## Convergence of solutions of nonhomogeneous linear difference systems with delays.(English)Zbl 1204.39003

Authors’ abstract: Sufficient conditions are given for the asymptotic constancy of the solutions of a system of linear difference equations with delays. Moreover, it is shown that the limits of the solutions, as $$t\to\infty$$, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation.

### MSC:

 39A06 Linear difference equations 39A22 Growth, boundedness, comparison of solutions to difference equations
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### References:

 [1] Agarwal, R., Bohner, M., Grace, S., O’Regan, D.: Discrete Oscillation Theory. Hindawi Publishing Corporation (2005) · Zbl 1084.39001 [2] Arino, O., Pituk, M.: More on linear differential systems with small delays. J. Differ. Equ. 170, 381–407 (2001) · Zbl 0989.34053 [3] Atkinson, F.V., Haddock, J.B.: Criteria for asymptotic constancy of solutions of functional differential equations. J. Math. Anal. Appl. 91, 410–423 (1983) · Zbl 0529.34065 [4] Bellman, R., Cooke, K.: Differential Difference Equations. Academic Press, Boston (1993) · Zbl 0105.06402 [5] Bereketoglu, H., Karakoc, F.: Asymptotic constancy for impulsive delay differential equations. Dyn. Syst. Appl. 17, 71–84 (2008) [6] Bereketoglu, H., Pituk, M.: Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays. Discrete Contin. Dyn. Syst. (Supplement Volume) 100–107 (2003) · Zbl 1071.34080 [7] Diblik, J.: Asymptotic representation of solutions of equation y ’(t)={$$\beta$$}(t)[y(t)(t(t))]. J. Math. Anal. Appl. 217, 210–215 (1998) · Zbl 0892.34067 [8] Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Wather, H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995) · Zbl 0826.34002 [9] Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977) · Zbl 0374.34001 [10] Elaydi, S.: An Introduction to Difference Equations. Springer, New York (1996) · Zbl 0840.39002 [11] El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Argument. Holden Day, San Francisco (1966) [12] El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Argument. Academic Press, New York (1973) [13] Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992) · Zbl 0752.34039 [14] Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations. Clarendon, Oxford (1991) [15] Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977) · Zbl 0352.34001 [16] Jaros, J., Stavroulakis, I.P.: Necessary and sufficient conditions for oscillations of difference equations with several delays. Util. Math. 45, 187–195 (1994) · Zbl 0808.39004 [17] Karakoc, F., Bereketoglu, H.: Some results for linear impulsive delay differential equations. Dyn. Contin. Discrete Impuls. Syst. (to appear) · Zbl 1196.34104 [18] Kelley, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, New York (1991) · Zbl 0733.39001 [19] Kolmanovski, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, New York (1986) [20] Koplatadze, R., Kvinikadze, G., Stavroulakis, I.P.: Oscillation of second-order linear difference equations with deviating arguments. Adv. Math. Sci. Appl. 12, 217–226 (2002) · Zbl 1033.39011 [21] Krisztin, T.: A note on the convergence of the solutions of a linear functional differential equation. J. Math. Anal. Appl. 145, 17–25 (1990) · Zbl 0693.45012 [22] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993) · Zbl 0777.34002 [23] Lakshmikhantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Academic Press, New York (1988) [24] MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989) · Zbl 0669.92001 [25] Murakami, K.: Asymptotic constancy for systems of delay differential equations. Nonlinear Anal. 30, 4595–4606 (1997) · Zbl 0959.34058 [26] Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) · Zbl 0261.34040 [27] Shen, J., Stavroulakis, I.P.: Oscillation criteria for delay difference equations. Electron. J. Differ. Equ. 2001(10), 1–15 (2001) · Zbl 0964.39009 [28] Stavroulakis, I.P.: Oscillations of delay difference equations. Comput. Math. Appl. 29, 83–88 (1995) · Zbl 0832.39002
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