editorial preserver problems on function spaces, operator

EditorialPreserver Problems on Function Spaces, Operator Algebras, andRelated Topics

Antonio M. Peralta,1 Chi-Keung Ng,2 Ngai-Ching Wong,3 and Jen-Chih Yao4

1 Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain2 Chern Institute of Mathematics, Nankai University, Tianjin 300071, China3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Correspondence should be addressed to Antonio M. Peralta; [email protected]

Received 21 January 2014; Accepted 21 January 2014; Published 27 February 2014

Copyright © 2014 Antonio M. Peralta et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This special volume of the journalAbstract andAppliedAnal-ysis was originally conceived to provide authors and readersa publication to present themost recent advances in the studyof those (linear) mappings preserving ranks, spectra, spectralradii, numerical ranges, numerical radii, orthogonality, ordisjointness between different, though related, structuresstudied in Matrix theory, Functional Analysis, and Operatortheory, including Function spaces, Operator algebras, andC∗-algebras.

This volume includes 23 original research papers by 66authors. The selection of the papers included in this volume,which has been based on a strict international peer reviewprocedure, contains a representative list of manuscripts withnewfangled results which covers the different topics consid-ered in our original proposal.

In the setting of preservers between C∗-algebras andMatrix spaces, the volume includes a characterization of𝜉-Lie multiplicative isomorphisms between von Neumannalgebras by Y. Song et al.; a study on linear maps on uppertriangular matrices spaces preserving idempotent tensorproducts by L. Yang et al. M. Burgos et al. study approx-imate versions of Hua’s theorem and other related resultsfor Banach algebras and C∗-algebras, including a study oflinear maps approximately preserving the conorm betweenunital C∗-algebras. Pioneering achievements on orthogo-nally additive and orthogonality preserving holomorphic

mappings between C∗-algebras together with a newfangledinvestigation on orthogonality preserving pairs of operatorsbetween C∗-algebras are presented by J. J. Garces et al. Astudy on linear maps between positive operators of Schatten𝑝-classes (1 < 𝑝 < +∞) which preserve the 𝑝-norms ofconvex combinations is developed by D. L.-W. Kuo et al. Newmethods and results to solve systems of operator equationson Hilbert C∗-modules are given by X. Fang et al.

Y. Song et al. characterize allmaps between vonNeumannalgebras that satisfy the equality Φ(𝑎𝑏 − 𝜉𝑏𝑎) = Φ(𝑎)Φ(𝑏) −𝜉Φ(𝑏)Φ(𝑎). M. A. Alghamdi et al. obtain basic properties ofthe solutions of the Q-lasso as a function of a tuning param-eter 𝛾. They also discuss iterative methods for solving theQ-lasso which include the proximal gradient algorithm andthe projection-gradient algorithm. L. Yang et al. characterize,for a general field 𝐹 and 𝑚, 𝑛 ≥ 3, maps of the form {𝑓

𝑖𝑗}

from𝑀𝑚𝑛(𝐹) to itself that preserves similarity and those that

preserves inverses.Concerning the applications of the fixed point theorems

and various techniques developed in preserver problems, H.Liu et al. establish some existence theorems for semilinearnonlocal functional differential equations. D.-G. Yang and J.-L. Liu study new subclasses of multivalent analytic functionsassociated with a linear operator. Moreover, T. Wang and F.Y. Lu obtain some new results on Lie triple derivations onstandard subalgebras of lattice algebras.

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 723685, 2 pageshttp://dx.doi.org/10.1155/2014/723685

2 Abstract and Applied Analysis

The paper by L. C. Ceng and C. F. Wen proposes andanalyzes hybrid implicit and explicit extragradient methodsfor finding a zero of an accretive operator and solving ageneral system of variational inequalities and a fixed pointproblem of an infinite family of nonexpansive self-mappingsin a uniformly convex Banach space 𝑋 which has a uni-formly Gateaux differentiable norm. Several strong conver-gence theorems for hybrid implicit and explicit extragradientalgorithms under suitable assumptions are established. Animplicit algorithm for the split fixed point and convexfeasibility problems is proposed in the paper by F. Guo etal. A strong convergence theorem is obtained. The paperby Y. Ling et al. is concerned with the semilocal as wellas the local convergence problems of Newton-Steffensen’smethod to solve nonlinear operator equations in Banachspaces. Under very mild assumptions, they establish theconvergence criterion and convergence ball for Newton-Steffensen’s method.

Mann-type extragradient methods for a general systemof variational inequalities with solutions of a multivaluedvariational inclusion and common fixed points of a countablefamily of nonexpansive mappings in real smooth Banachspaces are introduced in the paper by L. C. Ceng et al. Weakand strong convergence results are derived. The paper byN. Hussain et al. introduces modified weak 𝛼-𝜓-contractivemappings, where fixed point results for such mappingsdefined onpartialmetric spaces using the notion of triangular𝛼-admissibility are obtained.The well-posedness for general-ized set equilibrium problems is studied in the paper by Y. C.Lin, where two types of well-posed concepts for these prob-lems in topological vector spaces setting are proposed and therelationship between these well-posednesses is obtained.

The paper by M. Alshahrani et al. introduces nons-mooth vector quasi-variational-like inequalities by means ofa bifunction, where some existence results for solutions ofthese inequalities are established by using Fan-KKM theoremand a maximal element theorem. The paper by D. H. Fanand J. F. Bao considers a generalized equilibrium probleminvolving DC functions. By using the properties of the epi-graph of the conjugate functions, they provide some sufficientand/or necessary conditions for the weak and strong dualityresults and optimality conditions for generalized equilibriumproblems. The paper by Z. Chen and Zhe Chen considers aclass of nonsmooth generalized semi-infinite programmingproblems, where estimates for the value functions of thelower level problems and necessary optimality conditionsfor generalized semi-infinite programming problems arederived.

Finally, hybrid and relaxed Mann iteration methods for ageneral systemof variational inequalities with solutions beingalso common solutions of a countable family of variationalinequalities and common fixed points of a countable familyof nonexpansive mappings in real smooth and uniformlyconvex Banach spaces are proposed in the paper by L. C. Cenget al., where they derive some strong convergence theoremsfor hybrid and relaxed Mann iteration algorithms not onlyin the setting of uniformly convex and 2-uniformly smoothBanach space but also in a uniformly convex Banach spacehaving a uniformly Gateaux differentiable norm.

Acknowledgments

The editors of this special issue would like to expresstheir gratitude to the authors who have submitted theirmanuscripts for consideration and to the many expert refer-ees who were involved in the reviewing process and providedus an invaluable help.

Antonio M. PeraltaChi-Keung Ng

Ngai-Ching WongJen-Chih Yao

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