Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems

Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon ()
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Sotiris K. Ntouyas: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Bashir Ahmad: Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Jessada Tariboon: Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

Mathematics, 2025, vol. 13, issue 13, 1-25

Abstract: This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving ( k 1 , ψ 1 ) -Hilfer and ( k 2 , ψ 2 ) -Caputo fractional derivative operators, and ( k 2 , ψ 2 ) - Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the ( k 1 , ψ 1 ) -Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ψ 1 and the parameter β . Also the ( k 2 , ψ 2 ) -Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ψ 2 -Riemann–Liouville, k 2 -Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and L 1 -Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study.

Keywords: Hilfer and Caputo fractional derivative operators; inclusions; integral boundary conditions; existence; fixed-point theorems for multivalued maps (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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