Oscillation of deviating differential equations

George E. Chatzarakis

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 4, page 435-448
  • ISSN: 0862-7959

Abstract

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Consider the first-order linear delay (advanced) differential equation x ' ( t ) + p ( t ) x ( τ ( t ) ) = 0 ( x ' ( t ) - q ( t ) x ( σ ( t ) ) = 0 ) , t ≥ t 0 , where p ( q ) is a continuous function of nonnegative real numbers and the argument τ ( t ) ( σ ( t ) ) is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions lim sup t → ∞ ∫ τ ( t ) t p ( s ) d s > 1 lim sup t → ∞ ∫ t σ ( t ) q ( s ) d s > 1 and lim inf t → ∞ ∫ τ ( t ) t p ( s ) d s > 1 e lim inf t → ∞ ∫ t σ ( t ) q ( s ) d s > 1 e are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.

How to cite

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Chatzarakis, George E.. "Oscillation of deviating differential equations." Mathematica Bohemica 145.4 (2020): 435-448. <http://eudml.org/doc/297359>.

@article{Chatzarakis2020,
abstract = {Consider the first-order linear delay (advanced) differential equation\[ x^\{\prime \}(t)+p(t)x( \tau (t)) =0\quad (x^\{\prime \}(t)-q(t)x(\sigma (t)) =0),\quad t\ge t\_\{0\}, \] where $p$$(q)$ is a continuous function of nonnegative real numbers and the argument $\tau (t)$$(\sigma (t))$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions\[ \limsup \limits \_\{t\rightarrow \infty \}\int \_\{\tau (t)\}^\{t\}p(s) \{\rm d\}s>1\quad \biggl (\limsup \limits \_\{t\rightarrow \infty \}\int \_\{t\}^\{\sigma (t)\}q(s) \{\rm d\}s>1\bigg ) \] and \[ \liminf \_\{t\rightarrow \infty \}\int \_\{\tau (t)\}^\{t\}p(s) \{\rm d\}s>\frac\{1\}\{\rm e\}\quad \biggl (\liminf \_\{t\rightarrow \infty \}\int \_\{t\}^\{\sigma (t)\}q(s) \{\rm d\}s>\frac\{1\}\{\rm e\}\bigg ) \] are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.},
author = {Chatzarakis, George E.},
journal = {Mathematica Bohemica},
keywords = {differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution; Grönwall inequality},
language = {eng},
number = {4},
pages = {435-448},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Oscillation of deviating differential equations},
url = {http://eudml.org/doc/297359},
volume = {145},
year = {2020},
}

TY - JOUR
AU - Chatzarakis, George E.
TI - Oscillation of deviating differential equations
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 4
SP - 435
EP - 448
AB - Consider the first-order linear delay (advanced) differential equation\[ x^{\prime }(t)+p(t)x( \tau (t)) =0\quad (x^{\prime }(t)-q(t)x(\sigma (t)) =0),\quad t\ge t_{0}, \] where $p$$(q)$ is a continuous function of nonnegative real numbers and the argument $\tau (t)$$(\sigma (t))$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions\[ \limsup \limits _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>1\quad \biggl (\limsup \limits _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>1\bigg ) \] and \[ \liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>\frac{1}{\rm e}\quad \biggl (\liminf _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>\frac{1}{\rm e}\bigg ) \] are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.
LA - eng
KW - differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution; Grönwall inequality
UR - http://eudml.org/doc/297359
ER -

References

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