Open Access

A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

Fixed Point Theory and Applications20102010:262691

DOI: 10.1155/2010/262691

Received: 10 February 2010

Accepted: 15 July 2010

Published: 2 August 2010

Abstract

We introduce a new general iterative method by using the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq1_HTML.gif -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq2_HTML.gif be a real Hilbert space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq3_HTML.gif be a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq4_HTML.gif . A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq5_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq6_HTML.gif into itself is called nonexpansive if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq7_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq8_HTML.gif A point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq9_HTML.gif is called a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq10_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq11_HTML.gif . We denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq12_HTML.gif the set of fixed points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq13_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq14_HTML.gif ). Recall that a self-mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq15_HTML.gif is a contraction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq16_HTML.gif , if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq17_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq18_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq19_HTML.gif A bounded linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq20_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq21_HTML.gif is called strongly positive with coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq22_HTML.gif if there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq23_HTML.gif with the property
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ1_HTML.gif
(1.1)
In 1953, Mann [1] introduced a well-known classical iteration to approximate a fixed point of a nonexpansive mapping. This iteration is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ2_HTML.gif
(1.2)

where the initial guess https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq24_HTML.gif is taken in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq25_HTML.gif arbitrarily, and the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq26_HTML.gif is in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq27_HTML.gif . But Mann's iteration process has only weak convergence, even in a Hilbert space setting. In general for example, Reich [2] showed that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq28_HTML.gif is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq29_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq30_HTML.gif , then the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq31_HTML.gif generated by process (1.2) converges weakly to a point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq32_HTML.gif . Therefore, many authors try to modify Mann's iteration process to have strong convergence.

In 2005, Kim and Xu [3] introduced the following iteration process:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ3_HTML.gif
(1.3)

They proved in a uniformly smooth Banach space that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq33_HTML.gif defined by (1.3) converges strongly to a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq34_HTML.gif under some appropriate conditions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq36_HTML.gif .

In 2008, Yao et al. [4] alsomodified Mann's iterative scheme 1.2 to get a strong convergence theorem.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq37_HTML.gif be a finite family of nonexpansive mappings with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq38_HTML.gif There are many authors introduced iterative method for finding an element of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq39_HTML.gif which is an optimal point for the minimization problem. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq41_HTML.gif is understood as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq42_HTML.gif with the mod function taking values in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq43_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq44_HTML.gif be a fixed element of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq45_HTML.gif

In 2003, Xu [5] proved that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq46_HTML.gif generated by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ4_HTML.gif
(1.4)
converges strongly to the solution of the quadratic minimization problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ5_HTML.gif
(1.5)
under suitable hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq47_HTML.gif and under the additional hypothesis
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ6_HTML.gif
(1.6)
In 1999, Atsushiba and Takahashi [6] defined the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq48_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq49_HTML.gif This mapping is called the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq50_HTML.gif mapping generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq52_HTML.gif .

In 2000, Takahashi and Shimoji [7] proved that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq53_HTML.gif is strictly convex Banach space, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq54_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq55_HTML.gif .

In 2007,Shang et al.[8] introduced a composite iteration scheme as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ8_HTML.gif
(1.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq56_HTML.gif is a contraction, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq57_HTML.gif is a linear bounded operator.

Note that the iterative scheme (1.8) is not well-defined, because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq58_HTML.gif may not lie in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq59_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq60_HTML.gif is not defined. However, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq61_HTML.gif , the iterative scheme (1.8) is well-defined and Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq62_HTML.gif [8] is obtained. In the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq63_HTML.gif , we have to modify the iterative scheme (1.8) in order to make it well-defined.

In 2009, Kangtunyakarn and Suantai [9] introduced a new mapping, called https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq64_HTML.gif -mapping, for finding a common fixed point of a finite family of nonexpansive mappings. For a finite family of nonexpansive mappings https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq65_HTML.gif and sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq66_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq67_HTML.gif , the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq68_HTML.gif is defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ9_HTML.gif
(1.9)

The mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq69_HTML.gif is called the K-mapping generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq71_HTML.gif .

In this paper, motivated by Kim and Xu [3], Marino and Xu [10], Xu [5], Yao et al. [4], andShang et al. [8], we introduce a composite iterative scheme as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ10_HTML.gif
(1.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq72_HTML.gif is a contraction, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq73_HTML.gif is a bounded linear operator. We prove, under certain appropriate conditions on the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq75_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq76_HTML.gif defined by (1.10) converges strongly to a common fixed point of the finite family of nonexpansive mappings https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq77_HTML.gif , which solves a variational inequaility problem.

In order to prove our main results, we need the following lemmas.

Lemma 1.1.

For all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq78_HTML.gif there holds the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ11_HTML.gif
(1.11)

Lemma 1.2 (see [11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq80_HTML.gif be bounded sequences in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq81_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq82_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq83_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq84_HTML.gif . Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ12_HTML.gif
(1.12)
for all integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq85_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ13_HTML.gif
(1.13)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq86_HTML.gif

Lemma 1.3 (see [5]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq87_HTML.gif is a sequence of nonnegative real numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq88_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq89_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq91_HTML.gif is a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq92_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq93_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq94_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq95_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq96_HTML.gif .

Lemma 1.4 (see [10]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq97_HTML.gif be a strongly positive linear bounded operator on a Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq98_HTML.gif with coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq100_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq101_HTML.gif .

Lemma 1.5 (see [10]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq102_HTML.gif be a Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq103_HTML.gif be a strongly positive linear bounded operator with coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq104_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq105_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq106_HTML.gif be a nonexpansive mapping with a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq107_HTML.gif of the contraction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq108_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq109_HTML.gif converges strongly as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq110_HTML.gif to a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq111_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq112_HTML.gif , which solves the variational inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ14_HTML.gif
(1.14)

Lemma 1.6 (see [1]).

Demiclosedness principle. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq113_HTML.gif is nonexpansive self-mapping of closed convex subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq114_HTML.gif of a Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq115_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq116_HTML.gif has a fixed point, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq117_HTML.gif is demiclosed. That is, whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq118_HTML.gif is a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq119_HTML.gif weakly converging to some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq120_HTML.gif and the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq121_HTML.gif strongly converges to some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq122_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq123_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq124_HTML.gif is identity mapping of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq125_HTML.gif .

Lemma 1.7 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq126_HTML.gif be a nonempty closed convex subset of a strictly convex Banach space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq127_HTML.gif be a finite family of nonexpansive mappings of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq128_HTML.gif into itself with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq129_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq130_HTML.gif be real numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq131_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq133_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq134_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq135_HTML.gif -mapping of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq136_HTML.gif into itself generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq138_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq139_HTML.gif .

By using the same argument as in [9, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq140_HTML.gif ], we obtain the following lemma.

Lemma 1.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq141_HTML.gif be a nonempty closed convex subset of Banach space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq142_HTML.gif be a finite family of nonexpanxive mappings of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq143_HTML.gif into itself and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq144_HTML.gif sequences in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq145_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq146_HTML.gif Moreover, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq147_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq149_HTML.gif be the K -mappings generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq151_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq153_HTML.gif , respectively. Then, for every bounded sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq154_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq155_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq156_HTML.gif be real Hilbert space with inner product https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq157_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq158_HTML.gif a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq159_HTML.gif . Recall that the metric (nearest point) projection https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq160_HTML.gif from a real Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq161_HTML.gif to a closed convex subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq162_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq163_HTML.gif is defined as follows. Given that   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq165_HTML.gif is the only point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq166_HTML.gif with the property https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq167_HTML.gif . Below Lemma 1.9 can be found in any standard functional analysis book.

Lemma 1.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq168_HTML.gif be a closed convex subset of a real Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq169_HTML.gif . Given that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq171_HTML.gif then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq172_HTML.gif if and only if the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq173_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq174_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq175_HTML.gif is nonexpansive,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq176_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq177_HTML.gif ,

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq178_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq180_HTML.gif .

2. Main Result

In this section, we prove strong convergence of the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq181_HTML.gif defined by the iteration scheme (1.10).

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq182_HTML.gif be a Hilbert space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq183_HTML.gif a closed convex nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq184_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq185_HTML.gif be a strongly positive linear bounded operator with coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq186_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq187_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq188_HTML.gif be a finite family of nonexpansive mappings of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq189_HTML.gif into itself, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq190_HTML.gif be defined by (1.9). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq192_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq193_HTML.gif , given that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq195_HTML.gif are sequences in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq196_HTML.gif , and suppose that the following conditions are satisfied:

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq197_HTML.gif

(C2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq198_HTML.gif

(C3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq199_HTML.gif

(C4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq201_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq202_HTML.gif ;

(C5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq203_HTML.gif

(C6) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq204_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq205_HTML.gif is the composite process defined by (1.10), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq206_HTML.gif converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq207_HTML.gif , which also solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ15_HTML.gif
(2.1)

Proof.

First, we observe that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq208_HTML.gif is bounded. Indeed, take a point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq209_HTML.gif , and notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ16_HTML.gif
(2.2)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq210_HTML.gif , we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq211_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq212_HTML.gif . By Lemma 1.4, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq213_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq214_HTML.gif .

It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ17_HTML.gif
(2.3)
By simple inductions, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ18_HTML.gif
(2.4)
Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq215_HTML.gif is bounded, so are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq217_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq218_HTML.gif is nonexpansive and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq219_HTML.gif , we also have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ19_HTML.gif
(2.5)
By using the inequalities (2.6) and (2.11) of [9, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq220_HTML.gif ], we can conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ20_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq221_HTML.gif .

By (2.5) and (2.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ21_HTML.gif
(2.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq223_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq224_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq225_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq226_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq227_HTML.gif , by Lemma 1.3, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq228_HTML.gif . It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ22_HTML.gif
(2.8)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq229_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq231_HTML.gif are bounded, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq232_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq233_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ23_HTML.gif
(2.9)

it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq234_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq235_HTML.gif .

On the other hand, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ24_HTML.gif
(2.10)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq236_HTML.gif .

From condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq238_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq239_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ25_HTML.gif
(2.11)
By (C4), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq240_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq241_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq242_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq243_HTML.gif -mapping generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq245_HTML.gif . Next, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ26_HTML.gif
(2.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq246_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq247_HTML.gif being the fixed point of the contraction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq248_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq249_HTML.gif solves the fixed point equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq250_HTML.gif . By Lemma 1.5 and Lemma 1.7, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq252_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq253_HTML.gif . It follows by (2.11) and Lemma 1.8 that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq254_HTML.gif Thus, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq255_HTML.gif . It follows from Lemma 1.1 that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq256_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ27_HTML.gif
(2.13)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ28_HTML.gif
(2.14)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ29_HTML.gif
(2.15)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq257_HTML.gif in (2.15) and (2.14), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ30_HTML.gif
(2.16)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq258_HTML.gif is a constant such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq259_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq261_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq262_HTML.gif in (2.16), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ31_HTML.gif
(2.17)
On the other hand, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ32_HTML.gif
(2.18)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ33_HTML.gif
(2.19)
Therefore, from   (2.17) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq263_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ34_HTML.gif
(2.20)
Hence (2.12) holds. Finally, we prove that   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq264_HTML.gif . By using (2.2) and together with the Schwarz inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ35_HTML.gif
(2.21)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq266_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq267_HTML.gif are bounded, we can take a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq268_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ36_HTML.gif
(2.22)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq269_HTML.gif . It then follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ37_HTML.gif
(2.23)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq270_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq271_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq272_HTML.gif . By applying Lemma 1.3 to (2.23), we can conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq273_HTML.gif . This completes the proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq275_HTML.gif in Theorem 2.1, we obtain the following result.

Corollary 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq276_HTML.gif be a Hilbert space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq277_HTML.gif a closed convex nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq278_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq279_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq280_HTML.gif be a finite family of nonexpansive mappings of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq281_HTML.gif into itself, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq282_HTML.gif be defined by (1.9). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq283_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq284_HTML.gif , given that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq285_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq286_HTML.gif are sequences in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq287_HTML.gif , and suppose that the following conditions are satisfied:

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq288_HTML.gif

(C2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq289_HTML.gif

(C3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq290_HTML.gif

(C4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq292_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq293_HTML.gif ;

(C5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq294_HTML.gif

(C6) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq295_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq296_HTML.gif is the composite process defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ38_HTML.gif
(2.24)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq297_HTML.gif converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq298_HTML.gif , which also solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ39_HTML.gif
(2.25)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq299_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq300_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq301_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq302_HTML.gif is a constant in Theorem 2.1, we get the results of Kim and Xu [3].

Corollary 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq303_HTML.gif be a Hilbert space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq304_HTML.gif a closed convex nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq305_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq306_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq307_HTML.gif be a nonexpansive mapping of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq308_HTML.gif into itself. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq309_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq310_HTML.gif , given that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq311_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq312_HTML.gif are sequences in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq313_HTML.gif , and suppose that the following conditions are satisfied:

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq314_HTML.gif

(C2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq315_HTML.gif

(C3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq316_HTML.gif

(C4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq317_HTML.gif

(C5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq318_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq319_HTML.gif is the composite process defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ40_HTML.gif
(2.26)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq320_HTML.gif converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_IEq321_HTML.gif , which also solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ41_HTML.gif
(2.27)

Declarations

Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and thank the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate School of Chiang Mai University for financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University
(2)
PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University

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Copyright

© Urailuk Singthong and Suthep Suantai. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.