Existence of Weak Solutions to Fractional Differential Equations

Abstract

Fractional calculus generalizes the integer order integration and differentiation concepts to an arbitrary (real or complex) order. The main reason of this treatise is to generalize some classes of differential equations from integer order to fractional order with the main focus of interest at the existence theory of their weak solutions. Especially, we targeted such types of differential equations of fractional order which involve right and left hand fractional operators for which the corresponding integral equation is not possible to find. We also introduced a new generalized fractional derivative space along with some relevant results for the classes of fractional boundary value problems involving generalized differential operators of fractional orders to examine the existence of their solutions. We mainly investigated three different classes of boundary value problems with fractional order. The thesis starts with the introduction of some basic definitions and notations from the field of fractional calculus and nonlinear analysis. Later on, getting aid from a fixed point theorem, variational methods and critical point theory, we setup conditions which are sufficient for the existence of at least one unique weak solution to a class of nonlinear impulsive problems of fractional-order, linear impulsive problems of fractional-order with non-instantaneous impulses and for a class of generalized fractional boundary value problems. Some examples are also included in this work to highlight the effectiveness of our results and to make them interesting for the readers.