Multipoint BVP for the Langevin Equation under ϕ -Hilfer Fractional Operator

Abstract

In this research paper, we consider a class of boundary value problems for a nonlinear Langevin equation involving two generalized Hilfer fractional derivatives supplemented with nonlocal integral and infinite-point boundary conditions. At first, we derive the equivalent solution to the proposed problem at hand by relying on the results and properties of the generalized fractional calculus. Next, we investigate and develop sufficient conditions for the existence and uniqueness of solutions by means of semigroups of operator approach and the Krasnoselskii fixed point theorems as well as Banach contraction principle. Moreover, by means of Gronwall's inequality lemma and mathematical techniques, we analyze Ulam-Hyers and Ulam-Hyers-Rassias stability results. Eventually, we construct an illustrative example in order to show the applicability of key results.