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Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions

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Abstract

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay, in which the initial value belongs to the abstract space ℬ. We establish the existence and uniqueness of mild solutions for SNSEEIs under global and local Carathéodory conditions by means of the successive approximation. An application to the stochastic nonlinear wave equations with infinite delay is given to illustrate the theory.

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References

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

    Book  MATH  Google Scholar 

  2. Kolmanovskii, V.B., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic, Norwell (1992)

    Google Scholar 

  3. Liu, K.: Stability of Infinite Dimensional Stochastic Differential Equations with Applications. Chapman & Hall, London (2006)

    MATH  Google Scholar 

  4. Mahmudov, N.I.: Existence and uniqueness results for neutral SDEs in Hilbert spaces. Stoch. Anal. Appl. 24, 79–95 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)

    MATH  Google Scholar 

  6. Caraballo, T.: Asymptotic exponential stability of stochastic partial differential equations with delay. Stochastics 33, 27–47 (1990)

    MATH  MathSciNet  Google Scholar 

  7. Caraballo, T., Liu, K.: Exponential stability of mild solutions of stochastic partial differential equations with delays. Stoch. Anal. Appl. 17, 743–763 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Govindan, T.E.: Exponential stability in mean-square of parabolic quasilinear stochastic delay evolution equations. Stoch. Anal. Appl. 17, 443–461 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ren, Y., Chen, L.: A note on the neutral stochastic functional differential equations with infinite delay and Poisson jumps in an abstract space. J. Math. Phys. 50, 082704 (2009)

    Article  MathSciNet  Google Scholar 

  10. Ren, Y., Xia, N.: Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Appl. Math. Comput. 210, 72–79 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ren, Y., Xia, N.: A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay. Appl. Math. Comput. 214, 457–461 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hino, Y., Murakami, S., Naito, T.: Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)

    MATH  Google Scholar 

  13. Hale, J.K., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkc. Ekvacioj. 21, 11–41 (1978)

    MATH  MathSciNet  Google Scholar 

  14. Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North Holland Mathematics Studies, vol. 108. Elsevier Science, North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  15. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hung. 32, 76–96 (1978)

    Article  MathSciNet  Google Scholar 

  16. Travis, C.C., Webb, G.F.: Second order differential equations in Banach space. In: Proceedings International Symposium on Nonlinear Equations in Abstract Spaces, pp. 331–361. Academic Press, New York (1987)

    Google Scholar 

  17. McKibben, M.A.: Second-order damped functional stochastic evolution equations in Hilbert space. Dyn. Syst. Appl. 12, 467–487 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Mahmudov, N.I., McKibben, M.A.: Abstract second-order damped McKean-Vlasov stochastic evolution equations. Stoch. Anal. Appl. 24, 303–328 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. McKibben, M.A.: Second-order neutral stochastic evolution equations with heredity. J. Appl. Math. Stoch. Anal. 2004, 177–192 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Balasubramaniam, P., Muthukumar, P.: Approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. J. Optim. Theory Appl. 70, 1023–1039 (2009)

    MathSciNet  Google Scholar 

  21. Taniguchi, T.: Successive approximations to solutions of stochastic differential equations. J. Differ. Equ. 96, 152–169 (1992)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Anhui Normal University, Wuhu, 241000, P.R. China

    Y. Ren & D. D. Sun

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  1. Y. Ren
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  2. D. D. Sun
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Correspondence to Y. Ren.

Additional information

Y. Ren supported by the National Natural Science Foundation of China (Project 10901003) and the Great Research Project of Natural Science Foundation of Anhui Provincial Universities (Project KJ2010ZD02).

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Ren, Y., Sun, D.D. Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions. J Optim Theory Appl 147, 569–582 (2010). https://doi.org/10.1007/s10957-010-9727-9

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  • DOI: https://doi.org/10.1007/s10957-010-9727-9

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