Existence results for nonlocal Hilfer-type boundary value problems

Abstract

In this thesis, we have studied some nonlocal nonlinear boundary value problems involving Hilfer fractional derivative operators. As a first problem, we investigated the existence and uniqueness of solutions for a nonlinear Hilfer-type boundary value problem with mixed nonlinearities equipped with integral-multipoint boundary conditions. In the second problem, we extended our study of the first problem to its multivalued variant. The third problem was devoted to the investigation of a nonlinear coupled system of Hilfer fractional differential equations involving Riemann-Liouville mixed operators with nonlocal integro-multipoint boundary conditions. In the last problem, we discussed the existence and Ulam-Hyers stability of solutions for a coupled system of sequential Hilfer fractional differential equations with integro-multistrip-multipoint boundary conditions. The work accomplished in this thesis produced three research articles published in ISI journals: AIMS MATHEMATICS, QUALITATIVE THEORY OF DYNAMICAL SYSTEMs, and FILOMAT. The study of the problems considered in this thesis was based on the tools of modern analysis such as Banach contraction mapping principle, Krasnosel’skii’s fixed point theorem, Leray-Schauder nonlinear alternative, and some fixed point theorem for multivalued maps.