TURK 2025

This study focuses on the existence and uniqueness of solutions for multi-point boundary value problems involving differential equations with ψ-Caputo fractional derivatives. In recent years, fractional differential equations have attracted significant interest due to their ability to model various complex phenomena in physics, engineering, economics, and biology more accurately than classical integer-order models. In this context, the ψ-Caputo fractional derivative, which generalizes many classical fractional operators, is combined with the nonlinear p-Laplacian operator to examine a specific class of boundary value problems.
The study begins by presenting fundamental definitions, lemmas, and theorems related to fractional calculus, particularly those involving the ψ-Caputo derivative and the ψ-Riemann–Liouville integral. Subsequently, a multi-point boundary value problem governed by a p-Laplacian ψ-Caputo fractional differential equation is formulated. This problem is reduced to an equivalent integral equation by utilizing the intrinsic properties of fractional operators.
To establish the existence and uniqueness of solutions, two classical fixed point theorems—Banach’s Fixed Point Theorem and Schaefer’s Fixed Point Theorem—are applied under suitable assumptions. The necessary conditions for the existence of a solution are rigorously derived. Additionally, uniqueness is guaranteed by demonstrating the formation of a contraction mapping within the proposed framework.
To illustrate the applicability of the theoretical findings, two example problems are presented, showing that the proposed method yields valid and meaningful solutions. This study contributes to the ongoing research in fractional differential equations by providing analytical tools to solve multi-point boundary value problems involving nonlinear operators and fractional derivatives. Furthermore, it emphasizes the power and flexibility of fixed point theory in the field of fractional analysis.