In this work, we discuss the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. First of all, we give some definitions, theorems, and lemmas that are necessary for the understanding of the manuscript. Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. Third of all, in the inclusion case, to obtain the existence result, we use the Leray–Schauder alternative. Last but not least, we give an illustrative example.

This paper deals with the existence and uniqueness of solution for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions. The novelty of this work is that it is more general than the works based on the derivative of Caputo and Riemann-Liouville, because when ? = 0 we find the Riemann-Liouville fractional derivative and when ? = 1 we find the Caputo fractional derivative. Initially, we give some definitions and notions that will be used throughout the work, after that we will establish the existence and uniqueness results by employing the fixed point theorems. Finaly, we investigate different kinds of stability such as Ulam-Hyers stability, generalized Ulam-Hyers stability.

In this paper, we deal with the existence and uniqueness of solution for ψ -Hilfer Langevin fractional pantograph differential equation and inclusion; these types of pantograph equations are a special class of delay differential equations. The existence and uniqueness results are obtained by making use of the Krasnoselskii fixed-point theorem and Banach contraction principle, and for the inclusion version, we use the Martelli fixed-point theorem to get the existence result. In the end, we are giving an example to illustrate our results.

The aim of this paper deals with the existence results for a class of fractional langevin inclusion with multi-point boundary conditions. To prove the main results, we use the fixed theoreme for condensing multivalued maps, which is applicable to completely continuous operators. Our results extend and generalize several corespending results from the existing literature.