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Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. (English) Zbl 1167.34389

Summary: We obtain the existence and uniqueness of solutions to neutral 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. Furthermore, we give the continuous dependence of solutions on the initial value by means of the Corollary of Bihari inequality.

MSC:

34K50 Stochastic functional-differential equations
34K40 Neutral functional-differential equations
34K07 Theoretical approximation of solutions to functional-differential equations
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