Existence and Uniqueness of Solutions for Impulsive Fractional Partial Differential Equation Boundary Value Problems with theψ−Caputo Derivative under the Weak Topology.

Abstract

In this thesis, we study existence, uniqueness, and stability results for several classes of fractional differential equations in Banach spaces. We consider initial value problems for sequentialψ-Caputo fractional Langevin equations, boundary value problems involving impulses with power law kernels, a variable order Caputo thermistor model, and variable order Riemann-Liouville boundary value problems with multi-point data. The analysis is carried out in the weak topology framework using the Pettis integral, the De Blasi mea- sure of weak noncompactness, and fixed-point theorems of M¨onch, Schauder, Banach con- traction, and Krasnoselskii. Existence is proved via M¨onch’s theorem, uniqueness via the Banach contraction principle, and generalized Ulam-Hyers-Rassias stability is established. For the variable order thermistor problem, existence and uniqueness are obtained by split- ting the order into piecewise constant subintervals and applying Schauder’s and Banach’s theorems. For the Riemann-Liouville problem, the method of upper and lower solutions combined with Schauder’s theorem yields positive solutions in a fractional Sobolev space. Numerical examples illustrate the theoretical findings