Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

Abstract

In this paper, we establish necessary and sufficient conditions for the solutions of a second‐order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.

Nonoscillatory solutions to fourth-order neutral dynamic equations on time scales

“…It follows that r 3 z and z are both eventually positive or eventually negative, and z is always eventually monotonic, which implies that z is eventually positive or eventually negative. Similarly, we also obtain (7). In view of (C3), we see that one of cases (A1), (A2), and (A4) holds.…”

“…Hence, R 1 (or r 3 z ) is eventually strictly decreasing. Similarly, we always have (7), and one of cases (A1), (A2), and (A4) holds.…”

“…In this paper, we consider the existence of nonoscillatory solutions to fourth-order nonlinear neutral dynamic equations of the form r 1 (t) r 2 (t) r 3 (t) x(t) + p(t)x g (t) + f t, x h(t) = 0 (1) on a time scale T satisfying sup T = ∞, where t ∈ [t 0 , ∞) T with t 0 ∈ T. The oscillation and nonoscillation of nonlinear differential and difference equations have been developed rapidly in the recent decades. Afterwards, the theory of time scale united the differential and difference ones, and since then many researchers have investigated the oscillation and nonoscillation criteria of nonlinear dynamic equations on time scales; see, for instance, the papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. The majority of the scholars obtained the sufficient conditions to ensure that the solutions of the equations oscillate or tend to zero by using the generalized Riccati transformation and integral averaging technique.…”

“…Generally, a functional space and a fixed point theorem would be employed to analyze it. We refer the reader to [20][21][22][23][24][25] for details of the theory of time scale and to [4][5][6][7][8][9][10][11][12][13][14][15][16]19] for the studies on the existence of nonoscillatory solutions to nonlinear neutral dynamic equations on time scales. Definition 1.1 A solution x to (1) is defined to be eventually positive (or eventually negative) if there exists c ∈ T such that x(t) > 0 (or x(t) < 0) on [c, ∞) T .…”

An Asymptotic Criterion for Third-Order Dynamic Equations with Positive and Negative Coefcients

“…In recent years, there has been an increasing interest in obtaining sufficient conditions for the oscillatory and asymptotic behavior of solutions to various classes of differential and dynamic equations on time scales. We refer the reader to [1,2,4,[6][7][8][9][10][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and the references cited therein. For the study of asymptotic properties of third-order dynamic equations, Agarwal et al [1] and Erbe et al [8] established Hille and Nehari type criteria for third-order dynamic equations (a(rx ∆ ) ∆ ) ∆ (t) + p(t)x(τ (t)) = 0 and…”