【作者】 陈晓莉；

【导师】 朱传喜；

【作者基本信息】 南昌大学 ， 应用数学， 2007， 硕士

【摘要】 随机非线性算子理论是目前正在迅速发展的随机非线性泛函分析理论的重要组成部分，它与近代数学的许多分支有着紧密的联系，特别是在建立各类随机方程解的存在唯一性问题中起着重要的作用。本文主要用随机拓扑度和随机不动点指数方法以及半序和迭代方法研究Banach空间中随机非线性算子的随机不动点和随机算子方程．全文分为三章。第一章介绍了非线性算子随机不动点理论的历史背景、现状以及研究Banach空间中非线性算子的随机不动点需要的相关知识。第二章在实可分Banach空间中利用随机拓扑度和随机不动点指数方法，讨论了随机半闭1-集压缩算子、随机定点1-集压缩算子和随机算子方程A(ω，χ)=μχ(μ≥1)，在不同边界条件下随机不动点或随机解的存在情况，得到了几个新的结论，同时推广了一系列重要定理。第三章在由锥导出的半序空间中，利用半序和迭代方法，首先，研究了两点拉伸型随机混合单调算子、随机非混合单调算子随机不动点的存在性、唯一性以及迭代序列的收敛性。其次，讨论了随机非线性算子对的随机公共不动点问题。最后，在由泛函导出的半序空间中探讨随机单调及混合单调算子随机不动点的唯一性情况。更多还原

【Abstract】 Random nonlinear operator theory is becoming more and more important intergradient of random nonlinear functional analysis. It has much closed relation to the other mathematical branches. Especially, it plays an essential role in establishing the existence and uniqueness of the solutions of several kinds of random equations. In this thesis, random nonlinear operators and random operator equations in Banach Space are studied by applying the random topological degree, random fixed point index method, partial order and iterative methods. The material in this book is presented in three chapters.In chapter one, the backgrounds and current situation of random fixed-point theories of nonlinear operators are introduced, and the preliminaries in Banach spaces is given, which is needed to study the random fixed points of nonlinear operators.In chapter two, utilizing the method of random topological degree and random fixed-index, we discuss the existence of random fixed points and random solutions on random semi-closed 1-set-contractive operators, random constant 1-set-contractive operators and random operator equations such as A(ω, x)=μx(μ≥1) under different boundary conditions. We also obtain several new results and extend a serial of important theorems.In chapter three, by applying partial order and iterative methods, we first study the existence, uniqueness and convergence of iterative series of two point extend mixed random monotone operators and random fixed points of random nonmixed monotone operators in the partial order spaces induced by cones. Secondly, we discuss the problem on random common fixed points of random nonlinear operator pairs. Finally, we study the uniqueness of random fixed points of random monotone and that of mixed monotone operator in the partial order spaces induced by functions.更多还原