【摘要】：本文回顧總結了賦范線性空間以及m-賦范線性空間上(m = 2,n)的Aleksandrov問題、Mazur-Ulam定理及其Aleksandrov-Rassias問題的提出和研究現(xiàn)狀..引入了擬凸n-賦范線性空間的定義,并在其上深入探究了上述三個方面的問題,得到一些結論.在第一章節(jié)中,我們主要回顧總結了 Aleksandrov問題、Mazur-Ulam定理在賦范線性空間及和m-賦范線性空間上(m=2,n)的提出和已經(jīng)取得的成果,我們要特別關注的是H.Y.Chu等研究者在文獻[13]和[14]中關于m-賦范線性空間上的Aleksandrov問題和Mazur-Ulam定理的研究和已經(jīng)取得的結論.在第二章節(jié)中,我們主要討論了在擬凸n-賦范線性空間上的Aleksandrov問題和Mazur-Ulam定理.我們證明了f僅在滿足(nDOPP)和保共線的條件下即為n-等距映射以及在擬凸n-賦范線性空間上的等距映射即為仿射的結論.在第三章節(jié)中,我們主要研究的是擬凸n-賦范線性空間上的Aleksandrov-Rassias問題.我們在基于文獻[33]和文獻[44]中對賦范線性空間及n-賦范線性空間上獲得的有關Aleksandrov-Rassias問題的已有結論的基礎上,證明了在擬凸n-賦范線性空間上將條件:||x_1-y_1,x_2-y_2,…,x_n-y_n||≥1(?)||f(x_1)-f(y_1),f(x_2)-f(y_2),…,f(x_n)-f(y_n)||≥_1替換成:||x_1-y_1,x_2-y_2,…,x_n-y_n||≤1(?)||f(x_1)-f(y_1),f(x_2)-f(y_2),…,f(x_n)-f(y_n)||≤_1其相關結論仍然成立.
[Abstract]:In this paper, the Aleksandrov problem, Mazur-Ulam theorem and Aleksandrov-Rassias problem on normed linear space and m normed linear space (m = 2n) are reviewed and summarized. The definition of quasi convex n- normed linear space is introduced, on which the above three problems are deeply discussed, and some conclusions are obtained. In the first chapter, we review and summarize the Aleksandrov problem, the Mazur-Ulam theorem in normed linear space and m- normed linear space (mG2n) and the results obtained. We should pay special attention to the study of Aleksandrov problem and Mazur-Ulam theorem on m-normed linear spaces by H.Y.Chu and other researchers in [13] and [14]. In the second chapter, we mainly discuss the Aleksandrov problem and Mazur-Ulam theorem on quasiconvex n-normed linear space. We prove that f is an affine only if it satisfies (nDOPP) and collinear preserving conditions, that is, n-isometric mapping and isometric mapping on quasi-convex n-normed linear space. In the third chapter, we mainly study the Aleksandrov-Rassias problem on quasiconvex n-normed linear spaces. On the basis of the existing conclusions on Aleksandrov-Rassias problems on normed linear spaces and n- normed linear spaces in references [33] and [44], In this paper, we prove that the condition:\;\ , x_n-y_n 鈮，

本文編號：2351369