Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay. (English) Zbl 1152.34388

Summary: We obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((\(-\infty ,0]; \mathbb R^d\)) which denotes the family of bounded continuous \(\mathbb R^d\)-value functions \(\varphi\) defined on (\(-\infty ,0\)] with norm \(\| \varphi \| = \sup _{-\infty <\theta \leqslant 0}|\varphi(\theta)|\) under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.


34K50 Stochastic functional-differential equations
34K07 Theoretical approximation of solutions to functional-differential equations
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[1] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta Math. Acad. Sci. Hungar., 7, 71-94 (1956) · Zbl 0070.08201
[2] Boukfaoui, Y. El.; Erraoui, M., Remarks on the existence and approximation for semilinear stochastic differential in Hilbert spaces, Stochastic Anal. Appl., 20, 495-518 (2002) · Zbl 1002.60058
[3] Govindan, T. E., Stability of mild solution of stochastic evolution equations with variable delay, Stochastic Anal. Appl., 21, 1059-1077 (2003) · Zbl 1036.60052
[4] Liu, K., Lyapunov functionals and asymptotic of stochastic delay evolution equations, Stochastics and Stochastic Rep., 63, 1-26 (1998) · Zbl 0947.93037
[5] Mao, X. R., Stochastic Differential Equations and Applications (1997), Horwood Publication: Horwood Publication Chichester · Zbl 0892.60057
[6] Taniguchi, T., Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96, 152-169 (1992) · Zbl 0744.34052
[7] Wei, F. Y.; Wang, K., The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331, 516-531 (2007) · Zbl 1121.60064
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