Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation

Abstract

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations
 
 where in and are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration method are used to obtain the required.

Analysis of stochastic pantograph differential equations with generalized derivative of arbitrary order

“…Lemma 2.8. ( [9]) The function K(t, x(•), x(•)) is locally integrable in t for each fixed x ∈ [0, ∞) and is continuous non-decreasing in x for each fixed t ∈ J, for K(t, 0, 0) = 0 and γ > 0, if a non-negative continuous function φ(t) satisfies…”

Initial Value Problem for Stochastic Hyprid Hadamard Fractional Differential Equation

“…is the Hadamard fractional integral. In recent years, stochastic differential equations have become more and more important and interesting to researchers due to their successful and potential applications in various fields ( [24], [25], [26], [27], [28], [29], [31], [32]), and the basic theories and results of stochastic differential equations can be found in [24].…”