Asymptotic behaviour of solutions of two-dimensional linear differential systems with deviating arguments

Roman Koplatadze; N. L. Partsvania; Ioannis P. Stavroulakis

Archivum Mathematicum (2003)

  • Volume: 039, Issue: 3, page 213-232
  • ISSN: 0044-8753

Abstract

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Sufficient conditions are established for the oscillation of proper solutions of the system u 1 ' ( t ) = p ( t ) u 2 ( σ ( t ) ) , u 2 ' ( t ) = - q ( t ) u 1 ( τ ( t ) ) , where p , q : R + → R + are locally summable functions, while τ and σ : R + → R + are continuous and continuously differentiable functions, respectively, and lim t → + ∞ τ ( t ) = + ∞ , lim t → + ∞ σ ( t ) = + ∞ .

How to cite

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Koplatadze, Roman, Partsvania, N. L., and Stavroulakis, Ioannis P.. "Asymptotic behaviour of solutions of two-dimensional linear differential systems with deviating arguments." Archivum Mathematicum 039.3 (2003): 213-232. <http://eudml.org/doc/249141>.

@article{Koplatadze2003,
abstract = {Sufficient conditions are established for the oscillation of proper solutions of the system \begin\{align\} u\_1^\{\prime \}(t) & =p(t)u\_2(\sigma (t))\,, \\ u\_2^\{\prime \}(t) & =-q(t)u\_1(\tau (t))\,, \end\{align\} where $p,\,q: R_\{+\}\rightarrow R_\{+\}$ are locally summable functions, while $\tau $ and $\sigma : R_\{+\}\rightarrow R_\{+\}$ are continuous and continuously differentiable functions, respectively, and $\lim \limits _\{t\rightarrow +\infty \} \tau (t)=+\infty $, $\lim \limits _\{t\rightarrow +\infty \} \sigma (t)=+\infty $.},
author = {Koplatadze, Roman, Partsvania, N. L., Stavroulakis, Ioannis P.},
journal = {Archivum Mathematicum},
keywords = {two-dimensional differential system; proper solution; oscillatory system; two-dimensional differential system; proper solution; oscillatory system},
language = {eng},
number = {3},
pages = {213-232},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic behaviour of solutions of two-dimensional linear differential systems with deviating arguments},
url = {http://eudml.org/doc/249141},
volume = {039},
year = {2003},
}

TY - JOUR
AU - Koplatadze, Roman
AU - Partsvania, N. L.
AU - Stavroulakis, Ioannis P.
TI - Asymptotic behaviour of solutions of two-dimensional linear differential systems with deviating arguments
JO - Archivum Mathematicum
PY - 2003
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 039
IS - 3
SP - 213
EP - 232
AB - Sufficient conditions are established for the oscillation of proper solutions of the system \begin{align} u_1^{\prime }(t) & =p(t)u_2(\sigma (t))\,, \\ u_2^{\prime }(t) & =-q(t)u_1(\tau (t))\,, \end{align} where $p,\,q: R_{+}\rightarrow R_{+}$ are locally summable functions, while $\tau $ and $\sigma : R_{+}\rightarrow R_{+}$ are continuous and continuously differentiable functions, respectively, and $\lim \limits _{t\rightarrow +\infty } \tau (t)=+\infty $, $\lim \limits _{t\rightarrow +\infty } \sigma (t)=+\infty $.
LA - eng
KW - two-dimensional differential system; proper solution; oscillatory system; two-dimensional differential system; proper solution; oscillatory system
UR - http://eudml.org/doc/249141
ER -

References

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  8. Koplatadze R., Partsvania N., Oscillatory properties of solutions of two-dimensional differential systems with deviated arguments, (Russian) Differentsial’nye Uravneniya 33 (1997), No. 10, 1312–1320; translation in Differential Equations 33 (1997), No. 10, 1318–1326 (1998). (1997) MR1668129
  9. Lomtatidze A., Oscillation and nonoscillation criteria for second order linear differential equation, Georgian Math. J. 4 (1997), No. 2, 129–138. (1997) MR1439591
  10. Lomtatidze A., Partsvania N., Oscillation and nonoscillation criteria for two-dimensional systems of first order linear ordinary differential equations, Georgian Math. J. 6 (1999), No. 3, 285–298. (1999) Zbl0930.34025MR1679448
  11. Mirzov J. D., Asymptotic behavior of solutions of systems of nonlinear non-autonomous ordinary differential equations, (Russian) Maikop 1993. (1993) 
  12. Nehari Z., Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428–445. (1957) Zbl0078.07602MR0087816
  13. Partsvania N., On oscillation of solutions of second order systems of deviated differential equations, Georgian Math. J. 3 (1996), No. 6, 571–582. (1996) Zbl0868.34054MR1419836

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