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 $\psi$ -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 $\psi$ -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 $\psi$ , we recover most of all the preceding results on fractional hybrid equations. Finally, two examples are constructed to make our main findings effective.