The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces. The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.
Hussain et al.Fixed Point Theory and Applications2012,2012:160 http://www.fixedpointtheoryandapplications.com/content/2012/1/160
R E S E A R C HOpen Access Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces 1 2*3 Nawab Hussain, Arif Rafiqand Ljubomir B Ciric
* Correspondence: aarafiq@gmail.com 2 Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan Full list of author information is available at the end of the article
Abstract The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces. Keywords:Ishikawa iteration scheme with errors; strictly hemicontractive operators; Lipschitz operators; Banach space
1 Preliminaries ∗ LetKbe a nonempty subset of an arbitrary Banach spaceEandEbe its dual space. The symbolsD(T),R(T) andF(T) stand for the domain, the range and the set of fixed points ofTrespectively (for a single-valued mapT:X→X,x∈Xis called a fixed point ofTiff ∗ E T(x) =x). We denote byJthe normalized duality mapping fromEto defined by
∗ ∗∗∗ J(x) =f∈E:x,f=x=f.
LetTbe a self-mapping ofK.
Definition ThenTis calledLipshitzianif there existsL> such that
Tx–Ty ≤Lx–y
(.)
for allx,y∈K. IfL= , thenTis callednon-expansive, and if ≤L< ,Tis calledcon-traction.
Definition [, ] . The mappingTis said to bepseudocontractiveif the inequality
x–y ≤x–y+t(I–T)x– (I–T)y
(.)
holds for eachx,y∈Kand for allt> . As a consequence of a result of Kato [], it follows from the inequality (.) thatTispseudocontractiveif and only if there existsj(x–y)∈