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CONTROLLABILITY OF SECOND ORDER SEMI-LINEAR NEUTRAL IMPULSIVE DIFFERENTIAL INCLUSIONS ON UNBOUNDED DOMAIN WITH INFINITE DELAY IN BANACH SPACES

  • Chalishajar, Dimplekumar N. (Department of Mathematics and Computer Science Virginia Military Institute (VMI)) ;
  • Acharya, Falguni S. (Department of Applied Sciences and Humanities Institute of Technology and Management (ITM) Universe)
  • Received : 2009.12.18
  • Published : 2011.07.31
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Abstract

In this paper, we prove sufficient conditions for controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces using the theory of strongly continuous Cosine families. We shall rely on a fixed point theorem due to Ma for multi-valued maps. The controllability results in infinite dimensional space has been proved without compactness on the family of Cosine operators.

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References

  1. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel-Dekker, New York, 1980.
  2. M. Benchohra and S. K. Ntouyas, Controllability for functional differential and integrodifferential inclusions in Banach spaces, J. Optim. Theory Appl. 113 (2002), no. 3, 449-472. https://doi.org/10.1023/A:1015352503233
  3. M. Benchohra and S. K. Ntouyas, Controllability results for multivalued semilinear neutral functional equations, Math. Sci. Res. J. 6 (2002), no. 2, 65-77.
  4. M. Benchohra and S. K. Ntouyas, Existence and controllability results for nonlinear differential inclusions with nonlocal conditions in Banach spaces, J. Appl. Anal. 8 (2002), no. 1, 33-48. https://doi.org/10.1515/JAA.2002.33
  5. M. Benchohra and S. K. Ntouyas, On second order impulsive functional differential equations in Banach spaces, J. Appl. Math. Stoch. Anal. 15 (2002), no. 1, 47-55.
  6. J. Bochenek, An abstract nonlinear second-order differential equation, Ann. Polon. Math. 54 (1991), no. 2, 155-166. https://doi.org/10.4064/ap-54-2-155-166
  7. D. N. Chalishajar, Controllability of nonlinear integro-differential third order dispersion system, J. Math. Anal. Appl. 348 (2008), no. 1, 480-486. https://doi.org/10.1016/j.jmaa.2008.07.047
  8. D. N. Chalishajar and F. S. Acharya, Controllability of nonlinear integro-differential third order dispersion inclusions in Banach spaces, Nonl. Anal. TMA, Communicated.
  9. D. N. Chalishajar, R. K. George, and A. K. Nandakumaran, Exact controllability of the nonlinear third-order dispersion equation, J. Math. Anal. Appl. 332 (2007), no. 2, 1028-1044. https://doi.org/10.1016/j.jmaa.2006.10.084
  10. K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.
  11. J. Dugunji and A. Granas, Fixed Point Theory. I, Monografic. Math. PWN Warsaw, 1982.
  12. R. K. George, D. N. Chalishajar, and A. K. Nandakumaran, Contollability of second order semi-linear neutral functional differential inclusions in Banach spaces, Mediterr. J. Math. 99 (2004), 1-16.
  13. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Uni. Press, New-York, 1985.
  14. A. Granas and J. Dugunji, Fixed Point Theory, Springer-Verlag, New-York, 2003.
  15. E. M. Hernandez, M. Rabello, and H. Henriquez, Existence of solutions of impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 331 (2007), 1135-1158. https://doi.org/10.1016/j.jmaa.2006.09.043
  16. Sh. Hu and N. S. Papageogiou, Handbook of Multivalued Analysis, Kluwer Dordrecht, 1997.
  17. J. Kisynski, On second order Cauchy problem in a Banach space, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 18 (1970), 371-374.
  18. A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.
  19. B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005), no. 8, 1533-1552. https://doi.org/10.1016/j.na.2004.11.022
  20. T. W. Ma, Topological degrees for set-valued compact vector fields in locally convex spaces, Dissertationess Math. 92 (1972), 1-43.
  21. S. K. Ntouyas and P. Ch. Tsamatos, Global existence of semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1996), 679-687.
  22. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New-York, 1983.
  23. M. D. Quinn and N. Carmichael, An introduction to nonlinear control problems using fixed point methods, degree theory and pseudo-inverse, Numer. Funct. Anal. Optim. 7 (1984/1985), 197-219. https://doi.org/10.1080/01630568508816189
  24. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differntial Equations, World Scientific, Singapore, 1995.
  25. H. Schaefer, Uber die method der a priori Schranken, Math. Anal. 129 (1955), 415-416. https://doi.org/10.1007/BF01362380
  26. 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
  27. C. C. Travis and G. F. Webb, Second order differential equations in Banach spaces, Proc. Int. Symp. on Nonlinear Equations in Abstract spaces, 331-361, Academic Press New-York, 1978.
  28. R. Triggani, Addendum : A note on lack of exact controllability for mild solution in Banach spaces, SIAM J. cont. Opti. 15 (1977), 407-411. https://doi.org/10.1137/0315028
  29. R. Triggani, Addendum : A note on lack of exact controllability for mild solution in Banach spaces, SIAM J. Cont. Opti. 18 (1980), no. 1, 98-99. https://doi.org/10.1137/0318007
  30. K. Yosida, Functional Analysis, 6th Edition, Springer-Verlag, Berlin, 1980.

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