拟类b-度量空间中非线性算子不动点问题及其应用研究

Fixed point theory is an important tool of analyzing and solving nonlinear problems. With the emergence of various nonlinear problems in applied science, the classical fixed point theory is unable to meet the needs of people to solve practical problems, and further development of fixed point theory has become an important problem to be solved. In this project, we investigate the problem of fixed points for nonlinear operators in quasi b-metric-like spaces which is generalized metric spaces with broad conditions and its applications. Firstly, on the basis of previous work, the basic properties of quasi b-metric-like spaces are studied. Secondly, classic fixed point theorems in metric spaces are extended to quasi b-metric-like spaces, and obtain some new fixed point theorems which are identified with some method.Thirdly, some new fixed point theorems are obtained by researching on a series of new types of compression operators and expansion operators and operators satisfying some conditions. Finally, as applications, the existence and uniqueness of solutions and stability problem of solution for nonlinear operator equation,differential equation and integral equation are studied. At the same time, we use the results obtained in this project to get some new decision-making methods which can provide theoretical guidance for practical decision-making activities. The research of this project contributes important academic value and scientific significance to the enrichment and development of the theory of spaces and nonlinear functional analysis in both its theoretical and applicable aspects.

不动点理论是分析及解决非线性问题的重要工具。随着应用学科中各种非线性问题大量涌现，经典的不动点理论已远不能满足人们解决实际问题的需要，进一步发展不动点理论成为亟待解决的重要问题。本项目拟在条件更宽泛的度量空间—拟类b-度量空间中研究非线性算子的不动点及其应用问题。首先，在前期工作基础上继续对拟类b-度量空间的基本性质进行研究。其次，把度量空间中经典的不动点定理加以鉴别地推广到拟类b-度量空间，得到真正推广的新型不动点定理。再次，在拟类b-度量空间中对一系列新型压缩算子和扩张算子以及满足一定条件的算子进行研究，得到创新性的结果。最后，作为应用，研究拟类b-度量空间中非线性算子方程、微分方程和积分方程解的存在唯一性及稳定性问题，同时利用所得结果探讨决策新方法，为实际决策活动提供理论指导。本项目的研究对丰富和发展空间理论和非线性泛函分析理论及指导实践活动具有十分重要的理论价值和科学意义。

四年以来，本项目在泛函分析和模糊决策理论方向开展了诸多研究，取得了一些重要研究成果，主要包括：首先，在类b-度量空间通过把函数F 满足的条件改为F连续，提出了（A）型F-g弱压缩和（B）型F-g弱压缩的概念，并在类b-度量空间中讨论了映射对的公共不动点问题，得到了一些新的不动点定理。在提出拓展的Geraghty压缩-(T, g)F压缩的概念的基础上，得到了算子对存在公共不动点的条件；其次，在Gb度量空间、偏度量空间、omega完备模空间等广义度量空间中，基于一些条件的情况下，利用迭代等方法得到了一系列非线性算子的不动点定理；再次，利用变量变换方法、Moser 迭代法、山路定理研究了几类非线性微分方程的解的问题，得到了不同方程解的存在性条件，分析了这些方程存在正解、基态解、多解需要满足的条件，研究了一些方程的解的性质；最后，通过综合运用模糊集理论、粗糙集理论，在犹豫模糊集、直觉模糊集等广义模糊集上研究了距离测度、相关系数等信息度量问题；同时提出了概率中智立方集、概率对偶犹豫模糊集等广义模糊集，并在这些广义模糊集的基础上提出了一些新的模糊多属性决策方法。以上结果极大地丰富了泛函分析理论和决策分析理论。