The Fixed Point Approach to the Stability of Fractional Differential Equations with Causal Operators

Abstract

In this paper, the Hyers-Ulam (HU) stability and Hyers-Ulam-Rassias (HUR) stability of fractional differential equations with causal operators (FDEwCO) are investigated. The techniques rely on a fixed point theorem which is employed to study the HUR stability for FDEwCO on both bounded and unbounded time intervals as well as HU stability on bounded time interval. Finally, two typical examples are given to demonstrate the applications of theoretical results proposed.

Existence and Ulam–Hyers stability of a kind of fractional-order multiple point BVP involving noninstantaneous impulses and abstract bounded operator

“…This type of stability has been widely studied by many scholars since it was proposed. There have been many papers on this this stability (see [9][10][11][12][13][14][15][16][17][18][19][20]).…”

Hyers-Ulam Stability of a Nonlinear Multi-Term Fractional Differential Equation on Bounded Time Interval

“…Besides, in many papers, the Ulam stability of classical differential equations has been extended to the several types of fractional differential equations. For more details, one can see [17][18][19][20][21][22].…”

Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions

“…The study of a class of evolution equations with causal operators was approached in [1,15]. For some recent contribution to the study of autonomous fractional evolution equations with causal operators we refer to the following papers [29,30,54]. In this paper we bring some contributions to the study of a class of non-autonomous evolution equations involving causal operators.…”