Mild Solutions of Neutral Semilinear Stochastic Functional Dynamic Systems with Local Non-Lipschitz Coefficients

Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics and engineering. In this paper, we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional dynamic systems under local non-Lipschitz conditions on the coefficients by means of the stopping time technique. We especially generalize and improve the results that appeared in Govinadan (2005), Bao and Hou (2010), and Jiang and Shen (2011). 1. Introduction Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics, biology engineering, and so forth [1]. Recently, for the case where the coefficients satisfy the global Lipschitz condition and the linear growth condition, many results are known [1–3]. However, the global Lipschitz condition, even the local Lipschitz condition, is seemed to be considerably strong when one discusses variable applications in real world. Reference [4] discussed the existence of mild solution to neutral semilinear stochastic functional systems with non-Lipschitz coefficients. Reference [5] discussed the existence and uniqueness of solutions to neutral stochastic functional systems with infinite delay under the local Lipschitz conditions in the Euclidean space. We focus on neutral semilinear stochastic functional dynamic systems for the case where the coefficients do not necessarily satisfy the global Lipschitz condition. Thus we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional systems with the condition, which was investigated by [6–8] as the Carathéodory-type conditions for strong solutions. Motivated by the above papers, in this paper, we will extend the existence and uniqueness of mild solutions of (2) under the non-Lipschitz conditions to the local non-Lipschitz conditions. The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3 we prove the existence and uniqueness of the mild solution. 2. Preliminaries Throughout this paper, let be a complete probability space with a normal filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Moreover, let be two real separable Hilbert spaces, and we denote by their inner products and by their vector norms, respectively. We denote that denotes the space of all bounded linear operators from into , equipped with the usual operator norm . In this paper, we always use the same symbol to denote norms of operators