Well Posedness and Stability for the Nonlinear φ-Caputo Hybrid Fractional Boundary Value Problems with Two-Point Hybrid Boundary Conditions

Abstract

This article investigates into the study of nonlinear hybrid fractional boundary value problems, which involve ϕ-Caputo derivatives of fractional order and two-point hybrid boundary conditions. The author utilizes a fixed point theorem of Dhage to provide evidence for the existence and uniqueness of solutions, taking into consideration mixed Lipschitz and Caratheodory conditions. Additionally, the Ulam-Hyers types of stability are established in this context. The article concludes by introducing a class of fractional boundary value problems, which are dependent on the arbitrary values of ϕ and the boundary conditions chosen. The research presented in this article has the potential to be useful in various fields, such as engineering and science, where fractional differential equations are frequently used to model complex phenomena.

Key words and phrases. Hybrid fractional differential equation, Boundary value problem, Green’s function, Dhage fixed point theorem, ϕ-Caputo fractional derivatives, Existence Results, Hyers-Ulam stability of solutions.

2010 Mathematics Subject Classification. 34A40, 34A12, 34A99, 45D05