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.
The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the nth-commutativity degree of a finite algebraic structure S, denoted by P n (S) , is the probability that for chosen randomly two elements x and y of S, the relator x n y = yx n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.
Abstract. Discrete inequalities, in particular the discrete analogues of the Gronwall-Bellman inequality, have been extensively used in the analysis of finite difference equations. The aim of the present paper is to establish some fractional difference inequalities of Gronwall-Bellman type which provide explicit bounds for the solutions of fractional difference equations.
In the present paper, we define the nabla discrete Sumudu transform (S-transform) and present some of its basic properties. We obtain the nabla discrete Sumudu transform of fractional sums and differences. We apply this transform to solve some fractional difference equations with initial value problems. Finally, using S-transforms, we prove that discrete Mittag-Leffler function is the eigen function of Caputo type fractional difference operator $\nabla^\alpha$.
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