Stability analysis of Caputo q-fractional Langevin differential equations under q-fractional integral conditions

Abstract

Abstract The primary objective of the paper is exploring the Ulam stability ( US ) $(\mathcal{US})$ of a Caputo q-fractional Langevin differential equation ( FLDE ) $(\mathcal{FLDE})$ under q-fractional integral boundary conditions ( FIBC s ) $(\mathcal{FIBC}s)$ . The novelty of this work stands out for its broader generality compared to the existing research focused on the Caputo q-fractional derivative. We apply the Banach contraction principle ( BCP ) $(\mathcal{BCP})$ for checking the existence and uniqueness of solutions of Caputo q − FLD $q-\mathcal {FLD}$ equations. The framework of the study integrates fundamental principles from both fractional calculus and quantum calculus. Additionally, we discuss various forms of Ulam stability, namely UHS $\mathcal{UHS}$ , GUHS $\mathcal{GUHS}$ , UHRS $\mathcal{UHRS}$ , and GUHRS $\mathcal{GUHRS}$ . We validate our theoretical findings through illustrative examples.

Keywords

• Quantum calculus
• Langevin differential equation
• Caputo q-fractional derivative
• Green’s function
• Ulam stability
• Fixed point theorem