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Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

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  • Athasit Wongcharoen
  • Bashir Ahmad
  • Sotiris K. Ntouyas
  • Jessada Tariboon

Abstract

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications . This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

Suggested Citation

  • Athasit Wongcharoen & Bashir Ahmad & Sotiris K. Ntouyas & Jessada Tariboon, 2020. "Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative," Advances in Mathematical Physics, Hindawi, vol. 2020, pages 1-11, May.
    • Handle: RePEc:hin:jnlamp:9606428
    DOI: 10.1155/2020/9606428
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