The Research Of Generalized Convex Function And Cone Pseudo-Metric Space Contraction Mapping

Optimization theory is a part of the optimization which is a historical topic, as well as an important theoretical basis in operational research. The optimization method is used to provide people with the best technology, design, decision-making and management program through scientific method. With the rapid development of the contemporary science, the requirement of optimization theory is increasingly widespread. Convex sets and convex mapping, which are fundamental properties in optimization theory, are widely applied in various fields of mathematics. In early1950s, scientists began to lucubrate convex sets, convex cone and convex function. In1970, the writings of Rockafellar promoted the development of convex analysis. Hanson defined the invex convex function in the year1981. Moreover, the uniform convex function was defined by Bector, Gupta and Duneja in1992. Seven years later, the definition of E-convex sets and E-convex function were introduced by Youness. The primary introduction in this paper is (F,A)-affine invex sets,(F,A)-affine invex mapping, semi-(F,A)-affine invex mapping and quasi-(F,A)-affine invex mapping, which are followed by the discussion of their properties. The relative property study of cone and convex cone has also become a focus of attention in both domestic and abroad research fields as a study tool of optimization method. So this article is focus on the metrization of topological vector spaces by cone. In recent years, fixed point theorem plays an important role in many branches of analysis and topology. So for the purpose of obtaining wider range of applications of contraction mapping principle, it is discussed on the basis of topological vector space cone pseudo-metric space in this article.This thesis consists of two parts:Firstly, the concepts of (F,A)-affine invex sets,(F,A)-affine invex mapping, semi-(F,A)-affine invex mapping and quasi-(F,A)-affine invex mapping are given on the theoretical basis of semi-E-invex set and semi-E-affine invex function, and the related properties are studied.Secondly, Topological vector space-cone pseudo metric is proposed.And some properties of topological vector space-cone pseudo metric, the convergent sequence and complete sequence in topological vector space-cone pseudo metric are also investigated.At last,the contraction mapping in topological vector space-cone pseudo metric is proved.