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

Abstract

In this article, we deal with the existence and Hyers‐Ulam stability of solution to a class of implicit fractional differential equations (FDEs), having some initial and impulsive conditions. Some adequate conditions for the required results are obtained by utilizing fixed point theory and nonlinear functional analysis. At the end, we provide an illustrative example to demonstrate the applications of our obtained results.

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

“…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 the impulsive implicit Ψ‐Hilfer fractional differential equations with delay

“…13,15,16,[27][28][29][30][31] Further, looking towards the multidimensional utilization of impulsive FDES, 1,2 it is important to consider the new class implicit FDEs with impulse condition that incorporates a wide class of impulsive FDEs as particular cases. [16][17][18][19][20][21][22][23][24][25] The paper is organized as follows. If we take Ψ(t) = t and = 1, then the problem (1)-(4) reduces to implicit impulsive FDEs with the Caputo fractional derivative.…”

“…The existence and uniqueness of solutions and the Ulam-Hyers-Mittag-Leffler (UHML) stability of different kinds of fractional differential and integral equations with time delay have been investigated in Eghbali et al, 17 Wang et al, 18 and Niazi et al 19 by using Picard operator theory and abstract Gronwall lemma. Then again, there are many fascinating research papers involving Hilfer fractional derivative, which incorporates the Riemann-Liouville and Caputo fractional derivative as special cases.…”

“…32 Further, the results obtained in the present paper extends previous studies. [16][17][18][19][20][21][22][23][24][25] The paper is organized as follows. In Section 2, we collect definitions and some basic results related to the Ψ-Hilfer derivative.…”

Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative

“…A Laplace transform method is applied to show the Hyers-Ulam stability for integer order differential equations in [25,26] and Wang and Li [27] adopted the idea and applied a Laplace transform method to show the Hyers-Ulam stability for fractional order differential equations involving Caputo derivatives. There are many papers on differential equations involving fractional derivatives-see, for example, [28][29][30][31][32][33][34][35][36]. However, there are only a few papers on the Hyers-Ulam stability for differential equations with the Caputo-Fabrizio fractional derivative.…”