<i>q</i>-Fractional Langevin Differential Equation with <i>q</i>-Fractional Integral Conditions

Abstract

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer’s fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples.

Keywords

• Langevin equations
• fractional <i>q</i>-differential equation
• Caputo derivative
• green function
• Ulam stability