Nonlinear sequential fractional boundary value problems involving generalized $\psi$-Caputo fractional derivatives

TL;DR

This paper studies nonlinear fractional boundary value problems with generalized $$-Caputo derivatives, deriving new inequalities, eigenvalue bounds, and existence results, even for problems with singular source functions.

Contribution

It introduces a new Lyapunov-type inequality and eigenvalue bounds for nonlinear fractional problems involving generalized $$-Caputo derivatives, extending previous results to nonlocal boundary conditions.

Findings

Derived a new Lyapunov-type inequality

Established lower bounds for eigenvalues

Proved existence of solutions even with singular source functions

Abstract

This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $\psi$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.