( k , φ ) $(\mathtt{k},\varphi )$ -Hilfer fractional Langevin differential equation having multipoint boundary conditions

Abstract

Abstract The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin ( k , φ ) $(\mathtt{k},\varphi )$ -Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as ( k , φ ) $(\mathtt{k},\varphi )$ -Hilfer fractional differential equation. We provide some significant outcomes about ( k , φ ) $(\mathtt{k},\varphi )$ -Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to ( k , φ ) $(\mathtt{k},\varphi )$ -Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.

Keywords

• Fractional Langevin equation
• ( k , φ ) $({\mathtt{k}},\varphi )$ -Hilfer fractional derivative
• Existence and uniqueness
• Fixed point theorem
• Ulam
• Hyers stability