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Keywords:
delayed differential equation; asymptotic behaviour; boundedness of solutions; two-dimensional systems; Lyapunov method; Wa.zewski topological principle
delayed differential equation; asymptotic behaviour; boundedness of solutions; two-dimensional systems; Lyapunov method; Wa.zewski topological principle
Summary:
The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system $x^{\prime }(t)={\mathsf A}(t)x(t)+{\mathsf B}(t)x(t-r)+h(t,x(t),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $\mathsf A$, $\mathsf {B}$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\rightarrow \infty $ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wa.zewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].
References:
[1] J. Kalas: Asymptotic behaviour of a two-dimensional differential system with delay under the conditions of instability. Nonlinear Anal. 62 (2005), 207–224. DOI 10.1016/j.na.2005.03.015 | MR 2145603 | Zbl 1078.34055
[2] J. Kalas, L. Baráková: Stability and asymptotic behaviour of a two-dimensional differential system with delay. J. Math. Anal. Appl. 269 (2002), 278–300. DOI 10.1016/S0022-247X(02)00023-9 | MR 1907886
[3] J. Kalas, J. Osička: Bounded solutions of dynamical systems in the plane under the condition of instability. Math. Nachr. 170 (1994), 133–147. MR 1302371
[4] J. Mawhin: Periodic solutions of some planar nonautonomous polynomial differential equations. Differ. Integral Equ. 7 (1994), 1055–1061. MR 1270118
[5] R.Manásevich, J. Mawhin, F. Zanolin: Hölder inequality and periodic solutions of some planar polynomial differential equations with periodic coefficients. Inequalities and Applications. World Sci. Ser. Appl. Anal. 3 (1994), 459–466. MR 1299575
[6] R. Manásevich, J. Mawhin, F. Zanolin: Periodic solutions of complex-valued differential equations with periodic coefficients. J. Differ. Equations 126 (1996), 355–373. DOI 10.1006/jdeq.1996.0054 | MR 1383981
[7] M. Ráb, J. Kalas: Stability of dynamical systems in the plane. Differ. Integral Equ. 3 (1990), 127–144. MR 1014730
[8] K. P. Rybakowski: Wa.zewski principle for retarded functional differential equations. J. Differ. Equations 36 (1980), 117–138. DOI 10.1016/0022-0396(80)90080-7 | MR 0571132
[9] K. P. Rybakowski: A topological principle for retarded functional differential equations of Carathéodory type. J. Differ. Equations 39 (1981), 131–150. DOI 10.1016/0022-0396(81)90070-X | MR 0607779 | Zbl 0477.34048
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