Fixed points for convex continuous mappings in topological vector spaces

We prove the following result. Let be a convex compact subset in a topological vector space, and a convex continuous mapping. (See Definition 1.1.) Then has a fixed point. Moreover, continuous mappings that can be approximated by convex continuous mappings also have the fixed point property.

     

Extreme points of the distance function on convex surfaces

We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.

     

Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. Some results have been obtained which generalize and unify many important known results in recent literature.     

Extreme points of the closed convex hull of composition operators

We study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.     

Extreme points of subordination and weak subordination families of harmonic mappings

The aim of the paper is to discuss the extreme points of subordination and weak subordination families of harmonic mappings. Several necessary conditions and sufficient conditions for harmonic mappings to be extreme points of the corresponding families are established.     

Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces

In this paper, we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi-nonexpansive mappings in convex metric spaces. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results of Khan and Ahmed [4], Khan et al. [6], Sun [12], Wittmann [14] and Xu and Ori [15].     

Extreme points of convex sets of doubly stochastic matrices. I

Summary It is shown that there are no extreme elements (other than the permutation matrices) in certain naturally occurring convex sets of doubly stochastic matrices, including the set of all such matrices whose entries are all strictly less than (1), provided that is rational.It is conjectured that this last result is false for infinite matrices whenever is irrational.     

逼近一致凸Banach空间中渐近非扩张映象的不动点

借助于B ruck′s不等式,研究了一致凸Banach空间中渐近非扩张映象不动点的具误差的Ish ikaw a迭代序列的强收敛定理.所得的结果推广和改进了Schu,Rhoades,周海云,王绍荣等作者的相应结果.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. — I

Summary In this paper we consider several classes of mappings related to the class of contraction mappings by introducing a convexity condition with respect to the iterates of the mappings. Several fixed point theorems are proved for such mappings. Further, in a similar way we consider a related class of mappings satisfying a convexity condition with respect to diameters of bounded sets. In the last part we consider classes of mappings on PM- spaces (probabilistic metric spaces of K. Menger) and some fixed point theorems are given for such classes.