Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

Abstract

In this paper, we mainly consider the existence and finite-time stability of solutions for a kind of ψ-Hilfer fractional differential equations involving time-varying delays and non-instantaneous impulses. By Schauder's fixed point theorem, the contraction mapping principle and the Lagrange mean-value theorem, we present new constructive results as regards existence and uniqueness of solutions. In addition, under some new criteria and by applying the generalized Gronwall inequality, we deduce that the solutions of the addressed equation have finite-time stability. Some results in the literature can be generalized and improved. As an application, three typical examples are delineated to demonstrate the effectiveness of our theoretical results.

On the Novel Finite‐Time Stability Results for Uncertain Fractional Delay Differential Equations Involving Noninstantaneous Impulses

“…Based on the known results [21][22][23], we can clearly conclude that this fractional system (4) has at least one solution under some conditions. Now, we plan to investigate the specified stability.…”

“…In this paper, we mainly study the finitetime stability, which has been investigated by some researchers. For an extensive collection of such results, we refer the readers to the related literatures, such as the papers [19][20][21][22][23]. In detail, in [19], Wu et al studied the finite-time stability of Caputo delta fractional linear difference equations with the aid of Grönwall inequality.…”

“…To research the influence of impulses on finite-time stability of fractional differential system, D. F. Luo and Z. G. Luo in [23] investigated the following ψ-Hilfer fractional differential equation with time-varying delays and noninstantaneous impulses:…”

“…Mathematical Problems in Engineering Definition 1 [23]. One-parameter Mittag-Leffler function: the Mittag-Leffler function is given by the following series:…”

Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

“…In this section, we recall some notation, definitions and preliminaries about fractional calculus [6,7,21], ψ-Caputo fractional calculus [3][4][5]22,23], and Riesz or Riesz-Caputo fractional derivative [17][18][19]. Definition 1 ([6]).…”