# AN ESTIMATE OF THE SOLUTIONS FOR STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

• Kim, Young-Ho (Department of Mathematics, Changwon National University)
• Accepted : 2011.03.18
• Published : 2011.09.30

#### Abstract

In this paper, we give an estimate on the difference between $x^n(t)$ and x(t) and it clearly shows that one can use the Picard iteration procedure to the approximate solutions to stochastic functional differential equations with infinite delay at phase space BC(($-{\infty}$, 0] : $R^d$) which denotes the family of bounded continuous $R^d$-valued functions ${\varphi}$ defined on ($-{\infty}$, 0] with norm ${\parallel}{\varphi}{\parallel}={\sup}_{-{\infty}<{\theta}{\leq}0}{\mid}{\varphi}({\theta}){\mid}$ under non-Lipschitz condition being considered as a special case and a weakened linear growth condition.

#### Acknowledgement

Supported by : Changwon National University

#### References

1. R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10(1943), 643-647.
2. I. Bihari,A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 71-94. https://doi.org/10.1007/BF02022965
3. Y. El. Boukfaoui, M. Erraoui, Remarks on the existence and approximation for semilinear stochastic differential in Hilbert spaces, Stochastic Anal. Appl. 20 (2002), 495-518. https://doi.org/10.1081/SAP-120004113
4. T.E. Govindan, Stability of mild solution of stochastic evolution equations with variable delay, Stochastic Anal. Appl. 21 (2003), 1059-1077.
5. N. Halidias, Remarks and corrections on "An esistence theorem for stochastic functional differential equations with dealys under weak assumptions, Statistics and Probability Letters 78, 2008" by N. Halidias and Y. Ren, Stochastics and Probability Letters 79 (2009), 2220- 2222. https://doi.org/10.1016/j.spl.2009.07.021
6. D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific Publishing Co. 2006.
7. K. Liu, Lyapunov functionals and asymptotic of stochastic delay evolution equations, Stochastics and Stochastic Rep. 63 (1998) 1-26.
8. X. Mao, Stochastic Differential Equations and Applications, Horwood Publication Chich- ester, UK, 2007.
9. Y. Ren, S. Lu and N. Xia, Remarks on the existence and uniqueness of the solution to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math. 220 (2008), 364-372. https://doi.org/10.1016/j.cam.2007.08.022
10. T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992), 152-169. https://doi.org/10.1016/0022-0396(92)90148-G
11. F.Wei and K.Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl. 331 (2007), 516-531. https://doi.org/10.1016/j.jmaa.2006.09.020