Oscillation of Systems of Neutral Equations with Variable Coefficients

Abstract

Consider the system of the neutral delay differential equationswhere P ( t ) = (pij(t)), Q(t) = (qij(t)) and R ( t ) = (rij(t)) are n x n matrices for t 0 and the delays T, a and e are nonnegative numbers. We obtain sufficient conditions for the oscillation of all solutions of (1) under the following hypotheses: and IntroductionConsider the system of the neutral delay differential equations where P ( t ) = ( p i j ( t ) ) , Q ( t ) = (qij(t)) and R(t) = (rij(t)) are n x n continuous matrices for t 2 0 and the delays T , O and Q are nonnegative numbers.

Oscillation of Forced Nonlinear Neutral Delay Difference Equations of First Order

“…. , 0 are given, then (1) (or (2)) has a unique solution satisfying the initial conditions (3). A solution {y n } of (1) (or (2)) is said to be oscillatory if for every N > 0 there exists an n N such that y n y n+1 0; otherwise, it is called nonoscillatory.…”

“…A solution {y n } of (1) (or (2)) is said to be oscillatory if for every N > 0 there exists an n N such that y n y n+1 0; otherwise, it is called nonoscillatory. In recent years, several papers on oscillation of solutions of neutral delay difference equations have appeared (see [1]- [3], [5]- [7], [9], [10]). In [1], Cheng and Lin have provided a complete characterization of oscillation of solutions of (4) ∆(y n + py n−m ) + qy n−k = 0, n = 0, 1, 2, .…”

“…The method developed in this work is different from those in [3], [5], [6]. Our work heavily depends on a lemma which may be regarded as the discrete analogue of Lemma 1.5.2 in [4].…”

“…However, equation (1) or (2) does not follow from those equations for m = 1 due to their assumptions on the nonlinear term F (n, u). Our assumptions cannot always be compared with those in [3], [5], [6] because the approaches are different. However, some of our results extend the results in [3], [5], [6].…”

“…Our assumptions cannot always be compared with those in [3], [5], [6] because the approaches are different. However, some of our results extend the results in [3], [5], [6]. In an earlier work [7], we have studied (1) with p n = p. Equations (1) and (2) may be regarded as a discrete analogoue of…”