Existence and Stability of Implicit Fractional Differential Equations with Stieltjes Boundary Conditions Involving Hadamard Derivatives

Abstract

In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using the classical technique of functional analysis. At the end, the results are verified with the help of examples.

Analysis of implicit system of fractional order via generalized boundary conditions

“…Recently, researchers continuously focus on studying the solution of s in the sense of four main types of stability, that is, Ulam-Hyers stability ( ), generalized Ulam-Hyers stability ( ), Ulam-Hyers-Rassias stability ( ), and generalized Ulam-Hyers-Rassias stability ( ). Ulam's type stability was initially introduced by Ulam [18] in 1940 and has since been investigated and generalized by a number of mathematicians using various approaches [19][20][21][22][23][24].…”

“…Recently, researchers continuously focus on studying the solution of $$$ \mathcal{FDE}s $$$ in the sense of four main types of stability, that is, Ulam–Hyers stability ( $$$ \mathcal{UHS} $$$), generalized Ulam–Hyers stability ( $$$ \mathcal{GUHS} $$$), Ulam–Hyers–Rassias stability ( $$$ \mathcal{UHRS} $$$), and generalized Ulam–Hyers–Rassias stability ( $$$ \mathcal{GUHRS} $$$). Ulam's type stability was initially introduced by Ulam [18] in 1940 and has since been investigated and generalized by a number of mathematicians using various approaches [19–24].…”

A Two-Points Nonlocal Problem of an Implicit Delay Functional Integro-Differential Equation Malak. M. S. Ba-Ali

“…For papers studying such kind of problems (see [1,6,12,13]) and therein. Implicit differential equations have been studied in many papers and monographs [2,3,4,5,6,7,8] and [14,15,16,17,18]. Here, we are concerning with the delay implicit functional integro-differential equation dx dt = f t, dx dt , ϕ(t) 0 g(t, s, dx ds )ds , t ∈ (0, 1)…”

On a Coupled Impulsive Fractional Integrodifferential System with Hadamard Derivatives

“…In fact, there are not many applications of Hadamard integrals and derivatives in view of possibly the intrinsic intractability of these operators. However, one can find some recent results on Hadamard type fractional differential equations, for instance, see [17,21,25,39].…”

“…Coupled systems of fractional integrodifferential equations have also been extensively studied due to their applications. Some recent works dealing with coupled systems of Hadamard FDE s involving different kinds of boundary conditions can be found in [2,21,26,27].…”