Open Access
2010 Existence of Square-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations with Infinite Delay
Paul H. Bezandry, Toka Diagana
Commun. Math. Anal. 8(2): 103-124 (2010).
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.

References

1.

P. Acquistapace and B. Terreni, A Unified Approach to Abstract Linear Parabolic Equations, Tend. Sem. Mat. Univ. Padova 78 (1987) 47-107.  MR934508 P. Acquistapace and B. Terreni, A Unified Approach to Abstract Linear Parabolic Equations, Tend. Sem. Mat. Univ. Padova 78 (1987) 47-107.  MR934508

2.

P. Bezandry and T. Diagana, Existence of Almost Periodic Solutions to Some Stochastic Differential Equations. Applicable Analysis. 86 (2007), no. 7, pages 819 - 827.  MR2355540 10.1080/00036810701397788 P. Bezandry and T. Diagana, Existence of Almost Periodic Solutions to Some Stochastic Differential Equations. Applicable Analysis. 86 (2007), no. 7, pages 819 - 827.  MR2355540 10.1080/00036810701397788

3.

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.  MR2349945 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.  MR2349945

4.

C. Corduneanu, Almost Periodic Functions, 2nd Edition. Chelsea-New York, 1989. C. Corduneanu, Almost Periodic Functions, 2nd Edition. Chelsea-New York, 1989.

5.

G. Da Prato and C. Tudor, Periodic and Almost Periodic Solutions for Semilinear Stochastic Evolution Equations, Stoch. Anal. Appl. 13(1) (1995), 13–33.  MR1313204 10.1080/07362999508809380 G. Da Prato and C. Tudor, Periodic and Almost Periodic Solutions for Semilinear Stochastic Evolution Equations, Stoch. Anal. Appl. 13(1) (1995), 13–33.  MR1313204 10.1080/07362999508809380

6.

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.  MR2476888 10.1016/j.jmaa.2008.09.041 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.  MR2476888 10.1016/j.jmaa.2008.09.041

7.

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.  MR2276784 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.  MR2276784

8.

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.  MR1004452 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.  MR1004452

9.

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.  MR1621730 10.1006/jmaa.1997.5875 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.  MR1621730 10.1006/jmaa.1997.5875

10.

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.  MR1621738 10.1006/jmaa.1997.5899 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.  MR1621738 10.1006/jmaa.1997.5899

11.

E. Hernández, Existence Results for Partial Neutral Integrodifferential Equations with Unbounded Delay. J. Math. Anal. Appl 292 (2004), no. 1, pp. 194–210.  MR2050224 10.1016/j.jmaa.2003.11.052 E. Hernández, Existence Results for Partial Neutral Integrodifferential Equations with Unbounded Delay. J. Math. Anal. Appl 292 (2004), no. 1, pp. 194–210.  MR2050224 10.1016/j.jmaa.2003.11.052

12.

Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, 1991.  MR1122588 Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, 1991.  MR1122588

13.

A. Ichikawa, Stability of Semilinear Stochastic Evolution Equations. J. Math. Anal. Appl. 90 (1982), no.1, 12-44.  MR680861 10.1016/0022-247X(82)90041-5 A. Ichikawa, Stability of Semilinear Stochastic Evolution Equations. J. Math. Anal. Appl. 90 (1982), no.1, 12-44.  MR680861 10.1016/0022-247X(82)90041-5

14.

D. Kannan and A.T. Bharucha-Reid, On a Stochastic Integro-differential Evolution of Volterra Type. J. Integral Equations 10 (1985), 351-379.  MR831250 D. Kannan and A.T. Bharucha-Reid, On a Stochastic Integro-differential Evolution of Volterra Type. J. Integral Equations 10 (1985), 351-379.  MR831250

15.

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.  MR659491 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.  MR659491

16.

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.  MR1989580 10.1155/S1048953303000108 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.  MR1989580 10.1155/S1048953303000108

17.

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.  MR1776686 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.  MR1776686

18.

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.  MR2073526 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.  MR2073526

19.

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.  MR2045755 10.1016/S0893-9659(04)90092-4 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.  MR2045755 10.1016/S0893-9659(04)90092-4

20.

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE Vol. 16, Birkhäauser Verlag, Basel, 1995.  MR3012216 A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE Vol. 16, Birkhäauser Verlag, Basel, 1995.  MR3012216

21.

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.  MR2073753 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.  MR2073753

22.

C. Tudor, Almost Periodic Solutions of Affine Stochastic Evolutions Equations, Stochastics and Stochastics Reports 38 (1992), 251-266.  MR1274905 10.1080/17442509208833758 C. Tudor, Almost Periodic Solutions of Affine Stochastic Evolutions Equations, Stochastics and Stochastics Reports 38 (1992), 251-266.  MR1274905 10.1080/17442509208833758

23.

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.  MR2671928 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.  MR2671928
Copyright © 2010 Mathematical Research Publishers
Paul H. Bezandry and Toka Diagana "Existence of Square-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations with Infinite Delay," Communications in Mathematical Analysis 8(2), 103-124, (2010). https://doi.org/
Published: 2010
Vol.8 • No. 2 • 2010
Back to Top