Related Properties And Fixed Point Theory In Cone Metric Space

The fixed point theory in metric space is an important part of nonlinear function analysis. Since Longguang Huang and Xian Zhang recently introduced the concept of cone metric space, which replaces real number by ordered Banach space. The fixed point theorems in cone metric spaces become the topic that some mathematical researchers study about nonlinear function analysis in recent years. In this paper, we mainly study the existence and uniqueness of common fixed point theorems for contractive mappings and expansive mappings. The conclusions are not required with the condition of normal cone, and some corresponding corollaries are given. The conclusions of this paper improve and generalize some new conclusions of some authors. This paper is divided into five chapters. Now we will describe them briefly one by one.Chapter1, we introduce the history and the current research status of the fixed point theorems in cone metric spaces, at the same time, we also introduce the author’s main work in this paper.Chapter2, We mainly introduce the definition of cone and the basic theory of cone, moreover, we also introduce the definition of cone metric space and the basic theory of cone metric space.Chapter3, we obtain the existence and uniqueness of common fixed point theorems for contractive mappings without normal cone in cone metric spaces.Chapter4. we obtain the existence and uniqueness of common fixed point theorems for expansive mappings without normal cone in cone metric spaces.Chapter5. we summarize the contents of research in this paper, and the future’s research direction was prospected.