Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. II
type: Item
instance of: scholarly article
title: Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. II
zbMATH Open document ID: 0579.54033
DOI: 10.1007/BF01773511
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 3927894
zbMATH Keywords: complete metric space
zbMATH Keywords: contraction mappings
zbMATH Keywords: convex nonexpansive maps
zbMATH Keywords: complete inner product spaces
zbMATH Keywords: ergodic theorems
zbMATH Keywords: nonexpansive Tricomi mappings
zbMATH Keywords: uniformly convex space
author: Vasile I. Istratescu
MaRDI profile type: Publication
cites work: Q5786938
cites work: Asymptotic behavior of nonlinear contraction semigroups
cites work: Q5551838
cites work: On the Set of Subsequential Limit Points of Successive Approximations
cites work: Q5575314
cites work: Q5584384
cites work: Q4150675
cites work: Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. I
cites work: Q3917065
cites work: Q3324460
cites work: Q3226265
cites work: Q4148417
cites work: Asymptotic behavior of contractions in Hilbert space
cites work: On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space
cites work: Q5551837
cites work: LIMIT-COMPACT AND CONDENSING OPERATORS
cites work: A Fixed Point Criterion for Compact T 2 -Spaces
cites work: Fixed point theorems for nonexpansive mappings in linear topological spaces
cites work: Existence of common fixed points for commutative sets of nonlinear operators
cites work: Existence of common fixed points for abelian families of discontinuous operators
full work available at URL: https://doi.org/10.1007/bf01773511
OpenAlex ID: W4242824027
description: scientific article; zbMATH DE number 3927894