In this paper, sufficient conditions are established for the oscillatory and asymptotic behavior of higher–order half–linear delay difference equation of the form ∆(pn(∆ (xn + qnxτn)) ) + rnx β σn = 0, n ≥ n0, where it is assumed that ∑∞ s=n0 1/p 1/α s < ∞. The main theorem improves some existing results in the literature. An example is provided to demonstrate the effectiveness of the main result.

Necessary and sufficient conditions for the asymptotic stability of a class of difference equations with three parameters are obtained. These conditions are expressed in terms of subsets of the parameter space.

In this paper we put into relation the index of an infinite aperiodic word and its recurrence function. With the use of this relation, we then give a new characterization of Sturmian words. As a byproduct, we give a new proof of theorem of Damanik and Lenz describing the index of a Sturmian word in terms of the continued fraction expansion of its slope.

We give sufficient conditions for the boundedness of all solutions of some classes of systems of difference equations with delays, by comparison of their norms with the solution of certain auxiliary scalar difference equations.

We obtain necessary and sufficient conditions for the asymptotic stability of the linear delay difference equation x n+1 + p N j=1 x n−k+(j−1)l = 0, where n = 0,1,2,..., p is a real number, and k, l, and N are positive integers such that k > (N − 1)l.

We consider a difference equation involving the discrete p-Laplacian operator, depending on a positive real parameter k. We prove, under convenient assumptions, that for k big enough the equations admit at least one homoclinic constant sign solution in Z. Our method consists in two parts: first, we prove the existence of two Dirichlet-type solutions for the equation in the discrete interval ½Àn...

and Applied Analysis 3 We note that 1.1 in its general form involves some different types of differential and difference equations depending on the choice of the time scale T . For example: 1 for T R, we have σ t t, μ t 0, and xΔ t x′ t , and 1.1 becomes the Cauchy integrodifferential equation: x′ t f ( t, x t , ∫ t 0 k t, s, x s ds ) , t ∈ R,