Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. II
Let (X,d) be a complete metric space with the metric d. In the first part [ibid. 130, 89-104 (1982; Zbl 0477.54033)] we considered several classes of mappings related to contraction mappings. Here our aim is to present further results about these classes of mappings in different settings. Firstly, we consider the case when X is a compact space and T is a surjection convex nonexpansive mapping of order 2 and we get that it is in fact an isometry. This basic fact is used to obtain some extension of certain results of C. Dafermos and M. Slemrod, W. G. Dotson, L. L. Talman, etc., to the case of convex nonexpansive maps of order 2. As is known, in the recent years an increasing effort was made to extend several results about fixed points from the case of metric spaces (or Banach spaces) to uniform spaces (to topological linear Hausdorff spaces (locally convex spaces)). In {\S} 4 we present some results about convex nonexpansive mappings in uniform spaces. In the second section of this paper we give some results about convex nonexpansive mappings on closed convex sets in complete inner product spaces. Essentially we show that for convex nonexpansive mappings hold certain ergodic theorems which are known for nonexpansive mappings. Using these results we obtain that for any convex nonexpansive mapping of order 2 the set of fixed points is nonempty, closed and convex. In the third section of the paper we present results for the class of Tricomi mappings. Some results about nonexpansive Tricomi mappings defined on closed, convex and bounded subsets of a uniformly convex space are given in the author's book on fixed point theory. Here we are interested with the extension of some results for the case of convex nonexpansive mappings.
scientific article; zbMATH DE number 3927894
scientific article