Abstract
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.
Highlights
Fractional derivatives give an excellent description of memory and hereditary properties of different processes
Several researchers in the recent years have employed the fractional calculus as a way of describing natural phenomena in different fields such as physics, biology, finance, economics, and bioengineering
With the recent outstanding development in fractional differential equations, the Langevin equation has been considered a part of fractional calculus, and important results have been elaborated [11–15]
Summary
Fractional derivatives give an excellent description of memory and hereditary properties of different processes. We investigate the existence and uniqueness criteria for the solutions of the following nonlocal boundary value problem: We prove the existence of solution for problem (2) by applying the nonlinear alternative of the Leray–Schauder [19].
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