Communications in Mathematical Analysis

Existence of Square-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations with Infinite Delay

Paul H. Bezandry and Toka Diagana

Full-text: Open access

Abstract

In this paper, we make extensive use of the well-known Krasnoselskii fixed point theorem to obtain the existence of square-mean almost periodic solutions to some classes of hyperbolic stochastic evolution equations with infinite delay. Next, the existence of square-mean almost periodic solutions to not only the heat equation but also to a boundary value problem with infinite delay arising in control systems are studied.

Article information

Source
Commun. Math. Anal. Volume 8, Number 2 (2010), 103-124.

Dates
First available in Project Euclid: 21 April 2010

Permanent link to this document
http://projecteuclid.org/euclid.cma/1271890671

Mathematical Reviews number (MathSciNet)
MR2576914

Zentralblatt MATH identifier
05707213

Subjects
Primary: 34K50: Stochastic functional-differential equations [See also , 60Hxx] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 34B05: Linear boundary value problems 34C27: Almost and pseudo-almost periodic solutions 42A75: Classical almost periodic functions, mean periodic functions [See also 43A60] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 35L90: Abstract hyperbolic equations

Keywords
Stochastic differential equation stochastic processes square-mean almost periodicity sectorial operator hyperbolic semigroup infinite delay Brownian motion

Citation

Bezandry, Paul H.; Diagana, Toka. Existence of Square-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations with Infinite Delay. Commun. Math. Anal. 8 (2010), no. 2, 103--124. http://projecteuclid.org/euclid.cma/1271890671.


Export citation

References

  • P. Acquistapace and B. Terreni, A Unified Approach to Abstract Linear Parabolic Equations, Tend. Sem. Mat. Univ. Padova 78 (1987) 47-107.
  • P. Bezandry and T. Diagana, Existence of Almost Periodic Solutions to Some Stochastic Differential Equations. Applicable Analysis. 86 (2007), no. 7, pages 819 - 827.
  • P. Bezandry and T. Diagana, Square-mean almost periodic solutions nonautonomous stochastic differential equations. Electron. J. Diff. Eqns. Vol. 2007(2007), No. 117, pp. 1-10.
  • C. Corduneanu, Almost Periodic Functions, 2nd Edition. Chelsea-New York, 1989.
  • G. Da Prato and C. Tudor, Periodic and Almost Periodic Solutions for Semilinear Stochastic Evolution Equations, Stoch. Anal. Appl. 13(1) (1995), 13–33.
  • T. Diagana, Existence of Weighted Pseudo Almost Periodic Solutions to Some Classes of Hyperbolic Evolution Equations, J. Math. Anal. Appl. 350(2009), No. 1, pp. 18-28.
  • T. Diagana, Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. E. J. Qualitative Theory of Diff. Equ. No. 3. (2007), pp. 1-12.
  • A. Ya. Dorogovtsev and O. A. Ortega, On the Existence of Periodic Solutions of a Stochastic Equation in a Hilbert Space. Visnik Kiiv. Univ. Ser. Mat. Mekh. No. 30 (1988), 21-30, 115.
  • E. Hernández and H. R. Henríquez, Existence results for partial neutral functional differential equations with unbounded delay. J. Math. Anal. Appl 221 (1998), no. 2, pp. 452–475.
  • E. Hernández and H. R. Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. J. Math. Anal. Appl 221 (1998), no. 2, pp. 499–522.
  • E. Hernández, Existence Results for Partial Neutral Integrodifferential Equations with Unbounded Delay. J. Math. Anal. Appl 292 (2004), no. 1, pp. 194–210.
  • Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, 1991.
  • A. Ichikawa, Stability of Semilinear Stochastic Evolution Equations. J. Math. Anal. Appl. 90 (1982), no.1, 12-44.
  • D. Kannan and A.T. Bharucha-Reid, On a Stochastic Integro-differential Evolution of Volterra Type. J. Integral Equations 10 (1985), 351-379.
  • T. Kawata, Almost Periodic Weakly Stationary Processes. Statistics and probability: essays in honor of C. R. Rao, pp. 383–396, North-Holland, Amsterdam-New York, 1982.
  • D. Keck and M. McKibben, Functional Integro-differential Stochastic Evolution Equations in Hilbert Space. J. Appl. Math. Stochastic Analy. 16, no.2 (2003), 141-161.
  • J. Liang and T. J. Xiao, The Cauchy problem for nonlinear abstract functional differential equations with infinite delay, Comput. Math. Appl.40(2000), nos. 6-7, pp. 693-703.
  • J. Liang and T. J. Xiao, Solvability of the Cauchy problem for infinite delay equations. Nonlinear Anal. 58 (2004), nos. 3-4, pp. 271-297.
  • J. Liang, T. J. Xiao and J. van Casteren, A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Appl. Math. Lett. 17(2004), no. 4, pp. 473-477.
  • A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE Vol. 16, Birkhäauser Verlag, Basel, 1995.
  • L. Maniar and R. Schnaubelt, Almost Periodicity of Inhomogeneous Parabolic Evolution Equations, Lecture Notes in Pure and Appl. Math. Vol. 234, Dekker, New York, 2003, pp. 299-318.
  • C. Tudor, Almost Periodic Solutions of Affine Stochastic Evolutions Equations, Stochastics and Stochastics Reports 38 (1992), 251-266.
  • T. J. Xiao and J. Liang, Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces, Nonlinear Anal. 71 (2009), pp. 1442-1447.