Existence uniqueness of mild solutions for ψ-Caputo fractional stochastic evolution equations driven by fBm

Abstract

In this paper, we investigate the existence uniqueness of mild solutions for a class of ψ-Caputo fractional stochastic evolution equations with varying-time delay driven by fBm, which seems to be the first theoretical result of the ψ-Caputo fractional stochastic evolution equations. Alternative conditions to guarantee the existence uniqueness of mild solutions are obtained using fractional calculus, stochastic analysis, fixed point technique, and noncompact measure method. Moreover, an example is presented to illustrate the effectiveness and feasibility of the obtained abstract results.

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

“…Using both fractional and stochastic generalizations in one model offers a more flexible tool for modeling systems that exhibit both memory effects and randomness. Research on stochastic fractional differential equations (SFDEs) is a growing area in applied mathematics, e.g., [20][21][22][23][24][25][26][27].…”

Ulam–Hyers Stability for a Class of Caputo-Type Fractional Stochastic System with Delays

“…In [12], the authors employed two methods to prove the existence and uniqueness of fractional stochastic neutral differential equations, in which the contraction mapping principle was used under the Lipschitz condition at first, and then, the Picard approximation was applied under the non-Lipschitz condition. In [13], the author proved existence and uniqueness of ψ-Caputo fractional stochastic evolution equations with varying-time delay driven by fBm using the Schauder fixed-point theorem [14] and the Banach contraction principle, separately. In [15], Picard iterations were used to explore the existence and uniqueness of multidimensional FSDEs with variable order under non-Lipschitz condition.…”

“…In [15], Picard iterations were used to explore the existence and uniqueness of multidimensional FSDEs with variable order under non-Lipschitz condition. Although [13,15] both used Picard approximation, the mild solution in [13] was solved by Laplace transform and its inverse, while the mild solution in [15] was obtained by integrating both sides of the differential equations. In addition, Yang [13] investigated the constant fractional order, while Moualkia [15] explored variable fractional order.…”

“…Although [13,15] both used Picard approximation, the mild solution in [13] was solved by Laplace transform and its inverse, while the mild solution in [15] was obtained by integrating both sides of the differential equations. In addition, Yang [13] investigated the constant fractional order, while Moualkia [15] explored variable fractional order. In [16], the authors studied the existence of fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space by using fractional calculus, operator semigroup, and fixed-point theorem.…”