Existence And Uniqueness Of The Solution To Neutral Stochastic Functional Differential Equations With Infinite Delay

Neutral stochastic functional diferential equations with infinite delay have the follow-ing form:d[X（t） D（X_t）]=f（t, X_t）dt+g（t, X_t）dB（t）, X_t0=ξ, t∈[t₀, T],（0.0.2）where X_t={X（t+θ）,∞<θ≤0}.For the equation, in this dissertation, the author mainly discusses the existence anduniqueness of its solution. The work of this paper is as follows:Chapter1. First introducing the background of neutral stochastic functional difer-ential equations, abstract space and the Z-algorithm, the present situation of the researchand what we have do in this paper. Then giving some useful notations and lemmas.Chapter2. Hale and Kato [13][14] introduced an abstract space B(（∞,0];Rd）, onwhich Xu and Hu [11] obtained the existence and uniqueness of the solution to equation（0.0.2） under the Lipschitz condition. In this chapter, under the non-Lipschitz condition,the author proves the existence and uniqueness of the solution to equation （0.0.2） on theabstract space B(（∞,0];Rd） using the classic Picard iteration.Chapter3. Primarily, Zuber [18][19] posed an analytic iterative method–the Z-algorithm for the Cauchy problem of the ordinary diferential equations. In this chapter,the author uses the Z-algorithm to verify the existence of the solution to equation （0.0.2）on the space BC(（∞,0];Rd）.Chapter4. Basing on the chapter2and chapter3, the author, using the Z-algorithm,obtains the existence of the solution to equation （0.0.2） on the abstract space B(（∞,0];Rd）.