Existence and Ulam–Hyers stability of nonlinear fractional differential equations via enriched (g, alpha, beta, k)-functional contractions

G. Sudhaamsh Mohan Reddy

doi:10.46481/jnsps.2026.3775

Authors

G. Sudhaamsh Mohan Reddy [email protected] Department of Mathematics, Faculty of Science and Technology(IcfaiTech), Icfai Foundation for Higher Education, Hyderabad 501203, India

Keywords:

Fractional differential equations, Caputo derivative, Enriched contractions, Functional contractions, Ulam–Hyers stability

Abstract

This study investigates the existence, uniqueness, and Ulam--Hyers stability of solutions to nonlinear fractional differential equations (NFDEs) involving the Caputo fractional derivative. We extend and combine established fixed-point concepts into a class of contractive mappings on Banach spaces called enriched (g, alpha, beta, k)-functional contractions. A bounded linear operator g supplements the metric structure, while the functional contraction rates alpha and beta allow non-constant contraction behavior; a parameter k controls the weight of the auxiliary mapping. We prove a fixed-point theorem for this class and apply it to initial value problems for NFDEs. Under Lipschitz and continuity assumptions on the nonlinearity, we establish existence and uniqueness of a solution in a space of continuous functions. We also derive explicit sufficient conditions for Ulam--Hyers stability, together with a computable upper bound. The applicability of the results is discussed for fractional relaxation--oscillation models and fractional population dynamics, and several examples are used to examine the hypotheses and illustrate the theorems. The framework links fixed-point theory with the qualitative analysis of fractional systems through a unified approach to stability analysis.

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