We derive explicit stability conditions for delay difference equations in C n (the set of n complex vectors) and estimates for the size of the solutions are derived. Applications to partial difference equations, which model diffusion and reaction processes, are given. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see G...

Abstract This article introduces the basic qualitative and basic quantitative theory of Volterra integral equations on time scales and thus may be considered as a foundation for future advanced studies in the field. New sufficient conditions are introduced that guarantee: existence; uniqueness; approximation; boundedness and certain growth rates of solutions to both linear and nonlinear problem...

Let G be a unital group with a finite unit interval E, let n be the number of atoms in E, and let κ be the number of extreme points of the state spaceΩ(G). We introduce canonical order-preserving group homomorphisms ξ : Zn → G and ρ : G → Zκ linking G with the simplicial groups Zn and Zκ . We show that ξ is a surjection and ρ is an injection if and only if G is torsion-free. We give an explicit...

A language (resp. a relation) is expressible by a word equation e if it is dened as the set of all values of an unknown (resp. a set of unknowns) over all solutions of the equation e. We show rst tools that a language or a relation is not expressible by using a certain number of auxiliary unknowns. As a consequence an in nite hierarchy based on the number of auxiliary unknowns of expressible la...

In this paper, we establish a Lyapunov-type inequality for a fractional q-difference equation subject to Dirichlet-type boundary conditions. The obtained inequality generalizes several existing results from the literature including the standard Lyapunov inequality. We use that result to provide an interval, where a certain Mittag-Leffler function has no real zeros. We present also another appli...

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Lague...

Abstract. The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to “in between”. In this paper we present time scales versions of the inequ...

where {pi(n)} are sequences of nonnegative real numbers and not identically equal to zero, and ki is positive integer, i = 1,2, . . . , and is the first-order forward difference operator, xn = xn+1− xn, and xn = l−1( xn) for l ≥ 2. By a solution of (1.1) or inequality (1.2), we mean a nontrival real sequence {xn} satisfying (1.1) or inequality (1.2) for n ≥ 0. A solution {xn} is said to be osci...

Fix α ∈ [0, 1). Consider the random walk on the circle S1 which proceeds by repeatedly rotating points forward or backward, with probability 1 2 , by an angle 2πα. This paper analyzes the rate of convergence of this walk to the uniform distribution under “discrepancy” distance. The rate depends on the continued fraction properties of the number ξ = 2α. We obtain bounds for rates when ξ is any i...

Assume that Nm(x) denotes the density of the population at a point x at the beginning of the reproductive season in the mth year. We study the following impulsive reaction-diffusion model for any m ∈ Z+  u (m) t = div(A∇u(m) − au(m)) + f(u(m)) for (x, t) ∈ Ω× (0, 1] u(m)(x, 0) = g(Nm(x)) for x ∈ Ω Nm+1(x) := u(m)(x, 1) for x ∈ Ω for functions f, g, a drift a and a diffusion matrix A and Ω ⊂...