A New Result for ψ -Hilfer Fractional Pantograph-Type Langevin Equation and Inclusions

Abstract

In this paper, we deal with the existence and uniqueness of solution for $\psi$ -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.