Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

Abstract

ABSTRACT. In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

“…In this paper since we formulate our results with ( 1 ) and do not assume either ( 8 ) or ( 9 ) therefore our work improve some of the results of [1], [5], [6], [7], [8], [9]. Further one may observe an important point that the authors have found positive solutions of neutral delay differential equation…”

“…However since is decreasing the existing results of [1], [2], [5], [6], [7], [8], [9] are not applicable to this NDDE (3.5).…”

“…We don't find such a result in [1], [2] or [5]- [9]. To achieve our result we need the following Lemma.…”

“…We observe that the even order neutral differential equations are not so often studied as the odd order neutral differential equations have been. The authors have proved the existence of a bounded positive solution of neutral delay differential equations of various order in [1], [2], [5], [6], [7], [8], [9]. For that the authors assume the following hypothesis.…”

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

“…This is mainly because of the technical difficulties arising in the analysis. In addition, the works [7,9,10] provide sufficient conditions for nonoscillation of delay differential and difference equations, respectively (see also [5,[11][12][13][14][15]). To the best of our knowledge, there is no paper including results for nonlinear delay differential/difference equations; hence, this is the first paper considering the nonlinear case.…”