Implicit Fractional Differential Equations in Banach Spaces via Picard and Weakly Picard Operator Theory

TL;DR

This paper establishes existence and uniqueness results for nonlinear implicit fractional differential equations in Banach spaces using fixed-point methods, Picard, and weakly Picard operator theory, along with solution dependence analysis.

Contribution

It introduces novel fixed-point approaches for implicit fractional differential equations in Banach spaces, extending classical methods with Pompeiu--Hausdorff functional.

Findings

Existence and uniqueness of solutions proven

Solution dependence on initial conditions analyzed

Application of Picard and weakly Picard operator theory demonstrated

Abstract

In this paper, by employing fixed-point methods, we obtain the existence and uniqueness results for the nonlinear implicit fractional differential equations in Banach spaces. Further, we obtain the uniqueness, dependence of the solution on the initial condition as well as on the functions involved on the right-hand side by means of Picard and weakly Picard operator theory and Pompeiu--Hausdorff functional.