Abstract
The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.
Highlights
(H1) The operator A is a sectorial operator of type π < 0 for some M > 0 and 0 ≤ θ < π(1 − α/2), and there exists C > 0 such that
(H5) The operator A is a sectorial operator of type π with 0 ≤ θ < π(1 − α/2), and there exists φ(⋅) ∈ L1(R+)
First we prove that G(t) ∈ AA(R; L2(P, H))
Summary
This paper is mainly concerned with the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the following stochastic fractional partial integrodifferential equation in the form d [x (t) − f (t, x (t))]. The concept of asymptotically almost automorphic functions was firstly introduced by N’Guerekata in [12]. In a very recent paper [28], the authors introduced a new notation of square-mean asymptotically almost automorphic stochastic processes including a composition theorem. To the best of our knowledge, the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1) is an untreated topic. Abstract and Applied Analysis the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the problem (1). We give an example as an application of our abstract results
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