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

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving (k1,ψ1)-Hilfer and (k2,ψ2)-Caputo fractional derivative operators, and (k2,ψ2)- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the (k1,ψ1)-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ψ1 and the parameter β. Also the (k2,ψ2)-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ψ2-Riemann–Liouville, k2-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 L1-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.