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A Convex Structure of Abstract Metric Spaces and Its Fixed Points
Zheng Quan, Ding Zuohua
1991, 10(2):
59-62.
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((WA(x,x0),(WA,(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 (WA, x0) convex, m is a positive integral number such that fm is compact on {WA(x,x0;x∈f(K)) and r(fx,fy)≤r(x,y) for x,y∈K. Then f has a fixed point in K.
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