Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces

Hussain Nawab, Rafiq Arif, Ciric

Description

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.

Sujets

• Banach space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 2
• Langue: English

Extrait

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: aarafi[e-mail] 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 ≤Lx–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 (.) that Tispseudocontractiveif and only if there existsj(x–y)∈

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