概率度量空间在分形图理论中的应用

给出广义概率度量空间上的随机压缩映射的新定义,统一了概率度量空间中的概率压缩,E-空间中的强压缩,随机度量空间中的几乎处处压缩和均匀压缩的定义.在广义概率度量空间上给出几个新的不动点定理,将概率度量空间中的一些熟知的不动点定理作为推论得到.利用这些不动点定理,得到分形图理论中随机迭代函数系统的遍历性定理.     

半序概率度量空间中压缩算子的不动点定理

半序方法是研究非线性算子方程问题的主要方法之一.在概率度量空间中引入半序,并且利用半序方法研究了非线性算子的不动点问题,推广了度量空间中序压缩算子的不动点定理,获得若干新的结果.     

概率逻辑学基本定理在多值命题逻辑系统中的推广

通过引入概率测度空间,在n值Lukasiewicz命题逻辑系统中提出了满足Kolmogorov公理的命题公式的概率;证明了概率逻辑学基本定理,并将概率逻辑学基本定理推广到了更一般的形式,改进了对推理结论的不可靠度上界的估计;将概率逻辑学的基本方法引入计量逻辑学,建立了更一般的逻辑度量空间;通过概率逻辑学基本定理,证明了逻辑度量空间中概率MP,HS规则,它是真度MP,HS规则的推广.     

概率度量空间中若干新的不动点定理*

本文提出了Z-M-PN空间的概念,在概率度量空间中我们得到了若干新的不动点定理。同时,一些着名的不动点定理在概率度量空间中得到了推广,诸如:Schauder不动点定理、郭大钧不动点定理和Petryshyn不动点定理被推广到M-PN空间;Altman不动点定理被推广到Z-M-PN空间。     

概率度量空间的基本理论及应用(Ⅱ)*

本文是作者文章[1]的继续.得出了概率度量空间的集合的各种概率有界性的表征.借助于这些结果及[1]中所得结果,讨论了概率线性赋范空间中的线性算子理论及概率度量空间映象的不动点定理.     

广义H-空间的理论及其应用*

本文给出了广义H-空间的完备性特征性质和紧性特征性质,同时也研究了这一空间的度量化定理.作为这些理论的应用.我们得到了Menger概率度量空间的完备性特征和紧性特征.给出了该空间的度量化函数的具体形式.     

关于概率度量空间中压缩型映象对的不动点定理

Menger 1942 年提出概率度量空间的概念,近年来,Sehgal,Bharucha-Reid,Istratescu,林等对概率度量空间中压缩型映象不动点定理进行了研究。本文对概率度量空间压缩型映象对给出了几个新的不动点定理,这些结果统一和发展了[2,3,4]中的某些主要结果。     

关于任意随机变量序列泛函的一类强偏差定理

通过概率空间上的任意随机变量的分布与独立分布的比较.研究任意随机变量序列泛函的强偏差定理,即小偏差定理.将已有的某些连续型及离散随机变量序列的强偏差定理加以推广.     

FC-度量空间中新的不动点定理及其对变分不等式和鞍点的应用(英文)

在非紧FC-度量空间中建立了一个新的不动点定理.作为应用,获得了一个极大元定理,新建了FC-度量空间中的变分不等式和鞍点定理.     

FC-度量空间中的R-KKM定理及其对变分不等式和不动点的应用

引入了FC-度量空间,建立了非紧FC-度量空间中的R-KKM定理.作为应用,研究了非紧FC-度量空间中的变分不等式的解集、相交点集、Ky Fan截口和极大元集的性质,获得了FC-度量空间中的Fan-Browder不动点定理.     

Extension of contractive maps in the Menger probabilistic metric space

In this article, the topological properties of the Menger probabilistic metric spaces and the mappings between these spaces are studied. In addition, contractive and k-contractive mappings are introduced. As an application, a new fixed point theorem in a chainable Menger probabilistic metric space is proved.     

Ekeland's variational principle and Caristi's fixed point theorem in probabilistic metric space

The main purpose of this paper is to establish the Ekeland's variational principle and Caristi's fixed point theorem in probabilistic metric spaces and to give a direct simple proof of the equivalence between these two theorems in the probabilistic metric space. The results presented in this paper generalize the corresponding results of [9–12].The project is supported by National Natural Science Foundation of China.     

Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces

The probabilistic version of the classical Banach Contraction Principle was proved in 1972 by Sehgal and Bharucha-Reid [V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on PM spaces. Math. Syst. Theory 6, 97–102]. Their fixed point theorem is further generalized by many authors. In the intervening years many others have proved the probabilistic versions of the other known metric fixed point theorems. However, the problem to prove the probabilistic versions of the very important generalization of the Banach Contraction Principle, obtained in 1969 by Boyd and Wong [D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969) 458–464], who proved the fixed point theorem for a self-mapping of a metric space, satisfying the very general nonlinear contractive condition, is unsolved in these days. Similarly, as in the metric space case, to prove a fixed point theorem for a mapping, satisfying the general probabilistic nonlinear contractive condition, it was necessary to find a new approach, substantially different from the previous technique for cases where a mapping satisfies the probabilistic linear contraction condition, introduced by Sehgal and Bharucha-Reid and further used by many authors. So, the problem to obtain a truthful probabilistic version of the Banach fixed point principle for general nonlinear contractions existed unsolved for over 35 years. We have solved this problem in this paper.     

A Cyclic Probabilistic C-Contraction Results using Hadzic and Lukasiewicz T-Norms in Menger Spaces

In this paper, we introduce generalized cyclic C-contractions through p num-ber of subsets of a probabilistic metric space and establish two fixed point results for such contractions. In our first theorem we use the Hadzic type t-norm. In our next theorem we use Lukasiewicz t-norm. Our results generalize the results of Choudhury and Bhandari [11]. A control function [3] has been utilized in our second theorem. The results are illustrated with some examples.     

概率区间空间中的新型KKM定理及应用*

本文建立了概率区间空间的概念,并在此框架下建立了一个新型的KKM定理。作为应用我们得到了概率区间空间中的一个新的极大极小定理和截口定理,匹配定理及一些重合点定理。所得结果均是全新的,它们不仅包含了Vom Neumann[7]中的主要结果,而且将[1][3～4][6],[8]中的相应结果推广到概率区间空间。     

Edelstein's fixed point theorem in topological spaces

An extension to topological spaces of a wellknown fixed point theorem of M. Edelstein for contractive mappings on metric spaces is presented. Results based on the generalized Edelstein's theorem are also established concerning the existence of fixed points of continuous selfmaps on a topological space. As a special case a compact starshaped subset of a linear topological space is considered. The results extend the fixed point theoremsfor nonexpansive mappings on a compact metric space of L.F.Guseman, Jr. and B.C. Peters, Jr.     

概率度量空间中的Ekeland变分原理与集值映象的Caristi重合定理

借助偏序方法,本文得到概率度量空间中之一推广形式的Ekeland变分原理及一集值形式的Caristi重合定理,同时证明了这两个定理之间的等价性.本文结果是[1,2,5,6,7,9]中相应结果的改进和推广.     

随机泛函分析中可交换映象的随机不动点定理

随机不动点定理在随机泛函分析中是一重要问题.在可分完备的度量空间中的随机不动点定理Bharucha-Reid,王梓坤,?pa?ek,Han?,Itoh及作者等都曾进行过讨论(见[1-5,15-20,21]).在本文中我们对概率分析中可交换映象的随机不动点定理得出了几个新的结果,它推广了前述诸人工作中某些重要结果.在确定性情形也推广了Jungck[6,7,8],Das,Naik[9],Rhoades[10],及Ciric[11]的结果.