An asymptotic theorem for a class of nonlinear neutral differential equations

Naito, Manabu

Czechoslovak Mathematical Journal, Tome 48 (1998) no. 3, pp. 419-432.

Résumé

The neutral differential equation (1.1) $$ \frac{{\mathrm{d}}^n}{{\mathrm{d}} t^n} [x(t)+x(t-\tau)] + \sigma F(t,x(g(t))) = 0, $$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma =\pm 1$, $F(t,u)$ is nonnegative on $[t_0,\mathrm{\infty })\times (0,\mathrm{\infty })$ and is nondecreasing in $u\in (0,\mathrm{\infty })$, and $limg(t)=\mathrm{\infty }$ as $t\to \mathrm{\infty }$. It is shown that equation (1.1) has a solution $x(t)$ such that (1.2) $$ \lim_{t\rightarrow \infty} \frac{x(t)}{t^k}\ \text{exists and is a positive finite value if and only if} \int^{\infty}_{t_0} t^{n-k-1} F(t,c[g(t)]^k){\mathrm{d}} t \infty\text{ for some }c > 0. $$ Here, $k$ is an integer with $0\le k\le n-1$. To prove the existence of a solution $x(t)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

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Classification : 34K25, 34K40