On some existence results for nonlinear fractional differential equations with hybrid proportional Caputo derivatives

Abstract

The purpose of this manuscript is to establish the existence and uniqueness of some solutions to a class of boundary value problems for a generalized nonlinear hybrid Caputo fractional differential equation with nonlocal boundary conditions. The proofs of our main results are derived from mixed Lipshitz and Caratheodory conditions and some basic techniques of $\Psi$-fractional calculus. Some fundamental $\Psi$-fractional differential inequalities are also established. As an application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.