| The fixed point theory are widely used in many areas, such as nonlinear differential equation,partial differential equations, economic equilibrium theory and game theory. But existence conditions of the fixed point are very strong. For example, when the domain of mapping is not tight or the mapping is discontinuous, the fixed point of mapping may not exist. The actual applications, most of the time, we just need to get approximate fixed point. However, existence conditions of the approximate fixed point are weak. Thus, the research for this topic has important theoretical value and practical significance.We study the existence problems of approximate fixed points of three kinds of non-linear mappings. It mainly includes the approximate coupled fixed point of non-expansive mapping, approximate fixed point of monotone operator and approximate fixed point of set-valued mappings.This paper is divided into four chapters.In chapter1, We briefly summarizes overview of the research significance of the approximate fixed point, and introduce the structure of this article.In chapter2, In metric space, the approximate coupled fixed point theorems of con-traction mapping are proved, thereafter, under the condition of the domain of mapping is a bounded set, the approximation coupled fixed point theorems of the non-expansive map-ping are discussed in the normed spaced, we mainly discuss two kinds of non-expansive mappings, which are average non-expansive mapping and Kannan non-expansive map-ping.In chapter3, In the partially ordered metric spaces, firstly the approximate fixed point theorems of increasing operator are discussed; Second the existence problem of the approximation coupled fixed point of mixed monotone operator are discussed under the condition of some generalized contractive, and the approximate coupled fixed point theorems of mixed monotone operator are obtained under partially ordered interval.In chapter4, We firstly discuss existence problem of the approximate fixed point of set-valued mappings, and prove the approximate fixed point theorems of the generalized compression condition. Secondly under the condition of the domain of mapping is a bounded convex set,Proved the approximate fixed point theorem of ε-semi-continuous set-valued mappings. |
