Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings

Authors:
H. Zegeye;N. Shahzad
Affiliations:
Department of Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana;Department of Mathematics, King Abdul Aziz University, P.O.B. 80203, Jeddah 21589, Saudi Arabia
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T"i:C-H,i=1,2,...,N, be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann's type method to a common fixed point of {T"i:i=1,2,...,N} provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on T or on C. As a consequence, it is proved that Mann's method converges for a fixed point of nonexpansive mapping provided that interior of F(T)0@?. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.