Fixed point and approximate fixed point theorems for non-affine maps

L. Gajek, J. Jachymski and D. Zagrodny

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 $x_{0}\in C$ such that $Tx_{0}\in x_{0}+V$, where $V$ is a neighborhood of the origin). Our theorems generalize the recent results of M. Edelstein and K.-K. Tan. 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).

Keywords: Fixed point, $V$-fixed point, convex set, close range, affine map,commuting maps, common fixed point, common $V$-fixed point

Classification (MSC2000): 47H10, 47A99, 47H99

Full text of the article:

• Compressed PostScript file (131 kilobytes)
• PDF file (169 kilobytes)