Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

On the Cayley Graphs of Brandt Semigroups

In this article, the Cayley graphs of Brandt semigroups are investigated. The basic structures and properties of this kind of Cayley graphs are given, and a necessary and sufficient condition is given for the components of Cayley graphs of Brandt semigroups to be strongly regular. As an application, the generalized Petersen graph and k-partite graph, which cannot be obtained from the Cayley graphs of groups, can be constructed as a component of the Cayley graphs of Brandt semigroups.     

Context equivalence of semigroups

Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.     

Banach空间中非Lipschitz映象的连续半群的不动点定理

设E是一实的p一致凸Banach空间.利用渐近中心,E中范数不等式和逐次逼近的思想,本文证明了E中非Lipschitz映象的连续半群的不动点存在的定理.该定理本质上把Lipschitz半群的不动点存在性的研究推广到了非Lipschitz映象的连续半群的情况.另一方面,应用该定理,还得到了非Lipschitz映象的渐近非扩张型半群的渐近行为方面的一些结果.     

Generalized Cayley graphs of semigroups I

We introduce the concept of generalized Cayley graphs of semigroups and discuss their fundamental properties, and then study a special case, the universal Cayley graphs of semigroups so that some general results are given and the universal Cayley graph of a -partial order of complete graphs with loops is described.     

Convergence of almost-orbits of nonexpansive semigroups in Banach spaces

The purpose of this paper is to show that the study of mean ergodic theorems for almost-orbits of semigroups of nonexpansive mappings on closed convex subsets of a Banach space can be reduced to the study of orbits for semigroups of nonexpansive mappings. This provides a unified approach to various mean ergodic theorems for almost-orbits in the literature and new applications.

     

由格蕴涵代数诱导的伴随半群

给出由格蕴涵代数诱导出的伴随半群及有关概念 ,详细讨论伴随半群中的元素即格蕴涵代数的左映射的性质 ,得到它们的几个等价条件。最后讨论由格蕴涵代数诱导的两个双格半群与伴随半群之间的关系 ,并证明这些半群是幂等的当且仅当它们是由格 H蕴涵代数所诱导     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

Approximations to fixed points of contraction semigroups in hilbert spaces

Sequences (or curves) are constructed to approximate common fixed points of a pair of nonex-pansive mappings (or contraction semigroups) in Hilbert spaces. The obtained results extend the previously known results from a single mapping to a family of mappings.     

Commuting graphs of boundedly generated semigroups

Araújo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups.We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems.We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two.We also address a problem concerning knit degree.     

Generalized Cayley graphs of semigroups II

Following Zhu (Semigroup Forum, 2011, doi:), we study generalized Cayley graphs of semigroups. The Cayley D-saturated property, a particular combinatorial property, of generalized Cayley graphs of semigroups is considered and most of the results in Kelarev and Quinn (Semigroup Forum 66:89–96, 2003), Yang and Gao (Semigroup Forum 80:174–180, 2010) are extended. In addition, for some basic graphs and their complete fission graphs, we describe all semigroups whose universal Cayley graphs are isomorphic to these graphs.     

Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups

In this paper, we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a Hilbert space. Moreover, we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the recent ones announced by Kim and Xu [T.W. Kim, H.W. Xu, Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152], Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], and many others.     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.     

无限族非扩张映象及非扩张半群的强收敛定理

目的是研究Banach空间中无限族非扩张映象和非扩张半群的强收敛问题.为此提出一个改进的迭代序列,在适当条件下,某些强收敛定理被证明.结果改进和推广了一些人的最新结果.     

Actions and partial actions of inductive constellations

Constellations were recently introduced by the authors as one-sided analogues of categories: a constellation is equipped with a partial multiplication for which ‘domains’ are defined but, in general, ‘ranges’ are not. Left restriction semigroups are the algebraic objects modelling semigroups of partial mappings, equipped with local identities in the domains of the mappings. Inductive constellations correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids.     

Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces

In this paper, we introduce two modifications of the Ishikawa iteration, by using the hybrid methods, for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in a Hilbert space. Then, we prove that such two sequences converge strongly to common fixed points of two symptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our main result is connected with the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear. Anal. 67(2007) 2306-2315], Martinez-Yanes and Xu [C. Martinez-Yanes, H.K. Xu, Strong convergence of CQ method for fixed point iteration processes, Nonlinear. Anal. 64 (2006) 2400-2411] and many others.     

Kœnigs-type linearization models and asymptotic behavior of one-parameter semigroups

In this paper, we study linear-fractional models for one-parameter semigroups of holomorphic mappings via Schröder’s and Abel’s functional equations. By using some limit schemes in the spirit of K?nigs to solve those equation, we obtain new results on the asymptotic behavior of one-parameter semigroups having a boundary Denjoy-Wolff fixed point. In addition, we establish infinitesimal versions of the Burns-Krantz rigidity theorem for semigroups and their generators.