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

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October 2014, 19(8): 2447-2459. doi: 10.3934/dcdsb.2014.19.2447

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

Josef Diblík ^1, , Radoslav Chupáč ^2, and Miroslava Růžičková ^2,

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, Czech Republic
2.	Department of Mathematics, University of Žilina, Žilina, Slovak Republic, Slovak Republic

Received October 2013 Revised May 2014 Published August 2014

Abstract
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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.

Keywords: exponential solution, unbounded solution, delay., sufficient condition, System of differential equations.

Mathematics Subject Classification: Primary: 34K12, 34K2.

Citation: Josef Diblík, Radoslav Chupáč, Miroslava Růžičková. Existence of unbounded solutions of a linear homogenous system of differential equations with two delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2447-2459. doi: 10.3934/dcdsb.2014.19.2447

References:

[1]	O. Arino and M. Pituk, More on linear differential systems with small delays, J. Diff. Equat., 170 (2001), 381-407. doi: 10.1006/jdeq.2000.3824.
[2]	F. V. Atkinson and J. R. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl., 91 (1983), 410-423. doi: 10.1016/0022-247X(83)90161-0.
[3]	H. Bereketoǧlu and A. Huseynov, Convergence of solutions of nonhomogeneous linear difference systems with delays, Acta Appl. Math., 110 (2010), 259-269. doi: 10.1007/s10440-008-9404-2.
[4]	H. Bereketoǧlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dynam. Systems Appl., 17 (2008), 71-83.
[5]	H. Bereketoglu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
[6]	J. Diblík, Asymptotic convergence criteria of solutions of delayed functional differential equations, J. Math. Anal. Appl., 274 (2002), 349-373. doi: 10.1016/S0022-247X(02)00311-6.
[7]	J. Diblík, Asymptotic representation of solutions of equation $\dot y(t)=\beta (t)[y(t)-y(t-\tau(t))]$, J. Math. Anal. Appl., 217 (1998), 200-215. doi: 10.1006/jmaa.1997.5709.
[8]	J. Diblík, R. Chupáč and M. Růžičková, Unbounded solutions of the equation $\dot y(t)=\sum_{i=1}^n\beta_i(t)[y(t-\delta_i)-y(t-\tau_i)]$, Appl. Math. Comput., 221 (2013), 610-619. doi: 10.1016/j.amc.2013.07.001.
[9]	J. Diblík and M. Růžičková, Convergence of the solutions of the equation $\dot y(t)=\beta (t)[y(t-\delta)-y(t-\tau)]$ in the critical case, J. Math. Anal. Appl., 331 (2007), 1361-1370. doi: 10.1016/j.jmaa.2006.10.001.
[10]	J. Diblík and M. Růžičková, Exponential solutions of equation $\dot y(t)=\beta (t)[y(t-\delta)-y(t-\tau)]$, J. Math. Anal. Appl., 294 (2004), 273-287. doi: 10.1016/j.jmaa.2004.02.036.
[11]	J. Diblík, M. Růžičková and Z. Šutá, Asymptotic convergence of the solutions of a discrete system with delays, Appl. Math. Comput., 219 (2012), 4036-4044. doi: 10.1016/j.amc.2012.10.040.
[12]	I. Györi and L. Horváth, Asymptotic constancy in linear difference equations: Limit formulae and sharp conditions, Adv. Difference Equ., 2010, Art. ID 789302, 20 pp.
[13]	I. Györi, F. Karakoç and H. Bereketoǧlu, Convergence of solutions of a linear impulsive differential equations system with many delays, Dyn. Contin. Discrete Impuls. Syst., 18 (2011), 191-202.
[14]	H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, 1995.

show all references

References:

[1]	O. Arino and M. Pituk, More on linear differential systems with small delays, J. Diff. Equat., 170 (2001), 381-407. doi: 10.1006/jdeq.2000.3824.
[2]	F. V. Atkinson and J. R. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl., 91 (1983), 410-423. doi: 10.1016/0022-247X(83)90161-0.
[3]	H. Bereketoǧlu and A. Huseynov, Convergence of solutions of nonhomogeneous linear difference systems with delays, Acta Appl. Math., 110 (2010), 259-269. doi: 10.1007/s10440-008-9404-2.
[4]	H. Bereketoǧlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dynam. Systems Appl., 17 (2008), 71-83.
[5]	H. Bereketoglu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
[6]	J. Diblík, Asymptotic convergence criteria of solutions of delayed functional differential equations, J. Math. Anal. Appl., 274 (2002), 349-373. doi: 10.1016/S0022-247X(02)00311-6.
[7]	J. Diblík, Asymptotic representation of solutions of equation $\dot y(t)=\beta (t)[y(t)-y(t-\tau(t))]$, J. Math. Anal. Appl., 217 (1998), 200-215. doi: 10.1006/jmaa.1997.5709.
[8]	J. Diblík, R. Chupáč and M. Růžičková, Unbounded solutions of the equation $\dot y(t)=\sum_{i=1}^n\beta_i(t)[y(t-\delta_i)-y(t-\tau_i)]$, Appl. Math. Comput., 221 (2013), 610-619. doi: 10.1016/j.amc.2013.07.001.
[9]	J. Diblík and M. Růžičková, Convergence of the solutions of the equation $\dot y(t)=\beta (t)[y(t-\delta)-y(t-\tau)]$ in the critical case, J. Math. Anal. Appl., 331 (2007), 1361-1370. doi: 10.1016/j.jmaa.2006.10.001.
[10]	J. Diblík and M. Růžičková, Exponential solutions of equation $\dot y(t)=\beta (t)[y(t-\delta)-y(t-\tau)]$, J. Math. Anal. Appl., 294 (2004), 273-287. doi: 10.1016/j.jmaa.2004.02.036.
[11]	J. Diblík, M. Růžičková and Z. Šutá, Asymptotic convergence of the solutions of a discrete system with delays, Appl. Math. Comput., 219 (2012), 4036-4044. doi: 10.1016/j.amc.2012.10.040.
[12]	I. Györi and L. Horváth, Asymptotic constancy in linear difference equations: Limit formulae and sharp conditions, Adv. Difference Equ., 2010, Art. ID 789302, 20 pp.
[13]	I. Györi, F. Karakoç and H. Bereketoǧlu, Convergence of solutions of a linear impulsive differential equations system with many delays, Dyn. Contin. Discrete Impuls. Syst., 18 (2011), 191-202.
[14]	H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, 1995.

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