Abstract
In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath e ´ odory approximation. Furthermore, with the help of H o ¨ lder’s inequality, Jensen’s inequality, It o ^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.
Highlights
We study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays
E research on the existence and uniqueness of solutions to fractional differential equations is an important content of differential equations
In [28], by using fractional calculus, the properties of classical and generalized Mittag–Leffler functions and the Ulam–Hyers stability of linear fractional differential equations were proved by utilizing the Laplace transform method. e authors investigated the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions in [29]
Summary
We intend to recommend a few basic definitions, lemmas, and some necessary assumptions that will play a key role in the paper. For any continuous function f: [0, T] × Rd × Rd ⟶ Rd, the Caputo derivative of fractional-order α> 0 is defined by. An Rd-value stochastic process {X(t)}− τ≤t≤T is called a solution to equation (1) if it satisfies the following conditions:. System (1) is Ulam–Hyers stable if there exists a real number δ > 0 such that ∀ε > 0 and for each continuously differentiable function Z(t) ∈ ([0, T], Rd) satisfying. A function Z(t) ∈ ([0, T], Rd) is a solution of equation (7) if and only if there exists a function h(t) ∈ ([0, T], Rd), such that (i) E(sup‖h(t)‖20 ≤ t ≤ T) ≤ ε (ii) CDα0+ Z(t) f(t, Z(t), Z(t − τ)) + g(t, Z(t), Z(t − τ)). Where f and g are uniformly continuous functions and ∨ is defined as Y1∨Y2 max Y1, Y2. Let us set c max(k1, k2). e proof is complete
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