- Research Article
- Open access
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On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings
Fixed Point Theory and Applications volume 2009, Article number: 798319 (2009)
Abstract
We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others.
1. Introduction
Let be a nonempty closed convex subset of a Hilbert space
. A mapping
is said to be nonexpansive if for all
we have
. It is said to be asymptotically nonexpansive [1] if there exists a sequence
with
and
such that
for all integers
and for all
. The set of fixed points of
is denoted by
.
Let be a bifunction, where
is the set of real number. The equilibrium problem for the function
is to find a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by . In 2005, Combettes and Hirstoaga [2] introduced an iterative scheme of finding the best approximation to the initial data when
is nonempty, and they also proved a strong convergence theorem.
For a bifunction and a nonlinear mapping
, we consider the following equilibrium problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ2_HTML.gif)
The set of such that is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ3_HTML.gif)
In the case of ,
. In the case of
,
is denoted by
. The problem (1.2) is very general in the sense that it includes, as special cases, some optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [3, 4]).
Recall that a mapping is called monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ4_HTML.gif)
A mapping of
into
is called
-inverse strongly monotone, see [5–7], if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ5_HTML.gif)
for all . It is obvious that any
inverse strongly monotone mapping
is monotone and Lipschitz continuous.
Construction of fixed points of nonexpansive mappings and asymptotically nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [1, 8–10]). Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann [11] and Ishikawa [12] iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities; see, for example, [11–13]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [11, 12]).
Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made. In 2003, Nakajo and Takahashi [14] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ6_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
. They proved that if the sequence
bounded above from one, then
defined by (1.6) converges strongly to
.
Recently, Kim and Xu [15] adapted the iteration (1.6) to an asymptotically nonexpansive mapping in a Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ7_HTML.gif)
where , as
. They proved that if
for all
and for some
, then the sequence
generated by (1.7) converges strongly to
.
Very recently, Inchan and Plubtieng [16] introduced the modified Ishikawa iteration process by the shrinking hybrid method [17] for two asymptotically nonexpansive mappings and
, with
a closed convex bounded subset of a Hilbert space
. For
and
, define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ8_HTML.gif)
where , as
and
and
for all
. They proved that the sequence
generated by (1.8) converges strongly to a common fixed point of two asymptotically nonexpansive mappings
and
.
Zegeye and Shahzad [18] established the following hybrid iteration process for a finite family of asymptotically nonexpansive mappings in a Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ9_HTML.gif)
where , as
. Under suitable conditions strong convergence theorem is proved which extends and improves the corresponding results of Nakajo and Takahashi [14] and Kim and Xu [15].
On the other hand, for finding a common element of , Tada and Takahashi [19] introduced the following iterative scheme by the hybrid method in a Hilbert space:
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ10_HTML.gif)
for every , where
for some
and
satisfies
. Further, they proved that
and
converge strongly to
, where
.
Inspired and motivated by the above facts, it is the purpose of this paper to introduce the Mann iteration process for finding a common element of the set of common fixed points of an infinite family of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem. Then we prove some strong convergence theorems which extend and improve the corresponding results of Tada and Takahashi [19], Inchan and Plubtieng [16], Zegeye and Shahazad [18], and many others.
2. Preliminaries
We will use the following notations:
(1)"" for weak convergence and "
" for strong convergence;
(2) denotes the weak
-limit set of
.
Let be a real Hilbert space. It is well known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ11_HTML.gif)
for all .
It is well known that satisfies Opial's condition [20], that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ12_HTML.gif)
holds for every with
. Hilbert space
satisfies the Kadec-Klee property [21, 22], that is, for any sequence
with
and
together imply
.
We need some facts and tools in a real Hilbert space which are listed as follows.
Lemma 2.1 ([23]).
Let be an asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset
of a Hilbert space
. If
is a sequence in
such that
and
, then
.
Lemma 2.2 ([24]).
Let be a nonempty closed convex subset of
and also give a real number
. The set
is convex and closed.
Lemma 2.3 ([22]).
Let be a nonempty closed convex subset of
and let
be the (metric or nearest) projection from
onto
i.e.,
is the only point in
such that
. Given
and
. Then
if and only if it holds the relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ13_HTML.gif)
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see [3]):
(A1) for all
;
(A2) is monotone, that is,
for any
;
(A3) is upper-hemicontinuous, that is, for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ14_HTML.gif)
(A4) is convex and weakly lower semicontinuous for each
.
The following lemma appears implicity in [3].
Lemma 2.4 ([3]).
Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)–(A4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ15_HTML.gif)
The following lemma was also given in [2].
Lemma 2.5 ([2]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ16_HTML.gif)
for all . Then, the following holds
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any
,
.
This implies that , that is,
is a nonexpansive mapping:
(3);
(4) is a closed and convex set.
Definition 2.6 (see [25]).
Let be a nonempty closed convex subset of
. Let
be a family of asymptotically nonexpansive mappings of
into itself, and let
be a sequence of real numbers such that
for every
with
. For any
define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ17_HTML.gif)
Such a mapping is called the modified
-mapping generated by
and
.
Lemma 2.7 ([10, Lemma 4.1]).
Let be a nonempty closed convex subset of
. Let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
and let
be a sequence of real numbers with
for all
and
for every
and
for some
. Let
be the modified
-mapping generated by
and
. Let
for every
and
. Then, the followings hold:
-
(i)
for all
,
and
;
(ii) if is bounded and
, for every sequence
in C,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ18_HTML.gif)
(iii) if ,
and
is closed convex.
Lemma 2.8 ([10, Lemma 4.4]).
Let be a nonempty closed convex subset of
. Let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
. Let
for every
, where
for every
and
with
for every
and
for every
and let
for every
. Then, the following holds:
(i) for all
,
and
;
-
(ii)
if
is bounded and
, for every sequence
in C,
(2.9)
-
(iii)
if
,
and
is closed convex.
3. Main Results
In this section, we will introduce two iterative schemes by using hybrid approximation method for finding a common element of the set of common fixed points for a family of infinitely asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert space. Then we show that the sequences converge strongly to a common element of the two sets.
Theorem 3.1.
Let be a nonempty bounded closed convex subset of a real Hilbert space
, let
be a bifunction satisfying the conditions (A1)–(A4), let
be an
-inverse strongly monotone mapping of
into
, let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
, where
and let
be a sequence of real numbers with
for all
and
for every
and
for some
. Let
be the modified
-mapping generated by
and
. Assume that
for every
and
such that
. Let
and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ20_HTML.gif)
where and
and
and
. Then
and
converge strongly to
.
Proof.
We show first that the sequences and
are well defined.
We observe that is closed and convex by Lemma 2.2. Next we show that
for all
. we prove first that
is nonexpansive. Let
. Since
is
-inverse strongly monotone and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ21_HTML.gif)
Thus is nonexpansive.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ22_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ23_HTML.gif)
By Lemma 2.5, we have ,
.
Let , it follows the definition of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ24_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ25_HTML.gif)
Again by Lemma 2.5, we have ,
.
Since and
are nonexpansive, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ26_HTML.gif)
Then using the convexity of and Lemma 2.7 we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ27_HTML.gif)
So for all
and hence
for all
. This implies that
is well defined. From Lemma 2.4, we know that
is also well defined.
Next, we prove that ,
,
,
, as
.
It follows from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ28_HTML.gif)
So, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ29_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ30_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ31_HTML.gif)
Since is bounded, then
and
are bounded.
From and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ32_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ33_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ34_HTML.gif)
Hence, is nodecreasing, and so
exists.
Next, we can show that . Indeed, From (2.1) and (3.13), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ35_HTML.gif)
Since exists, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ36_HTML.gif)
On the other hand, it follows from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ37_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ38_HTML.gif)
Next, we claim that . Let
, it follows from (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ39_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ40_HTML.gif)
It follows from (3.19) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ41_HTML.gif)
From Lemma 2.5, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ42_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ43_HTML.gif)
By (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ44_HTML.gif)
Substituting (3.24) into (3.25), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ45_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ46_HTML.gif)
Noticing that and (3.19), it follows from (3.27) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ47_HTML.gif)
From (3.17) and (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ48_HTML.gif)
Similarly, from (3.19) and (3.28), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ49_HTML.gif)
Noticing that the condition , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ50_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ51_HTML.gif)
Next, we prove that there exists a subsequence of
which converges weakly to
, where
.
Since is bounded and
is closed, there exists a subsequence
of
which converges weakly to
, where
. From (3.28), we have
. Noticing (3.29) and (3.32), it follows from Lemma 2.7 that
. Next we prove that
. Since
, for any
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ52_HTML.gif)
From (A2), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ53_HTML.gif)
Replacing by
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ54_HTML.gif)
Put for all
and
. Then, we have
. So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ55_HTML.gif)
Since , we have
. Further, from monotonicity of
, we have
. So, from (A4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ56_HTML.gif)
as . From (A1) and (A4), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ57_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ58_HTML.gif)
Letting , we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ59_HTML.gif)
This implies that . Therefore
.
Finally we show that ,
, where
.
Putting and consider the sequence
. Then we have
and by the weak lower semicontinuity of the norm and by the fact that
for all
which is implied by the fact that
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ60_HTML.gif)
This implies that (hence
by the uniqueness of the nearest point projection of
onto
) and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ61_HTML.gif)
It follows that , and hence
. Since
is an arbitrary (weakly convergent) subsequence of
, we conclude that
. From (3.28), we know that
also. This completes the proof of Theorem 3.1.
Theorem 3.2.
Let be a nonempty bounded closed convex subset of a real Hilbert space
, let
be a bifunction satisfying the conditions (A1)–(A4), let
be an
-inverse strongly monotone mapping of
into
, and let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
, where
. Let
for every
, where
for every
and
with
for each
and
for every
and assume that
for every
such that
. Let
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ62_HTML.gif)
where and
and
. Then
and
converge strongly to
.
Proof.
We divide the proof of Theorem 3.2 into four steps.
(i)We show first that the sequences and
are well defined.
From the definition of and
, it is obvious that
is closed and
is closed and convex for each
. We prove that
is convex. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ63_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ64_HTML.gif)
it follows that is convex. So,
is a closed convex subset of
for any
.
Next, we show that . Indeed, let
and let
be a sequence of mappings defined as in Lemma 2.5. Similar to the proof of Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ65_HTML.gif)
By virtue of the convexity of norm , (3.46), and Lemma 2.8, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ66_HTML.gif)
Therefore, for all
.
Next, we prove that ,
. For
, we have
. Assume that
. Since
is the projection of
onto
, by Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ67_HTML.gif)
In particular, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ68_HTML.gif)
for each and hence
. Hence
,
. Therefore, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ69_HTML.gif)
This implies that is well defined. From Lemma 2.4, we know that
is also well defined.
(ii)We prove that ,
,
,
, as
.
Since is a nonempty closed convex subset of
, there exists a unique
such that
.
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ70_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ71_HTML.gif)
Since is bounded, we have
,
and
are bounded. From the definition of
, we have
, which together with the fact that
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ72_HTML.gif)
This shows that the sequence is nondecreasing. So,
exists.
It follows from (2.1) and (3.53) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ73_HTML.gif)
Noticing that exists, this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ74_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ75_HTML.gif)
So, we have . It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ76_HTML.gif)
Similar to the proof of Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ77_HTML.gif)
From (3.55) and (3.58), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ78_HTML.gif)
Similarly, from (3.57) and (3.58), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ79_HTML.gif)
Noticing the condition , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ80_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ81_HTML.gif)
-
(iii)
We prove that there exists a subsequence
of
which converges weakly to
, where
.
Since is bounded and
is closed, there exists a subsequence
of
which converges weakly to
, where
. From (3.58), we have
. Noticing (3.59) and (3.62), it follows from Lemma 2.8 that
. By using the same method as in the proof of Theorem 3.1, we easily obtain that
.
-
(iv)
Finally we show that
,
, where
.
Since and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ82_HTML.gif)
It follows from and the weak lower-semicontinuity of the norm that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ83_HTML.gif)
Thus, we obtain that . Using the Kadec-Klee property of
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ84_HTML.gif)
Since is an arbitrary subsequence of
, we conclude that
converges strongly to
. By (3.58), we have
also. This completes the proof of Theorem 3.2.
Corollary 3.3.
Let be a nonempty bounded closed convex subset of a real Hilbert space
, let
be a bifunction satisfying the conditions (A1)–(A4), let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
, where
and let
be a sequence of real numbers with
for all
and
for every
and
for some
. Let
be the modified
-mapping generated by
and
. Assume that
for every
and
such that
. Let
and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ85_HTML.gif)
where and
and
and
such that
. Then
and
converge strongly to
.
Proof.
Putting , the conclusion of Corollary 3.3 can be obtained as in the proof of Theorem 3.1.
Remark 3.4.
Corollary 3.3 extends the Theorem of Tada and Takahashi [19] in the following senses:
(1)from one nonexpansive mapping to a family of infinitely asymptotically nonexpansive mappings;
-
(2)
from computation point of view, the algorithm in Corollary 3.3 is also simpler and, more convenient to compute than the one given in [19].
Corollary 3.5.
Let be a nonempty bounded closed convex subset of a real Hilbert space
, let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
and let
be a sequence of real numbers with
for all
and
for every
and
for some
. Let
be the modified
-mapping generated by
and
. Assume that
for every
and
such that
. Let
be a sequence generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ86_HTML.gif)
where and
. Then
converges strongly to
.
Proof.
Putting ,
,
and
, for all
in Theorem 3.1, we have
, therefore
. The conclusion of Corollary 3.5 can be obtained from Theorem 3.1 immediately.
Remark 3.6.
Corollary 3.5 extends Theorem 3.1 of Inchan and Plubtieng [16] from two asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
Corollary 3.7.
Let be a nonempty bounded closed convex subset of a real Hilbert space
, and let
be a family of asymptotically nonexpansive mappings of
into itself with Lipschitz constants
, that is,
(
) such that
. Let
for every
, where
for every
and
with
for each
and
for every
and assume that
for every
such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F798319/MediaObjects/13663_2009_Article_1179_Equ87_HTML.gif)
where and
. Then
converges strongly to
.
Proof.
Putting ,
,
and
, for all
in Theorem 3.2, we have
, therefore
. The conclusion of Corollary 3.7 can be obtained from Theorem 3.2.
Remark 3.8.
Corollary 3.7 extends Theorem 3.1 of Zegeye and Shahzad [18] from a finite family of asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
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Acknowledgments
This research is supported by the National Science Foundation of China under Grant (10771175), and by the key project of chinese ministry of education(209078) and the Natural Science Foundational Committee of Hubei Province (D200722002).
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Cai, G., Hu, C. On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings. Fixed Point Theory Appl 2009, 798319 (2009). https://doi.org/10.1155/2009/798319
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DOI: https://doi.org/10.1155/2009/798319