Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces

In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067-2071).     

Best and coupled best approximation theorems in abstract convex metric spaces

In this paper, we first give a best approximation theorem in abstract convex metric spaces. As applications, we then derive some best and coupled best approximations and coupled coincidence point results in normed spaces and hyperconvex metric spaces.     

Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.     

关于度量投影的连续性

本文引入的Ｂａｎａｃｈ空间的（Ｃ－Ｉ）、（Ｃ－Ⅱ），（Ｃ－Ⅲ）等几何性质，证明了如下结果。设Ｍ是Ｂａｎａｃｈ空间的逼近凸子集，如果Ｂａｎａｃｈ空间有性质（Ｃ－Ｉ），（Ｃ－Ⅱ）（Ｃ－Ⅲ），则度量投影ＰＭ连续（范数－范数上半连续，范数－弱上半连续）。这些结果推广了文（４，７，８）相应的定理。最近，Ｄ．Ｋｕｔｚａｒｏｖａ，Ｂｏｒ－Ｌｕｈ Ｌｉｎ等引入了一些新的凸性空间，本文还研究了这些凸性空间中度量投影     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.     

Best Approximation on Convex Sets in Metric Linear Spaces

In this paper the concepts of strictly convex and uniformly convex normed linear spaces are extended to metric linear spaces. A relationship between strict convexity and uniform convexity is established. Some existence and uniqueness theorems on best approximation in metric linear spaces under different conditions are proved.     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。     

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.     

New results on the proximal point algorithm in non-positive curvature metric spaces

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.     

Giuseppe Tallini: His life and work

It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained.     

Strong additivity of metric isosceles orthogonality

It is known that the property of additivity of isosceles orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper it is shown that suitably metrized concepts of additivity of metric isosceles orthogonality characterize euclidean or hyperbolic spaces among complete, convex, externally convex metric spaces.     

凸距离空间内星形子集非扩张型映射不动点*

在本文中.我们给出包含一个集合的某种星形集的刻划及其性质.然后利用这些刻划和性质讨论凸距离空间的星形子集上非扩张型映射的不动点的存在问题,推广了丁协平、Beg和Azam的某些最近结果.最后还给出一个例子说明以上推广是本质上的推广.     

Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors, which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces. Furthermore, we obtain the corresponding results in Banach spaces. Our results extend and generalize many known results.     

Stability results for Jungck-type iterative processes in convex metric spaces

In this paper, we obtain some stability results in complete convex metric spaces for nonselfmappings satisfying certain general contractivity condition. We prove our results for Jungck-Mann and Jungck-Ishikawa iterations. Our results extend several stability results in the literature.     

一致凸度量线性空间的自反性及其应用

为研究度量线性空间中凸集的逼近性质，Ｇ．Ｃ．Ａｈｕｊａ等引起了度量线性空间的严格凸性及一致凸性的定义。本文证明了完备的一致凸的度量线性空间是自反的。同时，作为应用，研究了最佳联合逼近元的存在性与唯一性问题。     

Isometries of spaces of compact or compact convex subsets of metric manifolds

The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.     

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.     

On fuzzy metric spaces

In this paper we introduce the concept of a fuzzy metric space. The distance between two points in a fuzzy metric space is a non-negative, upper semicontinuous, normal and convex fuzzy number. Properties of fuzzy metric spaces are studied and some fixed point theorems are proved.     

Stronger convergence theorems for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces

The purpose of this paper is to study the Ishikawa type iterative scheme to approximate a common fixed point of infinite families of uniformly quasi-Lipschitzian mappings and nonexpansive mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved. The results presented in the paper generalize, improve and unify some recent results.