Isometric Theories In N-Normed Spaces And Fixed Point Theorems In Cone Metric Spaces

The theories of isometric mappings and fixed point are ones of the most active fields in functional analysis, it has very important meanings for other mathematic branches. In chapter 1, firstly we introduced Aleksandrov problem and some conclusions about this problem in the normed space. Then we take the research of isometric theories on the n -normed space, prove that the Riesz theorem holds when X is a linear n -normed spaces. And we give the definition of the linear ( n ,p )-normed spaces and strickly p -convex, we have proved that if X is a linear n -normed space and Y is a linear p -strictly convex ( n ,p )-normed space, what conditions the mapping f :X→Yis a n-isometry contains.In chapter 2, we firstly describe the concept and some properties of non-Archimedean normed space, establish the definitions of non-Archimedean p -normed spaces( 0 < p≤1) and Non-Achimedean ( 2, p )-normed spaces, obtain Marzur-Ulam theorem in non- Archimedean p -normed spaces: Let X ,Y be two non-Archimedean p -normed spaces over a non-Archimedean field K satisfying 2 = 1 and Y is p -strictly convex, if f :X→Ystatisfies , thenis additive, and the midpoint of a segment is f -invariant. And in Non-Achimedean ( 2, p )-normed spaces we proved that: if X ,Y is non-Archimedean ( 2, p )-normed spaces over a non-Archimedean field K satisfying 2 = 3 = 1 and Y is strictly convex, let f :X→Y be a 2-isometry and . Then ,y and z in XIn chapter 3, we show the definition and some properties of cone metric space which have been proved by Huang.L.G and Zhang.X. Then more properties and fixed point theorems were established by other scholars. In this part, we prove the completeness of cone metric space: Let( X ,d )be a cone metric space, P be a normal cone with normal constant K , Let B and ,. Then X is complete if and only if if rn→0, then (it is a single point set). And prove some fixed point theorems in cone metric space.