Previous |  Up |  Next

Article

Keywords:
Differential equation; state-dependent delay; fixed point; impulses; infinite delay
Summary:
In this paper we study the existence of solutions for impulsive differential equations with state dependent delay. Our results are based on the Leray–Schauder nonlinear alternative and Burton–Kirk fixed point theorem for the sum of two operators.
References:
[1] Abada, N., Agarwal, R. P., Benchohra, M., Hammouche, H.: Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay. Asian-Eur. J. Math. 1, 4 (2008), 449–468. DOI 10.1142/S1793557108000382 | MR 2474181 | Zbl 1179.34070
[2] Adimy, M., Bouzahir, H., Ezzinbi, K.: Local existence and stability for some partial functional differential equations with unbounded delay. Nonlinear Anal. 48 (2002), 323–348. DOI 10.1016/S0362-546X(00)00184-X | MR 1869515
[3] Adimy, M, Ezzinbi, K.: A class of linear partial neutral functional-differential equations with nondense domain. J. Differential Equations 147 (1998), 285–332. DOI 10.1006/jdeq.1998.3446 | MR 1633941 | Zbl 0915.35109
[4] Ait Dads, E., Ezzinbi, K.: Boundedness and almost periodicity for some state-dependent delay differential equations. Electron. J. Differential Equations 2002, 67 (2002), 1–13. MR 1921140
[5] Aiello, W. G., Freedman, H. I., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52, 3 (1992), 855–869. DOI 10.1137/0152048 | MR 1163810 | Zbl 0760.92018
[6] Anguraj, A., Arjunan, M. M., Hernàndez, E. M.: Existence results for an impulsive neutral functional differential equation with state-dependent delay. Appl. Anal. 86, 7 (2007), 861–872. DOI 10.1080/00036810701354995 | MR 2355543
[7] Bainov, D. D., Simeonov, P. S.: Systems with Impulsive effect. Horwood, Chichister, 1989. MR 1010418
[8] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, Vol 2, New York, 2006. MR 2322133 | Zbl 1130.34003
[9] Burton, T. A., Kirk, C.: A fixed point theorem of Krasnoselskiii–Schaefer type. Math. Nachr. 189 (1998), 23–31. DOI 10.1002/mana.19981890103 | MR 1492921
[10] Cao, Y., Fan, J., Gard, T. C.: The effects of state-dependent time delay on a stage-structured population growth model. Nonlinear Anal. 19, 2 (1992), 95–105. DOI 10.1016/0362-546X(92)90113-S | MR 1174461 | Zbl 0777.92014
[11] Corduneanu, C., Lakshmikantham, V.: Equations with unbounded delay. Nonlinear Anal. 4 (1980), 831–877. DOI 10.1016/0362-546X(80)90001-2 | MR 0586852 | Zbl 0449.34048
[12] Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On equations with delay depending on solution. Nonlinear Anal. 49, 5 (2002), 689–701. DOI 10.1016/S0362-546X(01)00132-8 | MR 1894304 | Zbl 1012.34066
[13] Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York, 2003. MR 1987179 | Zbl 1025.47002
[14] Hale, J., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvac. 21 (1978), 11–41. MR 0492721 | Zbl 0383.34055
[15] Hale, J. K., Verduyn Lunel, S. M.: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99, Springer, New York, 1993. MR 1243878 | Zbl 0787.34002
[16] Hartung, F.: Linearized stability in periodic functional differential equations with state-dependent delays. J. Comput. Appl. Math. 174 (2005), 201–211. DOI 10.1016/j.cam.2004.04.006 | MR 2106436 | Zbl 1077.34074
[17] Hartung, F.: Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study. Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000), Nonlinear Anal. 47 (2001), 4557–4566. MR 1975850 | Zbl 1042.34582
[18] Hartung, F., Turi, J.: Identification of parameters in delay equations with state-dependent delays. Nonlinear Anal. 29 (1997), 1303–1318. DOI 10.1016/S0362-546X(96)00100-9 | MR 1472420 | Zbl 0894.34071
[19] Hernández, E., Prokopczyk, A., Ladeira, L.: A note on partial functional differential equations with state-dependent delay. Nonlinear Anal. Real World Appl. 7 (2006), 510–519. MR 2235215 | Zbl 1109.34060
[20] Hernández, E., Sakthivel, R., Tanaka Aki, S.: Existence results for impulsive evolution differential equations with state-dependent delay. Electron. J. Differential Equations 2008, 28 (2008), 1–11. MR 2390434
[21] Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Unbounded Delay. Springer-Verlag, Berlin, 1991. MR 1122588
[22] Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional-Differential Equations. Kluwer Academic Publishers, Dordrecht, 1999. MR 1680144 | Zbl 0917.34001
[23] Lakshmikantham, V., Bainov D. D., Simeonov, P. S.: Theory of Impulsive Differntial Equations. World Scientific, Singapore, 1989. MR 1082551
[24] Lakshmikantham, V., Wen, L., Zhang, B.: Theory of Differential Equations with Unbounded Delay. Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1994. MR 1319339 | Zbl 0823.34069
[25] Liang, J., Xiao, Ti-jun: The Cauchy problem for non linear abstract fuctionnal differential with infinte delay. Comput. Math. Appl. 40 (2000), 693–707. DOI 10.1016/S0898-1221(00)00188-7 | MR 1776686
[26] Rezounenko, A. V., Wu, J.: A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors. J. Comput. Appl. Math. 190 (2006), 99–113. DOI 10.1016/j.cam.2005.01.047 | MR 2209496 | Zbl 1082.92039
[27] Samoilenko, A. M., Perestyuk, N. A.: Impulsive Differential Equations. World Scientific, Singapore, 1995. MR 1355787 | Zbl 0837.34003
[28] Willé, D. R., Baker, C. T. H.: Stepsize control and continuity consistency for state-dependent delay-differential equations. J. Comput. Appl. Math. 53 (1994), 163–170. DOI 10.1016/0377-0427(94)90043-4 | MR 1306123
[29] Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York, 1996. MR 1415838 | Zbl 0870.35116
Partner of
EuDML logo