Bounded oscillation of nonlinear neutral differential equations of arbitrary order

Yeter Ş. Yilmaz; Ağacik Zafer

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 1, page 185-195
  • ISSN: 0011-4642

Abstract

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The paper is concerned with oscillation properties of n -th order neutral differential equations of the form [ x ( t ) + c x ( τ ( t ) ) ] ( n ) + q ( t ) f x ( σ ( t ) ) = 0 , t ≥ t 0 > 0 , where c is a real number with | c | ≠ 1 , q ∈ C ( [ t 0 , ∞ ) , ℝ ) , f ∈ C ( ℝ , ℝ ) , τ , σ ∈ C ( [ t 0 , ∞ ) , ℝ + ) with τ ( t ) < t and lim t → ∞ τ ( t ) = lim t → ∞ σ ( t ) = ∞ . Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which c is a function of t and a certain type of a forcing term is present.

How to cite

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Yilmaz, Yeter Ş., and Zafer, Ağacik. "Bounded oscillation of nonlinear neutral differential equations of arbitrary order." Czechoslovak Mathematical Journal 51.1 (2001): 185-195. <http://eudml.org/doc/30626>.

@article{Yilmaz2001,
abstract = {The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form \[ [x(t)+cx(\tau (t))]^\{(n)\}+q(t)f\bigl (x(\sigma (t))\bigr )=0,\quad t\ge t\_0>0, \] where $c$ is a real number with $|c|\ne 1$, $q\in C([t_0,\infty ),\mathbb \{R\})$, $f\in C(\mathbb \{R\},\mathbb \{R\})$, $\tau ,\sigma \in C([t_0,\infty ),\mathbb \{R\}_+)$ with $\tau (t)<t$ and $\lim _\{t\rightarrow \infty \}\tau (t)=\lim _\{t\rightarrow \infty \}\sigma (t)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which $c$ is a function of $t$ and a certain type of a forcing term is present.},
author = {Yilmaz, Yeter Ş., Zafer, Ağacik},
journal = {Czechoslovak Mathematical Journal},
keywords = {oscillation; positive solutions; neutral equation; oscillation; positive solutions; neutral equation},
language = {eng},
number = {1},
pages = {185-195},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bounded oscillation of nonlinear neutral differential equations of arbitrary order},
url = {http://eudml.org/doc/30626},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Yilmaz, Yeter Ş.
AU - Zafer, Ağacik
TI - Bounded oscillation of nonlinear neutral differential equations of arbitrary order
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 1
SP - 185
EP - 195
AB - The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form \[ [x(t)+cx(\tau (t))]^{(n)}+q(t)f\bigl (x(\sigma (t))\bigr )=0,\quad t\ge t_0>0, \] where $c$ is a real number with $|c|\ne 1$, $q\in C([t_0,\infty ),\mathbb {R})$, $f\in C(\mathbb {R},\mathbb {R})$, $\tau ,\sigma \in C([t_0,\infty ),\mathbb {R}_+)$ with $\tau (t)<t$ and $\lim _{t\rightarrow \infty }\tau (t)=\lim _{t\rightarrow \infty }\sigma (t)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which $c$ is a function of $t$ and a certain type of a forcing term is present.
LA - eng
KW - oscillation; positive solutions; neutral equation; oscillation; positive solutions; neutral equation
UR - http://eudml.org/doc/30626
ER -

References

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  9. 10.1016/0893-9659(93)90010-K, Appl. Math. Lett. 6 (1993), 43–46. (1993) MR1347773DOI10.1016/0893-9659(93)90010-K
  10. 10.1016/S0893-9659(98)00028-7, Appl. Math. Lett. 11 (1998), 21–25. (1998) Zbl0933.34075MR1628987DOI10.1016/S0893-9659(98)00028-7
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