The Qualitative Theory of Fractional Difference Equations

Abstract

Focus of this study is to cover up some gaps and further build up the theory of discrete fractional calculus. This dissertation starts with brief introduction and de nitions to discrete fractional calculus. Two new de nitions of generalized fractional di erence operator are introduced namely Hilfer fractional di erence operator and substantial fractional di erence operator. A missing property in the literature for delta Laplace transform i.e. delta exponential shift is established. The delta Laplace transform is presented for the newly introduced Hilfer and substantial fractional di erences. The double Laplace transform in a delta discrete setting is introduced, and its existence, uniqueness and basic properties are discussed. The delta double Laplace transform is presented for integer and non-integer order partial di erences. Another goal of this study is to establish the existence and UHR stability for various classes of fractional di erence equations. Conditions are acquired for RL, Caputo, Hilfer and substantialtype fractional di erence equations. Moreover we establish a technique to transforming arbitrary real order delta di erence equations with impulses to corresponding summation equations. Existence results are built up for impulsive delta fractional di erence equation with nonlocal initial condition and two-point and four-point boundary conditions. The conditions for existence and UHR stability of the solution to multi-point summation boundary value problem are established.