【摘要】：本文在Menger PM-空間中提出了許多新的概念.在此基礎上,利用半序方法、迭代方法研究了Menger PM-空間中非線性映射不動點的存在性和唯一性問題,獲得許多新的結(jié)果.同時,我們給出了許多例子對主要結(jié)果進行了說明.本文的結(jié)果推廣和改進了近期相關文獻中的不動點定理和重合點定理.全文共分為3章.每章的內(nèi)容如下:第1章介紹非線性泛函分析中Menger PM-空間的發(fā)展狀況、最新前沿、所要研究的相關問題,以及給出Menger PM-空間中一些基本概念.第2章在完備的序Menger PM-空間中引入一廣義循環(huán)-弱壓縮映射的概念,并獲得了該映射的一些重合點定理.第3章在廣義Menger PM-空間中定義了(H,Ψ,Φ)-壓縮映射和概率(α,)-壓縮映射,獲得了這兩類映射的一些不動點定理,所得結(jié)果推廣和改進了許多最近相關文獻中的結(jié)論.給出了一些具體例子.同時,作為主要結(jié)果的應用,研究了一類積分方程解的存在性問題.
[Abstract]:In this paper, many new concepts are proposed in Menger PM- spaces. On this basis, the existence and uniqueness of fixed points of nonlinear mappings in Menger PM- spaces are studied by means of the semi-ordered method and iterative methods, and many new results are obtained. At the same time, we give many examples to illustrate the main results. The results of this paper extend and improve the fixed point theorem and coincidence point theorem in recent literature. The full text is divided into three chapters. The contents of each chapter are as follows: chapter 1 introduces the development of Menger PM- space in nonlinear functional analysis, the latest frontier, the related problems to be studied, and some basic concepts in Menger PM- space. In chapter 2, we introduce a concept of generalized circulation-weakly contractive mapping in a complete ordered Menger PM- space, and obtain some coincidence point theorems of the mapping. In chapter 3, we define (H, 蠄, 桅) -contractive mappings and probabilistic (偽,) -contractive mappings in generalized Menger PM- spaces, and obtain some fixed point theorems for these mappings. The results extend and improve the conclusions in many recent literatures. Some concrete examples are given. At the same time, the existence of solutions for a class of integral equations is studied.
【學位授予單位】：南昌大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O177.91

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本文編號：2351440