Abstract
The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in Hilbert space. First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by using the successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The obtained existence result in this article is new in the sense that it generalizes some of the existing results in the literature. Furthermore, we discuss the averaging principle for the proposed neutral fractional stochastic system in view of the convergence in mean square between the solution of the standard INFSEEs and that of the simplified equation. Finally, the obtained averaging theory is validated with an example.
Highlights
The subject of fractional calculus has received considerable critical attention due to its applications in widespread areas of engineering and science
Derived the existence and optimal control for delay neutral fractional stochastic differential equations (NFSDEs) driven with Poisson jumps by using successive approximations under non-Lipschitz condition
This section is devoted to the establishment of an averaging principle for INFSEEs driven by fractional Brownian motion (fBm) and Poisson jumps
Summary
The subject of fractional calculus has received considerable critical attention due to its applications in widespread areas of engineering and science. Cui and Yan [37] proved the existence result for fractional neutral stochastic integro-differential equations with infinite delay. Derived the existence and optimal control for delay neutral fractional stochastic differential equations (NFSDEs) driven with Poisson jumps by using successive approximations under non-Lipschitz condition. Dineshkumar et al [40,41] derived the approximate controllability for Hilfer fractional neutral stochastic delay integro-differential equations driven by Brownian motion. Existence and uniqueness problem of INFSEEs driven by fBm and Poisson jumps under Carathéodory conditions is desired. The local and global existence and uniqueness results for Equation (1), under local and global Carathéodory conditions by means of successive approximation and stopping time techniques, are rarely available in the literature, which is the key inspiration to our research work in this article and seems to be new to our knowledge.
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