On Solutions of Hybrid–Sturm-Liouville–Langevin Equations with Generalized Versions of Caputo Fractional Derivatives

Abstract

The main intention of this research article is to introduce a new class of generalized fractional differential equations that fall into the categories of Sturm-Liouville’s, Langevin’s, and hybrid’s problems involving Y -Caputo fractional derivatives. The existence of the solutions of the proposed equations is discussed by using the technique of the measure of noncompactness related to the fixed point theorem, which is a generalization of Darbo’s fixed point theorem. Additionally, pertinent examples are provided along with the different values of the function Y to confirm the validity of the reported results.

Coupled Fixed Point Theorem for the Generalized Langevin Equation with Four-Point and Strip Conditions

“…The stochastic nonlinear fractional Langevin equation with a multiplicative noise has been studied by [28]. The hybrid Sturm-Liouville-Langevin equations with new versions of Caputo fractional derivatives have been investigated in [29].…”