# Article

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Keywords:
Positive solution; oscillating solution; convergent solution; linear differential equation with delay; topological principle of Ważewski (Rybakowski’s approach)
Summary:
This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form $\dot{x}(t)= -c(t)x(t-\tau (t)) \qquad \mathrm {{(^*)}}$ with positive function $c(t).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation $\dot{y}(t)=\beta (t)[y(t)-y(t-\tau (t))]$ where the function $\beta (t)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation $x^{\prime \prime }(t)+a(t)x(t)=0$ for positive function $a(t)$ in critical case is considered.
References:
[1] O. Arino M. Pituk: Convergence in asymptotically autonomous functional differential equations. University of Veszprém, Department of Mathematics and Computing, preprint No. 065 (1997). MR 1708180
[2] F. V. Atkinson J. R. Haddock: Criteria for asymptotic constancy of solutions of functional differential equations. J. Math. Anal. Appl. 91 (1983), 410-423. MR 0690880
[3] Li Bingtuan: Oscillation of first order delay differential equations. Proc. Amer. Math. Soc. 124 (1996), 3729-3737. MR 1363175 | Zbl 0865.34057
[4] Li Bingtuan: Oscillations of delay differential equations with variable coefficients. J. Math. Anal. Appl. 192 (1995), 312-321. MR 1329426
[5] J. Čermák: On the asymptotic behaviour of solutions of certain functional differential equations. to appear in Math. Slovaca.
[6] J. Diblík: A criterion for convergence of solutions of homogeneous delay linear differential equations. submitted.
[7] J. Diblík: A criterion for existence of positive solutions of systems of retarded functional differential equations. to appear in Nonlin. Anal., T. M. A. MR 1705781
[8] 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. MR 1492085
[9] J. Diblík: Existence of solutions with prescribed asymptotic for certain systems retarded functional differential equations. Siberian Mathematical Journal, 32 (1991), (No 2), 222–226. MR 1138440
[10] J. Diblík: Existence of decreasing positive solutions of a system of linear differential equations with delay. Proceedings of the Fifth International Colloquium on Differential Equations, VSP, Eds.: D. Bainov and V. Covachov, 83–94, 1995. MR 1458348
[11] J. Diblík: Positive and oscillating solutions of differential equations with delay in critical case. J. Comput. Appl. Math., 88 (1998), 185–202. MR 1609086
[12] A. Domoshnitsky M. Drakhlin: Nonoscillation of first order differential equations with delay. J. Math. Anal. Appl., 206 (1997), 254–269. MR 1429290
[13] Y. Domshlak: Sturmian Comparison Method in Investigation of the Behavior of Solutions for Differential-Operator Equations. ELM, Baku, USSR, 1986. (In Russian) MR 0869589
[14] Y. Domshlak I. P. Stavroulakis: Oscillation of first-order delay differential equations in a critical case. Applicable Analysis, 61 (1996), 359–371. MR 1618248
[15] Á. Elbert I. P. Stavroulakis: Oscillation and non-oscillation criteria for delay differential equations. Proc. Amer. Math. Soc., 123 (1995), 1503–1510. MR 1242082
[16] L. H. Erbe Q. Kong B. G. Zhang: Oscillation Theory for Functional Differential Equations. Marcel Dekker, Inc., 1995. MR 1309905
[17] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, 1992. MR 1163190 | Zbl 0752.34039
[18] I. Györi G. Ladas: Oscillation Theory of Delay Differential Equations. Clarendon Press, 1991. MR 1168471
[19] I. Györi M. Pituk: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dynamic Systems and Application, 5 (1996), 277–302. MR 1396192
[20] J. K. Hale S. M. V. Lunel: Introduction to Functional Differential Equations. Springer-Verlag, 1993. MR 1243878
[21] P. Hartman: Ordinary Differential Equations. John Wiley & Sons, 1964. MR 0171038 | Zbl 0125.32102
[22] E. Hille: Non-oscillation theorems. Trans. Amer. Math. Soc., 64 (1948), 232–252. MR 0027925 | Zbl 0031.35402
[23] J. Jaroš I. P. Stavroulakis: Oscillation tests for delay equations. Technical Report No 265, University of Ioannina, Department of Mathematics, June 1996.
[24] A. Kneser: Untersuchungen über die reelen Nullstellen der Integrale linearen Differentialgleichungen. Math. Ann., 42 (1893), 409–503; J. Reine Angew. Math., 116 (1896), 178–212. MR 1510784
[25] R. G. Koplatadze T. A. Chanturija: On the oscillatory and monotonic solutions of first order differential equations with deviating arguments. Differencial’nyje Uravnenija, 18 (1982), 1463–1465. (In Russian) MR 1034096
[26] E. Kozakiewicz: Conditions for the absence of positive solutions of a first order differential inequality with a single delay. Archivum Mathematicum (Brno), 31 (1995), 291–297. MR 1390588 | Zbl 0849.34054
[27] E. Kozakiewicz: Conditions for the absence of positive solutions of a first order differential inequality with one continuously retarded argument. Berlin, preprint, (1997), 1–6. MR 1669773
[28] E. Kozakiewicz: Über das asymptotische Verhalten der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument. Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 13:4 (1964), 577–589. Zbl 0277.34085
[29] E. Kozakiewicz: Zur Abschätzung des Abklingens der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument. Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 15:5 (1966), 675–676. MR 0216262 | Zbl 0221.34025
[30] E. Kozakiewicz: Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument. Math. Nachr., 32: 1/2 (1966), 107–113. MR 0204801 | Zbl 0182.15504
[31] T. Krisztin: A note on the convergence of the solutions of a linear functional differential equation. J. Math. Anal. Appl., 145 (1990), 17–25, (1990). MR 1031171 | Zbl 0693.45012
[32] A. D. Myshkis: Linear Differential Equations with Retarded Arguments. (2nd Ed.), Nauka, 1972. [In Russian] MR 0352648
[33] F. Neuman: On equivalence of linear functional-differential equations. Results in Mathematics, 26 (1994), 354–359. MR 1300618 | Zbl 0829.34054
[34] F. Neuman: On transformations of differential equations and systems with deviating argument. Czechoslovak Mathematical Journal, 31 (106) (1981), 87–90. MR 0604115 | Zbl 0463.34051
[35] M. Pituk: Asymptotic characterization of solutions of functional differential equations. Bolletino U. M. I., 7, 7-B, (1993), 653–683. MR 1244413 | Zbl 0809.34087
[36] K. P. Rybakowski: Ważewski’s principle for retarded functional differential equations. Journal of Differential Equations, 36 (1980), 117–138. MR 0571132 | Zbl 0407.34056
[37] C. A. Swanson: Comparison and Oscillating Theory of Linear Differential Equations. Academic Press, 1968. MR 0463570
[38] T. Ważewski: Sur un principle topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles. Ann. Soc. Polon. Math., 20 (1947), 279–313. MR 0026206
[39] J. Werbowski: Oscillations of first order linear differential equations with delay. Proceedings of the Conference on Ordinary Differential Equations, Poprad (Slovak Republic), 87–94, 1996. Zbl 0911.34062
[40] Yu Jiang, Yan Jurang: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. J. Math. Anal. Appl., 207 (1997), 388–396. MR 1438921 | Zbl 0877.34054
[41] S. N. Zhang: Asymptotic behaviour and structure of solutions for equation $\dot{x}(t)=p(t)[x(t)-x(t-1)],$. J. Anhui University (Natural Science Edition), 2 (1981), 11–21. (In Chinese)
[42] D. Zhou: On a problem of I. Györi. J. Math. Anal. Appl., 183 (1994), 620–623. MR 1274862 | Zbl 0803.34070

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