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Keywords:
oscillation; non-oscillation; neutral equation; asymptotic behaviour
oscillation; non-oscillation; neutral equation; asymptotic behaviour
Summary:
In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation \[ \big (y(t)- \sum _{i=1}^k p_i(t) y(r_i(t))\big )^{(n)}+ v(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) \] oscillates or tends to zero as $t\rightarrow \infty $, where, $n \ge 1$ is any positive integer, $p_i$, $r_i\in C^{(n)}([0,\infty ),\mathbb{R})$ and $p_i$ are bounded for each $i=1,2,\dots ,k$. Further, $f\in C([0, \infty ), \mathbb{R})$, $g$, $h$, $v$, $u \in C([0, \infty ), [0, \infty ))$, $G$ and $H \in C(\mathbb{R},\mathbb{R})$. The functional delays $r_i(t)\le t$, $g(t)\le t$ and $h(t)\le t$ and all of them approach $\infty $ as $t\rightarrow \infty $. The results hold when $u\equiv 0$ and $f(t)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.
References:
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