Theory of Fractional Hybrid Problems in the Frame of ψ -Hilfer Fractional Operators

Abstract

In the present manuscript, we develop and extend a qualitative analysis for two classes of boundary value problems for nonlinear hybrid fractional differential equations with hybrid boundary conditions involving a ψ-Hilfer fractional order derivative introduced by Sousa and de Oliveira (2018). First, we derive the equivalent fractional integral equations to the proposed problems from some properties of the ψ-fractional calculus. Next, we establish the existence theorems in the weighted spaces via equivalent fractional integral equations with the help of Dhage's fixed-point theorem (2004). Besides, for an adequate choice of the kernel ψ, we recover most of all the preceding results on fractional hybrid equations. Finally, two examples are constructed to make our main findings effective.