Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXISTENCE AND CONTROLLABILITY RESULTS FOR NONDENSELY DEFINED STOCHASTIC EVOLUTION DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS

  • Ni, Jinbo (Department of Mathematics Anhui University of Science and Technology) ;
  • Xu, Feng (Department of Mathematics Anhui University of Science and Technology) ;
  • Gao, Juan (Department of Mathematics Anhui University of Science and Technology)
  • Received : 2011.11.18
  • Published : 2013.01.01
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Abstract

In this paper, we investigate the existence and controllability results for a class of abstract stochastic evolution differential inclusions with nonlocal conditions where the linear part is nondensely defined and satisfies the Hille-Yosida condition. The results are obtained by using integrated semigroup theory and a fixed point theorem for condensing map due to Martelli.

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References

  1. N. Abada, M. Benchohra, and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), no. 10, 3834-3863. https://doi.org/10.1016/j.jde.2009.03.004
  2. M. Adimy, H. Bouzahir, and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal. 46 (2001), no. 1, 91-112. https://doi.org/10.1016/S0362-546X(99)00447-2
  3. N. U. Ahmed, Nonlinear stochastic differential inclusions on Bananch space, Stochastic Anal. Appl. 12 (1994), 1-10. https://doi.org/10.1080/07362999408809334
  4. W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), no. 3, 327-352. https://doi.org/10.1007/BF02774144
  5. W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3) 54 (1987), no. 2, 321-349. https://doi.org/10.1112/plms/s3-54.2.321
  6. P. Balasubramaniam, Existence of solutions of functional stochastic differential inclusions, Tamkang J. Math. 33 (2002), no. 1, 35-43.
  7. P. Balasubramaniam, S. K. Ntouyas, and D. Vinayagam, Existence of solutions of semi-linear stochastic delay evolution inclusions in a Hilbert space, J. Math. Anal. Appl. 305 (2005), no. 2, 438-451. https://doi.org/10.1016/j.jmaa.2004.10.063
  8. P. Balasubramaniam and S. K. Ntouyas, Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space, J. Math. Anal. Appl. 324 (2006), no. 1, 161-176. https://doi.org/10.1016/j.jmaa.2005.12.005
  9. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Dekker, New York, 1980.
  10. M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, Nondensely defined evolution impulsive differential inclusions with nonlocal conditions, J. Math. Anal. Appl. 286 (2003), no. 1, 307-325. https://doi.org/10.1016/S0022-247X(03)00490-6
  11. T. Caraballo, K. Liu, and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 1999, 1755-1082.
  12. G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 2, 285-344.
  13. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
  14. K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin, 1992.
  15. K. Ezzinbi and J. Liu, Nondensely defined evolution equations with nonlocal conditions, Math. Comput. Modelling 36 (2002), no. 9-10, 1027-1038. https://doi.org/10.1016/S0895-7177(02)00256-X
  16. E. P. Gatsori, Controllability results for nondensely defined evolution differential inclu-sions with nonlocal conditions, J. Math. Anal. Appl. 297 (2004), no. 1, 194-211. https://doi.org/10.1016/j.jmaa.2004.04.055
  17. V. Kavitha and M. Mallika Arjunan, Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 3, 441-450. https://doi.org/10.1016/j.nahs.2009.11.002
  18. H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), no. 1, 160-180. https://doi.org/10.1016/0022-1236(89)90116-X
  19. 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.
  20. Y. Li and B. Liu, Existence of solution of nonlinear neutral stochastic differential in-clusions with infinite delay, Stochastic Anal. Appl. 25 (2007), no. 2, 397-415. https://doi.org/10.1080/07362990601139610
  21. M. Martelli, A Rothe's type theorem for non-compact acyclic-valued map, Boll, Un. Mat. Ital. (4) 11 (1975), no. 3, 70-76.
  22. H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990), no. 2, 416-477. https://doi.org/10.1016/0022-247X(90)90074-P
  23. H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3 (1990), no. 6, 1035-1066.
  24. R. Subalakshmi and K. Balachandran, Approximate controllability of nonlinear sto-chastic impulsive integrodifferential systems in Hilbert spaces, Chaos Solitons Fractals 42 (2009), no. 4, 2035-2046. https://doi.org/10.1016/j.chaos.2009.03.166
  25. R. Subalakshmi, K. Balachandran, and J. Y. Park, Controllability of semilinear stochastic functional integrodifferential systems in Hilbert spaces, Nonlinear Anal. Hybrid Syst. 3 (2009), no. 1, 39-50. https://doi.org/10.1016/j.nahs.2008.10.004
  26. F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl. 331 (2007), no. 1, 516-531. https://doi.org/10.1016/j.jmaa.2006.09.020