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Boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. (English) Zbl 1497.34009

Summary: 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.

MSC:

34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A60 Ordinary differential inclusions
26A33 Fractional derivatives and integrals
47N20 Applications of operator theory to differential and integral equations
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