Existence of unbounded solutions of a linear homogenous systemof differential equations with two delays

Josef Diblík; Radoslav Chupáč; Miroslava Růžičková

doi:10.3934/dcdsb.2014.19.2447

Article Contents

2014, Volume 19, Issue 8: 2447-2459. Doi: 10.3934/dcdsb.2014.19.2447

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Existence of unbounded solutions of a linear homogenous system of differential equations with two delays

1.
Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering and Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno
2.
Department of Mathematics, University of Žilina, Žilina

Received: September 30, 2013

Revised: April 30, 2014

Published: September 2014

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Abstract

Behavior of solutions of a linear homogeneous system of differential equations with deviating arguments in the form \begin{equation*} \dot y(t)=\beta(t)\left[y(t-\delta)-y(t-\tau)\right] \end{equation*} is discussed for $t\to\infty$. It is assumed that $y$ is an $n$-dimensional column vector, $n\geq 1$ is an integer, $\delta,\tau\in{\mathbb{R}}$, $\tau>\delta>0$, and $\beta(t)$ is an $n\times n$ matrix defined for $t\geq t_{0}$, $t_{0}\in\mathbb{R}$, and such that its elements are nonnegative, continuous functions and in every row of this matrix at least one element is nonzero. The existence of solutions in an exponential form under certain assumptions is proved. Sufficient conditions for the existence of unbounded solutions and the estimations for a solution are derived. A comparison with the known results and an illustrative example are given.

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Mathematics Subject Classification: Primary: 34K12, 34K25.

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