几类空间中不动点定理的统一及其应用

Fixed point theorems are wildly applied to many branches of mathematics. Fixed point theory, including its application, has always been an active area of mathematical researches. This project will unify and generalize various fixed point theorems in several kinds of spaces, and give their applications to differential equations and integral equations. First, we will consider unified versions and generalized versions of the following several kinds of fixed point theorems: common fixed point theorems for nonlinear contraction of Ciric type, common fixed point theorems for four self-maps in b-metric spaces, fixed point theorems satisfying a (φ,ψ)-contraction in generalized metric space and common fixed point theorems in locally convex spaces. Then, we will introduce some notions, such as b-quasi-metric-like spaces and generalized b-metric spaces, and discuss topological properties of these spaces. In particular, by using these new spaces and their properties, we will establish several general fixed point theorems in the framework of these spaces. These new theorems will unify and improve the related known results. Moreover, we will apply these new fixed point theorems to solving problems of some differential equations and integral equations, and obtain some new sufficient conditions of existence and uniqueness of solutions, which are weaker than ones in literature.

不动点定理在数学的许多分支具有广泛的应用。不动点理论及其应用一直是数学研究中一个活跃的领域。本项目拟研究几类空间中各种不动点定理的统一和推广，并给出它们在微分方程和积分方程中的应用。首先，我们拟考虑以下几种不动点定理的统一和推广：非线性Ciric型公共不动点定理，b-距离空间中含有四个自映射的公共不动点定理，广义距离空间中的(φ,ψ)压缩不动点定理，局部凸空间中的公共不动点定理。然后，我们拟引入一些概念，例如类b-拟距离空间和广义b–距离空间等，并讨论这些空间的拓扑性质。特别地，利用这些新空间及其性质，我们将在新空间的框架下，建立几个一般形式的不动点定理。这些新定理将统一和改进已有的相关结果。更进一步，我们拟应用这些新的不动点定理来解某些微分方程和积分方程问题，并获得这些问题解的存在性和唯一性的充分条件，它们弱于已有文献中的条件。

不动点定理和Ekeland变分原理是非线性分析中非常重要的结果。由于它们的广泛应用，其各种形式的推广不断出现。按照研究目标，我们主要讨论Ciric型不动点定理及相关的Ekeland变分原理。在b-距离空间中，我们给出循环Ciric型不动点定理成立的条件，解决了推广的Banach压缩猜想，给出Ciric型不动点定理的压缩系数由[0,1/s)扩大为[0,1)的条件。在距离空间中，建立了Ciric-Pata型不动点定理，这一结果统一了Ciric型和Pata型不动点定理。在类拟b-距离空间中，建立了一类Ciric型不动点定理，这一结果统一了Fan的三个结果。特别地，我们解决了不动点领域专家提出的几个公开问题。我们还建立了一种改进的准序原理，它不但蕴涵了我们已知的绝大多数仅涉及通常距离的集值型变分原理，还可导出扰动项包含集合和各类广义距离的 Ekeland 变分原理。我们进一步研究了 Ekeland 变分原理在均衡问题、向量优化、行为科学和行为动力系统的应用。