In 1.1 , the given real sequences {pn}, {qn} satisfy pn T pn > 0, qn T qn for any n ∈ Z, f : Z×R → R is continuous in the second variable, and f n T, z f n, z for a given positive integer T and for all n, z ∈ Z×R. −1 δ −1, δ > 0, and δ is the ratio of odd positive integers. By a solution of 1.1 , we mean a real sequence x {xn}, n ∈ Z, satisfying 1.1 . In 1, 2 , the qualitative behavior of linea...

Keywords: Systems of functional-difference equations Bounded continuous solutions on half-intervals a b s t r a c t Some sufficient conditions for the existence of families of bounded continuous solutions of some classes of systems of functional-difference equations on the intervals ½0; þ1Þ and ðÀ1; 0Š are given.

We show that the p-periodic logistic equation xn+1 = μn mod pxn(1 − xn) has cycles (periodic solutions) of minimal periods 1, p, 2p, 3p, .... Then we extend Singer’s theorem to periodic difference equations, and use it to show the p-periodic logistic equation has at most p stable cycles. Also, we present computational methods investigating the stable cycles in case p = 2 and 3.

For the equation with a distributed delay

Table of contents §9.0. Foreword. §9.1. My more general auxiliary functions in the case k = 1, 2, 3. §9.2. Some relations for the functions, considered in §9.1 in the case l = 0. §9.3. Passing to the system of difference equations for the functions, considered in §9.2. §9.4. Passing from (τ, μ), to (α, ν) in the case l = 0. §9.0. Foreword. Let |z| ≥ 1,−3π/2 < arg(z) ≤ π/2, log(z) = ln(|z|) + i ...

We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and those with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non-)inversions, squares and p-repetitions. We give recurrence formulas in the general case, th...