Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

Abstract This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 1 H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.


Introduction
The fractional Brownian motion (fBm) is one of the natural generalizations of the Brownian motion. It is a family of centered and continuous Gaussian processes with Hurst parameter ∈ ( ) H 0, 1 . It is reduced to the standard Brownian motion if = H 1 2 . But if ≠ H 1 2 , fBm is different from a Markov process and martingale; therefore, the classical stochastic analysis is not possible to be used. The fractional Brownian motion was introduced by Kolmogorov [1] in 1940 and has very important properties such as self-similarity and non-stationary. Mandelbrot and Van Ness [2] made it famous by introducing it into financial models and studying its properties. These proprieties allow fBm to be used in several domains such as telecommunication, biology, finance, and engineering. For that it is beneficial and important to investigate stochastic differential equations driven by an fBm. Recently, stochastic partial functional differential equations driven by a fractional Brownian motion have drawn the interest of many researchers (see [3][4][5][6][7][8][9][10]). For example, under the global Lipschitz condition, Caraballo et al. [11] showed the existence, uniqueness and stability of mild solutions for stochastic partial differential equations (SPDEs) with finite delays driven by an fBm; under the global Lipschitz condition, Boufoussi and Hajji [12] considered the existence and uniqueness of mild solutions to neutral SPDEs with finite delays driven by an fBm; Boufoussi et al. [13] obtained the existence and uniqueness result of mild solution to a class of timedependent stochastic functional differential equations driven by an fBm; Ren et al. [9] proved the existence and uniqueness of the mild solution for a class of time-dependent stochastic evolution equations with finite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion.
For more details on the fractional Brownian motion, see [11,12,[14][15][16] and references therein. In addition, the theory of nonlocal evolution equations has become an important area of investigation in recent years due to their applications to various problems arising in physics, biology, aerospace and medicine. Nonlocal conditions are known to give a better description of real models than classical initial ones, e.g., the condition allows taking additional measurements instead of solely initial datum. The first result and physical significance for nonlocal problems are given by Byszewski's work [17]. It developed greater interest in various nonlocal issues related to differential equations and stochastic differential equations. Many of the basic results for nonlocal problems have been obtained, see [18][19][20][21][22][23][24] and references therein for more comments and citations. Recently, some authors have drawn attention to the Cauchy problems driven by differential equations. One can see the studies of Balachandran et al. [25], Balasubramaniam and Park [26], Balasubramaniam et al. [27], Deng [28][29][30][31], Liang and Xiao [32] and references therein.
Motivated by the previously mentioned problems, in this paper, we will extend some such results of mild solutions for the following nonlocal integro-differential stochastic equations driven by a fractional Brownian motion of the following form: where A is the infinitesimal generator of a strongly continuous semi-group The aim of our paper is to study the solvability of (1) and present the results on the existence of mild solutions of (1) based on the Krasnoselskii-Schaefer-type fixed point theorem combined with the theory of resolvent operator for integro-differential equations in the sense of Grimmer. We know that many existence results of stochastic differential equations with nonlocal conditions are under the compact assumptions on nonlocal terms. In this paper, we are interested in weakening these hypotheses regarding nonlocal terms.
The remainder of this paper is organized as follows. In Section 2, we recall briefly the notations, concepts and basic results about the Wiener process and deterministic integro-differential equations. The main results in Section 3 are devoted to the study of the existence and uniqueness of mild solutions for system (1) with their proofs. An example is given in Section 4 to illustrate the obtained results. Section 5 concludes the paper and presents future work.

Wiener process
Let and be two real separable Hilbert spaces and ( ) Ω, , b be a complete probability space with a normal filtration ∈[ ] t b 0, . We denote by b the predicable σ-field on . Space is equipped with a Borel σ-field ( ) X . Introduce the following Banach spaces: is a bounded linear operator , Ω, , : , Ω, , : Ω, , is a continuous mapping from into Ω, , Before continuing, let us give the definition of one-dimensional fBm.
H H with the covariance function , the fractional Brownian motion is then a standard Brownian motion. In this paper, we assume that ∈ H , 1 is a Wiener process and where c H is a non-negative constant with respect to H.
Denote by ϵ the linear space of step functions on J of the form , ∈ n , ∈ a i and the closure of ϵ with respect to the scalar product The Wiener integral of ( ∈ ) ϕ ϕ ϵ with respect to β H is given by is an isometry between ϵ and the linear space span { ( ) ∈ } β t t J , H viewed as a subspace of ( ) L Ω 2 , which can be extended to an isometry between and the first Wiener chaos of the fBm The image on an element ∈ h by this isometry is called the Wiener integral of h with respect to β H . that can be extended to . We have the following relation between Wiener integral with respect to fBm and Itô integral with respect to the Wiener process: For > H 1 2 , we have (see [11]) Next, we define the infinite dimensional fBm and give the definition of the corresponding stochastic integral.
Let where : is a Hilbert Schmidt operator . (3) Then its stochastic integral with respect to the fBm B H is defined as follows: , we have where C H is a constant depending on H. If, in additional,

Integro-differential equations
In this subsection, we recall some knowledge on partial integro-differential equations and the related resolvent operators. Let and be two Banach spaces such that ∥ ∥ ≔ ∥ ∥ + ∥ ∥ ∈ z A z z z , for .
; ; In the following, we give some results for the existence of solutions for the following integro-differential equation: Now, we give the definition of mild solution for (1).

Definition 2.5.
A -valued stochastic process ∈ Y C is said to be a mild solution of system (1) if ( ) + ( ) = Y GY Y 0 0 and for any ∈ t J, it satisfies the following integral equation: ; (ii) 1 is a contraction; (iii) 2 is completely continuous. Then, the equation has a solution on V.

Existence of mild solutions
This part is devoted to state and prove our main results. We define the operator on C by .
. Moreover, it has linear growth in the variable y uniformly into t; that is, there exists a positive constant > c 0 1 such that , , almost all Ω.
Therefore, we only need to check that I i tends to zero when → = t t i , 1, 2, 3 2 1 . For I 1 , by using strong continuity of ( ) t , we have For I 2 , we can get by direct calculations    For The aforementioned arguments show that By application of ( ) H3 , Definition 2.3 and Hölder inequality, we get Proof. We show that is a contraction mapping. For any ∈ Y Y C ,  It follows from (12) that is a contraction mapping. According to the contraction principle, we know By ( ) H9 , we deduce that the right hand side of (15) tends to zero independently of ∈ Y D r as → t t  be fixed and ϵ be real number ∈ ( ) ϵ t 0, and for ∈ Y D r , we define the operators

Conclusion
Existence results of nonlocal stochastic integro-differential equations driven by a fractional Brownian motion have been investigated. First, by using the contraction principle, the existence and uniqueness of mild solutions are given. Next, the existence of mild solutions is investigated based on Krasnoselskii's fixed point theorem. Finally, the obtained theoretical results have been verified by an illustrative example.
As further direction, we will investigate the existence results of nonlocal stochastic integro-differential equations driven by a fractional Brownian motion via Kuratowski measure of noncompactness.