Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces

• Autores: Pengyu Chen, Yongxiang Li
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 1, 2015, págs. 63-76
• Idioma: inglés
• DOI: 10.1007/s13348-014-0106-y
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• Resumen

  • This paper is concerned with the existence of \alpha-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \mathbb {H} and the nonlocal term satisfies some local growth conditions in fractional power space \mathbb {H}_\alpha. The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

• Referencias bibliográficas
  • Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solutions for fractional differential equations with uncertainly. Nonlinear...
  • Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space....
  • Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal....
  • Byszewski, L.: Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems....
  • Cui, J., Yan, L., Wu, X.: Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces. J. Korean Stat. Soc....
  • Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A 44, 335201...
  • Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
  • Darwish, M.A., Ntouyas, S.K.: On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. Nonlinear...
  • EI-Borai, M.M.: On some stochastic fractional integro-differential equations. Advan. Dynam. Syst. Appl. 1, 49–57 (2006)
  • Eidelman, S.D., Kochubei, A.N.: Cauchy problem for fractional diffusion equations. J. Differ. Equat. 199, 155–211 (2004)
  • El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons Fractals 14, 433–440...
  • Ezzinbi, K., Fu, X., Hilal, K.: Existence and regularity in the -norm for some neutral partial differential equations with nonlocal conditions....
  • Grecksch, W., Tudor, C.: Stochastic Evolution Equations: a Hilbert space approach. Akademic Verlag, Berlin (1995)
  • Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Math, vol. 840. Springer, New York (1981)
  • Ichikawa, A.: Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90, 12–44 (1982)
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics...
  • Liang, J., Liu, J., Xiao, T.J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57, 183–189 (2004)
  • Liu, H., Chang, J.: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70, 3076–3083 (2009)
  • Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, Berlin (1983)
  • Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
  • Ren, Y., Sakthivel, R.: Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with...
  • Sakthivel, R., Revathi, P., Mahmudov, N.I.: Asymptotic stability of fractional stochastic neutral differential equations with infinite delays....
  • Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81,...
  • Sakthivel, R., Luo, J.: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl....
  • Sakthivel, R., Ren, Y.: Exponential stability of second-order stochastic evolution equations with Poisson jumps. Commun. Nonlinear Sci. Numer....
  • Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real. World Appl. 12, 263–272 (2011)
  • Wang, R.N., Xiao, T.J., Liang, J.: A note on the fractional Cauchy problems with nonlocal conditions. Appl. Math. Lette. 24, 1435–1442 (2011)
  • Yan, Z., Yan, X.: Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces. Z....
  • Yan, Z., Yan, X.: Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay....