Bulletin of the Korean Mathematical Society (대한수학회보)

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CONTROLLABILITY OF SECOND-ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

  • Received : 2010.07.22
  • Published : 2011.11.30
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Abstract

The purpose of this paper is to investigate the controllability of certain types of second order nonlinear impulsive systems with statedependent delay. Sufficient conditions are formulated and the results are established by using a fixed point approach and the cosine function theory Finally examples are presented to illustrate the theory.

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References

  1. W. G. Aiello, H. I. Freedman, and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math. 52 (1992), no. 3, 855-869. https://doi.org/10.1137/0152048
  2. A. Anguraj, M. M. Arjunan, and E. Hernandez, Existence results for an impulsive neutral functional differential equation with state-dependent delay, Appl. Anal. 86 (2007), no. 7, 861-872. https://doi.org/10.1080/00036810701354995
  3. K. Balachandran and S. M. Anthoni, Controllability of second-order semilinear neutral functional differential systems in Banach spaces, Comput. Math. Appl. 41 (2001), no. 10-11, 1223-1235. https://doi.org/10.1016/S0898-1221(01)00093-1
  4. M. Bartha, Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Anal. 53 (2003), no. 6, 839-857. https://doi.org/10.1016/S0362-546X(03)00039-7
  5. Y. Cao, J. Fan, and T. C. Gard, The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear Anal. 19 (1992), no. 2, 95-105. https://doi.org/10.1016/0362-546X(92)90113-S
  6. Y.-K. Chang, M. M. Arjunan, and V. Kavitha, Existence results for a second order impulsive functional differential equation with state-dependent delay, Differ. Equ. Appl. 1 (2009), no. 3, 325-339.
  7. F. Chen, D. Sun, and J. Shi, Periodicity in a food-limited population model with toxicants and state-dependent delays, J. Math. Anal. Appl. 288 (2003), no. 1, 136-146. https://doi.org/10.1016/S0022-247X(03)00586-9
  8. A. Domoshnitsky, M. Drakhlin, and E. Litsyn, On equations with delay depending on solution, Nonlinear Anal. 49 (2002), no. 5, 689-701. https://doi.org/10.1016/S0362-546X(01)00132-8
  9. R. D. Driver, A neutral system with state-dependent delay, J. Differential Equations 54 (1984), no. 1, 73-86. https://doi.org/10.1016/0022-0396(84)90143-8
  10. H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland, Amsterdam, 1985.
  11. F. Hartung, 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), no. 7, 4557-4566. https://doi.org/10.1016/S0362-546X(01)00569-7
  12. F. Hartung, T. L. Herdman, and J. Turi, Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear Anal. 39 (2000), no. 3, 305-325. https://doi.org/10.1016/S0362-546X(98)00169-2
  13. F. Hartung, T. Krisztin, H.-O. Walther, and J. Wu, Functional Differential Equations with State-Dependent Delays: Theory and Applications, in: A. Canada, P. Drabek, A. Fonda (Eds.), Handbook of Differential Equations : Ordinary Differential Equations, Elsevier B. V, 2006.
  14. F. Hartung and J. Turi, Identification of parameters in delay equations with state-dependent delays, Nonlinear Anal. 29 (1997), no. 11, 1303-1318. https://doi.org/10.1016/S0362-546X(96)00100-9
  15. E. Hernandez, Existence of solutions for a second order abstract functional differential equation with state-dependent delay, Electron. J. Differential Equations 2007 (2007), no. 21, 10 pp.
  16. E. Hernandez and M. McKibben, On state-dependent delay partial neutral functional differential equations, Appl. Math. Comput. 186 (2007), no. 1, 294-301. https://doi.org/10.1016/j.amc.2006.07.103
  17. E. Hernandez, M. A. McKibben, and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling 49 (2009), no. 5-6, 1260-1267. https://doi.org/10.1016/j.mcm.2008.07.011
  18. E. Hernandez, M. Pierri, and G. Uniao, Existence results for an impulsive abstract partial differential equation with state-dependent delay, Comput. Math. Appl. 52 (2006), no. 3-4, 411-420. https://doi.org/10.1016/j.camwa.2006.03.022
  19. E. Hernandez, A. Prokopczyk, and L. A. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real World Appl. 7 (2006), no. 4, 510-519. https://doi.org/10.1016/j.nonrwa.2005.03.014
  20. E. Hernandez, R. Sakthivel, and S. Tanaka, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations 2008 (2008), no. 28, 11 pp.
  21. Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.
  22. J. M. Jeong and H. G. Kim, Controllability for semilinear functional integrodifferential equations, Bull. Korean Math. Soc. 46 (2009), no. 3, 463-475. https://doi.org/10.4134/BKMS.2009.46.3.463
  23. J. Kisynski, On cosine operator functions and one parameter group of operators, Studia Math. 49 (1972), 93-105.
  24. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  25. W. S. Li, Y.-K. Chang, and J. J. Nieto, Solvability of impulsive neutral evolution differential inclusions with state-dependent delay, Math. Comput. Modelling 49 (2009), no. 9-10, 1920-1927. https://doi.org/10.1016/j.mcm.2008.12.010
  26. S. K. Ntouyas and D. O' Regan, Some remarks on controllability of evolution equations in Banach spaces, Electron. J. Differential Equations 2009 (2009), no. 79, 6 pp.
  27. J. Y. Park and H. K. Han, Controllability for some second order differential equations, Bull. Korean Math. Soc. 34 (1997), no. 3, 411-419.
  28. J. Y. Park, S. H. Park, and Y. H. Kang, Controllability of second-order impulsive neutral functional differential inclusions in Banach spaces, Math. Methods Appl. Sci. 33 (2010), no. 3, 249-262.
  29. B. N. Sadovskii, On a fixed point principle, Funct. Anal. Appl. 1 (1967), no. 2, 74-76.
  30. R. Sakthivel, N. I. Mahmudov, and J. H. Kim, On controllability of second-order nonlinear impulsive differential systems, Nonlinear Anal. 71 (2009), no. 1-2, 45-52. https://doi.org/10.1016/j.na.2008.10.029
  31. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  32. J. G. Si and X. P. Wang, Analytic solutions of a second-order functional differential equation with a state dependent delay, Results Math. 39 (2001), no. 3-4, 345-352. https://doi.org/10.1007/BF03322694
  33. C. C. Travis and G. F.Webb, Compactness, regularity and uniform continuity properties of strongly continuous cosine families, Houston J. Math. 3 (1977), no. 4, 555-567.
  34. C. C. Travis and G. F.Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 1-2, 75-96. https://doi.org/10.1007/BF01902205
  35. Z. Yang and J. Cao, Existence of periodic solutions in neutral state-dependent delay equations and models, J. Comput. Appl. Math. 174 (2005), no. 1, 179-199. https://doi.org/10.1016/j.cam.2004.04.007

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