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

In this paper, we research on Aleksandrov problem and Mazur-Ulam theorem in linear non-Archimedean normed spaces and linear (2, p)-normed spaces and the fixed point theorem of cone ultrametric space and cone2-metric space. Moreover, we show some definitions and properties in cone2-metric 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 cone2-metric 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 Mazur-Ulam 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 preserving2-isometry, then f is an affine.In the second chapter, we show the example of a new non-Archimedean valuation, then give the definitions of isometry, general2-isometry and general n-isometry on the non-Archimedean normed space, the non-Archimedean2-normed space and the non-Archimedean n-normed space the last in the new the Archimedes domain the study of Archimedes normed space, the Archimedes2-normed space and the Archimedes n-normed space isometry problem. Get the following main conclusion, Let X and Y be non-Arichimedean 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)‖x-y‖for all x,y∈X (2)f is a surjective mapping satisfying (SDnPP) and‖f(x)-f(y)‖=‖x-y‖when‖x-y‖≤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 cone2-metric space, then research sequence convergence, Cauchy sequence and space convergence property.And we get that Let (X, d) be a cone2-metric 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.