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Asymptotic representation of solutions of the equation $\dot y(t)=\beta(t)[y(t)-y(t-\tau(t))]$. (English) Zbl 0892.34067
The equation considered is the following differential difference equation $$ \dot{y}=\beta(t) [y(t)-y(t-\tau(t))], \tag1$$ where $\tau\in C(I_{-1},{\bbfR}^+)$, $I_{-1}=[t_{-1},\infty)$, $t_{-1}=t_0-\tau(t_0)$, $t_0\in{\bbfR}$, $t\mapsto t-\tau(t)$ is increasing, $\tau(t)\geq\widetilde{\tau}$ for all $t$, $\widetilde{\tau}= \text{ const}>0$, $\beta\in C([t_0,\infty),{\bbfR}^+)$. The author proves theorems on the existence of solutions of (1) tending to infinity as $t\to\infty$, and of solutions with the range in $[C_1,C_2]$ for any $C_1<C_2$. Exponential estimates and comparison theorems are also discussed.

MSC:
34K25Asymptotic theory of functional-differential equations
34C11Qualitative theory of solutions of ODE: growth, boundedness
34K99Functional-differential equations
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References:
[1] Arino, O.; Györi, I.; Pituk, M.: Asymptotically diagonal delay differential systems. J. math. Anal. appl. 204, 701-728 (1996) · Zbl 0876.34078
[2] Atkinson, F. V.; Haddock, J. R.: Criteria for asymptotic constancy of solutions of functional differential equations. J. math. Anal. appl. 91, 410-423 (1983) · Zbl 0529.34065
[3] Bellman, R.; Cooke, K. L.: Differential-difference equations. (1963) · Zbl 0105.06402
[4] Diblı\acute{}k, J.: Bounded solutions with positive coordinates of functional-differential equations of the retarded type. Proceedings of the conference on ordinary differential equations, 13-22 (1994)
[5] Diblı\acute{}k, J.: Asymptotic behaviour of solutions of linear differential equations with delay. Ann. polon. Math. 2, 131-137 (1993) · Zbl 0784.34053
[6] Erbe, L. H.; Kong, Qingkai; Zhang, B. G.: Oscillation theory for functional differential equations. (1995) · Zbl 0821.34067
[7] Györi, I.; Pituk, M.: L2. J. math. Anal. appl. 195, 415-427 (1995)
[8] Györi, I.; Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dynamic systems and applications 5, 277-302 (1996) · Zbl 0859.34053
[9] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[10] Hale, J.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[11] Kozakiewicz, E.: On the asymptotic behaviour of positive solutions of two different inequalities with retarded argument. Colloquia Mathematica societatis jános bolyai 15, differential equations, 309-319 (1975)
[12] Krisztin, T.: Asymptotic estimation for functional differential equations via Lyapunov functions, qualitative theory of differential equations. Colloq. math. Soc. jános bolyai 53, 365-376 (1988)
[13] Neuman, F.: On equivalence of linear functional-differential equations. Results in mathematics 26, 354-359 (1994) · Zbl 0829.34054
[14] Neuman, F.: On transformations of differential equations and systems with deviating argument. Czechoslovak math. J. 31, 87-90 (1981) · Zbl 0463.34051
[15] Razumikhin, B. S.: Stability of hereditary systems. (1988)
[16] Rybakowski, K. P.: Wa\dot{}zewski’s principle for retarded functional differential equations. J. differential equations 36, 117-138 (1980) · Zbl 0407.34056
[17] Wa.Zewski, T.: Sur un principe topologique de le’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires. Ann. soc. Polon. math. 20, 279-313 (1947)
[18] Zhang, S. N.: Asymptotic behaviour and structure of solutions for equationx\dot{}(tptxtxt. J. anhui university (Natural science edition) 2, 11-21 (1981)
[19] Zhou, Detang: Negative answer to a problem of gÿori. J. shandong university 24, 117-121 (1989)