Approximation of common fixed points of a finite family of &amp;#216; - demicontractive mappings by an implicit iteration method.

Di Igbokwe,Ue Udofia

doi:10.4314/gjmas.v12i1.15

Abstract

We prove that the Implicit Iteration process of Xu and Ori (2001) converges strongly to the commonfixed pointsof a finite family of Ø - demicontractive mappings in real Hilbertand Banachspaces. Our results extend the results of Osilike (2004a) from strictl pseudocontractive maps to the much more general Ø - demicontractive maps;complement and generalize several others in the literature.KEY WORDS AND PHRASES: Ø - Demicontractive Maps, Implicit Iteration Process, Fixed Points,Strong Convergence.

Highlights

AND PRELIMINARIESLet be a real Banach space
The class of demicontractive maps was first introduced in Hilbert space by Hicks and Kubicek (1979)
This class of maps was first studied in arbitrary real Banach space by Isiogugu (2005)

Summary

Introduction

AND PRELIMINARIESLet be a real Banach space. Let J denote the normalized duality mapping from into 2 * given byLet K be a nonempty subset of E. An example of a demicontractive map which is not strictly pseudocontractive is given by A mapping T : K K is said to be demicontractive (Isiogugu, 2005) if F T and there exists a strictly increasing continuous function : 0, ) 0, ) with (0) 0 such that x Tx, j(x p) (|| x Tx ||), (3)

Results

Conclusion