Existence of positive solutions for a class of higher order neutral functional differential equations

Satoshi Tanaka

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 3, page 573-583
  • ISSN: 0011-4642

Abstract

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The higher order neutral functional differential equation d n d t n x ( t ) + h ( t ) x ( τ ( t ) ) + σ f t , x ( g ( t ) ) = 0 ( 1 ) is considered under the following conditions: n ≥ 2 , σ = ± 1 , τ ( t ) is strictly increasing in t ∈ [ t 0 , ∞ ) , τ ( t ) < t for t ≥ t 0 , lim t → ∞ τ ( t ) = ∞ , lim t → ∞ g ( t ) = ∞ , and f ( t , u ) is nonnegative on [ t 0 , ∞ ) × ( 0 , ∞ ) and nondecreasing in u ∈ ( 0 , ∞ ) . A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).

How to cite

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Tanaka, Satoshi. "Existence of positive solutions for a class of higher order neutral functional differential equations." Czechoslovak Mathematical Journal 51.3 (2001): 573-583. <http://eudml.org/doc/30656>.

@article{Tanaka2001,
abstract = {The higher order neutral functional differential equation \[ \frac\{\mathrm \{d\}^n\}\{\mathrm \{d\}t^n\} \bigl [x(t) + h(t) x(\tau (t))\bigr ] + \sigma f\bigl (t,x(g(t))\bigr ) = 0 \qquad \mathrm \{(1)\}\] is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau (t)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau (t)<t$ for $t\ge t_0$, $\lim _\{t\rightarrow \infty \} \tau (t)= \infty $, $\lim _\{t\rightarrow \infty \} g(t) = \infty $, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u \in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).},
author = {Tanaka, Satoshi},
journal = {Czechoslovak Mathematical Journal},
keywords = {neutral differential equation; positive solution; neutral differential equation; positive solution},
language = {eng},
number = {3},
pages = {573-583},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of positive solutions for a class of higher order neutral functional differential equations},
url = {http://eudml.org/doc/30656},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Tanaka, Satoshi
TI - Existence of positive solutions for a class of higher order neutral functional differential equations
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 573
EP - 583
AB - The higher order neutral functional differential equation \[ \frac{\mathrm {d}^n}{\mathrm {d}t^n} \bigl [x(t) + h(t) x(\tau (t))\bigr ] + \sigma f\bigl (t,x(g(t))\bigr ) = 0 \qquad \mathrm {(1)}\] is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau (t)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau (t)<t$ for $t\ge t_0$, $\lim _{t\rightarrow \infty } \tau (t)= \infty $, $\lim _{t\rightarrow \infty } g(t) = \infty $, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u \in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).
LA - eng
KW - neutral differential equation; positive solution; neutral differential equation; positive solution
UR - http://eudml.org/doc/30656
ER -

References

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