On Ulam's type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions

Abstract

In this manuscript, using Schaefer's fixed point theorem, we derive some sufficient conditions for the existence of solutions to a class of fractional differential equations (FDEs). The proposed class is devoted to the impulsive FDEs with nonlinear integral boundary condition. Further, using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss various kinds of Ulam-Hyers stability. Finally to illustrate the established results, we provide an example.

Hyers‐Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions

“…Here, we mention some physical and evolution processes which suffer from sudden changes like mechanical systems subjected to impacts, function of pendulum clock, function of heart, operation of damper subjected to percussive effects, electromechanical systems with relaxation type oscillation, dynamical systems with automatic regulations and stock exchange, etc. For some of the applications of impulsive differential equations, we refer …”

“…For some of the applications of impulsive differential equations, we refer. [18][19][20][21][22][23][24] In the systems of differential equations, optimization theory, numerical analysis, economics, etc, the stability analysis plays an important role. In literature, various concepts of stability like exponential stability, Layponove stability, Mittag-Lefler stability, and Hyers-Ulam stability have been adopted to investigate the stability of different systems of differential equations.…”

Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

“…A lot of fractional differential equations and coupled systems have been studied widely; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]24] and the references therein. As is well known, coupled systems with boundary conditions appear in the investigations of many problems such as mathematical biology (see [9,30]), natural sciences and engineering; for example, we can see beam deformation and steady-state heat flow (see [25,26]) and heat equations (see [18,24]).…”

“…So the subject of coupled systems is gaining much attention and importance. There are a large number of articles dealing with the existence or multiplicity of solutions or positive solutions for some nonlinear coupled systems with boundary conditions; for details, see [7,8,10,11,20,21,27,29,32,33,[35][36][37][38][39][40][41].…”

On uniform exponential stability of linear switching system

“…In recent years, research work is specially focused on the analysis of dynamic behaviors of linear and nonlinear systems, such as stability, controllability, reachability, and observability . Stability analysis is a fundamental issue among numerous interesting topics on switched systems.…”