| Title | ASYMPTOTIC STABILITY OF MILD SOLUTIONS OF STOCHASTIC EVOLUTION EQUATIONS |
| Publication Type | Journal Article |
| Year of Publication | 2006 |
| Authors | Govindan, TE |
| Secondary Title | Communications in Applied Analysis |
| Volume | 10 |
| Issue | 3 |
| Start Page | 345 |
| Pagination | 360 |
| Date Published | 08/2006 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 34K50, 60H20, 93E15 |
| Abstract | In this paper, we study the existence and stability problems associated with stochastic evolution equations in Hilbert spaces. To be precise, we first consider an existence result for a mild solution and then the exponential stability of the moments of such solutions and also of its sample paths. Such results are established employing the theory of a stochastic convolution integral and a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms. Moreover, we consider asymptotic stability in probability of the sample paths of the solution process. The results obtained here generalize the corresponding main results from Taniguchi [18] and also the classical results of Ichikawa [7, 8] all established under the Lipschitz hypothesis. |
| URL | http://www.acadsol.eu/en/articles/10/3/6.pdf |
| Short Title | Asymptotic Stability of Mild Solutions |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.
[2] L.I. Galcuk and M.H.A. Davis, A note on a comparison theorem for equations with different diffusions, Stochastics, 6 (1982), 147-149. [3] I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations, SpringerVerlag, 1972. [4] T.E. Govindan, Existence and stability of solutions of stochastic semilinear evolution equations, Comm. Applied Anal., 7 (2003), 101-114. [5] R.Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980. [6] M. He, Global existence and stability of solutions for reaction diffusion functional differential equations, J. Math. Anal. Appl., 199 (1996), 842-858. [7] A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90 (1982), 12-44. [8] A. Ichikawa, Semilinear stochastic evolution equations: boundedness, stability and invariant measures, Stochastics, 12 (1984), 1–39. [9] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633. [10] V. Laksmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vols. I & II, Academic Press, NY, 1969. [11] G. Leha, B. Maslowski, and G. Ritter, Stability of solutions to semilinear stochastic evolution equations, Stochastic Anal. Appl., 17 (1999), 1009-1051. [12] K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, to appear.
[13] R. Liu and V. Mandrekar, Stochastic semilinear evolution equations: Lyapunov function, stability and ultimate boundedness, J. Math. Anal. Appl., 212 (1997), 537–553. [14] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, NY, 1994. [15] A.E. Rodkina, On existence and uniqueness of solution of stochastic differential equations with heredity, Stochastics, 12 (1984), 187–200. [16] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Eqns., 96 (1992), 152–169. [17] T. Taniguchi, On the estimate of solutions of perturbed linear differential equations, J. Math. Anal. Appl., 153 (1990), 288-300. [18] T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stoch. Stoch. Reports, 53 (1995), 41-52. [19] Y. Ting, Comparison method of stability analysis of semilinear time-delay stochastic evolution equation, Stochastic Anal. Appl., 22 (2004), 403-427. [20] T. Yamada, On comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ., 13 (1973), 497-521. [21] T. Yamada, On the successive approximations of solutions of stochastic differential equations, J. Math. Kyoto Univ., 21 (1981), 501-515. |