Mild Solution Existence for Nonlinear φ–Caputo Fractional Differential Equations in Banach Spaces

This paper investigates a nonlinear initial value problem involving the φ–Caputo fractional derivative within the framework of a Banach space. Fractional differential equations of this type have gained significant attention due to their ability to model complex dynamical systems with memory and hereditary properties. To establish the existence of mild solutions for the considered problem, we employ the Meir–Keeler fixed point theorem in combination with the concept of measure of noncompactness. This approach allows us to derive sufficient conditions under which the problem admits at least one mild solution, extending the applicability of fractional analysis to abstract Banach space settings. The theoretical framework developed herein provides a general methodology for analyzing nonlinear fractional differential equations, and the results are applicable to a wide class of nonlinear operators. Furthermore, to illustrate the practical relevance and applicability of the theoretical findings, a representative example is presented, demonstrating how the derived conditions ensure solvability in a concrete scenario. Overall, the paper contributes to the understanding of nonlinear φ–Caputo fractional differential equations by offering both rigorous existence results and a clear methodological approach that can be adapted to related problems in fractional calculus and functional analysis. The results provide a foundation for further studies on qualitative behavior, stability, and numerical approximations of such equations.