Semistability of iterations in cone spaces

Yadegarnegad A, Jahedi S, Yousefi B, Vaezpour, Vaezpour Sm

Description

The aim of this work is to prove some iteration procedures in cone metric spaces. This extends some recent results of T-stability. Mathematics Subject Classification 47J25; 26A18. The aim of this work is to prove some iteration procedures in cone metric spaces. This extends some recent results of T-stability. Mathematics Subject Classification 47J25; 26A18.

Sujets

• Contraction
• Stability
• Affine

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 9
• Langue: English

Extrait

Yadegarnegadet al.Fixed Point Theory and Applications2011,2011:70 http://www.fixedpointtheoryandapplications.com/content/2011/1/70

R E S E A R C HOpen Access Semistability of iterations in cone spaces 1 2*1 3 A Yadegarnegad , S Jahedi, B Yousefiand SM Vaezpour

* Correspondence: jahedi@sutech. ac.ir 2 Department of Mathematics, Shiraz University of Technology, P. O. Box: 71555313, Shiraz, Iran Full list of author information is available at the end of the article

Abstract The aim of this work is to prove some iteration procedures in cone metric spaces. This extends some recent results of Tstability. Mathematics Subject Classification:47J25; 26A18. Keywords:Cone metric, contraction, stability, nonexpansive, affine, semicompact

1. Introduction LetEbe a real Banach space. A subsetP⊂Eis called a cone inEif it satisfies in the following conditions: (i)Pis closed, nonempty andP≠{0}. (ii)a,bÎR,a,b≥0 andx,yÎPimply thatax+byÎP. (iii)xÎPand xÎPimply thatx= 0. The spaceEcan be partially ordered by the coneP⊂E, by defining;x≤yif and only ifyxÎP, Also, we writex≪yifyxÎintP, where intPdenotes the interior ofP. A conePis called normal if there exists a constantk> 1 such that 0≤x≤y implies ||x||≤k||y||. In the following we suppose thatEis a real Banach space,Pis a cone inEand≤is a partial ordering with respect toP. Definition 1.1. ([1]) LetXbe a nonempty set. Assume that the mappingd:X×X® Esatisfies in the following conditions: (i) 0≤d(x,y) for allx,yÎXandd(x,y) = 0 if and only ifx=y, (ii)d(x,y) =d(y,x) for allx,yÎX. (iii)d(x,y)≤d(x,z) +d(z,y) for allx,y,zÎX. Thendis called a cone metric onXand (X,d) is called a cone metric space. IfTis a selfmap ofX, then byF(T) we mean the set of fixed points ofT. Also,N 0 denotes the set of nonnegative integers, i.e.,N 0 =N∪{0}. 1 Definition 1.2. ([2]) If 0 <a< 1, 0 <b, γ < we say that a mapT:X®Xis 2 Zamfirescu with respect to (a,b,g), if for each pairx,yÎX,Tsatisfies at least one of the following conditions: Z(1).d(Tx, Ty)≤ad(x,y), Z(2).d(Tx,Ty)≤b(d(x,Tx) +d(y,Ty)), Z(3).d(Tx,Ty)≤g(d(x, Ty) +d(y,Tx)). Usually for simplicity,Tis called a Zamfirescu operator ifTis Zamfirescu with respect to some (a,b,g), for some scalarsa,b,gwith above restrictions. Also,Tis

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