Guest Editors: Qamrul Hasan Ansari, Mohamed Amine Khamsi, Abdul Latif, and Jen-Chih Yao

Variational Analysis, Optimization, and Fixed Point Theory

Abstract and Applied Analysis

Variational Analysis, Optimization,and Fixed Point Theory

Abstract and Applied Analysis

Variational Analysis, Optimization,and Fixed Point Theory

Guest Editors:QamrulHasanAnsari,MohamedAmineKhamsi,Abdul Latif, and Jen-Chih Yao

Copyright © 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Ravi P. Agarwal, USABashir Ahmad, Saudi ArabiaM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyPablo Amster, ArgentinaDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalMohamed Kamal Aouf, EgyptNarcisa C. Apreutesei, RomaniaNatig M. Atakishiyev, MexicoFerhan M. Atici, USAIvan G. Avramidi, USASoohyun Bae, KoreaChuanzhi Bai, ChinaZhanbing Bai, ChinaDumitru Baleanu, TurkeyJózef Banaś, PolandGerassimos Barbatis, GreeceMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Başar, TurkeyA. Bellouquid, MoroccoDaniele Bertaccini, ItalyMichiel Bertsch, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin J. Bohner, USAJulian F. Bonder, ArgentinaGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, KoreaFabio M. Camilli, ItalyJinde Cao, ChinaAnna Capietto, ItalyJianqing Chen, ChinaW.-S. Cheung, Hong KongMichel Chipot, SwitzerlandChangbum Chun, KoreaSoon Y. Chung, Korea

Jaeyoung Chung, KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, ItalyDiego Córdoba, SpainJuan Carlos Cortés López, SpainGraziano Crasta, ItalyB. Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranLuis F. P. de Castro, PortugalT. Diagana, USAJesús I. Dı́az, SpainJosef Dibĺık, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainA. I. Domoshnitsky, IsraelMarco Donatelli, ItalyBo-Qing Dong, ChinaOndǔej DošlÝ, Czech RepublicWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USAAhmed El-Sayed, EgyptLuca Esposito, ItalyJose A. Ezquerro, SpainKhalil Ezzinbi, MoroccoDashan Fan, USAAngelo Favini, ItalyMarcia Federson, BrazilS. Filippas, Equatorial GuineaAlberto Fiorenza, ItalyTore Flåtten, NorwayIlaria Fragala, ItalyBruno Franchi, ItalyXianlong Fu, ChinaMassimo Furi, ItalyGiovanni P. Galdi, USAIsaac Garcia, SpainJesús Garćıa Falset, SpainJ. A. Garćıa-Rodŕıguez, SpainLeszek Gasinski, PolandGyörgy Gát, HungaryVladimir Georgiev, Italy

Lorenzo Giacomelli, ItalyJaume Giné, SpainValery Y. Glizer, IsraelLaurent Gosse, ItalyJean P. Gossez, BelgiumJose L. Gracia, SpainMaurizio Grasselli, ItalyQian Guo, ChinaYuxia Guo, ChinaChaitan P. Gupta, USAUno Hämarik, EstoniaFerenc Hartung, HungaryBehnam Hashemi, IranNorimichi Hirano, JapanJafari Hossein, IranJiaxin Hu, ChinaChengming Huang, ChinaZhongyi Huang, ChinaGennaro Infante, ItalyIvan Ivanov, BulgariaJaan Janno, EstoniaAref Jeribi, TunisiaUn C. Ji, KoreaZhongxiao Jia, ChinaL. Jódar, SpainJ. S. Jung, Republic of KoreaHenrik Kalisch, NorwayHamid Reza Karimi, NorwayS. Kichenassamy, FranceTero Kilpeläinen, FinlandS. G. Kim, Republic of KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, FinlandVictor Kovtunenko, AustriaRen-Jieh Kuo, TaiwanPavel Kurasov, SwedenMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USAMatti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, China

Elena Litsyn, IsraelShengqiang Liu, ChinaYansheng Liu, ChinaCarlos Lizama, ChileMilton C. Lopes Filho, BrazilJulian López-Gómez, SpainGuozhen Lu, USAJinhu Lü, ChinaGrzegorz Lukaszewicz, PolandWanbiao Ma, ChinaShiwang Ma, ChinaEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesús Maŕın-Solano, SpainJose M. Martell, SpainM. MastyThlo, PolandMing Mei, CanadaTaras Mel’nyk, UkraineAnna Mercaldo, ItalyChangxing Miao, ChinaStanislaw Migorski, PolandMihai Mihǎilescu, RomaniaFeliz Minhós, PortugalDumitru Motreanu, FranceRoberta Musina, ItalyG. M. N’Guérékata, USAMaria Grazia Naso, ItalySylvia Novo, SpainMicah Osilike, NigeriaM. Ôtani, JapanTurgut Ôziş, TurkeyFilomena Pacella, ItalyN. S. Papageorgiou, GreeceSehie Park, KoreaAlberto Parmeggiani, ItalyKailash C. Patidar, South AfricaKevin R. Payne, ItalyAdemir Fernando Pazoto, BrazilJ. E. Pečarić, CroatiaShuangjie Peng, ChinaS. V. Pereverzyev, AustriaMaria Eugenia Perez, SpainJosefina Perles, SpainAllan Peterson, USAAndrew Pickering, SpainCristina Pignotti, Italy

Somyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, MexicoI. Rachůnková, Czech RepublicMaria A. Ragusa, ItalySimeon Reich, IsraelAbdelaziz Rhandi, ItalyHassan Riahi, MalaysiaJuan P. Rincón-Zapatero, SpainLuigi Rodino, ItalyYuriy V. Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalyMarco Sabatini, ItalySatit Saejung, ThailandStefan G. Samko, PortugalMartin Schechter, USAJavier Segura, SpainSigmund Selberg, NorwayValery Serov, FinlandNaseer Shahzad, Saudi ArabiaAndrey Shishkov, UkraineStefan Siegmund, GermanyA.-M. A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalyS. Staněk, Czech RepublicStevo Stevic, SerbiaAntonio Suárez, SpainWenchang Sun, ChinaRobert Szalai, UKSanyi Tang, ChinaChun-Lei Tang, ChinaYoushan Tao, ChinaGabriella Tarantello, ItalyN.-e. Tatar, Saudi ArabiaSusanna Terracini, ItalyGerd Teschke, GermanyAlberto Tesei, ItalyBevanThompson, AustraliaSergey Tikhonov, SpainClaudia Timofte, RomaniaThanh Tran, Australia

Juan J. Trujillo, SpainCiprian A. Tudor, FranceGabriel Turinici, FranceMilan Tvrdy, Czech RepublicMehmet Unal, TurkeyCsaba Varga, RomaniaCarlos Vazquez, SpainGianmaria Verzini, ItalyJesus Vigo-Aguiar, SpainYushun Wang, ChinaQing-WenWang, ChinaJing Ping Wang, UKShawn X. Wang, CanadaQing Wang, USAYouyu Wang, ChinaPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeZili Wu, ChinaYong Hong Wu, AustraliaShanhe Wu, ChinaTie-cheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaGongnan Xie, ChinaFuding Xie, ChinaNaihua Xiu, ChinaDaoyi Xu, ChinaXiaodong Yan, USAZhenya Yan, ChinaNorio Yoshida, JapanBeong In Yun, KoreaVjacheslav Yurko, RussiaAgacik Zafer, TurkeySergey V. Zelik, UKJianming Zhan, ChinaChengjian Zhang, ChinaWeinian Zhang, ChinaMeirong Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaTianshou Zhou, ChinaYong Zhou, ChinaChun-Gang Zhu, ChinaQiji J. Zhu, USAMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

Variational Analysis, Optimization, and Fixed PointTheory, Qamrul Hasan Ansari,Mohamed Amine Khamsi, Abdul Latif, and Jen-Chih YaoVolume 2013, Article ID 823070, 2 pages

Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings inHilbert Spaces, Sy-Ming Guu and Wataru TakahashiVolume 2013, Article ID 904164, 10 pages

Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces,Zhi-bin Liu, Yi-shen Chen, Xue-song Li, and Yi-bin XiaoVolume 2013, Article ID 503198, 8 pages

Paratingent Derivative Applied to the Measure of the Sensitivity in Multiobjective DifferentialProgramming, F. Garćıa and M. A. Melguizo PadialVolume 2013, Article ID 812125, 11 pages

Some Results on Fixed and Best Proximity Points of Multivalued Cyclic Self-Mappings with a PartialOrder, M. De la SenVolume 2013, Article ID 968492, 11 pages

Strong ConvergenceTheorems for Semigroups of Asymptotically Nonexpansive Mappings in BanachSpaces, D. R. Sahu, Ngai-Ching Wong, and Jen-Chih YaoVolume 2013, Article ID 202095, 8 pages

Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities,Zhao-Rong Kong, Lu-Chuan Ceng, Qamrul Hasan Ansari, and Chin-Tzong PangVolume 2013, Article ID 718624, 17 pages

Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems andAsymptotically Strict Pseudocontractive Mappings in the Intermediate Sense, Lu-Chuan Ceng,Qamrul Hasan Ansari, and Ching-Feng WenVolume 2013, Article ID 854297, 14 pages

Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly SmoothBanach Spaces, Ming Tian and Xin JinVolume 2013, Article ID 264910, 9 pages

Existence of Solutions for a Periodic Boundary Value Problem via GeneralizedWeakly Contractions,Sirous Moradi, Erdal Karapınar, and Hassen AydiVolume 2013, Article ID 704160, 7 pages

Meir-Keeler Type Multidimensional Fixed PointTheorems in Partially Ordered Metric Spaces,Erdal Karapınar, Antonio Roldán, Juan Mart́ınez-Moreno, and Concepcion RoldánVolume 2013, Article ID 406026, 9 pages

ANon-NP-Complete Algorithm for a Quasi-Fixed Polynomial Problem,Yi-Chou Chen and Hang-Chin LaiVolume 2013, Article ID 893045, 10 pages

Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points ofNonexpansive Semigroups, Li-Jun ZhuVolume 2013, Article ID 604369, 8 pages

Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces,N. I. MahmudovVolume 2013, Article ID 502839, 9 pages

TheMann-Type Extragradient Iterative Algorithms with Regularization for Solving VariationalInequality Problems, Split Feasibility, and Fixed Point Problems, Lu-Chuan Ceng, Himanshu Gupta,and Ching-Feng WenVolume 2013, Article ID 378750, 19 pages

A New Fixed PointTheorem and Applications, Min Fang and Xie Ping DingVolume 2013, Article ID 432402, 5 pages

Neural Network Dynamics without Minimizing Energy, Mau-Hsiang Shih and Feng-Sheng TsaiVolume 2013, Article ID 496217, 4 pages

Ordered Variational Inequalities and Ordered Complementarity Problems in Banach Lattices,Jinlu Li and Ching-Feng WenVolume 2013, Article ID 323126, 9 pages

Berinde-Type Generalized Contractions on Partial Metric Spaces, Hassen Aydi, Sana Hadj Amor,and Erdal KarapınarVolume 2013, Article ID 312479, 10 pages

General Split Feasibility Problems in Hilbert Spaces, Mohammad Eslamian and Abdul LatifVolume 2013, Article ID 805104, 6 pages

Existence Solutions of Vector Equilibrium Problems and Fixed Point of Multivalued Mappings,Kanokwan Sitthithakerngkiet and Somyot PlubtiengVolume 2013, Article ID 952021, 6 pages

Best Proximity Point Results for MK-Proximal Contractions, Mohamed Jleli, Erdal Karapınar,and Bessem SametVolume 2012, Article ID 193085, 14 pages

AMaximum Principle for Controlled Time-Symmetric Forward-Backward Doubly StochasticDifferential Equation with Initial-Terminal Sate Constraints, Shaolin Ji, Qingmeng Wei,and Xiumin ZhangVolume 2012, Article ID 537376, 29 pages

The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors forUniformly Continuous GeneralizedWeak Φ-Pseudocontractive Mappings in Banach Spaces,Guiwen Lv and Haiyun ZhouVolume 2012, Article ID 169410, 14 pages

Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems,Zhao-Rong Kong, Lu-Chuan Ceng, and Ching-Feng WenVolume 2012, Article ID 975981, 32 pages

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 823070, 2 pageshttp://dx.doi.org/10.1155/2013/823070

EditorialVariational Analysis, Optimization, and Fixed Point Theory

Qamrul Hasan Ansari,1 Mohamed Amine Khamsi,2 Abdul Latif,3 and Jen-Chih Yao4

1 Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India2Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia4Center of General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Correspondence should be addressed to Qamrul Hasan Ansari; qhansari@gmail.com

Received 5 September 2013; Accepted 5 September 2013

Copyright © 2013 Qamrul Hasan Ansari 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.

In the last two decades, the theory of variational analysisincluding variational inequalities (VI) emerged as a rapidlygrowing area of research because of its applications innonlinear analysis, optimization, economics, game theory,and so forth; see, for example, [1] and the references therein.In the recent past, many authors devoted their attentionto study the VI defined on the set of fixed points of amapping, called hierarchical variational inequalities. Veryrecently, several iterative methods have been investigatedto solve VI, hierarchical variational inequalities, and triplehierarchical variational inequalities. Since the origin of theVI, it has been used as a tool to study optimization problems.Hierarchical variational inequalities are used to study thebilevel mathematical programming problems. A triple levelmathematical programming problem can be studied by usingtriple hierarchical variational inequalities.

The Ekeland’s variational principle provides the existenceof an approximate minimizer of a bounded below and lowersemicontinuous function. It is one of the most importantresults from nonlinear analysis and it has applications indifferent areas of mathematics and mathematical sciences,namely, fixed point theory, optimization, optimal controltheory, game theory, nonlinear equations, dynamical systems,and so forth, for example, [2–9] and the references therein.During the last decade, it has been used to study the existenceof solutions of equilibrium problems in the setting of metricspaces, for example, [2, 3] and the references therein.

Banach’s contraction principle is remarkable in its sim-plicity, yet it is perhaps the most widely applied fixed point

theory in all of analysis. This is because the contractivecondition on the mapping is simple and easy to verify, andbecause it requires only completeness of the metric space.Although, the basic idea was known to others earlier, theprinciple first appeared in explicit form inBanach’s 1922 thesiswhere it was used to establish the existence of a solution to anintegral equation.

Caristi’s fixed point theorem [10, 11] has found manyapplications in nonlinear analysis. It is shown, for example,that this theorem yields essentially all the known inwardnessresults of geometric fixed point theory in Banach spaces.Recall that inwardness conditions are the ones which assertthat, in some sense, points from the domain are mappedtoward the domain. This theorem is amazing equivalent toEkeland’s variational principle.

Qamrul Hasan AnsariMohamed Amine Khamsi

Abdul LatifJen-Chih Yao

References

[1] Q. H. Ansari, C. S. Lalitha, and M. Mehta, GeneralizedConvexity, Nonsmooth Variational Inequalities, and NonsmoothOptimization, CRC Press, Taylor & Francis Group, Boca Raton,Fla, USA, 2014.

[2] Q. H. Ansari, Metric Spaces: Including Fixed Point Theory andSet-Valued Maps, Narosa Publishing House, New Delhi, India,2010.

2 Abstract and Applied Analysis

[3] M. Bianchi, G. Kassay, and R. Pini, “Existence of equilibriavia Ekeland’s principle,” Journal of Mathematical Analysis andApplications, vol. 305, no. 2, pp. 502–512, 2005.

[4] D. G. de Figueiredo, Lectures on the Ekeland VariationalPrinciple with Applications and Detours, vol. 81 of Tata Instituteof Fundamental Research Lectures on Mathematics and Physics,Tata Institute of Fundamental Research, Mumbai, India, 1989.

[5] I. Ekeland, “Sur les problèmes variationnels,” Comptes Rendusde l’Académie des Sciences, vol. 275, pp. 1057–1059, 1972.

[6] I. Ekeland, “On the variational principle,” Journal of Mathemat-ical Analysis and Applications, vol. 47, pp. 324–353, 1974.

[7] I. Ekeland, “Nonconvex minimization problems,” Bulletin of theAmerican Mathematical Society, vol. 1, no. 3, pp. 443–474, 1979.

[8] O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimiza-tion theorems and fixed point theorems in complete metricspaces,”Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.

[9] M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spacesand Fixed PointTheory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001.

[10] J. Caristi, “Fixed point theorems for mappings satisfyinginwardness conditions,” Transactions of the American Mathe-matical Society, vol. 215, pp. 241–251, 1976.

[11] J. Caristi and W. A. Kirk, “Geometric fixed point theory andinwardness conditions,” in The Geometry of Metric and LinearSpaces, vol. 490 of Lecture Notes in Mathematics, pp. 74–83,Springer, Berlin, Germany, 1975.

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 904164, 10 pageshttp://dx.doi.org/10.1155/2013/904164

Research ArticleExistence and Approximation of Attractive Points of the WidelyMore Generalized Hybrid Mappings in Hilbert Spaces

Sy-Ming Guu1 and Wataru Takahashi2

1 Graduate Institute of Business and Management, College of Management, Chang-Gung University, Kwei-Shan,Taoyuan Hsien 330, Taiwan

2Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Correspondence should be addressed to Sy-Ming Guu; iesmguu@mail.cgu.edu.tw

Received 30 January 2013; Accepted 5 June 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 S.-M. Guu and W. Takahashi.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

We study thewidelymore generalized hybridmappings which have been proposed to unify several well-known nonlinearmappingsincluding the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without theconvexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widelymore generalized hybridmappings in a Hilbert space. Moreover, we prove a weak convergence theorem ofMann’s type and a strongconvergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our resultscan be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalizedhybrid mappings.

1. Introduction

Let𝐻 be a real Hilbert space, and let 𝐶 be a nonempty subsetof𝐻. For amapping𝑇 : 𝐶 → 𝐻, we denote by𝐹(𝑇) and𝐴(𝑇)the sets of fixed points and attractive points of 𝑇, respectively,that is,

(i) 𝐹(𝑇) = {𝑧 ∈ 𝐶 : 𝑇𝑧 = 𝑧};(ii) 𝐴(𝑇) = {𝑧 ∈ 𝐻 : ‖𝑇𝑥 − 𝑧‖ ≤ ‖𝑥 − 𝑧‖, ∀𝑥 ∈ 𝐶}.

Amapping𝑇 : 𝐶 → 𝐻 is called nonexpansive [1] if ‖𝑇𝑥−𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖ for all 𝑥, 𝑦 ∈ 𝐶. A mapping 𝑇 : 𝐶 → 𝐻 iscalled nonspreading [2], hybrid [3] if

2𝑇𝑥 − 𝑇𝑦

2

≤𝑇𝑥 − 𝑦

2

+𝑇𝑦 − 𝑥

2

,

3𝑇𝑥 − 𝑇𝑦

2

≤𝑥 − 𝑦

2

+𝑇𝑥 − 𝑦

2

+𝑇𝑦 − 𝑥

2

(1)

for all 𝑥, 𝑦 ∈ 𝐶, respectively; see also [4, 5].These three termsare independent, and they are deduced from the notion offirmly nonexpansive mapping in a Hilbert space; see [3]. Amapping 𝐹 : 𝐶 → 𝐻 is said to be firmly nonexpansive if

𝐹𝑥 − 𝐹𝑦

2

≤ ⟨𝑥 − 𝑦, 𝐹𝑥 − 𝐹𝑦⟩ (2)

for all 𝑥, 𝑦 ∈ 𝐶; see, for instance, Goebel and Kirk [6]. Theclass of nonspreading mappings was first defined in a strictlyconvex, smooth, and reflexive Banach space.The resolvents ofa maximal monotone operator are nonspreading mappings;see [2] for more details. These three classes of nonlinearmappings are important in the study of the geometry ofinfinite dimensional spaces. Indeed, by using the fact that theresolvents of amaximalmonotone operator are nonspreadingmappings, Takahashi et al. [7] solved an open problem whichis related to Ray’s theorem [8] in the geometry of Banachspaces. Motivated by these mappings, Kocourek et al. [9]introduced a broad class of nonlinear mappings in a Hilbertspace which covers nonexpansive mappings, nonspreadingmappings, and hybrid mappings. A mapping 𝑇 : 𝐶 → 𝐻is said to be generalized hybrid if there exist 𝛼, 𝛽 ∈ R suchthat

𝛼𝑇𝑥 − 𝑇𝑦

2

+ (1 − 𝛼)𝑥 − 𝑇𝑦

2

≤ 𝛽𝑇𝑥 − 𝑦

2

+ (1 − 𝛽)𝑥 − 𝑦

2

(3)

for all 𝑥, 𝑦 ∈ 𝐶, where R is the set of real numbers. Wecall such a mapping an (𝛼, 𝛽)-generalized hybrid mapping.

http://dx.doi.org/10.1155/2013/904164

2 Abstract and Applied Analysis

An (𝛼, 𝛽)-generalized hybrid mapping is nonexpansive for𝛼 = 1 and 𝛽 = 0, nonspreading for 𝛼 = 2 and 𝛽 = 1, andhybrid for 𝛼 = 3/2 and 𝛽 = 1/2. They proved fixed pointtheorems for suchmappings; see also Kohsaka and Takahashi[10] and Iemoto and Takahashi [4]. Moreover, they provedthe following nonlinear ergodic theorem which generalizesBaillon’s theorem [11].

Theorem 1 (see [9]). Let 𝐻 be a real Hilbert space, let 𝐶 bea nonempty closed convex subset of 𝐻, let 𝑇 be a generalizedhybrid mapping from 𝐶 into itself with 𝐹(𝑇) ̸= 0, and let 𝑃 bethe metric projection of𝐻 onto 𝐹(𝑇). Then for any 𝑥 ∈ 𝐶,

𝑆𝑛𝑥 =

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥 (4)

converges weakly to 𝑝 ∈ 𝐹(𝑇), where 𝑝 = lim𝑛→∞

𝑃𝑇𝑛𝑥.

We see that the set 𝐶 needs to be closed and convexin Theorem 1. As a contrast, Takahashi and Takeuchi [12]proved the following theoremwhich establishes the existenceof attractive point and mean convergence property withoutthe convexity assumption in a Hilbert space; see also Lin andTakahashi [13] and Takahashi et al. [14].

Theorem 2. Let 𝐻 be a real Hilbert space, and let 𝐶 be anonempty subset of𝐻. Let 𝑇 be a generalized hybrid mappingfrom 𝐶 into itself. Let {V

𝑛} and {𝑏

𝑛} be sequences defined by

V1∈ 𝐶, V

𝑛+1= 𝑇V𝑛, 𝑏𝑛=1

𝑛

𝑛

∑

𝑘=1

V𝑘 (5)

for all 𝑛 ∈ N. If {V𝑛} is bounded, then the followings hold:

(1) 𝐴(𝑇) is nonempty, closed, and convex;(2) {𝑏𝑛} converges weakly to 𝑢

0∈ 𝐴(𝑇), where 𝑢

0=

lim𝑛→∞

𝑃𝐴(𝑇)

V𝑛and 𝑃

𝐴(𝑇)is the metric projection of

𝐻 onto 𝐴(𝑇).

Very recently Kawasaki and Takahashi [15] introduced aclass of nonlinear mappings in a Hilbert space which coverscontractive mappings [16] and generalized hybrid mappings.A mapping 𝑇 : 𝐶 → 𝐻 is called widely more generalizedhybrid if there exist 𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂 ∈ R such that

𝛼𝑇𝑥 − 𝑇𝑦

2

+ 𝛽𝑥 − 𝑇𝑦

2

+ 𝛾𝑇𝑥 − 𝑦

2

+ 𝛿𝑥 − 𝑦

2

+ 𝜀‖𝑥−𝑇𝑥‖2+𝜁

𝑦−𝑇𝑦

2

+𝜂(𝑥−𝑇𝑥)−(𝑦−𝑇𝑦)

2

≤0

(6)

for any 𝑥, 𝑦 ∈ 𝐶; see also Kawasaki and Takahashi [17].A mapping 𝑇 : 𝐶 → 𝐻 is called quasi-nonexpansive

if 𝐹(𝑇) ̸= 0 and ‖𝑇𝑥 − 𝑦‖ ≤ ‖𝑥 − 𝑦‖ for all 𝑥 ∈ 𝐶 and𝑦 ∈ 𝐹(𝑇). It is well known that if 𝐶 is closed and convexand 𝑇 : 𝐶 → 𝐻 is quasi-nonexpansive, then 𝐹(𝑇) is closedand convex; see Itoh and Takahashi [18]. For a simpler proofof such a result in a Hilbert space, see, for example, [19].A generalized hybrid mapping with a fixed point is quasi-nonexpansive. However, a widely more generalized hybridmapping is not quasi-nonexpansive generally even if it has

a fixed point. In [15], they proved fixed point theorems andnonlinear ergodic theorems of Baillon’s type for such newmappings in a Hilbert space.

In this paper, motivated by these results, we establishthe attractive point theorem and mean convergence theoremwithout the commonly required convexity for the widelymore generalized hybrid mappings in a Hilbert space. More-over, we prove a weak convergence theorem of Mann’s type[20] and a strong convergence theorem of Shimizu andTakahashi’s type [21] for such a class of nonlinear mappingsin a Hilbert space which generalize Kocourek et al. [9] andHojo and Takahashi [22] for generalized hybrid mappings,respectively.

2. Preliminaries

Throughout this paper, we denote by N the set of positiveintegers. Let𝐻 be a real Hilbert space with inner product ⟨⋅, ⋅⟩and norm ‖ ⋅ ‖. We denote the strong convergence and theweak convergence of {𝑥

𝑛} to 𝑥 ∈ 𝐻 by 𝑥

𝑛→ 𝑥 and 𝑥

𝑛⇀ 𝑥,

respectively. Let 𝐴 be a nonempty subset of𝐻. We denote byco𝐴 the closure of the convex hull of 𝐴. In a Hilbert space, itis known [1] that for any 𝑥, 𝑦 ∈ 𝐻 and 𝛼 ∈ R,

𝑦

2

− ‖𝑥‖2≤ 2 ⟨𝑦 − 𝑥, 𝑦⟩ , (7)

𝛼𝑥 + (1 − 𝛼) 𝑦

2

= 𝛼‖𝑥‖2+ (1 − 𝛼)

𝑦

2

− 𝛼 (1 − 𝛼)𝑥 − 𝑦

2

.

(8)

Furthermore, we have that

2 ⟨𝑥 − 𝑦, 𝑧 − 𝑤⟩ = ‖𝑥 − 𝑤‖2+𝑦 − 𝑧

2

− ‖𝑥 − 𝑧‖2−𝑦 − 𝑤

2

(9)

for any 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝐻.Let 𝐷 be a nonempty closed convex subset of 𝐻 and

𝑥 ∈ 𝐻. Then we know that there exists a unique nearest point𝑧 ∈ 𝐷 such that ‖𝑥 − 𝑧‖ = inf

𝑦∈𝐷‖𝑥 − 𝑦‖. We denote such

a correspondence by 𝑧 = 𝑃𝐷𝑥. The mapping 𝑃

𝐷is called

the metric projection of 𝐻 onto 𝐷. It is known that 𝑃𝐷is

nonexpansive and

⟨𝑥 − 𝑃𝐷𝑥, 𝑃𝐷𝑥 − 𝑢⟩ ≥ 0 (10)

for any 𝑥 ∈ 𝐻 and 𝑢 ∈ 𝐷; see [1] for more details. For provinga nonlinear ergodic theorem in this paper, we also need thefollowing lemma proved by Takahashi and Toyoda [23].

Lemma 3. Let 𝐷 be a nonempty closed convex subset of 𝐻.Let 𝑃 be the metric projection from 𝐻 onto 𝐷. Let {𝑢

𝑛} be a

sequence in 𝐻. If ‖𝑢𝑛+1

− 𝑢‖ ≤ ‖𝑢𝑛− 𝑢‖ for any 𝑢 ∈ 𝐷 and

𝑛 ∈ N, then {𝑃𝑢𝑛} converges strongly to some 𝑢

0∈ 𝐷.

To prove a strong convergence theorem in this paper, weneed the following lemma.

Abstract and Applied Analysis 3

Lemma 4 (Aoyama et al. [24]). Let {𝑠𝑛} be a sequence of

nonnegative real numbers, let {𝛼𝑛} be a sequence of [0, 1] with

∑∞

𝑛=1𝛼𝑛

= ∞, let {𝛽𝑛} be a sequence of nonnegative real

numbers with ∑∞𝑛=1

𝛽𝑛< ∞, and let {𝛾

𝑛} be a sequence of real

numbers with lim sup𝑛→∞

𝛾𝑛≤ 0. Suppose that

𝑠𝑛+1

≤ (1 − 𝛼𝑛) 𝑠𝑛+ 𝛼𝑛𝛾𝑛+ 𝛽𝑛 (11)

for all 𝑛 = 1, 2, . . .. Then lim𝑛→∞

𝑠𝑛= 0.

Let 𝑙∞ be the Banach space of bounded sequences withsupremum norm. Let 𝜇 be an element of (𝑙∞)∗ (the dualspace of 𝑙∞). Then we denote by 𝜇(𝑓) the value of 𝜇 at 𝑓 =(𝑥1, 𝑥2, 𝑥3, . . .) ∈ 𝑙

∞. Sometimes, we denote by 𝜇𝑛(𝑥𝑛) the

value 𝜇(𝑓). A linear functional 𝜇 on 𝑙∞ is called a mean if𝜇(𝑒) = ‖𝜇‖ = 1, where 𝑒 = (1, 1, 1, . . .). A mean 𝜇 is called aBanach limit on 𝑙∞ if 𝜇

𝑛(𝑥𝑛+1

) = 𝜇𝑛(𝑥𝑛). We know that there

exists a Banach limit on 𝑙∞. If 𝜇 is a Banach limit on 𝑙∞, thenfor 𝑓 = (𝑥

1, 𝑥2, 𝑥3, . . .) ∈ 𝑙

∞,

lim inf𝑛→∞

𝑥𝑛≤ 𝜇𝑛(𝑥𝑛) ≤ lim sup𝑛→∞

𝑥𝑛. (12)

In particular, if 𝑓 = (𝑥1, 𝑥2, 𝑥3, . . .) ∈ 𝑙

∞ and 𝑥𝑛

→

𝑎 ∈ R, then we have 𝜇(𝑓) = 𝜇𝑛(𝑥𝑛) = 𝑎. See [25]

for the proof of existence of a Banach limit and its otherelementary properties. Using means and the Riesz theorem,we can obtain the following result; see [25, 26].

Lemma 5. Let𝐻 be a real Hilbert space, let {𝑥𝑛} be a bounded

sequence in𝐻, and let 𝜇 be a mean on 𝑙∞. Then there exists aunique point 𝑧

0∈ co{𝑥

𝑛| 𝑛 ∈ N} such that

𝜇𝑛(⟨𝑥𝑛, 𝑦⟩) = ⟨𝑧

0, 𝑦⟩ (13)

for any 𝑦 ∈ 𝐻.

The following result obtained by Takahashi and Takeuchi[12] is important in this paper.

Lemma6. Let𝐻 be aHilbert space, let𝐶 be a nonempty subsetof 𝐻, and let 𝑇 be a mapping from 𝐶 into 𝐻. Then 𝐴(𝑇) is aclosed and convex subset of𝐻.

We also know the following result from [14].

Lemma7. Let𝐻 be aHilbert space, let𝐶 be a nonempty subsetof𝐻, and let 𝑇 be a quasi-nonexpansive mapping from 𝐶 into𝐻. Then 𝐴(𝑇) ∩ 𝐶 = 𝐹(𝑇).

3. Attractive Point Theorems

Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonemptysubset of 𝐻. Recall that a mapping 𝑇 from 𝐶 into 𝐻 issaid to be widely more generalized hybrid [15] if there exist𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂 ∈ R such that

𝛼𝑇𝑥 − 𝑇𝑦

2

+ 𝛽𝑥 − 𝑇𝑦

2

+ 𝛾𝑇𝑥 − 𝑦

2

+ 𝛿𝑥 − 𝑦

2

+ 𝜀‖𝑥 − 𝑇𝑥‖2+ 𝜁

𝑦 − 𝑇𝑦

2

+ 𝜂(𝑥 − 𝑇𝑥) − (𝑦 − 𝑇𝑦)

2

≤ 0

(14)

for any 𝑥, 𝑦 ∈ 𝐶. Such amapping𝑇 is called (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widely more generalized hybrid. An (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widelymore generalized hybridmapping is generalized hybrid in thesense of Kocourek et al. [9] if 𝛼 + 𝛽 = −𝛾 − 𝛿 = 1 and 𝜀 = 𝜁 =𝜂 = 0. We first prove an attractive point theorem for widelymore generalized hybrid mappings in a Hilbert space.

Theorem 8. Let𝐻 be a real Hilbert space, let𝐶 be a nonemptysubset of 𝐻, and let 𝑇 be an (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widely moregeneralized hybrid mapping from 𝐶 into itself which satisfieseither of the following conditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 > 0, 𝜀 + 𝜂 ≥ 0 and 𝜁 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 ≥ 0 and 𝜀 + 𝜂 ≥ 0.

Then 𝑇 has an attractive point if and only if there exists𝑧 ∈ 𝐶 such that {𝑇𝑛𝑧 | 𝑛 = 0, 1, . . .} is bounded.

Proof. Suppose that𝑇 has an attractive point 𝑧.Then ‖𝑇𝑛+1𝑥−𝑧‖ ≤ ‖𝑇

𝑛𝑥 − 𝑧‖ for all 𝑥 ∈ 𝐶 and 𝑛 ∈ N. Therefore {𝑇𝑛𝑧 | 𝑛 =

0, 1, . . .} is bounded.Conversely suppose that there exists 𝑧 ∈ 𝐶 such

that {𝑇𝑛𝑧 | 𝑛 = 0, 1, . . .} is bounded. Since 𝑇 is an(𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widely more generalized hybrid mappingfrom 𝐶 into itself, we obtain that

𝛼𝑇𝑥 − 𝑇

𝑛+1𝑧

2

+ 𝛽𝑥 − 𝑇𝑛+1

𝑧

2

+ 𝛾𝑇𝑥 − 𝑇

𝑛𝑧

2

+ 𝛿𝑥 − 𝑇

𝑛𝑧

2

+ 𝜀‖𝑥 − 𝑇𝑥‖2+ 𝜁

𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧

2

+ 𝜂(𝑥 − 𝑇𝑥) − (𝑇

𝑛𝑧 − 𝑇𝑛+1

𝑧)

2

≤ 0

(15)

for any 𝑛 ∈ N ∪ {0} and 𝑥 ∈ 𝐶. By (9) we obtain that

(𝑥 − 𝑇𝑥) − (𝑇

𝑛𝑧 − 𝑇𝑛+1

𝑧)

2

= ‖𝑥 − 𝑇𝑥‖2+𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧

2

− 2 ⟨𝑥 − 𝑇𝑥, 𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧⟩

= ‖𝑥−𝑇𝑥‖2+𝑇𝑛𝑧−𝑇𝑛+1

𝑧

2

+𝑥−𝑇

𝑛𝑧

2

+𝑇𝑥−𝑇

𝑛+1𝑧

2

−𝑥 − 𝑇𝑛+1

𝑧

2

−𝑇𝑥 − 𝑇

𝑛𝑧

2

.

(16)

Thus we have that

(𝛼 + 𝜂)𝑇𝑥 − 𝑇

𝑛+1𝑧

2

+ (𝛽 − 𝜂)𝑥 − 𝑇𝑛+1

𝑧

2

+ (𝛾 − 𝜂)𝑇𝑥 − 𝑇

𝑛𝑧

2

+ (𝛿 + 𝜂)𝑥 − 𝑇

𝑛𝑧

2

+ (𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2+ (𝜁 + 𝜂)

𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧

2

≤ 0.

(17)

From

(𝛾 − 𝜂)𝑇𝑥 − 𝑇

𝑛𝑧

2

= (𝛼+𝛾) (‖𝑥−𝑇𝑥‖2+𝑥−𝑇

𝑛𝑧

2

−2 ⟨𝑥−𝑇𝑥, 𝑥−𝑇𝑛𝑧⟩)

− (𝛼 + 𝜂)𝑇𝑥 − 𝑇

𝑛𝑧

2

,

(18)

4 Abstract and Applied Analysis

we have that

(𝛼 + 𝜂)𝑇𝑥 − 𝑇

𝑛+1𝑧

2

+ (𝛽 − 𝜂)𝑥 − 𝑇𝑛+1

𝑧

2

+ (𝛼+𝛾) (‖𝑥−𝑇𝑥‖2+𝑥−𝑇

𝑛𝑧

2

−2 ⟨𝑥−𝑇𝑥, 𝑥−𝑇𝑛𝑧⟩)

− (𝛼 + 𝜂)𝑇𝑥 − 𝑇

𝑛𝑧

2

+ (𝛿 + 𝜂)𝑥 − 𝑇

𝑛𝑧

2

+ (𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2+ (𝜁 + 𝜂)

𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧

2

≤ 0,

(19)

and hence

(𝛼 + 𝜂) (𝑇𝑥 − 𝑇

𝑛+1𝑧

2

−𝑇𝑥 − 𝑇

𝑛𝑧

2

)

+ (𝛽 − 𝜂)𝑥 − 𝑇𝑛+1

𝑧

2

− 2 (𝛼 + 𝛾) ⟨𝑥 − 𝑇𝑥, 𝑥 − 𝑇𝑛𝑧⟩

+ (𝛼 + 𝛾 + 𝛿 + 𝜂)𝑥 − 𝑇

𝑛𝑧

2

+ (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2

+ (𝜁 + 𝜂)𝑇𝑛𝑧 − 𝑇𝑛+1

𝑧

2

≤ 0.

(20)

By 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, we have that

− (𝛽 − 𝜂) = − (𝛽 + 𝛿) + 𝛿 + 𝜂 ≤ 𝛼 + 𝛾 + 𝛿 + 𝜂. (21)

From this inequality and 𝜁 + 𝜂 ≥ 0 we obtain that

(𝛼 + 𝜂) (𝑇𝑥 − 𝑇

𝑛+1𝑧

2

−𝑇𝑥 − 𝑇

𝑛𝑧

2

)

+ (𝛽 − 𝜂) (𝑥 − 𝑇𝑛+1

𝑧

2

−𝑥 − 𝑇

𝑛𝑧

2

)

− 2 (𝛼+𝛾) ⟨𝑥−𝑇𝑥, 𝑥−𝑇𝑛𝑧⟩+(𝛼+𝛾+𝜀+𝜂) ‖𝑥−𝑇𝑥‖

2

≤ 0.

(22)

Applying a Banach limit 𝜇 to both sides of this inequality, weobtain that

(𝛼 + 𝜂) (𝜇𝑛

𝑇𝑥 − 𝑇

𝑛+1𝑧

2

− 𝜇𝑛

𝑇𝑥 − 𝑇𝑛𝑧

2

)

+ (𝛽 − 𝜂) (𝜇𝑛

𝑥 − 𝑇𝑛+1

𝑧

2

− 𝜇𝑛

𝑥 − 𝑇𝑛𝑧

2

)

− 2 (𝛼 + 𝛾) 𝜇𝑛⟨𝑥 − 𝑇𝑥, 𝑥 − 𝑇

𝑛𝑧⟩

+ (𝛼 + 𝛾 + 𝜀 + 𝜂) 𝜇𝑛‖𝑥 − 𝑇𝑥‖

2≤ 0,

(23)

and hence

− 2 (𝛼 + 𝛾) 𝜇𝑛⟨𝑥 − 𝑇𝑥, 𝑥 − 𝑇

𝑛𝑧⟩ + (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖

2

≤ 0.

(24)

Since there exists 𝑝 ∈ 𝐻 by Lemma 5 such that

𝜇𝑛⟨𝑦, 𝑇𝑛𝑧⟩ = ⟨𝑦, 𝑝⟩ (25)

for any 𝑦 ∈ 𝐻, we obtain from (24) that

−2 (𝛼 + 𝛾) ⟨𝑥 − 𝑇𝑥, 𝑥 − 𝑝⟩ + (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2≤ 0.

(26)

We obtain from (9) that

− (𝛼 + 𝛾) (𝑥 − 𝑝

2

+ ‖𝑇𝑥 − 𝑥‖2−𝑇𝑥 − 𝑝

2

)

+ (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2≤ 0,

(27)

and hence

− (𝛼 + 𝛾) (𝑥 − 𝑝

2

−𝑇𝑥 − 𝑝

2

) + (𝜀 + 𝜂) ‖𝑥 − 𝑇𝑥‖2≤ 0.

(28)

Since 𝜀 + 𝜂 ≥ 0, we obtain that

− (𝛼 + 𝛾) (𝑥 − 𝑝

2

−𝑇𝑥 − 𝑝

2

) ≤ 0. (29)

Since 𝛼 + 𝛾 > 0, we obtain that𝑇𝑥 − 𝑝

2

≤𝑥 − 𝑝

2

. (30)

This implies that 𝑝 ∈ 𝐻 is an attractive point.In the case of 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 ≥ 0, and

𝜀 + 𝜂 ≥ 0, we can obtain the result by replacing the variables𝑥 and 𝑦. This completes the proof.

Using Theorem 8, we can show the following attractivepoint theorem for generalized hybrid mappings in a Hilbertspace.

Theorem9 (Takahashi and Takeuchi [12]). Let𝐻 be aHilbertspace, let 𝐶 be a nonempty subset of 𝐻, and let 𝑇 be ageneralized hybrid mapping from 𝐶 into 𝐶; that is, there existreal numbers 𝛼 and 𝛽 such that

𝛼𝑇𝑥 − 𝑇𝑦

2

+ (1 − 𝛼)𝑥 − 𝑇𝑦

2

≤ 𝛽𝑇𝑥 − 𝑦

2

+ (1 − 𝛽)𝑥 − 𝑦

2

(31)

for all 𝑥, 𝑦 ∈ 𝐶. Then 𝑇 has an attractive point if and only ifthere exists 𝑧 ∈ 𝐶 such that {𝑇𝑛𝑧 | 𝑛 = 0, 1, . . .} is bounded.

Proof. An (𝛼, 𝛽)-generalized hybrid mapping 𝑇 is an (𝛼, 1 −𝛼, −𝛽, −(1 − 𝛽), 0, 0, 0)-widely more generalized hybrid map-ping. Furthermore, the mapping satisfies the condition (2) inTheorem 8, that is,

𝛼 + (1 − 𝛼) − 𝛽 − (1 − 𝛽) = 0 ≥ 0, 𝛼 + (1 − 𝛼) = 1 > 0,

0 + 0 ≥ 0, 0 + 0 ≥ 0.

(32)

Then we have the desired result fromTheorem 8.

4. Nonlinear Ergodic Theorems

In this section, using the technique developed by Takahashi[26], we prove a mean convergence theorem without convex-ity for widely more generalized hybrid mappings in a Hilbertspace. Before proving the result, we need the following twolemmas.

Abstract and Applied Analysis 5

Lemma 10. Let 𝐶 be a nonempty subset of a real Hilbertspace𝐻. Let𝑇 be an (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widelymore generalizedhybrid mapping from 𝐶 into itself such that 𝐴(𝑇) ̸= 0. Supposethat it satisfies either of the following conditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 + 𝜀 + 𝜂 > 0 and 𝜁 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 + 𝜁 + 𝜂 > 0 and 𝜀 + 𝜂 ≥ 0.

For any 𝑥 ∈ 𝐶, define 𝑆𝑛𝑥 = (1/𝑛)∑

𝑛

𝑘=1𝑇𝑘𝑥. Then,

lim𝑛→∞

‖𝑆𝑛𝑥 −𝑇𝑆

𝑛𝑥‖ = 0. In particular, if 𝐶 is bounded, then

lim𝑛→∞

sup𝑥∈𝐶

𝑆𝑛𝑥 − 𝑇𝑆𝑛𝑥 = 0. (33)

Proof. Let 𝑥 ∈ 𝐶. Since 𝐴(𝑇) is nonempty, we obtain that𝑇𝑛+1

𝑥 − 𝑦≤𝑇𝑛𝑥 − 𝑦

(34)

for any 𝑛 ∈ N ∪ {0} and 𝑦 ∈ 𝐴(𝑇). Then we have that {𝑇𝑛𝑥} isbounded. Since

𝑆𝑛𝑥 − 𝑦 ≤

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥 − 𝑦

≤𝑥 − 𝑦

(35)

for any 𝑛 ∈ N ∪ {0} and 𝑦 ∈ 𝐴(𝑇), {𝑆𝑛𝑥 | 𝑛 = 0, 1, . . .} is also

bounded. Using 𝛼+𝛽+𝛾+𝛿 ≥ 0 and 𝜁+𝜂 ≥ 0, as in the proofof Theorem 8 we have that

(𝛼 + 𝜂) (𝑇𝑧 − 𝑇

𝑘+1𝑥

2

−𝑇𝑧 − 𝑇

𝑘𝑥

2

)

+ (𝛽 − 𝜂) (𝑧 − 𝑇𝑘+1

𝑥

2

−𝑧 − 𝑇𝑘𝑥

2

)

− 2 (𝛼+𝛾) ⟨𝑧−𝑇𝑧, 𝑧−𝑇𝑘𝑥⟩+(𝛼+𝛾+𝜀+𝜂) ‖𝑧−𝑇𝑧‖

2

≤ 0

(36)

for any 𝑘 ∈ N∪ {0} and 𝑧 ∈ 𝐶. Summing up these inequalitieswith respect to 𝑘 = 0, 1, . . . , 𝑛−1 and dividing by 𝑛, we obtainthat

𝛼 + 𝜂

𝑛(𝑇𝑧 − 𝑇

𝑛𝑥

2

− ‖𝑇𝑧 − 𝑥‖2)

+𝛽 − 𝜂

𝑛(𝑧 − 𝑇

𝑛𝑥

2

− ‖𝑧 − 𝑥‖2)

− 2 (𝛼 + 𝛾) ⟨𝑧 − 𝑇𝑧, 𝑧 − 𝑆𝑛𝑥⟩

+ (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑧 − 𝑇𝑧‖2≤ 0.

(37)

Replacing 𝑧 by 𝑆𝑛𝑥, we obtain that

𝛼 + 𝜂

𝑛(𝑇𝑆𝑛𝑥 − 𝑇

𝑛𝑥

2

−𝑇𝑆𝑛𝑥 − 𝑥

2

)

+𝛽 − 𝜂

𝑛(𝑆𝑛𝑥 − 𝑇

𝑛𝑥

2

−𝑆𝑛𝑥 − 𝑥

2

)

+ (𝛼 + 𝛾 + 𝜀 + 𝜂)𝑆𝑛𝑥 − 𝑇𝑆𝑛𝑥

2

≤ 0.

(38)

Since {𝑇𝑆𝑛𝑥}, {𝑆𝑛𝑥}, and {𝑇𝑛𝑥} are bounded, we have that

(𝛼 + 𝛾 + 𝜀 + 𝜂) lim sup𝑛→∞

𝑆𝑛𝑥 − 𝑇𝑆𝑛𝑥 ≤ 0. (39)

Since 𝛼+𝛾+𝜀+𝜂 > 0, we have that lim𝑛→∞

‖𝑆𝑛𝑥−𝑇𝑆

𝑛𝑥‖ = 0.

In particular, if 𝐶 is bounded, then

lim sup𝑛→∞

sup𝑥∈𝐶

𝑆𝑛𝑥 − 𝑇𝑆𝑛𝑥 ≤ 0, (40)

and hence lim𝑛→∞

sup𝑥∈𝐶

‖𝑆𝑛𝑥 − 𝑇𝑆

𝑛𝑥‖ = 0.

Similarly, we can obtain the desired result for the case of𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 + 𝜁 + 𝜂 > 0, and 𝜀 + 𝜂 ≥ 0. Thiscompletes the proof.

Lemma 11. Let𝐻 be a Hilbert space, and let 𝐶 be a nonemptysubset of 𝐻. Let 𝑇 : 𝐶 → 𝐻 be an (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widelymore generalized hybrid mapping. Suppose that it satisfieseither of the following conditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 > 0 and 𝜀 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0 and 𝜁 + 𝜂 ≥ 0.

If 𝑥𝑛⇀ 𝑧 and 𝑥

𝑛− 𝑇𝑥𝑛→ 0, then 𝑧 ∈ 𝐴(𝑇).

Proof. Let 𝑇 : 𝐶 → 𝐻 be an (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widely moregeneralized hybrid mapping, and suppose that 𝑥

𝑛⇀ 𝑧 and

𝑥𝑛− 𝑇𝑥𝑛→ 0. Replacing 𝑥 by 𝑥

𝑛in (14), we have that

𝛼𝑇𝑥𝑛 − 𝑇𝑦

2

+ 𝛽𝑥𝑛 − 𝑇𝑦

2

+ 𝛾𝑇𝑥𝑛 − 𝑦

2

+ 𝛿𝑥𝑛 − 𝑦

2

+ 𝜀𝑥𝑛−𝑇𝑥𝑛

2

+𝜁𝑦 − 𝑇𝑦

2

+𝜂(𝑥𝑛−𝑇𝑥𝑛)−(𝑦−𝑇𝑦)

2

≤ 0.

(41)

From this inequality, we have that

𝛼 (𝑇𝑥𝑛 − 𝑥𝑛

2

+𝑥𝑛 − 𝑇𝑦

2

+ 2 ⟨𝑇𝑥𝑛− 𝑥𝑛, 𝑥𝑛− 𝑇𝑦⟩)

+ 𝛽𝑥𝑛−𝑇𝑦

2

+𝛾 (𝑇𝑥𝑛−𝑥𝑛

2

+𝑥𝑛−𝑦

2

+ 2 ⟨𝑇𝑥𝑛−𝑥𝑛, 𝑥𝑛−𝑦⟩ )

+𝛿𝑥𝑛−𝑦

2

+𝜀𝑥𝑛−𝑇𝑥𝑛

2

+𝜁𝑦−𝑇𝑦

2

+ 𝜂(𝑥𝑛 − 𝑇𝑥𝑛) − (𝑦 − 𝑇𝑦)

2

≤ 0.

(42)

We apply a Banach limit 𝜇 to both sides of this inequality. Wehave that

𝛼𝜇𝑛(𝑇𝑥𝑛 − 𝑥𝑛

2

+𝑥𝑛 − 𝑇𝑦

2

+ 2 ⟨𝑇𝑥𝑛− 𝑥𝑛, 𝑥𝑛− 𝑇𝑦⟩)

+ 𝛽𝜇𝑛

𝑥𝑛−𝑇𝑦

2

+𝛾𝜇𝑛(𝑇𝑥𝑛−𝑥𝑛

2

+𝑥𝑛−𝑦

2

+ 2 ⟨𝑇𝑥𝑛−𝑥𝑛, 𝑥𝑛−𝑦⟩ )

+𝛿𝜇𝑛

𝑥𝑛−𝑦

2

+𝜀𝜇𝑛

𝑥𝑛−𝑇𝑥𝑛

2

+𝜁𝑦−𝑇𝑦

2

+ 𝜂𝜇𝑛

(𝑥𝑛 − 𝑇𝑥𝑛) − (𝑦 − 𝑇𝑦)

2

≤ 0,

(43)

and hence

𝛼𝜇𝑛

𝑥𝑛 − 𝑇𝑦

2

+𝛽𝜇𝑛

𝑥𝑛−𝑇𝑦

2

+𝛾𝜇𝑛

𝑥𝑛−𝑦

2

+𝛿𝜇𝑛

𝑥𝑛−𝑦

2

+ 𝜁𝑦 − 𝑇𝑦

2

+ 𝜂𝜇𝑛

(𝑥𝑛 − 𝑇𝑥𝑛) − (𝑦 − 𝑇𝑦)

2

≤ 0.

(44)

6 Abstract and Applied Analysis

Thus we have

(𝛼 + 𝛽) 𝜇𝑛

𝑥𝑛 − 𝑇𝑦

2

+ (𝛾 + 𝛿) 𝜇𝑛

𝑥𝑛 − 𝑦

2

+ (𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

≤ 0.

(45)

From ‖𝑥𝑛− 𝑇𝑦‖

2= ‖𝑥𝑛− 𝑦‖2+‖𝑦 − 𝑇𝑦‖

2+2⟨𝑥𝑛−𝑦, 𝑦−𝑇𝑦⟩,

we also have

(𝛼 + 𝛽) (𝜇𝑛

𝑥𝑛 − 𝑦

2

+𝑦 − 𝑇𝑦

2

+ 2𝜇𝑛⟨𝑥𝑛− 𝑦, 𝑦 − 𝑇𝑦⟩)

+ (𝛾 + 𝛿) 𝜇𝑛

𝑥𝑛 − 𝑦

2

+ (𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

≤ 0.

(46)

From 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0 we obtain that

(𝛼 + 𝛽)𝑦 − 𝑇𝑦

2

+ 2 (𝛼 + 𝛽) 𝜇𝑛⟨𝑥𝑛− 𝑦, 𝑦 − 𝑇𝑦⟩

+ (𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

≤ 0,

(47)

and hence

(𝛼 + 𝛽 + 𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

+ 2 (𝛼 + 𝛽) 𝜇𝑛⟨𝑥𝑛− 𝑦, 𝑦 − 𝑇𝑦⟩

≤ 0.

(48)

Since 𝑥𝑛⇀ 𝑧, we have that

(𝛼 + 𝛽 + 𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

+ 2 (𝛼 + 𝛽) ⟨𝑧 − 𝑦, 𝑦 − 𝑇𝑦⟩

≤ 0.

(49)

Using (9), we have that

(𝛼 + 𝛽 + 𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

+ (𝛼 + 𝛽) (𝑧 − 𝑇𝑦

2

−𝑧 − 𝑦

2

−𝑦 − 𝑇𝑦

2

) ≤ 0,

(50)

and hence

(𝜁 + 𝜂)𝑦 − 𝑇𝑦

2

+ (𝛼 + 𝛽) (𝑧 − 𝑇𝑦

2

−𝑧 − 𝑦

2

) ≤ 0.

(51)

Since 𝛼 + 𝛽 > 0 and 𝜁 + 𝜂 ≥ 0, we have that

𝑧 − 𝑇𝑦

2

−𝑧 − 𝑦

2

≤ 0 (52)

for all 𝑦 ∈ 𝐶. This implies that 𝑧 ∈ 𝐴(𝑇).Similarly, we can obtain the desired result for the case of

𝛼+𝛽+ 𝛾 + 𝛿 ≥ 0, 𝛼+ 𝛾 > 0, and 𝜀 + 𝜂 ≥ 0. This completes theproof.

Nowwe have the following nonlinear ergodic theorem forwidely more generalized hybrid mappings in a Hilbert space.

Theorem 12. Let 𝐻 be a real Hilbert space, let 𝐶 be anonempty subset of 𝐻, let 𝑇 be an (𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂)-widelymore generalized hybrid mapping from 𝐶 into itself such that

𝐴(𝑇) ̸= 0, and let 𝑃 be the metric projection of 𝐻 onto 𝐴(𝑇).Suppose that 𝑇 satisfies either of the conditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 > 0, 𝜀 + 𝜂 ≥ 0 and 𝜁 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 ≥ 0 and 𝜀 + 𝜂 ≥ 0.

Then for any 𝑥 ∈ 𝐶,

𝑆𝑛𝑥 =

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥 (53)

is weakly convergent to an attractive point 𝑝 of 𝑇, where 𝑝 =lim𝑛→∞

𝑃𝑇𝑛𝑥.

Proof. Let 𝑥 ∈ 𝐶. Since 𝐴(𝑇) is nonempty, we obtain that𝑇𝑛+1

𝑥 − 𝑦≤𝑇𝑛𝑥 − 𝑦

(54)

for any 𝑛 ∈ N ∪ {0} and 𝑦 ∈ 𝐴(𝑇). Thus {𝑇𝑛𝑥} is bounded.Since

𝑆𝑛𝑥 − 𝑦 ≤

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥 − 𝑦

≤𝑥 − 𝑦

(55)

for any 𝑛 ∈ N ∪ {0} and 𝑦 ∈ 𝐴(𝑇), {𝑆𝑛𝑥 | 𝑛 = 0, 1, . . .} is also

bounded.Therefore there exists a strictly increasing sequence{𝑛𝑖} and𝑝 ∈ 𝐻 such that {𝑆

𝑛𝑖𝑥 | 𝑖 = 0, 1, . . .} converges weakly

to 𝑝. Using 𝛼+𝛽+𝛾+𝛿 ≥ 0, 𝛼+𝛾 > 0, 𝜀+𝜂 ≥ 0, and 𝜁+𝜂 ≥ 0,we have from Lemma 10 that

lim𝑛→∞

𝑆𝑛𝑥 − 𝑇𝑆𝑛𝑥 = 0. (56)

We have from Lemma 11 that 𝑝 ∈ 𝐴(𝑇). Since 𝐴(𝑇) is closedand convex from Lemma 6, the metric projection 𝑃 from 𝐻onto 𝐴(𝑇) is well defined. By Lemma 3, there exists 𝑞 ∈ 𝐴(𝑇)such that {𝑃𝑇𝑛𝑥 | 𝑛 = 0, 1, . . .} converges strongly to 𝑞. Tocomplete the proof, we show that 𝑞 = 𝑝. Note that the metricprojection 𝑃 satisfies

⟨𝑧 − 𝑃𝑧, 𝑃𝑧 − 𝑢⟩ ≥ 0 (57)

for any 𝑧 ∈ 𝐻 and for any 𝑢 ∈ 𝐴(𝑇); see [25]. Therefore

⟨𝑇𝑘𝑥 − 𝑃𝑇

𝑘𝑥, 𝑃𝑇𝑘𝑥 − 𝑦⟩ ≥ 0 (58)

for any 𝑘 ∈ N ∪ {0} and 𝑦 ∈ 𝐴(𝑇). Since 𝑃 is the metricprojection from𝐻 onto 𝐴(𝑇) and 𝑃𝑇𝑛−1𝑥 ∈ 𝐴(𝑇), we obtainthat

𝑇𝑛𝑥 − 𝑃𝑇

𝑛𝑥 ≤

𝑇𝑛𝑥 − 𝑃𝑇

𝑛−1𝑥

≤𝑇𝑛−1

𝑥 − 𝑃𝑇𝑛−1

𝑥.

(59)

That is, {‖𝑇𝑛𝑥 − 𝑃𝑇𝑛𝑥‖ | 𝑛 = 0, 1, . . .} is nonincreasing.Therefore we obtain

⟨𝑇𝑘𝑥 − 𝑃𝑇

𝑘𝑥, 𝑦 − 𝑞⟩ ≤ ⟨𝑇

𝑘𝑥 − 𝑃𝑇

𝑘𝑥, 𝑃𝑇𝑘𝑥 − 𝑞⟩

≤𝑇𝑘𝑥 − 𝑃𝑇

𝑘𝑥⋅𝑃𝑇𝑘𝑥 − 𝑞

≤ ‖𝑥 − 𝑃𝑥‖ ⋅𝑃𝑇𝑘𝑥 − 𝑞

.

(60)

Abstract and Applied Analysis 7

Summing up these inequalities with respect to 𝑘 = 0, 1, . . . ,𝑛 − 1 and dividing by 𝑛, we obtain

⟨𝑆𝑛𝑥 −

1

𝑛

𝑛−1

∑

𝑘=0

𝑃𝑇𝑘𝑥, 𝑦 − 𝑞⟩ ≤

‖𝑥 − 𝑃𝑥‖

𝑛

𝑛−1

∑

𝑘=0

𝑃𝑇𝑘𝑥 − 𝑞

.

(61)

Since {𝑆𝑛𝑖𝑥 | 𝑖 = 0, 1, . . .} converges weakly to 𝑝 and {𝑃𝑇𝑛𝑥 |

𝑛 = 0, 1, . . .} converges strongly to 𝑞, we obtain that

⟨𝑝 − 𝑞, 𝑦 − 𝑞⟩ ≤ 0. (62)

Putting 𝑦 = 𝑝, we obtain

𝑝 − 𝑞

2

≤ 0, (63)

and hence 𝑞 = 𝑝. This completes the proof.Similarly, we can obtain the desired result for the case of

𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 > 0, and 𝜀 + 𝜂 ≥ 0.

As the proof of Theorem 9, we can prove Takahashi andTakeuchi’s mean convergence theorem for generalized hybridmappings in a Hilbert space.

Theorem 13. Let 𝐻 be a Hilbert space, let 𝐶 be a nonemptysubset of𝐻, and let 𝑇 be a generalized hybrid mapping from 𝐶into itself; that is, there exist 𝛼, 𝛽 ∈ R such that

𝛼𝑇𝑥 − 𝑇𝑦

2

+ (1 − 𝛼)𝑥 − 𝑇𝑦

2

≤ 𝛽𝑇𝑥 − 𝑦

2

+ (1 − 𝛽)𝑥 − 𝑦

2

(64)

for all 𝑥, 𝑦 ∈ 𝐶. Suppose that 𝐴(𝑇) ̸= 0, and let 𝑃 be the metricprojection from𝐻 onto 𝐴(𝑇). Then for any 𝑥 ∈ 𝐶,

𝑆𝑛𝑥 =

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥 (65)

converges weakly to 𝑝 ∈ 𝐴(𝑇), where 𝑝 = lim𝑛→∞

𝑃𝑇𝑛𝑥.

5. Weak Convergence Theorems ofMann’s Type

In this section, we prove a weak convergence theorem ofMann’s type [20] for widely more generalized hybrid map-pings in a Hilbert space by using Lemma 11 and the techniquedeveloped by Ibaraki and Takahashi [27, 28].

Theorem 14. Let 𝐻 be a Hilbert space, and let 𝐶 be a convexsubset of 𝐻. Let 𝑇 : 𝐶 → 𝐶 be a widely more generalizedhybrid mapping with𝐴(𝑇) ̸= 0 such that it satisfies either of theconditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 > 0 and 𝜀 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0 and 𝜁 + 𝜂 ≥ 0.

Let 𝑃 be the metric projection of 𝐻 onto 𝐴(𝑇). Let {𝛼𝑛}

be a sequence of real numbers such that 0 ≤ 𝛼𝑛≤ 1 and

lim inf𝑛→∞

𝛼𝑛(1 − 𝛼

𝑛) > 0. Suppose that {𝑥

𝑛} is the sequence

generated by 𝑥1= 𝑥 ∈ 𝐶 and

𝑥𝑛+1

= 𝛼𝑛𝑥𝑛+ (1 − 𝛼

𝑛) 𝑇𝑥𝑛, 𝑛 ∈ N. (66)

Then {𝑥𝑛} converges weakly to V ∈ 𝐴(𝑇), where V =

lim𝑛→∞

𝑃𝑥𝑛.

Proof. Let 𝑧 ∈ 𝐴(𝑇). We have that

𝑥𝑛+1 − 𝑧

2

=𝛼𝑛𝑥𝑛 + (1 − 𝛼𝑛) 𝑇𝑥𝑛 − 𝑧

2

≤ 𝛼𝑛

𝑥𝑛 − 𝑧

2

+ (1 − 𝛼𝑛)𝑇𝑥𝑛 − 𝑧

2

≤ 𝛼𝑛

𝑥𝑛 − 𝑧

2

+ (1 − 𝛼𝑛)𝑥𝑛 − 𝑧

2

=𝑥𝑛 − 𝑧

2

(67)

for all 𝑛 ∈ N. Hence lim𝑛→∞

‖𝑥𝑛− 𝑧‖2 exists. Then {𝑥

𝑛} is

bounded. We also have from (8) that𝑥𝑛+1 − 𝑧

2

=𝛼𝑛𝑥𝑛 + (1 − 𝛼𝑛) 𝑇𝑥𝑛 − 𝑧

2

= 𝛼𝑛

𝑥𝑛−𝑧

2

+(1−𝛼𝑛)𝑇𝑥𝑛−𝑧

2

−𝛼𝑛(1−𝛼𝑛)𝑇𝑥𝑛−𝑥𝑛

2

≤ 𝛼𝑛

𝑥𝑛−𝑧

2

+(1−𝛼𝑛)𝑥𝑛−𝑧

2

−𝛼𝑛(1−𝛼𝑛)𝑇𝑥𝑛−𝑥𝑛

2

=𝑥𝑛 − 𝑧

2

− 𝛼𝑛(1 − 𝛼

𝑛)𝑇𝑥𝑛 − 𝑥𝑛

2

.

(68)

Thus we have

𝛼𝑛(1 − 𝛼

𝑛)𝑇𝑥𝑛 − 𝑥𝑛

2

≤𝑥𝑛 − 𝑧

2

−𝑥𝑛+1 − 𝑧

2

. (69)

Since lim𝑛→∞

‖𝑥𝑛− 𝑧‖2 exists and lim inf

𝑛→∞𝛼𝑛(1−𝛼𝑛) > 0,

we have that𝑇𝑥𝑛 − 𝑥𝑛

→ 0. (70)

Since {𝑥𝑛} is bounded, there exists a subsequence {𝑥

𝑛𝑖} of {𝑥

𝑛}

such that 𝑥𝑛𝑖

⇀ V. By Lemma 11 and (70), we obtain thatV ∈ 𝐴(𝑇). Let {𝑥

𝑛𝑖} and {𝑥

𝑛𝑗} be two subsequences of {𝑥

𝑛}

such that 𝑥𝑛𝑖

⇀ V1and 𝑥

𝑛𝑗⇀ V2. To complete the proof,

we show V1= V2. We know that V

1, V2∈ 𝐴(𝑇), and hence

lim𝑛→∞

‖𝑥𝑛− V1‖2 and lim

𝑛→∞‖𝑥𝑛− V2‖2 exist. Put

𝑎 = lim𝑛→∞

(𝑥𝑛 − V1

2

−𝑥𝑛 − V2

2

) . (71)

Note that for 𝑛 = 1, 2, . . .,𝑥𝑛 − V1

2

−𝑥𝑛 − V2

2

= 2 ⟨𝑥𝑛, V2− V1⟩ +

V1

2

−V2

2

.

(72)

From 𝑥𝑛𝑖⇀ V1and 𝑥

𝑛𝑗⇀ V2, we have

𝑎 = 2 ⟨V1, V2− V1⟩ +

V1

2

−V2

2

, (73)

𝑎 = 2 ⟨V2, V2− V1⟩ +

V1

2

−V2

2

. (74)

8 Abstract and Applied Analysis

Combining (73) and (74), we obtain 0 = 2⟨V2− V1, V2− V1⟩.

Thus we obtain V2= V1. This implies that {𝑥

𝑛} converges

weakly to an element V ∈ 𝐴(𝑇). Since ‖𝑥𝑛+1

− 𝑧‖ ≤ ‖𝑥𝑛− 𝑧‖

for all 𝑧 ∈ 𝐴(𝑇) and 𝑛 ∈ N, we obtain from Lemma 3 that{𝑃𝑥𝑛} converges strongly to an element 𝑝 ∈ 𝐴(𝑇). On the

other hand, we have from the property of 𝑃 that

⟨𝑥𝑛− 𝑃𝑥𝑛, 𝑃𝑥𝑛− 𝑢⟩ ≥ 0 (75)

for all 𝑢 ∈ 𝐴(𝑇) and 𝑛 ∈ N. Since 𝑥𝑛⇀ V and 𝑃𝑥

𝑛→ 𝑝, we

obtain

⟨V − 𝑝, 𝑝 − 𝑢⟩ ≥ 0 (76)

for all 𝑢 ∈ 𝐴(𝑇). Putting 𝑢 = V, we obtain 𝑝 = V. This meansV = lim

𝑛→∞𝑃𝑥𝑛. This completes the proof.

Using Theorem 14, we can show the following weakconvergence theorem of Mann’s type for generalized hybridmappings in a Hilbert space.

Theorem 15 (Kocourek et al. [9]). Let 𝐻 be a Hilbert space,and let 𝐶 be a closed convex subset of 𝐻. Let 𝑇 : 𝐶 → 𝐶be a generalized hybrid mapping with 𝐹(𝑇) ̸= 0. Let {𝛼

𝑛} be

a sequence of real numbers such that 0 ≤ 𝛼𝑛

≤ 1 andlim inf

𝑛→∞𝛼𝑛(1 − 𝛼

𝑛) > 0. Suppose {𝑥

𝑛} is the sequence

generated by 𝑥1= 𝑥 ∈ 𝐶 and

𝑥𝑛+1

= 𝛼𝑛𝑥𝑛+ (1 − 𝛼

𝑛) 𝑇𝑥𝑛, 𝑛 ∈ N. (77)

Then the sequence {𝑥𝑛} converges weakly to an element V ∈

𝐹(𝑇).

Proof. As in the proof of Theorem 9, a generalized hybridmapping is a widely more generalized hybrid mapping. Since{𝑥𝑛} ⊂ 𝐶 and 𝐶 is closed and convex, we have from

Theorem 14 that V ∈ 𝐴(𝑇)∩𝐶. A generalized hybrid mappingwith 𝐹(𝑇) ̸= 0 is quasi-nonexpansive, we have from Lemma 7that 𝐴(𝑇) ∩ 𝐶 = 𝐹(𝑇). Thus {𝑥

𝑛} converges weakly to an

element V ∈ 𝐹(𝑇).

6. Strong Convergence Theorem

In this section, using an idea of mean convergence byShimizu andTakahashi [21, 29], we prove the following strongconvergence theorem for widely more generalized hybridmappings in a Hilbert space.

Theorem 16. Let 𝐶 be a nonempty convex subset of a realHilbert space 𝐻. Let 𝑇 be a widely more generalized hybridmapping of𝐶 into itself with𝐴(𝑇) ̸= 0 such that it satisfies eitherof the following conditions:

(1) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛾 > 0, 𝜀 + 𝜂 ≥ 0 and 𝜁 + 𝜂 ≥ 0;(2) 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 ≥ 0 and 𝜀 + 𝜂 ≥ 0.Let 𝑢 ∈ 𝐶, and define sequences {𝑥

𝑛} and {𝑧

𝑛} in 𝐶 as

follows: 𝑥1= 𝑥 ∈ 𝐶 and

𝑥𝑛+1

= 𝛼𝑛𝑢 + (1 − 𝛼

𝑛) 𝑧𝑛,

𝑧𝑛=1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥𝑛

(78)

for all 𝑛 = 1, 2, . . ., where 0 ≤ 𝛼𝑛≤ 1, 𝛼

𝑛→ 0 and ∑∞

𝑛=1𝛼𝑛=

∞. If 𝐴(𝑇) is nonempty, then {𝑥𝑛} and {𝑧

𝑛} converge strongly

to 𝑃𝑢, where 𝑃 is the metric projection of𝐻 onto 𝐴(𝑇).

Proof. Since 𝑇 : 𝐶 → 𝐶 is a widely more generalized hybridmapping, there exist 𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, 𝜂 ∈ R such that

𝛼𝑇𝑥 − 𝑇𝑦

2

+ 𝛽𝑥 − 𝑇𝑦

2

+ 𝛾𝑇𝑥 − 𝑦

2

+ 𝛿𝑥 − 𝑦

2

+ 𝜀‖𝑥 − 𝑇𝑥‖2+𝜁

𝑦 − 𝑇𝑦

2

+𝜂(𝑥 − 𝑇𝑥)−(𝑦 − 𝑇𝑦)

2

≤0

(79)

for any 𝑥, 𝑦 ∈ 𝐶. Since𝐴(𝑇) ̸= 0, we have that for all 𝑞 ∈ 𝐴(𝑇)and 𝑛 = 1, 2, 3, . . .,

𝑧𝑛 − 𝑞 =

1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥𝑛− 𝑞

≤1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥𝑛− 𝑞

≤1

𝑛

𝑛−1

∑

𝑘=0

𝑥𝑛 − 𝑞 =

𝑥𝑛 − 𝑞 .

(80)

Then we have𝑥𝑛+1 − 𝑞

=𝛼𝑛𝑢 + (1 − 𝛼𝑛) 𝑧𝑛 − 𝑞

≤ 𝛼𝑛

𝑢 − 𝑞 + (1 − 𝛼𝑛)

𝑧𝑛 − 𝑞

≤ 𝛼𝑛

𝑢 − 𝑞 + (1 − 𝛼𝑛)

𝑥𝑛 − 𝑞 .

(81)

Hence, by induction, we obtain𝑥𝑛 − 𝑞

≤ max {𝑢 − 𝑞

,𝑥 − 𝑞

} (82)

for all 𝑛 ∈ N. This implies that {𝑥𝑛} and {𝑧

𝑛} are bounded.

Since ‖𝑇𝑛𝑥𝑛− 𝑞‖ ≤ ‖𝑥

𝑛− 𝑞‖, we have also that {𝑇𝑛𝑥

𝑛} is

bounded.Let 𝑛 ∈ N. Using 𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0 and 𝜁 + 𝜂 ≥ 0, as in the

proof of Theorem 8 we have that

(𝛼 + 𝜂) (𝑇𝑧 − 𝑇

𝑘+1𝑥𝑛

2

−𝑇𝑧 − 𝑇

𝑘𝑥𝑛

2

)

+ (𝛽 − 𝜂) (𝑧 − 𝑇𝑘+1

𝑥𝑛

2

−𝑧 − 𝑇𝑘𝑥𝑛

2

)

− 2 (𝛼+𝛾) ⟨𝑧−𝑇𝑧, 𝑧−𝑇𝑘𝑥𝑛⟩+(𝛼+𝛾+𝜀+𝜂) ‖𝑧−𝑇𝑧‖

2

≤ 0

(83)

for any 𝑘 ∈ N∪ {0} and 𝑧 ∈ 𝐶. Summing up these inequalitieswith respect to 𝑘 = 0, 1, . . . , 𝑛−1 and dividing by 𝑛, we obtainthat𝛼 + 𝜂

𝑛(𝑇𝑧 − 𝑇

𝑛𝑥𝑛

2

−𝑇𝑧 − 𝑥𝑛

2

)

+𝛽 − 𝜂

𝑛(𝑧 − 𝑇

𝑛𝑥𝑛

2

−𝑧 − 𝑥𝑛

2

)

− 2 (𝛼 + 𝛾) ⟨𝑧 − 𝑇𝑧, 𝑧 − 𝑧𝑛⟩ + (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑧 − 𝑇𝑧‖

2

≤ 0.

(84)

Abstract and Applied Analysis 9

Since {𝑧𝑛} is bounded, there exists a subsequence {𝑧

𝑛𝑖} of {𝑧

𝑛}

such that 𝑧𝑛𝑖⇀ 𝑤 ∈ 𝐻. Replacing 𝑛 by 𝑛

𝑖, we have that

𝛼 + 𝜂

𝑛𝑖

(𝑇𝑧 − 𝑇

𝑛𝑖𝑥𝑛𝑖

2

−𝑇𝑧 − 𝑥

𝑛𝑖

2

)

+𝛽 − 𝜂

𝑛𝑖

(𝑧 − 𝑇𝑛𝑖𝑥𝑛𝑖

2

−𝑧 − 𝑥𝑛𝑖

2

)

− 2 (𝛼+𝛾) ⟨𝑧−𝑇𝑧, 𝑧−𝑧𝑛𝑖⟩+(𝛼+𝛾+𝜀+𝜂) ‖𝑧−𝑇𝑧‖

2≤0.

(85)

Since {𝑥𝑛} and {𝑇𝑛𝑥

𝑛} are bounded, we have that

−2 (𝛼 + 𝛾) ⟨𝑧 − 𝑇𝑧, 𝑧 − 𝑤⟩ + (𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑧 − 𝑇𝑧‖2≤ 0

(86)

as 𝑖 → ∞. Using (9), we have that

(𝛼 + 𝛾 + 𝜀 + 𝜂) ‖𝑧 − 𝑇𝑧‖2

+ (𝛼 + 𝛾) (‖𝑤 − 𝑇𝑧‖2− ‖𝑤 − 𝑧‖

2− ‖𝑧 − 𝑇𝑧‖

2) ≤ 0,

(87)

and hence

(𝜀 + 𝜂) ‖𝑧 − 𝑇𝑧‖2+ (𝛼 + 𝛾) (‖𝑤 − 𝑇𝑧‖

2− ‖𝑤 − 𝑧‖

2) ≤ 0.

(88)

Since 𝛼 + 𝛾 > 0 and 𝜀 + 𝜂 ≥ 0, we have that

‖𝑤 − 𝑇𝑧‖2− ‖𝑤 − 𝑧‖

2≤ 0 (89)

for all 𝑧 ∈ 𝐶. This implies that 𝑤 ∈ 𝐴(𝑇).On the other hand, since 𝑥

𝑛+1− 𝑧𝑛= 𝛼𝑛(𝑢 − 𝑧

𝑛), {𝑧𝑛} is

bounded, and 𝛼𝑛→ 0, we have lim

𝑛→∞‖𝑥𝑛+1

− 𝑧𝑛‖ = 0. Let

us showlim sup𝑛→∞

⟨𝑢 − 𝑃𝑢, 𝑥𝑛+1

− 𝑃𝑢⟩ ≤ 0. (90)

We may assume without loss of generality that there exists asubsequence {𝑥

𝑛𝑖+1} of {𝑥

𝑛+1} such that

lim sup𝑛→∞

⟨𝑢 − 𝑃𝑢, 𝑥𝑛+1

− 𝑃𝑢⟩ = lim𝑖→∞

⟨𝑢 − 𝑃𝑢, 𝑥𝑛𝑖+1

− 𝑃𝑢⟩

(91)

and 𝑥𝑛𝑖+1

⇀ V. From ‖𝑥𝑛+1

− 𝑧𝑛‖ → 0, we have 𝑧

𝑛𝑖⇀ V.

From the above argument, we have V ∈ 𝐴(𝑇). Since 𝑃 is themetric projection of𝐻 onto 𝐴(𝑇), we have

lim𝑖→∞

⟨𝑢 − 𝑃𝑢, 𝑥𝑛𝑖+1

− 𝑃𝑢⟩ = ⟨𝑢 − 𝑃𝑢, V − 𝑃𝑢⟩ ≤ 0. (92)

This implies

lim sup𝑛→∞

⟨𝑢 − 𝑃𝑢, 𝑥𝑛+1

− 𝑃𝑢⟩ ≤ 0. (93)

Since 𝑥𝑛+1

− 𝑃𝑢 = (1 − 𝛼𝑛)(𝑧𝑛− 𝑃𝑢) + 𝛼

𝑛(𝑢 − 𝑃𝑢), from (7)

and (80) we have𝑥𝑛+1 − 𝑃𝑢

2

=(1 − 𝛼𝑛) (𝑧𝑛 − 𝑃𝑢) + 𝛼𝑛 (𝑢 − 𝑃𝑢)

2

≤ (1−𝛼𝑛)2𝑧𝑛−𝑃𝑢

2

+ 2𝛼𝑛⟨𝑢−𝑃𝑢, 𝑥

𝑛+1−𝑃𝑢⟩

≤ (1−𝛼𝑛)𝑥𝑛−𝑃𝑢

2

+ 2𝛼𝑛⟨𝑢−𝑃𝑢, 𝑥

𝑛+1−𝑃𝑢⟩ .

(94)

Putting 𝑠𝑛= ‖𝑥𝑛− 𝑃𝑢‖

2,𝛽𝑛= 0, and 𝛾

𝑛= 2⟨𝑢−𝑃𝑢, 𝑥

𝑛+1−𝑃𝑢⟩

in Lemma 4, we have from ∑∞𝑛=1

𝛼𝑛= ∞ and (93) that

lim𝑛→∞

𝑥𝑛 − 𝑃𝑢 = 0. (95)

By lim𝑛→∞

‖𝑥𝑛+1

− 𝑧𝑛‖ = 0, we also obtain 𝑧

𝑛→ 𝑃𝑢 as 𝑛 →

∞.Similarly, we can obtain the desired result for the case of

𝛼 + 𝛽 + 𝛾 + 𝛿 ≥ 0, 𝛼 + 𝛽 > 0, 𝜁 + 𝜂 ≥ 0, and 𝜀 + 𝜂 ≥ 0.

Using Theorem 16, we can show the following resultobtained by Kurokawa and Takahashi [30].

Theorem 17 (Hojo and Takahashi [22]). Let𝐶 be a nonemptyclosed convex subset of a real Hilbert space 𝐻. Let 𝑇 be ageneralized hybrid mapping of 𝐶 into itself. Let 𝑢 ∈ 𝐶 anddefine two sequences {𝑥

𝑛} and {𝑧

𝑛} in 𝐶 as follows: 𝑥

1= 𝑥 ∈ 𝐶

and𝑥𝑛+1

= 𝛼𝑛𝑢 + (1 − 𝛼

𝑛) 𝑧𝑛,

𝑧𝑛=1

𝑛

𝑛−1

∑

𝑘=0

𝑇𝑘𝑥𝑛

(96)

for all 𝑛 = 1, 2, . . ., where 0 ≤ 𝛼𝑛≤ 1, 𝛼

𝑛→ 0 and ∑∞

𝑛=1𝛼𝑛=

∞. If 𝐹(𝑇) is nonempty, then {𝑥𝑛} and {𝑧

𝑛} converge strongly

to 𝑃𝑢 ∈ 𝐹(𝑇), where 𝑃 is the metric projection of𝐻 onto𝐴(𝑇).

Proof. As in the proof of Theorem 9, a generalized hybridmapping is a widely more generalized hybrid mapping. Since{𝑥𝑛}, {𝑧𝑛} ⊂ 𝐶 and 𝐶 is closed and convex, we have from

Theorem 16 that 𝑃𝑢 ∈ 𝐴(𝑇) ∩ 𝐶. A generalized hybridmapping with 𝐹(𝑇) ̸= 0 is quasi-nonexpansive, we have fromLemma 7 that 𝐴(𝑇) ∩ 𝐶 = 𝐹(𝑇). Thus {𝑥

𝑛} and {𝑧

𝑛} converge

strongly to an element 𝑃𝑢 ∈ 𝐹(𝑇).

Acknowledgments

Research for Sy-Ming Guu is partially supported by NSC100-2221-E-182-072-MY2. Wataru Takahashi is partially sup-ported by Grant-in-Aid for Scientific Research no. 23540188from Japan Society for the Promotion of Science.

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[24] K. Aoyama, Y. Kimura, W. Takahashi, andM. Toyoda, “Approx-imation of common fixed points of a countable family ofnonexpansive mappings in a Banach space,”Nonlinear Analysis:Theory, Methods and Applications A, vol. 67, no. 8, pp. 2350–2360, 2007.

[25] W. Takahashi, Nonlinear Functional Analysis. Fixed PointsTheory and Its Applications, Yokohama Publishers, Yokohama,Japan, 2000.

[26] W. Takahashi, “A nonlinear ergodic theorem for an amenablesemigroup of nonexpansive mappings in a Hilbert space,”Proceedings of the American Mathematical Society, vol. 81, no.2, pp. 253–256, 1981.

[27] T. Ibaraki and W. Takahashi, “Weak convergence theoremfor new nonexpansive mappings in Banach spaces and itsapplications,” Taiwanese Journal of Mathematics, vol. 11, no. 3,pp. 929–944, 2007.

[28] T. Ibaraki andW. Takahashi, “Fixed point theorems for nonlin-ear mappings of nonexpansive type in Banach spaces,” Journalof Nonlinear and Convex Analysis, vol. 10, no. 1, pp. 21–32, 2009.

[29] T. Shimizu andW. Takahashi, “Strong convergence to commonfixed points of families of nonexpansive mappings,” Journal ofMathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997.

[30] Y. Kurokawa andW. Takahashi, “Weak and strong convergencetheorems for nonspreading mappings in Hilbert spaces,” Non-linear Analysis: Theory, Methods and Applications A, vol. 73, no.6, pp. 1562–1568, 2010.

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 503198, 8 pageshttp://dx.doi.org/10.1155/2013/503198

Research ArticleImplicit Ishikawa Approximation Methods forNonexpansive Semigroups in CAT(0) Spaces

Zhi-bin Liu,1,2 Yi-shen Chen,1,3 Xue-song Li,4 and Yi-bin Xiao5

1 Southwest Petroleum University, Chengdu 610500, China2 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu 610500, China3 China Petroleum Engineering and Construct Corporation, Beijing 100120, China4Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China5 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China

Correspondence should be addressed to Xue-song Li; xuesongli78@hotmail.com

Received 29 November 2012; Accepted 12 June 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 Zhi-bin Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point ofnonexpansive semigroup in CAT(0) spaces. We obtain the Δ-convergence results of the implicit Ishikawa iteration sequences fora family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergencetheorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces.The results presented in thispaper extend and generalize some previous results.

1. Introduction

Let (𝑋, 𝑑)be ametric space and𝐾be a subset of𝑋. Amapping𝑇 : 𝐾 → 𝑋 is said to be nonexpansive if 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑑(𝑥, 𝑦)for all 𝑥, 𝑦 ∈ 𝐾. We denote the set of all nonnegative elementsinR byR+ and denote the set of all fixed points of 𝑇 by 𝐹(𝑇),that is,

𝐹 (𝑇) = {𝑥 ∈ 𝐾 : 𝑇𝑥 = 𝑥} . (1)

For each 𝑛 ∈ N, let 𝑇𝑛: 𝐾 → 𝐾 be nonexpansive mappings

and denote the common fixed points set of the family {𝑇𝑛} by

⋂∞

𝑛=1𝐹(𝑇𝑛). A family of mappings {𝑇

𝑛} is said to be uniformly

asymptotically regular if, for any bounded subset 𝐵 of𝐾,

lim𝑛→∞

sup𝑧∈𝐵

𝑑 (𝑇𝑛𝑧, 𝑇𝑖(𝑇𝑛𝑧)) = 0, (2)

for all 𝑖 ∈ N.A nonexpansive semigroup is a family,

Γ := {𝑇 (𝑡) : 𝑡 ≥ 0} , (3)

of mappings 𝑇(𝑡) on𝐾 such that

(1) 𝑇(𝑠 + 𝑡)𝑥 = 𝑇(𝑠)(𝑇(𝑡)𝑥) for all 𝑥 ∈ 𝐾 and 𝑠, 𝑡 ≥ 0;(2) 𝑇(𝑡) : 𝐾 → 𝐾 is nonexpansive for each 𝑡 ≥ 0;(3) for each 𝑥 ∈ 𝐾, the mapping 𝑇(⋅)𝑥 from R+ to 𝐾 is

continuous.

We denote by𝐹(Γ) the common fixed points set of nonex-pansive semigroup Γ, that is,

𝐹 (Γ)

= ⋂

𝑡∈R+

𝐹 (𝑇 (𝑡)) = {𝑥 ∈ 𝑋 : 𝑇 (𝑡) 𝑥 = 𝑥 for each 𝑡 ≥ 0} .

(4)

Note that, if 𝐾 is compact, then 𝐹(Γ) is nonempty (see [1, 2,28]).

A geodesic from 𝑥 to 𝑦 in𝑋 is a mappingΨ from a closedinterval [0, 𝑙] ⊂ R to 𝑋 such that Ψ(0) = 𝑥, Ψ(𝑙) = 𝑦, and𝑑(Ψ(𝑡), Ψ(𝑡

)) = |𝑡 − 𝑡

| for all 𝑡, 𝑡 ∈ [0, 𝑙]. In particular, Ψ

is an isometry and 𝑑(𝑥, 𝑦) = 𝑙. The image Θ of Ψ is called

2 Abstract and Applied Analysis

a geodesic (or metric) segment joining 𝑥 and 𝑦. The space(𝑋, 𝑑) is said to be a geodesic space if any two points of𝑋 arejoined by a geodesic, and 𝑋 is said to be uniquely geodesic ifthere is exactly one geodesic joining 𝑥 and 𝑦 for any 𝑥, 𝑦 ∈ 𝑋,which is denoted by [𝑥, 𝑦] and is called the segment joining 𝑥and 𝑦. A subset𝐾 of a geodesic space𝑋 is said to be convex iffor any 𝑥, 𝑦 ∈ 𝐾, [𝑥, 𝑦] ⊂ 𝐾.

A geodesic triangle Δ(𝑥1, 𝑥2, 𝑥3) in a geodesic metric

space (𝑋, 𝑑) consists of three points 𝑥1, 𝑥2, 𝑥3in 𝑋 (the ver-

tices of Δ) and a geodesic segment between each pair of ver-tices (the edges of Δ). A comparison triangle for the geodesictriangle Δ(𝑥

1, 𝑥2, 𝑥3) in (𝑋, 𝑑) is a triangle Δ(𝑥

1, 𝑥2, 𝑥3) =

Δ(𝑥1, 𝑥2, 𝑥3) in R2 such that 𝑑R2(𝑥𝑖, 𝑥𝑗) = 𝑑(𝑥𝑖, 𝑥𝑗) for all

𝑖, 𝑗 ∈ {1, 2, 3}. It is known that such a triangle always exists(see [3]). A geodesic space is said to be a CAT(0) space if allgeodesic triangles of appropriate size satisfy the followingcomparison axiom (CA).

Let Δ be a geodesic triangle in (𝑋, 𝑑), and let Δ ⊂ R2be a comparison triangle for Δ. Then, Δ is said to satisfy theCAT(0) inequality if, for all 𝑥, 𝑦 ∈ Δ and all comparisonpoints 𝑥, 𝑦 ∈ Δ,

𝑑 (𝑥, 𝑦) ≤ 𝑑 (𝑥, 𝑦) . (5)

The complete CAT(0) spaces are often called, Hadamardspaces (see [4]). For any 𝑥, 𝑦 ∈ 𝑋, we denote by 𝛼𝑥⊕ (1−𝛼)𝑦the unique point 𝑧 ∈ [𝑥, 𝑦] which satisfies

𝑑 (𝑥, 𝛼𝑥 ⊕ (1 − 𝛼) 𝑦) = (1 − 𝛼) 𝑑 (𝑥, 𝑦) ,

𝑑 (𝑦, 𝛼𝑥 ⊕ (1 − 𝛼) 𝑦) = 𝛼𝑑 (𝑥, 𝑦) .(6)

It is known that if (𝑋, 𝑑) is a CAT(0) space and 𝑥, 𝑦 ∈ 𝑋, thenfor any 𝛽 ∈ [0, 1], there exists a unique point 𝛽𝑥 ⊕ (1 − 𝛽)𝑦 ∈[𝑥, 𝑦]. For any 𝑧 ∈ 𝑋, the following inequality holds:

𝑑 (𝑧, 𝛽𝑥 ⊕ (1 − 𝛽) 𝑦) ≤ 𝛽𝑑 (𝑧, 𝑥) + (1 − 𝛽) 𝑑 (𝑧, 𝑦) , (7)

where 𝛽𝑥 ⊕ (1 − 𝛽)𝑦 ∈ [𝑥, 𝑦] (for metric spaces of hyperbolictype, see [5]).

Recently, Cho et al. [6] proved the strong convergence ofan explicit iterative sequence {𝑧

𝑛} in a CAT(0) space, where

{𝑧𝑛} is generated by the following iterative scheme for a non-

expansive semigroup Γ:

𝑧1∈ 𝐾, 𝑧

𝑛+1= 𝛼𝑧𝑛⊕ (1 − 𝛼) 𝑇 (𝑡

𝑛) 𝑧𝑛, ∀𝑛 ≥ 1, (8)

where 𝛼 ∈ (0, 1) and {𝑡𝑛} ⊂ R+. The existence of fixed points,

an invariant approximation, and convergence theorems forseveral mappings in CAT(0) spaces have been studied bymany authors (see [7–18]).

On the other hand, Thong [19] considered an implicitMann iteration process for a nonexpansive semigroup Γ ={𝑇(𝑡) : 𝑡 ∈ R+} on a closed convex subset𝐶 of a Banach spaceas follows:

𝑥1∈ 𝐶, 𝑥

𝑛+1= 𝛼𝑛𝑥𝑛+ (1 − 𝛼

𝑛) 𝑇 (𝑡𝑛) 𝑥𝑛+1

, 𝑛 ≥ 1. (9)

Under different conditions, Thong [19] proved the weakconvergence and strong convergence results of the implicitMann iteration scheme (9) for nonexpansive semigroups

in certain Banach spaces. Many authors have studied theconvergence of implicit iteration sequences for nonexpansivemappings, nonexpansive semigroups and pseudocontractivesemigroups in the Banach spaces (see [20–23]). Readers mayconsult [24, 25] for the convergence of the Ishikawa iterationsequences for nonexpansive mappings and nonexpansivesemigroups in certain Banach spaces. However, to our bestknowledge, there is no paper to study the convergence of theimplicit Ishikawa type iteration processes for nonexpansivesemigroups in CAT(0) spaces. Therefore, it is of interest toinvestigate the convergence of implicit Ishikawa type iterationprocesses for nonexpansive semigroups in CAT(0) spacesunder some suitable conditions.

Motivated and inspired by the work mentioned previ-ously, we consider the following implicit Ishikawa iterationscheme for a family of nonexpansive mappings in a CAT(0)space:

𝑥1∈ 𝐾,

𝑥𝑛+1

= 𝛼𝑛𝑥𝑛⊕ (1 − 𝛼

𝑛) ((1 − 𝜃

𝑛) 𝑇𝑛𝑥𝑛⊕ 𝜃𝑛𝑇𝑛𝑥𝑛+1

) ,

∀𝑛 ≥ 1,

(10)

where {𝛼𝑛} ⊂ (0, 1] and {𝜃

𝑛} ⊂ [0, 1] are given sequences.

We prove that {𝑥𝑛} generated by (10) is Δ-convergent to some

point in ⋂∞𝑛=1

𝐹(𝑇𝑛) under appropriate conditions. We also

consider the following implicit Ishikawa iteration process fora nonexpansive semigroup Γ = {𝑇(𝑡) : 𝑡 ∈ R+} in a CAT(0)space:

𝑥1∈ 𝐾,

𝑥𝑛+1

= 𝛼𝑛𝑥𝑛⊕ (1 − 𝛼

𝑛) ((1 − 𝜃

𝑛) 𝑇 (𝑡𝑛) 𝑥𝑛⊕ 𝜃𝑛𝑇 (𝑡𝑛) 𝑥𝑛+1

) ,

∀𝑛 ≥ 1,

(11)

where {𝛼𝑛} ⊂ (0, 1] and {𝜃

𝑛} ⊂ [0, 1] are given sequences.

Under various and appropriate conditions, we obtain that{𝑥𝑛} generated by (11) converges strongly to a common fixed

point of Γ.The results presented in this paper extend and gen-eralize some previous results in [6, 19].

2. Definitions and Lemmas

Let {𝑥𝑛} be a bounded sequence in a CAT(0) space (𝑋, 𝑑). For

any 𝑥 ∈ 𝑋, denote

𝑟 (𝑥, {𝑥𝑛}) = lim sup

𝑛→∞

𝑑 (𝑥, 𝑥𝑛) . (12)

Consider the following:

(i) 𝑟({𝑥𝑛}) = inf{𝑟(𝑥, 𝑥

𝑛) : 𝑥 ∈ 𝑋} is called the asymp-

totic radius of {𝑥𝑛};

(ii) 𝑟𝐾({𝑥𝑛}) = inf{𝑟(𝑥, 𝑥

𝑛) : 𝑥 ∈ 𝐾} is called the asymp-

totic radius of {𝑥𝑛} with respect to 𝐾;

(iii) the set 𝐴({𝑥𝑛}) = {𝑥 ∈ 𝑋 : 𝑟(𝑥, {𝑥

𝑛}) = 𝑟({𝑥

𝑛})} is

called the asymptotic center of {𝑥𝑛};

(iv) the set 𝐴𝐾({𝑥𝑛}) = {𝑥 ∈ 𝐾 : 𝑟(𝑥, {𝑥

𝑛}) = 𝑟

𝐾({𝑥𝑛})} is

called the asymptotic center of {𝑥𝑛}with respect to𝐾.

Abstract and Applied Analysis 3

Definition 1 (see [12, 26]). A sequence {𝑥𝑛} in a CAT(0) space

𝑋 is said to be Δ-convergent to a point 𝑥 in 𝑋, if 𝑥 is theunique asymptotic center of {𝑥

𝑛𝑗} for all subsequences {𝑥

𝑛𝑗} ⊆

{𝑥𝑛}. In this case, we write Δ-lim

𝑛→∞𝑥𝑛= 𝑥, and 𝑥 is called

the Δ-limit of {𝑥𝑛}.

For the sake of convenience, we restate the following lem-mas that shall be used.

Lemma 2 (see [10]). Let (𝑋, 𝑑) be a CAT(0) space. Then,

𝑑 ((1 − 𝑡) 𝑥 ⊕ 𝑡𝑦, 𝑧) ≤ (1 − 𝑡) 𝑑 (𝑥, 𝑧) + 𝑡𝑑 (𝑦, 𝑧) , (13)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑡 ∈ [0, 1].

Lemma 3 (see [10]). Let (𝑋, 𝑑) be a CAT(0) space. Then,

[𝑑 ((1 − 𝑡) 𝑥 ⊕ 𝑡𝑦, 𝑧)]2

≤ (1 − 𝑡) [𝑑 (𝑥, 𝑧)]2+ 𝑡[𝑑 (𝑦, 𝑧)]

2

− 𝑡 (1 − 𝑡) [𝑑 (𝑥, 𝑦)]2

,

(14)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑡 ∈ [0, 1].

Lemma 4 (see [10]). Let𝐾 be a closed convex subset of a com-plete CAT(0) space and 𝑇 : 𝐾 → 𝐾 be a nonexpansive map-ping. Suppose that {𝑥

𝑛} is a bounded sequence in 𝐾 such that

lim𝑛→∞

𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0 and {𝑑(𝑥

𝑛, 𝑝)} converges for all 𝑝 ∈

𝐹(𝑇). Then, 𝜔𝑤(𝑥𝑛) = ⋃𝐴({𝑥

𝑛𝑗}) ⊂ 𝐹(𝑇), where the union is

taken over all subsequences {𝑥𝑛𝑗} of {𝑥

𝑛}. Moreover, 𝜔

𝑤(𝑥𝑛)

consists of exactly one point.

Lemma 5 (see [6]). Let {𝑧𝑛} and {𝑤

𝑛} be bounded sequences in

a CAT(0) space𝑋. Let {𝛼𝑛} be a sequence in [0, 1] such that 0 <

lim inf𝑛→∞

𝛼𝑛≤ lim sup

𝑛→∞𝛼𝑛< 1. Define 𝑧

𝑛+1= 𝛼𝑛𝑧𝑛⊕

(1 − 𝛼𝑛)𝑤𝑛for all 𝑛 ∈ N and suppose that

lim sup𝑛→∞

[𝑑 (𝑤𝑛+1

, 𝑤𝑛) − 𝑑 (𝑧

𝑛+1, 𝑧𝑛)] ≤ 0. (15)

Then, lim𝑛→∞

𝑑(𝑤𝑛, 𝑧𝑛) = 0.

3. Main Results

It is necessary for us to show that the implicit Ishikawa iter-ation sequences generated by schemes (10) and (11) are welldefined, before providing the main results of this presentpaper.

Lemma 6. Let 𝐾 be a nonempty, closed, and convex subset ofa complete CAT(0) space𝑋 and 𝑇

𝑛: 𝐾 → 𝐾 be nonexpansive

mappings. Suppose that {𝛼𝑛} ⊂ (0, 1] and {𝜃

𝑛} ⊂ [0, 1] are

given parameter sequences. Then, the sequence {𝑥𝑛} generated

by the implicit Ishikawa iteration process (10) is well defined.

Proof. For each 𝑛 ∈ N and any given 𝑢 ∈ 𝐾, define a mapping𝑆𝑛+1

: 𝐾 → 𝐾 by

𝑆𝑛+1

𝑥 := 𝛼𝑛𝑢 ⊕ (1 − 𝛼

𝑛) ((1 − 𝜃

𝑛) 𝑇𝑛𝑢 ⊕ 𝜃𝑛𝑇𝑛𝑥) , ∀𝑛 ≥ 1.

(16)

It can be verified that for any fixed 𝑛 ∈ N, 𝑆𝑛+1

is a contractivemapping. Indeed, if setting 𝑝

𝑛= (1 − 𝜃

𝑛)𝑇𝑛𝑢 ⊕ 𝜃

𝑛𝑇𝑛𝑥 and

𝑞𝑛= (1 − 𝜃

𝑛)𝑇𝑛𝑢 ⊕ 𝜃

𝑛𝑇𝑛𝑦, then we have 𝑆

𝑛+1𝑥 = 𝛼

𝑛𝑢 ⊕ (1 −

𝛼𝑛)𝑝𝑛and 𝑆𝑛+1

𝑦 = 𝛼𝑛𝑢 ⊕ (1 −𝛼

𝑛)𝑞𝑛. It follows from Lemmas

3 and 2 that

[𝑑 (𝑝𝑛, 𝑞𝑛)]2

= [𝑑 (𝑝𝑛, (1 − 𝜃

𝑛) 𝑇𝑛𝑢 ⊕ 𝜃𝑛𝑇𝑛𝑦)]2

≤ 𝜃𝑛[𝑑 (𝑝𝑛, 𝑇𝑛𝑦)]2

+ (1 − 𝜃𝑛) [𝑑 (𝑝

𝑛, 𝑇𝑛𝑢)]2

− 𝜃𝑛(1 − 𝜃

𝑛) [𝑑 (𝑇

𝑛𝑢, 𝑇𝑛𝑦)]2

≤ 𝜃𝑛{(1 − 𝜃

𝑛) [𝑑 (𝑇

𝑛𝑢, 𝑇𝑛𝑦)]2

+ 𝜃𝑛[𝑑 (𝑇𝑛𝑥, 𝑇𝑛𝑦)]2

−𝜃𝑛(1 − 𝜃

𝑛) [𝑑 (𝑇

𝑛𝑢, 𝑇𝑛𝑥)]2

}

+ (1 − 𝜃𝑛) 𝜃2

𝑛[𝑑 (𝑇𝑛𝑢, 𝑇𝑛𝑥)]2

− 𝜃𝑛(1 − 𝜃

𝑛) [𝑑 (𝑇

𝑛𝑢, 𝑇𝑛𝑦)]2

= 𝜃2

𝑛[𝑑 (𝑇𝑛𝑥, 𝑇𝑛𝑦)]2

.

(17)

Consequently, 𝑑(𝑝𝑛, 𝑞𝑛) ≤ 𝜃

𝑛𝑑(𝑇𝑛𝑥, 𝑇𝑛𝑦) ≤ 𝜃

𝑛𝑑(𝑥, 𝑦), and

thus,

𝑑 (𝑆𝑛+1

𝑥, 𝑆𝑛+1

𝑦) ≤ (1 − 𝛼𝑛) 𝑑 (𝑝𝑛, 𝑞𝑛)

≤ (1 − 𝛼𝑛) 𝜃𝑛𝑑 (𝑥, 𝑦) ,

(18)

which shows that for each 𝑛 ∈ N, 𝑆𝑛+1

is a contractivemapping. By induction, Banach’s fixed theorem yields that thesequence {𝑥

𝑛} generated by (10) is well defined. This com-

pletes the proof.

We need the following lemma for our main results. Theanalogs of [6, Lemma 3.1] and [27, Lemma 2.2] are given inwhat follows.We sketch the proof here for the convenience ofthe reader.

Lemma 7. Let