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,

1 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Key Laboratory of Network Control and Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education, Chongqing 400065, China 3 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 4 Fundamental Department, Heb...

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We study branching random walks in random i.i.d. environment in Zd, d ≥ 1. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotic...