Ulam-Hyers Stability of Four-Point Boundary Value Problem for Caputo Fractional Differential Equations with a Parameter

Abstract

Fractional calculus is considered to be a powerful tool in describing complex systems with a~wide range of applicability in many fields of science and engineering. The behavior of many systems can be described by using fractional differential equations with boundary conditions. In this sense, the study on the stability of fractional boundary value problems is of high importance.\\ The main goal of this paper is to investigate the Ulam--Hyers stability and Ulam--Hyers--Rassias stability of a class of fractional four-point boundary value problem involving Caputo derivative and with a given parameter. By using contraction principles, sufficient conditions are obtained to guarantee the uniqueness of solution. Therefore, we obtain sufficient conditions for the stability of that class of nonlinear fractional boundary value problems on the space of continuous functions. The presented results improve and extend some previous research. Finally, we construct some examples to illustrate the theoretical results.