Abstract
Abstract Let C be a non–empty subset of a linear topological space X, and T be a selfmap of C such that the range of I – T is convex, where I denotes the identity map on X. We give conditions under which a map T has a fixed point or a V–fixed point (i.e. a point x 0 ∈ C such that Tx 0 ∈ X 0 + V, where V is a neighborhood of the origin). Our theorems generalize the recent results of M. Edelstein and K.-K. Tan ([Math. Japon. 38: 325–332, 1993], [J. Math. Anal. Appl. 181: 181–187, 1994]). As an application we provide a simple proof of the Markov–Kakutani theorem. We also establish a common V–fixed point theorem for commuting affine maps (possibly discontinuous).
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