Hybrid Mann–Halpern iteration methods for nonexpansive mappings and semigroups

Hybrid Mann–Halpern iteration methods for nonexpansive mappings

and semigroups

Nguyen Buong a,⇑

, Nguyen Duc Lang b

a Vietnamese Academy of Science and Technology, Institute of Information Technology,18, Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam

b College of Science, Thainguyen National University, Viet Nam

article info

Keywords:

Metric projection

Fixed point

Nonexpansive mappings and semigroups

abstract

In this paper, we introduce some new iteration methods based on the hybrid method in

mathematical programming, the Mann’s iterative method and the Halpern’s method for

finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpan￾sive semigroup Hilbert spaces.

2011 Elsevier Inc. All rights reserved.

1. Introduction

Let H be a real Hilbert space with the scalar product and the norm denoted by the symbols h,i and kk, respectively, and

let C be a nonempty closed and convex subset of H. Denote by PC(x) the metric projection from x 2 H onto C. Let T be a non￾expansive mapping on C, i.e., T : C ? C and kTx Tyk 6 kx yk for all x, y 2 C. We use F(T) to denote the set of fixed points of

T, i.e., F(T)={x 2 C : x = Tx}. We know that F(T) is nonempty, if C is bounded, for more details see [1].

Let {T(t) : t > 0} be a nonexpansive semigroup on C, that is,

(1) for each t > 0, T(t) is a nonexpansive mapping on C;

(2) T(0)x = x for all x 2 C;

(3) T(t1 + t2) = T(t1)  T(t2) for all t1, t2 > 0; and

(4) for each x 2 C, the mapping T()x from (0,1) into C is continuous.

Denote by F¼\t>0FðTðtÞÞ the set of common fixed points for the semigroup {T(t) : t > 0}. We know that F is a closed con￾vex subset in H and F – ; if C is compact (see, [2]).

For finding a fixed point of a nonexpansive mapping T on C, in 1953, Mann [3] proposed the following method:

x0 2 C any element;

xnþ1 ¼ anxn þ ð1 anÞTxn; ð1:1Þ

that converges only weakly, in general (see [4] for an example). In 1967, Halpern [5] firstly proposed the following iteration

process:

xnþ1 ¼ bnu þ ð1 bnÞTxn; n P 0; ð1:2Þ

where u,x0 are two fixed elements in C and {bn}  (0, 1). He pointed out that the conditions limn?1bn = 0 and P1

n¼0bn ¼ 1 are

necessary in the sense that, if the iteration (1.2) converges to a fixed point of T, then these conditions must be satisfied.

0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2011.07.059

⇑ Corresponding author.

E-mail addresses: [email protected] (N. Buong), [email protected] (N.D. Lang).

Applied Mathematics and Computation 218 (2011) 2459–2466

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Applied Mathematics and Computation

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