The Studies For Some Problems In Neutral Stochastic Differential Equations

This Ph. D. thesis is composed of four Chapters. The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness theorem for the solution of neutral stochastic functional differential equations with infinite delay are mainly discussed; The exponential stability and asymptotic stability in mean square for neutral stochastic neural networks with time-varying delays are investigated, respectively; And the existence and uniqueness, exponential stability and asymptotic stability in p (p≥2)-moment for mild solution of neutral stochastic partial differential equations with delays are also studied.In Chapter 1 The history background and developments about problems above, some main works and some preliminary knowledges in this context are briefly introduced.In Chapter 2 The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay are mainly studied under the space of bounded and continuous functions. Firstly, the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay under the uniformly Lipschitz condition, linear grown condition and contractive condition can be directly derived; And the moment estimate of the solution and the estimate for error between the approximate solution and the accurate solution can be both given; If the uniformly Lipschitz condition is replaced by the local Lipschitz condition, the existence and uniqueness theorem can be gained; Meanwhile, the existence and uniqueness of the global solution in the interval [0, +∞) can also be obtained; Secondly, L~p-exponential estimate of the solution for neutral stochastic functional differential equations with infinite delay can be studied; At length, the theorem of the local solution about neutral stochastic functional differential equations with infinite delay only under the local Lipschitz condition and the contractive condition can be established.In Chapter 3 Firstly, the exponential stability in mean square and almost sure exponential stability for neutral stochstic neural networks with time-varying delay can be discussed by using the linear matrix inequality (LMI) and the semimartingale convergence theorem; And our sufficient conditions are less conserative; Secondly, by utilizing the fixed point theorem, some sufficient and necessary conditions ensuring the asymp- totic stability in mean square for neutral stochastic neural networks with time-varying delays can be given.In Chapter 4 In first quarter, the existence and uniqueness of mild solution for stochastic partial differential equations with delays under the non-Lipschitz condition are mainly discussed; In second quarter, by establishing a Lemma, some sufficient conditions ensuring p (p≥2)-moment exponential stability and almost surely exponential stability for mild solution to neutral stochastic partial differential equations with delays can be given; Specially, when the neutral item is removed, some sufficient conditions about p-moment exponential stability and almost sure exponential stability for mild solution of stochastic partial differential equations with delays are weaker than some existing literatures by establishing a Lemma, too; Thus, we improve some results; In third quarter, similarly, by constructing a Lemma, some sufficient conditions about asymptotic stability in p-moment of mild solution for neutral stochastic partial differential equations with delays can be given; If the neutral item is dropped out, some sufficient conditions about asymptotic stability in p-moment of mild solution for stochastic partial differential equations with delays can be derived; These two results are new.