Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle

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 Caratheodory 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.

Automated summary

This article considers a class of neutral Caputo fractional stochastic evolution equations with infinite delay in Hilbert space, driven by fractional Brownian motion and Poisson jumps. The local and global existence and uniqueness theorems of mild solutions for this stochastic system are established using successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The averaging principle for the proposed neutral fractional stochastic system is discussed, and the obtained theory is validated with an example.