On the solvability and stability of nonlinear \(\varphi \)-Caputo fractional differential equations in Banach spaces

Abstract

This paper investigates a nonlinear initial value problem involving the \(\varphi \)-Caputo fractional derivative within the framework of a Banach space. To establish the existence of mild solutions, we utilize the Meir-Keeler fixed point theorem in conjunction with the concept of measure of noncompactness. The existence and uniqueness of solutions are further demonstrated via the Banach contraction principle. In addition, we explore the solution’s sensitivity to initial data and examine its Ulam-Hyers stability. To illustrate the theoretical findings, representative examples are presented.