Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications

Zhu Jin-Hua, Chang Shih-Sen, Liu, Liu Min

Description

In this article, an iterative sequence for relatively nonexpansive multi-valued mapping by modifying Halpern and Mann's iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also. AMS Subject Classification : 47H09; 47H10; 49J25. In this article, an iterative sequence for relatively nonexpansive multi-valued mapping by modifying Halpern and Mann's iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also. AMS Subject Classification : 47H09; 47H10; 49J25.

Sujets

• Fixed point

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 3
• Langue: English

Extrait

Zhuet al.Journal of Inequalities and Applications2012,2012:73 http://www.journalofinequalitiesandapplications.com/content/2012/1/73

R E S E A R C HOpen Access Strong convergence theorems by HalpernMann iterations for multivalued relatively nonexpansive mappings in Banach spaces with applications 1 2*1 Jinhua Zhu , Shihsen Changand Min Liu

* Correspondence: changss@yahoo. cn 2 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China Full list of author information is available at the end of the article

Abstract In this article, an iterative sequence for relatively nonexpansive multivalued mapping by modifying Halpern and Mann’s iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also. AMS Subject Classification: 47H09; 47H10; 49J25. Keywords:multivalued mapping, relatively nonexpansive, fixed point, iterative sequence

1 Introduction Throughout this article, we denote byNandℝthe sets of positive integers and real numbers, respectively. LetDbe a nonempty closed subset of a real Banach spaceE. A singlevalued mappingT:D®Dis called nonexpansive if∥TxTy∥≤∥xy∥for all x, yÎD. LetN(D) andCB(D) denote the family of nonempty subsets and nonempty closed bounded subsets ofD, respectively. The Hausdorff metric onCB(D) is defined by

H ( A 1 ,A 2 ) = max

supd(x,A 2 ), supd(y,A 1 ) x∈A 1 y∈A 2

forA 1 ,A 2 ÎCB(D), whered(x, A 1 ) = inf{∥xy∥,yÎA 1 }. The multivalued mapping T:D®CB(D) is called nonexpansive ifH(T(x),T(y))≤∥xy∥for allx, yÎD. An ele mentpÎDis called a fixed point ofT:D®N(D) ifpÎT(p). The set of fixed points ofTis represented byF(T). LetEbe a real Banach space with dualE*. We denote byJthe normalized duality E* mapping fromEdefined byto 2   2 ∗ ∗∗2∗  J(x) =x∈E:x,x=  x  =x,x∈E

where〈∙,∙〉denotes the generalized duality pairing.   x+y A Banach spaceEfor allis said to be strictly convex ifx, yÎU= {zÎE: < ∥z∥= 1} withx≠y.Eis said to be uniformly convex if, for eachÎ(0, 2], there exists x+y δ> 0 such thatfor allx, yÎUwith∥xy∥≥.Eis said to be smooth if <1−

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