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We consider difference equations with a single delay term and obtain sufficient conditions for both oscillation and nonoscillation. Moreover, our nonoscillation theorem improves the affirmative answer to Ladas’ corrected conjecture given by Tang and Yu in [Comput. Math. Appl. 38(11–12) (1999), pp. 229–237]. We also present a detailed discussion with examples to emphasize the significance and the sharpness of the new results. [ABSTRACT FROM AUTHOR]

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid's construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper. [ABSTRACT FROM AUTHOR]

Linear difference equations with variable delay are considered. The most important result of this paper is a new oscillation criterion, which should be looked upon as the discrete analogue of a well-known oscillation criterion for first order linear delay differential equations. This criterion constitutes a substantial improvement of an oscillation result due to the first author (Funkcial. Ekvac. 34 (1991), pp. 157-172). The results obtained extend the ones by the authors and Stavroulakis (J. Differ. Equ. Appl. 10 (2004), pp. 419-435) concerning the special case of linear difference equations with constant delay. [ABSTRACT FROM AUTHOR]

We study existence of nonoscillatory solutions and positiveness of the fundamental function for a scalar linear difference equation with several delays:[image omitted] where al can be either positive or negative. Coefficients of arbitrary signs are also considered. [ABSTRACT FROM AUTHOR]

The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [ J. Appl. Math. Simulation 2 (1989), 101-111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with the nonlinear delay difference equation x ( n +1)- x ( n )+ p ( n ) g ( x ( n - k ))=0, n =0,1,2,…, where p ( n ) is a nonnegative number for n ≥0, k is a positive integer and g ( x ) is a continuous function satisfying xg ( x )>0 if x ≠0. Sufficient conditions are established for all solutions of this equation to be oscillatory. Also, sufficient conditions are obtained for the existence of a nonoscillatory solution. Moreover, a nonoscillation theorem is given in the case that k =1. Our results are proved by use of phase plane analysis for an equivalent system to this equation. Examples and phase portraits are included to illustrate our theorems. [ABSTRACT FROM AUTHOR]

Some new oscillation criteria for quasilinear difference equations of the form Δ(a[SUBn](Δx[SUBn])[SUPα+q[SUBnf(x[SUBn-&rau;+1]) = 0 are established, where a is the ratio of odd positive integers. [ABSTRACT FROM AUTHOR]

In this paper, classification schemes are given for the nonoscillatory solutions of higher order neutral difference equations. Several necessary and/or sufficient conditions are obtained for the existence of each kind of solution. [ABSTRACT FROM AUTHOR]

We are interested in extensions of certain averaging techniques for the second-order scalar differential equation ( r ( t ) x Δ ( t )) Δ + q ( t ) x σ ( t )=0, on a time scale (measure chain) T . These results include some earlier criteria for the difference equations case. [ABSTRACT FROM AUTHOR]

A new nonoscillatory theory is presented for delay and neutral differential equations. In particular our nonlinearity may be singular in both the dependent and independent variables. [ABSTRACT FROM AUTHOR]

In this paper, the structure of the solution space of y n +3 + ry n +2 + qy n +1 + py n =0, n ≥0, is studied, keeping oscillatory/nonoscillatory behaviour of solutions of the equation in view, where p , q and r are constants. Some of these results are generalized partially to hold for y n +3 + r n y n +2 + q n y n +1 + p n y n =0, n ≥0, where { p n }, { q n } and { r n } are sequences of real numbers. [ABSTRACT FROM AUTHOR]