Mild solutions generated by a

Since the concept of almost automorphicity was introduced by Bochner [

For the last few decades, time-fractional differential equations have appeared as an essential tool for studying dynamics of various complex processes on anomalous phenomena in physics, finance, hydrology, and cell biology; see [

Recently, there has been an increasing interest in extending certain classical deterministic results to stochastic cases. The most important reason for such a consideration is that most problems in real life are inevitably subject to some random environmental effects. Thus, stochastic models are more natural and realistic to describe phenomena in the natural science; see [

In this paper, we investigate the existence and uniqueness of square-mean asymptotically almost automorphic solutions to the fractional stochastic relaxation equations of the linear form

The paper is organized as follows. In Section

Throughout this paper, we assume that

We denote by

A stochastic process

A mean-square continuous process

Let

If

there exists a constant

A function

Let

A continuous function

A continuous function

Let

Let

Let

In this section, based on the definitions and lemmas given in the previous section, we derive conditions for existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to fractional stochastic relaxation equations (

To begin with, we consider linear fractional stochastic relaxation equation (

An

In order to get main result in this section, we need the following lemmas.

Let

Since

Let

First, we claim that

Similarly,

Next, we prove that

There remains claiming that

Analogously,

Next, we show that

The main result concerning linear fractional stochastic relaxation equation (

If

From Definition

In this section, we consider the existence and uniqueness of a square-mean asymptotically almost automorphic mild solution of (

An

Here when we consider that

In order to establish the existence and uniqueness result for the semilinear case with more flexible initial conditions and coefficients, we need the following assumptions and lemma:

There exists an

The function

The function

Let

Similar to Lemma

Assume that the assumptions (1)–(4) hold. Then there exists a unique square-mean asymptotically almost automorphic mild solution to (

Let

First, let us check the fact that

For

In this paper, the square-mean asymptotically almost automorphic solutions to abstract fractional stochastic relaxation equations have been investigated. Based on

The author declares that there is no conflict of interests regarding the publication of this paper.

The author wishes to thank Dr. Marjorie Hahn for her help and advice and the author’s peer L. Chlebak for discussions and valuable suggestions after reading first version of this paper as well as Dr. Patricia Garmirian for discussions. The author would also like to thank the editor for his suggestion on the clarity of the paper.