Recessive and dominant solutions for the nonoscillatory half-linear difference equation are investigated. By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, we prove that recessive solutions are the “smallest solutions in a neighborhood of infinity,” like in the linear case. Other asymptotic properties of recessive and dominant solutions are tr...

The aim of this paper is to prove that given a linear recurrence sequence U = kb u„ +z = u„ + , + u„, n w 1, u t =1, uz > u l , then the set of positive integers can be partitioned uniquely into two disjoint subsets such that the sum of any two distinct members from any one set can never be in U and study related problems . We prove our main result in Section 2. In Section 3 we give a graph the...

For a Dirac particle in one dimension with random mass, the time evolution for the average wavefunction is considered. Using the supersymmetric representation of the average Green’s function, we derive a fourth order linear difference equation for the low-energy asymptotics of the average wavefunction. This equation is of Poincaré type, though highly critical and therefore not amenable to stand...

We study almost automorphic solutions of recurrence relations with values in a Banach space V for quasilinear almost automorphic difference systems. Its linear part is a constant bounded linear operator Λ defined on V satisfying an exponential dichotomy. We study the existence of almost automorphic solutions of the non-homogeneous linear difference equation and to quasilinear difference equatio...

It is known that input-output approaches based on scaled small-gain theorems with constant Dscalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input-output approaches provide, in general, suff...

A new method named rational expansion method of exponent function is presented to find exact traveling wave solutions of differential-difference equations. This method generalizes the so-called tanh-method and other similar methods. Some examples are dealt with and their abundant exact solutions which include solitary solutions and periodic solutions are obtained. Among them, many solutions are...

We announce proofs of a number of theorems concerning finite p-groups and nilpotent groups. These include: (1) the number of p-groups of class c on d generators of order pn satisfies a linear recurrence relation in n; (2) for fixed n the number of p-groups of order pn as one varies p is given by counting points on certain varieties mod p; (3) an asymptotic formula for the number of finite nilpo...

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation p t ψ ◦ x x Δ Δ q t f ◦ x 0 on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference...

The half-linear difference equations with the deviating argument Δ an|Δxn| sgn Δxn bn|xn q| sgn xn q 0 , q ∈ Z are considered. We study the role of the deviating argument q, especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation q 1 . ...

This paper deals with asymptotic properties of the solutions of a variable order linear difference equation. As the main result, we derive the effective asymptotic estimate valid for all solutions of this equation. Moreover, we are going to discuss some consequences of this theoretical result, especially with respect to the numerical analysis of the multi-pantograph differential equation.