Hybrid fixed points of multivalued operators in metric spaces with applications

This paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space. Our claim is also illustrated with applications to a class of integral inclusions for proving the existence results.     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II

This is an addendum to the paper [K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28-56]. We prove the tensorization property for the curvature-dimension condition, add some detailed calculations - including explicit dependence of constants - and comment on assumptions and conjectures concerning the local-to-global statement in Bacher and Sturm (2010) [1] and Villani (2009) [6], respectively.     

Common fixed points for generalized contractive multivalued operators in complete metric spaces

In this paper,we discuss some new fixed point theorems for a pair of multivalued operators which satisfy generalized contractive condition. Our results are the extension and improvement of corresponding results of single-valued operators. Finally, some examples are given to illustrate the usability of our results.     

复值度量空间中F-压缩映射的叠合点

将F-压缩的概念从度量空间推广到复值度量空间中,并引进F-拟压缩的概念.说明了每个F-压缩是F-拟压缩,但反之不真.运用序列逼近的方法,对两个满足某种F-压缩条件的映射,证明了 一些叠合点结果,同时得到了一些不动点结果,这些结果推广了 Wardowski的结果.给出了几个说明性例子以突出这种推广.     

Topologies on homeomorphism spaces of certain metric spaces

A new proof for the equivalence of several topologies on homeomorphism groups over certain metric spaces X is given, which is based on the metric of X.     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD^∗(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD^∗(K,N) has the local-to-global property. We also prove the tensorization property for CD^∗(K,N). As an application we conclude that the fundamental group π₁(M,x₀) of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.     

Common fixed point theorems for non-self-mappings in metric spaces of hyperbolic type

In this paper, the concept of a pair of non-linear contraction type mappings in a metric space of hyperbolic type is introduced and the conditions guaranteeing the existence of a common fixed point for such non-linear contractions are established. Presented results generalize and improve some of the known results. An example is constructed to show that our theorems are genuine generalizations of the main theorems of Assad, ?iri?, Khan et al., Rhoades and Imdad and Kumar. One of the possible applications of our results is also presented.     

Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces

In this paper, we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence, and existence and uniqueness theorems for contractive type mappings. Our results generalize and extend recent coupled fixed point theorems established by Gnana Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379-1393]. Examples to support our new results are given.     

Remarks on coarse embeddings of metric spaces into uniformly convex Banach spaces

We study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex Banach space. By using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex Banach space is determined by its finite subspaces.     

Fixed and periodic point theory in certain spaces

This paper proves the existence of periodic and fixed points for contractive conditions in metric (symmetric) space.     

Common fixed points of two maps in cone metric spaces

We prove the existence of points of coincidence and common fixed points of a pair of self-mappings satisfying a generalized contractive condition in cone metric spaces. Our results generalize several well-known recent and classical results.      

Operator norm localization property of metric spaces under finite decomposition complexity

The notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper we show that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator norm localization property. It follows that any metric space with finite decomposition complexity has the operator norm localization property. In particular, we obtain an alternative way to prove a very recent result by E. Guentner, R. Tessera and G. Yu that all countable linear groups have the operator norm localization property.     

Conjugate points in length spaces

In this paper we extend the concept of a conjugate point in a Riemannian manifold to geodesic spaces. In particular, we introduce symmetric conjugate points and ultimate conjugate points and relate these notions to prior notions developed for more restricted classes of spaces. We generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We close with applications of these new kinds of conjugate points to CBA(κ) spaces: proving both known and new theorems. In particular we prove a Rauch comparison theorem, a Relative Rauch Comparison Theorem, the fact that there are no ultimate conjugate points less than π apart in a CBA(1) space and a few facts concerning closed geodesics. This paper is written to be accessible to students and includes open problems.     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

A common fixed point theorem for compatible quasi contractive self mappings in metric spaces

A common fixed point theorem for weakly commuting quasi contractive self mappings with contracting orbital diameters in metric spaces [V. Berinde, A common fixed point theorem for quasi contractive type mappings, Ann. Univ. Sci. Budapest. 46 (2003) 81-90] is extended to the more general class of compatible quasi contractive self mappings. Our result does extend and generalize numerous related results in literature.     

Common fixed points of generalized contractions on partial metric spaces and an application

In this paper, common fixed point theorems for four mappings satisfying a generalized nonlinear contraction type condition on partial metric spaces are proved. Presented theorems extend the very recent results of I. Altun, F. Sola and H. Simsek [Generalized contractions on partial metric spaces, Topology and its applications 157 (18) (2010) 2778-2785]. As application, some homotopy results for operators on a set endowed with a partial metric are given.     

Some periodic point results in generalized metric spaces

We prove some fixed point and periodic point theorems for a map in generalized metric spaces. An example is provided to support our result. The results presented in this paper generalize several well known comparable results in the literature.     

完备Riemann流形具有闭的割空间

本文证明了完备的Ｒｉｅｍａｎｎ流形即拥有闭的割空间（ｃｕｔｓｐａｃｅ），这一结论不但完满解答了段海豹在［１］中提出的问题１．６，大大地改进了他的主要结果（［１］，定理１．３），而且作为一个推论，我们还得到了经典Ｂｏｒｓｕｋ－Ｕｌａｍ定理的一个进一步推广．     

Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces

In this paper, we extend the coupled fixed point theorems for mixed monotone operators F:X×X→X obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations.