摘要
不动点理论在数学及其应用中有重要作用.它的发展给解决诸如方程问题、物理问题带来很大的方便.不动点问题与优化问题、最大最小值问题、相补问题、变分不等式问题、均衡问题的研究也都有一些密切的联系.
本文在多种空间框架中研究几类压缩映射、扩张映射的不动点存在性,给出不动点的迭代算法,并探讨它们在微分方程和均衡问题中的应用.
本文主要分三个部分:
第一章介绍序言和一些预备知识.第二章我们分别在度量空间中讨论了一个推广的扩张映射不动点的存在性、利用压缩映射解决了微分方程问题中的带二点边界问题;在拓扑线性空间中应用已有的不动点定理解决了带上下界的均衡问题以及带限制条件的上下界均衡问题;在锥度量空间中讨论了一种推广的压缩映射不动点的存在性;在模糊度量空间中我们证明了不动点的存在性定理,且不动点可用PK迭代算法迭代求解.第三章我们首先针对渐进k严格伪压缩映射的不动点,研究其在不同迭代算法下迭代序列的弱收敛和强收敛性;接着利用修正的Ishikawa迭代算法,给出了关于不动点问题与均衡问题解集的弱收敛和强收敛定理.
本文在某种程度上改进和推广了有关学者的一些相应结果,并给出了有关结果的一些实例和应用.
The fixed point theory takes a great important space on math and its applications. The development of fixed point problem brought great convenience to solve the problems of the equation and physics. Fixed point problems have close relationship with optimization problems, maximum and minimum problems, complementarity problems, variational inequality problems and equilibrium problems.
In this paper,we studied the existence of the fixed point on some mappings, such as contraction mappings and expansive mappings in some spaces, got the iterative algorithm of the fixed point and studied the applications to differential equations and the equilibrium problems. This paper is divided into three parts:
In the first chapter, we showed the introductions and some preliminaries. In the second chapter, in metric space, we discussed the existence of fixed points with promoted expansive mapping and the contraction mapping application to differential equations with two ordinary point; in topological linear space, we used the fixed point existence theorem to solve the equilibrium problem with upper and lower bounded and the equilibrium problems with lower and upper bounds and restricted conditions; in cone metric spaces, we studied the existence of fixed points with a general contraction mapping; in fuzzy metric spaces, discussed the existence of the fixed points theorem and the fixed point can be found by PK algorithm. In the third chapter, at first, we studied the fixed point of asymptotically k-strict pseudo-contraction mapping, proved weak and strong convergence of its iterative sequence, then showed weak and strong convergence theorems of the intersection between the solution of the equilibrium problem and fixed point problem under modified Ishikawa algorithms.
In some way, the conclusions above have not only improved the conclusions some scholars have already got, but also we showed some examples and applications about relative conclusions.
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