2014.

A. Hamoud, “Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro differential equations”, ATNAA, vol. 4, no. 4, pp. 321–331, 2020.

E. Hernández, M. Rabello and H. R. Henríquez, “Existence of solutions for impulsive partial neutral functional differential equations”, J. Math. Anal. Appl., vol. 311, no. 2, pp. 1135–1158, 2007.

R. S. Jain and M. B. Dhakne, “On impulsive nonlocal integro-differential equations with finite delay”, Int. J. Math. Res., vol. 5, no. 4, pp. 361–373, 2013.

R. S. Jain, B. S. Reddy, and S. D. Kadam, “Approximate solutions of impulsive integro-differential equations”, Arab. J. Math., vol. 7, no. 4, pp. 273–279, 2018.

R. Kasinathan, R. Kasinathan, V. Sandrasekaran and J. J. Nieto, “Qualitative Behaviour of Stochastic Integro-differential Equations with Random Impulses”, Qual. Theory Dyn. Syst, vol. 22, no. 2, Art. ID 61, 2023.

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations. Singapore: World Scientific Publishing Company, 1989.

W. Lang , S. Deng, X.B. Shu and F. Xu, “Existence and Ulam-Hyers-Rassias stability of stochastic differential equations with random impulses”, Filomat, vol. 35, no. 2, pp. 399–407, 2021.

X. Mao, Stochastic Differential Equations and Applications. Chichester, England: Horwood Publishing Limited, 1997.

B. Radhakrishnan and M. Tanilarasi, “Existence of solutions for quasilinear random impulsive neutral differential evolution equation”, Arab J. Math. Sci., vol. 24, no. 2, pp. 235–246, 2018.

A. Vinodkumar, M. Gowrisankar and P. Mohankumar, “Existence, uniqueness and stability of random impulsive neutral partial differential equations”, J. Egyptian Math. Soc., vol. 23, no. 1, pp. 31–36, 2015.

A. Vinodkumar, K. Malar, M. Gowrisankar and P. Mohankumar, “Existence, uniqueness and stability of random impulsive fractional differential equations”, Acta Math. Sci. (English Ed.), vol. 36, no. 2, pp. 428–442, 2016.

S. Wu and X. Meng, “Boundedness of nonlinear differential systems with impulsive effect on random moments”, Acta Math. Appl. Sin., vol. 20, no. 1, pp. 147–154, 2004.

X. Yang, X. Li, Q. Xi and P. Duan, “Review of stability and stabilization for impulsive delayed systems”, Math. Biosci. Eng., vol. 15, no. 6, pp. 1495–1515, 2018.

X. Yang and Q. Zhu, “pth moment exponential stability of stochastic partial differential equations with Poisson jumps”, Asian J. Control, vol. 16, no. 5, pp. 1482–1491, 2014.

S. Zhang and W. Jiang,“The existence and exponential stability of random impulsive fractional differential equations”, Adv. Difference Equ., vol. 2018, Art. ID 404, 2018.