Fixed Point Theory In Cone Metric Spaces And Its Applications

The fixed point theory in metric spaces is an important part of nonlinear function anal-ysis. Since Longguang Huang and Xian Zhang [1] recently introduced the concept of conemetric space, replacing real number by ordered Banach space, the fixed point theorems incone metric spaces become the topic that people study in the recent years. In this paper, wemainly study the existence and uniqueness of fixed point theorems and common fixed pointtheorems for all kinds of contractive mappings and expanding mappings, with the conditionthat the cone is not assumed to be normal. Moreover, we extend fixed point theorems ofmonotone and weak (relative) monotone mappings. Examples are given to distinguish theseresults from the known ones. This paper includes five chapters. Now we will describe thembriefly one by one.Chapter1, as an introduction, we recall the history and the current research status of thefixed point theorems in cone metric spaces, and we also give a summary of this work.Chapter2, we obtain the existence and uniqueness of fixed point theorems and commonfixed point theorems for several kinds of contractive mappings in cone metric spaces, suchas φ-mappings, Lipschitz mappings, three and four mappings.Chapter3, we obtain the existence and uniqueness of fixed point theorems and commonfixed point theorems for expanding mappings in cone metric spaces.Chapter4, we study the fixed point theorems for monotone and weak (relative) mono-tone mappings on c-distance and common fixed point theorems for three mappings in theordered cone metric spaces.Finally, in Chapter5, corresponding examples of the former chapters are given to sup-port our new results.