Nonlinear Langevin time-delay differential equations with generalized Caputo fractional derivatives

Abstract

In a Banach space, we consider the nonlinear Langevin time-delay differential equations with $\psi$-Caputo fractional derivatives. Using weighted norms involving Mittag-Leffler functions, we obtain some existence and uniqueness of solutions of the problem. Besides, in some cases, the condition for the problem having a unique solution can be relaxed using Burton's method. We also obtain some Ulam-Hyers and Ulam-Hyers-RassiasMittag-Leffler stability results for the main equation. Two examples are given to illustrate our theoretical findings.