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Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces

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Abstract

This paper discusses a class of impulsive nonlocal stochastic functional integrodifferential inclusions in a real separable Hilbert space. The existence of mild solutions of these inclusions is determined under the mixed continuous and Carathéodory conditions by using Bohnenblust–Karlin’s fixed point theorem and fractional operators combined with approximation techniques. An example is provided to illustrate the theory.

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References

  1. Benchohra M., Henderson J., Ntouyas S.K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)

    Book  MATH  Google Scholar 

  2. Lakshmikanthan V., Bainov D.D., Simeonov P.S.: Theory of Impulsive Differential Equations. World Scientifi, Singapore (1989)

    Book  Google Scholar 

  3. Hernández E., Rabello M., Henríquez H.: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl. 331, 1135–1158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anguraj A., Karthikeyan K.: Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions. Nonlinear Anal. 70, 2717–2721 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hernández E., Henríquez H.: Impulsive partial neutral differential equations. Appl. Math. Lett. 19, 215–222 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mao X.: Stochastic Differential Equations and Their Applications. Horwood Publishing Ltd., Chichester (1997)

    MATH  Google Scholar 

  7. Grecksch W., Tudor C.: Stochastic Evolution Equations: A Hilbert Space Approach. Akademic Verlag, Berlin (1995)

    MATH  Google Scholar 

  8. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  9. Taniguchi T., Liu K., Truman A.: Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differ. Equ. 181, 72–91 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. El-Borai M.M., Mostafa O.L., Ahmed H.M.: Asymptotic stability of some stochastic evolution equations. Appl. Math. Comput. 144, 273–286 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Govindan T.E.: Stability of mild solutions of stochastic evolution equations with variable delay. Stoch. Anal. Appl. 21, 1059–1077 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balasubramaniam P., Ntouyas S.K.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324, 161–176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Balasubramaniam P., Vinayagam D.: Existence of solutions of nonlinear stochastic integrodifferential inclusions in a Hilbert space. Comput. Math. Appl. 50, 809–821 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Balasubramaniam P., Vinayagam D.: Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space. Stoch. Anal. Appl. 23, 137–151 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sakthivel R., Luo J.: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl. 356, 1–6 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sakthivel R., Sakthivel R.: Asymptotic stability of nonlinear impulsive stochastic differential equations. Stat. Probab. Lett. 79, 1219–1223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Anguraj A., Vinodkumar A.: Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional differential equations with infinite delays. Electron. J. Qual. Theory Differ. Equ 67, 1–13 (2009)

    MathSciNet  Google Scholar 

  18. Ren Y., Hu L., Sakthivel R.: Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay. J. Comput. Appl. Math. 235, 2603–2614 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Byszewski L., Lakshmikantham V.: Theorem about existence and uniqueness of a solutions of a nonlocal Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1990)

    Article  MathSciNet  Google Scholar 

  20. Byszewski L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Byszewski L., Akca H.: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear Anal. 34, 65–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Deng K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630–637 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin Y., Liu J.H.: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal. 26, 1023–1033 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ezzinbi K., Fu X., Hilal K.: Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal. 67, 1613–1622 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yan Z.: Nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces. Math. Commun. 14, 5–45 (2009)

    Google Scholar 

  26. Balasubramaniam P., Ntouyas S.K.: Global existence for semilinear stochastic delay evolution equations with nonlocal conditions. Soochow J. Math. 27, 331–342 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Balasubramaniam P., Park J.Y., Kumar A.V.A.: Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions. Nonlinear Anal. 71, 1049–1058 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Keck D.N., McKibben M.A.: Functional integro-differential stochastic evolution equations in Hilbert space. J. Appl. Math. Stoch. Anal. 16, 127–147 (2003)

    Article  MathSciNet  Google Scholar 

  29. Lin A., Hu L.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Comput. Math. Appl. 59, 64–73 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guo M., Xue X., Li R.: Controllability of impulsive evolution inclusions with nonlocal conditions. J. Optim. Theory Appl. 120, 355–374 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yan Z.: Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators. J. Comput. Appl. Math. 235, 2252–2262 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kree P.: Diffusion equation for multivalued stochastic differential equations. J. Funct. Anal. 49, 73–90 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kisielewicz M., Michta M., Motyl J.: Set-valued approach to stochastic control part I (existence and regularity properties). Dyn. Syst. Appl. 12, 405–432 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Kisielewicz M., Michta M., Motyl J.: Set-valued approach to stochastic control part II (viability and semimartingale issues). Dyn. Syst. Appl. 12, 433–466 (2003)

    MathSciNet  MATH  Google Scholar 

  35. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  36. Deimling K.: Multivalued Differential Equations. De Gruyter, Berlin (1992)

    Book  MATH  Google Scholar 

  37. Hu S., Papageorgiou N.: Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht (1997)

    MATH  Google Scholar 

  38. Bohnenblust, H.F., Karlin, S.: On a theorem of Ville. In: Contributions to the Theory of Games, vol. 1, pp. 155–160. Princeton University Press, Princeton (1950)

  39. Lasota A., Opial Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 13, 781–786 (1965)

    MathSciNet  MATH  Google Scholar 

  40. Travis C.C., Webb G.F.: Existence, stability and compactness with α-norm for partial functional differential equations. Trans. Am. Math. Soc. 240, 129–143 (1978)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Hexi University, Zhangye, 734000, Gansu, People’s Republic of China

    Zuomao Yan & Xingxue Yan

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  1. Zuomao Yan
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  2. Xingxue Yan
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Yan, Z., Yan, X. Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces. Z. Angew. Math. Phys. 64, 573–590 (2013). https://doi.org/10.1007/s00033-012-0249-1

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  • DOI: https://doi.org/10.1007/s00033-012-0249-1

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