Hilfer fractional stochastic integro-differential equations

Abstract

In this paper, we investigate the existence of mild solutions of Hilfer fractional stochastic integro-differential equations with nonlocal conditions. The main results are obtained by using fractional calculus, semigroups and Sadovskii fixed point theorem. In the end, an example is given to illustrate the obtained results.

Introduction

It is well-known that the fractional derivatives are valuable tools for the description of memory and hereditary properties of various materials and processes, which integer order derivatives cant characterize. Many problems in various fields can be described by fractional calculus such as material sciences, mechanics, wave propagation, signal processing, system identification and so on(see [1], [2], [3], [4]). The fractional differential equations have gained considerable importance during the past three decades. Hence, the theory of fractional differential equations has emerged as an active branch of applied mathematics. It has been used to construct many mathematical models in various fields, such as physics, chemistry, viscoelasticity, electrochemistry, control, porous media, electromagnetic and polymer rheology, etc. The recent works on the theory and application of fractional differential equations, we refer to the monographs [5], [6], [7], [8], [9]. Moreover, stochastic perturbation is unavoidable in nature and hence it is important and necessary to consider stochastic effect into the investigation of fractional differential equations (see [10], [11], [12], [13]). Hilfer proposed a generalized Riemann–Liouville fractional derivative for short, Hilfer fractional derivative, which includes Riemann–Liouville fractional derivative and Caputo fractional derivative (see [2], [14]). Subsequently, many authors studied the fractional differential equations involving Hilfer fractional derivatives (see [15], [16], [17], [18], [19]).

In this paper, we study the existence of mild solutions of Hilfer fractional stochastic integro-differential equations of the form

$$\begin{array}{c}\hfill \{\begin{array}{c}{D}_{0+}^{\nu ,\mu }[x\left(t\right)+F(t,v\left(t\right))]+Ax\left(t\right)={\int }_0^{t}G(s,\eta \left(s\right))d\omega \left(s\right),t\in J:=(0,b],\hfill \\ I_{0+}^{(1-\nu )(1-\mu )}x\left(0\right)-g\left(x\right)=x_0,\hfill \end{array}\end{array}$$

where $(t,v\left(t\right))=(t,x\left(t\right),x\left(b_1\left(t\right)\right),\dots ,x\left(b_m\left(t\right)\right))$ and $(t,\eta \left(t\right))=(t,x\left(t\right),x\left(a_1\left(t\right)\right),\dots ,x\left(a_n\left(t\right)\right)),$ $D_{0+}^{\nu ,\mu }$ is the Hilfer fractional derivative, 0 ≤ ν ≤ 1, 0 < µ < 1, $-A$ is the infinitesmal generator of an analytic semigroup of bounded linear operators S(t), t ≥ 0, on a separable Hilbert space H with inner product ⟨., .⟩ and norm ∥.∥. Let K be another separable Hilbert space with inner product ⟨., .⟩_K and norm ∥.∥_K. Suppose {ω(t)}_t ≥ 0 is given K-valued Wiener process with a finite trace nuclear covariance operator Q ≥ 0. We are also employing the same notation ∥.∥ for the norm L(K, H), where L(K, H) denotes the space of all bounded linear operators from K into H. The functions F, G and g are given functions to be defined later.