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

Archivum Mathematicum (2003)

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

top

## Abstract

top
Sufficient conditions are established for the oscillation of proper solutions of the system $\begin{array}{cc}\hfill {u}_{1}^{\text{'}}\left(t\right)& =p\left(t\right){u}_{2}\left(\sigma \left(t\right)\right)\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {u}_{2}^{\text{'}}\left(t\right)& =-q\left(t\right){u}_{1}\left(\tau \left(t\right)\right)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$ where $p,\phantom{\rule{0.166667em}{0ex}}q:{R}_{+}\to {R}_{+}$ are locally summable functions, while $\tau$ and $\sigma :{R}_{+}\to {R}_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty }{lim}\tau \left(t\right)=+\infty$, $\underset{t\to +\infty }{lim}\sigma \left(t\right)=+\infty$.

## How to cite

top

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>.

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 - 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

top
1. Chantladze T., Kandelaki N., Lomtatidze, A, Oscillation and nonoscillation criteria for a second order linear equation, Georgian Math. J. 6 (1999), No. 5, 401–414. (1999) Zbl0944.34025MR1692963
2. Chantladze T., Kandelaki N., Lomtatidze A., On oscillation and nonoscillation of second order half-linear equation, Georgian Math. J. 7 (2000), No. 1, 329–346. MR1779555
3. Coppell W. A., Stability and asymptotic behaviour of differential equations, Heat and Co., Boston, 1965. (1965)
4. Hille E., Non-oscillation theorems, Trans. Amer. Math. Soc.64 (1948), 234–252. (1948) Zbl0031.35402MR0027925
5. Koplatadze R. G., Criteria for the oscillation of solutions of second order differential inequalities and equations with a retarded argument, (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 17 (1986), 104–121. (1986) MR0853276
6. Koplatadze R., On oscillatory properties of solutions of functional differential equations, Mem. Differential Equations Math. Phys. 3 (1994), 1–179. (1994) Zbl0843.34070MR1375838
7. Koplatadze R., Kvinikadze G., Stavroulakis I. P., Oscillation of second order linear delay differential equations, Funct. Differ. Equ. 7 (2000), No. 1–2, 121–145. Zbl1057.34077MR1941863
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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.