Fractional Order Difference Equations

A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations. 1. Introduction Fractional calculus has gained importance during the past three decades due to its applicability in diverse fields of science and engineering. The notions of fractional calculus may be traced back to the works of Euler, but the idea of fractional difference is very recent. Diaz and Osler [1] defined the fractional difference by the rather natural approach of allowing the index of differencing, in the standard expression for the th difference, to be any real or complex number. Later, Hirota [2] defined the fractional order difference operator where is any real number, using Taylor’s series. Nagai [3] adopted another definition for fractional order difference operator by modifying Hirota’s [2] definition. Recently, Deekshitulu and Mohan [4] modified the definition of Nagai [3] for in such a way that the expression for does not involve any difference operator. The study of theory of fractional differential equations was initiated and existence and uniqueness of solutions for different types of fractional differential equations have been established recently [5]. Much of literature is not available on fractional integrodifferential equations also, though theory of integrodifferential equations [6] has been almost all developed parallel to theory of differential equations. Very little progress has been made to develop the theory of fractional order difference equations. The main aim of this paper is to establish theorems on existence and uniqueness of solutions of various classes of fractional order difference equations. Further we define autonomous and nonautonomous fractional order difference equations and find their solutions. 2. Preliminaries Throughout the present paper, we use the following notations: be the set of natural numbers including zero. for . Let . Then for all and , , and , that is, empty sums and products are taken to be 0 and 1, respectively. If and are in , then for this function , the backward difference operator is defined as . Now we introduce some basic definitions and results concerning nabla discrete fractional calculus. Definition 2.1. The extended binomial coefficient , ( ) is defined by Definition 2.2 (see [7]). For any complex numbers and , let be defined as follows: Remark 2.3. For any