Abstract
This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.
Highlights
Stochastic differential equations (SDEs) arise in many areas of science and engineering, wherein, quite often the future state of such systems depends on present state, and on its history leading to stochastic functional differential equations with delays rather than SDEs
The stability of impulsive differential equations has been discussed by several authors
Stochastic integrodifferential equations with delay are important for investigating several problems raised from natural phenomena
Summary
Stochastic differential equations (SDEs) arise in many areas of science and engineering, wherein, quite often the future state of such systems depends on present state, and on its history leading to stochastic functional differential equations with delays rather than SDEs. Regarding the fractional Brownian motion (fBm), one can find results involving existence, uniqueness and stability of solutions for stochastic functional differential equations; see [3,4,5,6, 20, 22,23,24, 28, 34]. We shall prove the existence, uniqueness and asymptotic behavior of mild solution for a class of impulsive stochastic neutral functional integrodifferential equation with delays driven by Rosenblatt process and Poisson jumps described in the form:.
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