弱拓扑下的非线性随机积分和微分方程组的解

Solutions for a System of Nonlinear Random Integral and Differential Equations under Weak Topology

在本文中，我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理．利用这一定理，我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则．作为应用，我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理．也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果．我们的定理改进和推广了Szep，Mitchell－Smith，Cramer－Lakshmikantham，Lakshmikantham—Leela和丁的相应结果．

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. Then, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach s paces are given, As applications, some existence theorems of weak random aolu,tions for the random rauchy problem of a system of nonlinear random differential equations are obtained,as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, MitchellSmith , Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.