Isometric Theories in NonArchimedean Normed Spaces and Fixed Point Theorems in Cone Metric Spaces 

Author  WangDanPing 
Tutor  LiuYuBo 
School  Tianjin University of Technology 
Course  Applied Mathematics 
Keywords  nonArchimedean normed space cone ultrametric space cone2metric space fixed point theorem 
CLC  O177.91 
Type  Master's thesis 
Year  2012 
Downloads  19 
Quotes  0 
In this paper, we research on Aleksandrov problem and MazurUlam theorem in linear nonArchimedean normed spaces and linear (2, p)normed spaces and the fixed point theorem of cone ultrametric space and cone2metric space. Moreover, we show some definitions and properties in cone2metric space and the definition of the Hausdorff metric on cone metric space, furthermore, we study the multivalued fixed point on spherically complete cone ultrametric space, convergence, complete and fixed point on cone2metric space. Four main achievements have been made as follows:In the first chapter, we discuss the results about the relationships between isometry mappings and linear mappings in linear (2, p)normed spaces. We proved MazurUlam theorem is valid in linear (2, p)normed spaces. That is, let X and Y be two linear (2, p)normed spaces. If mapping f:X→Y is an interior preserving2isometry, then f is an affine.In the second chapter, we show the example of a new nonArchimedean valuation, then give the definitions of isometry, general2isometry and general nisometry on the nonArchimedean normed space, the nonArchimedean2normed space and the nonArchimedean nnormed space the last in the new the Archimedes domain the study of Archimedes normed space, the Archimedes2normed space and the Archimedes nnormed space isometry problem. Get the following main conclusion, Let X and Y be nonArichimedean normed spaces and one of them has dimension greater than one. Suppose that (1)f:X→Y is a Lipschiz mapping with K=1,‖f(x)f(y)‖xy‖for all x,y∈X (2)f is a surjective mapping satisfying (SDnPP) and‖f(x)f(y)‖=‖xy‖when‖xy‖≤1for all x, y∈X. Then f is an isomety.In the third chapter, we show the definitions of cone ultrametric space, spherically complete cone ultrametric space and the Hausdorff metric on cone metric space. Then using space spherically complete and Zorn lemma to prove the multivalued fixed point on spherically complete cone ultrametric space.That is, Let(X,d) be a spherically complete cone ultrametric space, if T:X→2cX is such that for any x,y∈X, x≠y H(Tx, Ty)(?)max{d(x, y), m(x, Tx), m(y, Ty)}. Then T has a fixed point (i.e., there exists x∈X such that x∈Tx).In the fourth chapter, we introduce the definitions of cone2metric space, then research sequence convergence, Cauchy sequence and space convergence property.And we get that Let (X, d) be a cone2metric space, if P be a normal cone with normal constant K and E is a bounded Banach space. Suppose the mapping T:X→X satisfies the contractive condition d(Tx, Ty,a)(?)kd(x,y,a), for all x,y,a∈X, Where k∈[0,1) is a constant. Then T has a unique fixed point in X. And for any x∈X, iterative sequence {Tnx} converges to the fixed point.