Hilfer fractional stochastic integro-differential equations
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 where and is the Hilfer fractional derivative, 0 ≤ ν ≤ 1, 0 < µ < 1, 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.
Section snippets
Preliminaries
In order to derive the existence of mild solutions of Hilfer fractional stochastic integro-differential equations with nonlocal conditions, we need the following basic definitions and Lemmas.
Definition 2.1 (see [20], [21]). The Riemann–Liouville fractional integral operator of order µ > 0 for a function f can be defined as
where Γ( · ) is the Gamma function. Definition 2.2 (see [2]). The Hilfer fractional derivative of order 0 ≤ ν ≤ 1 and 0 < µ < 1 is defined as
Main result
Throughout this paper, we introduce the following hypotheses:
(H1) is a continuous function, and there exists a constant β ∈ (0, 1) and M1, M2 > 0 such that the function AβF satisfies the Lipschitz condition: for 0 ≤ s1, s2 ≤ b, xi, yi ∈ H, and the inequality holds for
(H2) The function satisfies the following conditions:
(i)
Example
Consider the following Hilfer fractional stochastic partial differential equation with nonlocal conditions in the form where b ≤ π, p is a positive integer, 0 < t0 < t1 < ⋅⋅⋅ < tp < b.
To write system (4.1) in the form of (1.1) let and A be defined by with domain
Conclusion
In this paper, we focused on Hilfer fractional stochastic integro-differential equations. By using Sadovskii fixed point theorem combined with the semigroup theory and fractional calculus, we obtained sufficient conditions for existence of mild solutions of Hilfer fractional stochastic integro-differential equations with nonlocal conditions. Finally an example is given to illustrate our results.
Our future work will be focused on investigate the null controllability for Sobolev-Type Hilfer
Acknowledgments
I would like to thank the referees and the editor for their careful reading and their valuable comments.
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