Oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients. (English) Zbl 1171.34043

Karpuz, Basak; Padhy, Laxmi Narayan; Rath, Radhanath

Electron. J. Differ. Equ. 2008, Paper No. 113, 15 p. (2008).

Summary: We obtain necessary and sufficient conditions so that every solution of
\[ \big(y(t)- p(t) y(r(t))\big)^{(n)}+ q(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) \]
oscillates or tends to zero as \(t \to \infty\), where \(n\) is an integer \(n \geq 2, q>0, u\geq 0\). Both bounded and unbounded solutions are considered in this paper. The results hold also when \(u\equiv 0, f(t)\equiv 0\), and \(G(u)\equiv u\). This paper extends and generalizes some recent results.

MSC:

34K11	Oscillation theory of functional-differential equations
34K40	Neutral functional-differential equations

Keywords:

oscillatory solution; neutral differential equation; asymptotic behaviour