Fixed point theorem of nonexpansive mappings in convex metric spaces

Let X be a convex metric space with the property that even decreasing sequence of nonempty closed subsets of X with diameters tending to zero has nonempty intersection This paper proved that if T is a mapping of a closed convex nonempty subset K of X into itself satisfying the inequality: d(Tx, Ty)ad(x, y)+b{d(x, Tx)+d(y, T _y )} +c{d(x, Ty)+d(y, Tx)} for all x, y in K, where 0a<1, b0, c0, a+c0 and a+2b+3c1, then T has a unique fixed point in K.The author is grateful to Professor Zhang Shi-seng of Sichuan University for his care and help in completion of this paper.