A Note on the Paper “Regularization Proximal Point Algorithm for Common Fixed Points of Nonexpansive Mappings in Banach Spaces”

J Optim Theory Appl (2012) 155:723–725

DOI 10.1007/s10957-012-0081-y

A Note on the Paper “Regularization Proximal Point

Algorithm for Common Fixed Points of Nonexpansive

Mappings in Banach Spaces”

Nguyen Thu Hang · Truong Minh Tuyen

Received: 8 April 2012 / Accepted: 30 April 2012 / Published online: 11 May 2012

© Springer Science+Business Media, LLC 2012

Abstract In this note, a small gap is corrected in the assumption of main theorem

of T.M. Tuyen (Theorem 3.1, Regularization proximal point algorithm for common

fixed points of nonexpansive mappings in Banach spaces, J. Optim. Theory Appl.,

152:351–365, 2012). We give another assumption, which allows us to obtain the

strong convergence of regularization algorithms.

Keywords Uniformly smooth and uniformly convex Banach space · Sunny

nonexpansive retraction · Weak sequential continuous mapping · Regularization

1 Errata Corrige

The paper [1] deals with the strong convergence of a regularization method for Rock￾afellar type proximal point algorithm. Particularly, T.M. Tuyen [1] introduced and

proved the following theorem:

Theorem 1.1 [1, Theorem 3.1] Suppose that E be a uniformly convex and uniformly

smooth Banach space which admits a weakly sequentially continuous normalized

duality mapping j from E to E∗. Let Ti : E −→ E, i = 1, 2,...,N be nonexpansive

mappings with S = N

i=1 Fix(Ti) = ∅. If the sequence {tn}⊂]0, 1[ satisfies

lim

n→∞ tn = 0; ∞

n=0

tn = +∞,

N.T. Hang · T.M. Tuyen ()

Department of Mathematics, Thainguyen University, Thainguyen, Vietnam

e-mail: [email protected]

N.T. Hang

e-mail: [email protected]