Abstract: Our purpose is to introduce the concept of a convex structure of abstract metric spaces, by virtue of which we prove the following fixed point theorem for nonex-pansive mappings.Theorem. Let (X,r) be a complete convex abstract metric spaces. Let K⊂X be a bounded closed star shaped set, xa be its center, satisfying r((W_A(x,x₀),(W_A,(y,x₀))≤AR(x,y),∀x,y∈K.where WA is the convex structure of X, and A, which satisfies some conditions, is a mapping in a partially ordered space. Suppose further that mapping f:K-K is (W_A, x₀) convex, m is a positive integral number such that f^m is compact on {W_A(x,x₀;x∈f(K)) and r(fx,fy)≤r(x,y) for x,y∈K. Then f has a fixed point in K.

Key words: abstract metric space, convex structure, fixed point

摘要： 本文在抽象度量空间中引入了凸结构的概念,并得到以下非扩张映象的不动点定理:定理,设(X,r)是完备的凸抽象度量空间,K⊂X是有界闭星形集,x₀为星形中心,K满足r((W<sub>A(x,x₀),(W_A,(y,x₀))≤AR(x,y),∀x,y∈K.这里W_A是X的凸结构,A是r取值的半序空间中的一个映象,并满足适当的条件。又设f:K→K是(W_A,x₀)凸的,且存在正整数m,使得f^m在{W_A(x,x₀);x∈f(K)}上紧及r(fx,fy)≤r(x,y),∀x,y∈K.则f在K中存在不动点。

关键词: 抽象度量空间, 凸结构, 不动点