On Stability Analysis of Fractional Difference Equations

Abstract

In this dissertation, we discussed existence, uniqueness and stability of di erent types of fractional di erence equations in delta and nabla sense. Existence and stability results for a class of non-linear Caputo nabla fractional di erence equations are obtained using xed point theorems including Schauder's xed point theorem, the Banach contraction principle and Krasnoselskii's xed point theorem. Furthermore, results depends on the structure of nabla discrete Mittag-Le er functions. Existence and uniqueness of solution for impulsive fractional di erence equation is investigated through xed point theorems including the Banach contraction principle, Schaefer's xed point theorem and nonlinear alternative Leray Schauder theorem. Moreover, Ulam's type stability of problem in delta sense is discussed using newly developed Gronwall inequality. Using existing q-fractional Gronwall inequality, Ulam-Hyers stability and the Ulam-Hyers-Rassias stability is discussed for a delay Caputo q-fractional di erence system. Using newly developed Gronwall-Bellman inequality, we discussed Ulam-Hyers stability of Caputo nabla fractional di erence system. Existence of solution of p-Laplacian fractional difference equations in nabla sense is discussed using Schaefer's xed point theorem and then Ulam-Hyers stability is examined. Furthermore, we discussed Ulam-Hyers-Mittag-Le er and Ulam-Hyers-Rassias-Mittag-Le er stability for a class of Caputo nabla fractional order delay di erence equation using Banach xed point theorem in generalized complete metric space and using Chebyshev norm. Moreover, we obtained existence and stability results for a fractional di erence Langevin equation within nabla Caputo fractional di erence and subject to non-local boundary conditions using xed point theorems.