ON THE NONLINEAR GENERALIZED LANGEVIN EQUATION INVOLVING ψ-CAPUTO FRACTIONAL DERIVATIVES

Abstract

This paper considers the generalized Langevin equation involving [Formula: see text]-Caputo fractional derivatives in a Banach space. The fractional derivative is generalized from the Caputo derivative ([Formula: see text]), the Caputo–Katugampola ([Formula: see text], the Hadamard derivative ([Formula: see text]). We investigate the existence of mild solutions [Formula: see text] of the problem, in which the source function is assumed to satisfy some weakly singular conditions. Before proceeding to the main results, we transform the problem into an integral equation. Based on the obtained integral equation, the main results are proved via the nonlinear Leray–Schauder alternatives and Banach fixed point theorems. To prove this end, a new generalized weakly Gronwall-type inequality is established. Further, we prove that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders and the friction constant. As a consequence, we deduce that the solution [Formula: see text] of the equation involving the Caputo–Katugampola derivative tends to the solution [Formula: see text] of the equation involving the Hadamard derivative as [Formula: see text].