Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the ψ-Caputo sense

ABSTRACT

The main aim of this paper is to prove the Ulam–Hyers stability of solutions for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the ψ-Caputo sense. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Next, we proceed to investigate the existence and uniqueness of the solution by applying Schauder and Banach fixed point theorems. Finally, we study the Ulam–Hyers stability criteria for the main fractional system. Illustrative examples are presented to demonstrate the validity of the obtained results. The results are new and provide extensions to some known results in the literature.

Keywords:

• ψ-Caputo fractional derivative
• fixed point theorems
• existence
• uniqueness
• Ulam stability
• Langevin equation

2010 Mathematics Subject Classifications:

• 34A08
• 34A128

Disclosure statement

No potential conflict of interest was reported by the author(s).