A Class Of Banach Spaces Having The Fixed Piont Property

Orthogonality is a very important concept in the geometric theory of Euclidean spaces, it is play a important role in some of the basic Euclidean geometric theorems. Since 20th century, many types of orthogonality, of which Birkhoff orthogonality and isosceles orthogonality are two well known types, have been introduced one after another into normed linear spaces and a large number of fundamental results have been obtained by several researchers. The reserches on orthogonalities focus mainly on the properties of orthogonalities, the relation between different types of orthogonalities, and geometric constants related to some orthogonalities. This thesis is organized into three parts as fllowing, some new conclusions are obtained in this paper.The content of this paper consists of three parts. The main results of this paper are as follows:First, it is evaluating and summarizing the pioneer's main research results, the background, significance of the theories, the internal, external actuality are showed, and the main results of are described.Second, some basic definitions and fundamental results of normed linear spaces, some geometric property, and geometric constants are introduced. The developments of different types of orthogonalities are presented and the previous researches on these orthogonalities are summarized. Some definations and conclusions which are relative to this paper are given.Last, a sufficient condition of uniform non-squareness for Banach spaces is shown. It is proved that every Banach space X with D ( X ) > 2( 2 ? 1)is uniformly non-square,and hence,it has fixed point property for non-expansive mappings. This chapter is the main part of the paper.