Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations

Abstract

This manuscript deals with the existence theory, uniqueness, and various kinds of Ulam–Hyers stability of solutions for a class and coupled system of fractional order differential equations involving Caputo derivatives. Applying Schaefer and Banach’s fixed point approaches, existence and uniqueness results are obtained for the proposed problems. Stability results are investigated by using the classical technique of nonlinear functional analysis. Examples are given with each problem to illustrate the main results.

Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations

“…Proof. To begin with, the operator T : P → P is described by (6). Next, we demonstrate that T maps bounded sets into bounded sets in C([1, T ], R).…”

“…Let us interpret B θ = {y ∈ P : y ≤ θ}, where θ ≥ ξ ∆. To prove the hypothesis of Krasnoselskii's fixed point result, we split the operator T given by (6) as…”

“…Let g : [1, T ] × R → R be a continuous function satisfying the assumption: Proof. Let us interpret the operator T : P → P as in (6) and the continuous nondecreasing function Φ :…”

“…The monographs [14,15,16,18,20] have ordinarily referred to the hypothesis of fractional derivatives and integrals and applications to differential equations of fractional order. For more points of interest and models, see [1,4,5,6,9,10,17,24,25,26,27,28,29,21,22,23,31], and the references in that.…”

Stability analysis of first‐order impulsive nonautonomous system on timescales

“…Furthermore, Hyers result was generalized by Rassias, so this notion is given a name known as Ulam‐Hyers‐Rassias stability. Ulam's stability has been well investigated for different differential equations with different approaches, we recommend the reader to previous studies …”