On the impulsive implicit Ψ‐Hilfer fractional differential equations with delay

Abstract

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam-Hyers-Mittag-Leffler stability results for impulsive implicit Ψ-Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam-Hyers and generalized Ulam-Hyers stability are the specific cases of Ulam-Hyers-Mittag-Leffler stability. Extended version of the Gronwall inequality, abstract Gronwall lemma, and Picard operator theory are the primary devices in our investigation. We provide an example to illustrate the obtained results.

A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

“…Definition 2 (see [21,23]). Problem (2) is called Hyers-Ulam-Rassias stable, with respect to ϕ ∈ C([0, a], R), if there exists a positive constant C ζ,ϕ such that, for any ε > 0 and for each ξ ∈ C 1− c,g ([0, a]) satisfying the inequality,…”

“…e stability theory for FDIs and IDEs via ψ− Hilfer fractional derivative have also been discussed (see [18][19][20][21][22]). In [23], by using Gronwall inequality and Picard operator theory, Kharade and Kucche proved the existence and uniqueness of solutions for impulsive implicit delay ψ− Hilfer fractional differential equations. e Ulam-Hyers-Mittag-Leffler stability was also considered.…”

“…Motivated by Tate el al. [14,15], Sousa et al [16], and Kharade et al [23], in this paper, we investigate the existence and uniqueness of solutions and some properties of solutions of the following fractional differential equation with the constant coefficient λ > 0:…”

Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

“…The Ulam-Hyers stability point of view, is the vital and special type of stability that attracts many researchers in the field of mathematical analysis. Moreover, the Ulam-Hyers and Ulam-Hyers-Rassias stability of linear, implicit and nonlinear fractional differential equations were examined in [17,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49].…”

Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations

“…The Hilfer version of the fractional derivative with another function called Ψ-Hilfer FDO has been presented by Sousa et al 44 The basic study about existence and uniqueness of the solution of a nonlinear Ψ-Hilfer FDEs with different kinds of initial conditions and the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions have been explored in previous studies. [45][46][47][48][49][50][51][52] The implicit FDEs involving Ψ-Hilfer derivative has been investigated in Sousa and Oliveira 53 for the existence and uniqueness of the solution and the Ulam-Hyers-Rassias stability.…”