An asymptotic theorem for a class of nonlinear neutral differential equations

Naito, Manabu. "An asymptotic theorem for a class of nonlinear neutral differential equations." Czechoslovak Mathematical Journal 48.3 (1998): 419-432. <http://eudml.org/doc/30430>.

@article{Naito1998,
abstract = {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 $[t_0, \infty ) \times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $\lim g(t) = \infty $ as $t\rightarrow \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.},
author = {Naito, Manabu},
journal = {Czechoslovak Mathematical Journal},
keywords = {neutral differential equation; asymptotic properties},
language = {eng},
number = {3},
pages = {419-432},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An asymptotic theorem for a class of nonlinear neutral differential equations},
url = {http://eudml.org/doc/30430},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Naito, Manabu
TI - An asymptotic theorem for a class of nonlinear neutral differential equations
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 3
SP - 419
EP - 432
AB - 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 $[t_0, \infty ) \times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $\lim g(t) = \infty $ as $t\rightarrow \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.
LA - eng
KW - neutral differential equation; asymptotic properties
UR - http://eudml.org/doc/30430
ER -