- Research Article
- Open access
- Published:
On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi-
-Asymptotically Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 157278 (2010)
Abstract
We prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi--asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and, a Fréchet differentiable norm under weaker conditions. The method of the proof is different from, S. Takahashi and W. Takahashi that by (2008) and that by Takahashi and Zembayashi (2008) and see references. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.
1. Introduction
Let be a Banach space and let
be a closed convex subsets of
. Let
be an equilibrium bifunction from
into
, let
be a real-valued function, and let
be a nonlinear mapping. The "so-called" generalized mixed equilibrium problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ2_HTML.gif)
Sepecial Examples
(i)If , then the problem (1.1) is equivalent to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ3_HTML.gif)
which is called the mixed equilibrium problem; see [1]. The set of solutions of (1.3) is denoted by MEP.
(ii)If , then the problem (1.1) is equivalent to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ4_HTML.gif)
which is called the mixed variational inequality of Browder type. The set of solutions of (1.4) is denoted by .
(iii)If , then the problem (1.1) is equivalent to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ5_HTML.gif)
which is called the generalized equilibrium problem; see [2]. The set of solutions of (1.5) is denoted by EP.
(iv)If , then the problem (1.1) is equivalent to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ6_HTML.gif)
which is called the equilibrium problem. The set of solutions of (1.6) is denoted by EP(F).
Recently, Tada and Takahashi [3] and S. Takahashi and W. Takahashi [4] considered iterative methods for finding an element of in Hilbert space. Very recently, S.Takahashi and W.Takahashi [2] introduced an iterative method for finding an element of
, where
is an inverse-strongly monotone mapping and
is nonexpansive mapping and then proved a strong convergence theorem in Hilbert space. On the other hand, Takahashi and Zembayashi [5] prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using the shrinking Projection method. Very recently, Kimura and Takahashi [6] prove a strong convergence theorem for a family of relatively nonexpansive mapping in a Banach space by using a hybrid method.
In this paper, motivated by Kimura and Takahashi [6], we prove a strong convergence theorem for finding an element of in Banach space by using a hybrid method, where
is a continuous and monotone operator and
is a family of quasi-
-asymptotically nonexpansive mapping. Moreover, the method of proof adopted in the paper is different from that of [2, 5].
2. Preliminaries
Throughout this paper, we assume that all the Banach spaces are real. We denote by and
the sets of positive integers and real numbers, respectively. Let
be a Banach space and let
be the topological dual of
. For all
and
, we denote the value of
at
by
. The duality mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ7_HTML.gif)
By Hahn-Banach theorem, is nonempty; see [7] for more details. We denote the weak convergence and the strong convergence of a sequence
to
in
by
and
, respectively. A Banach space
is said to be strictly convex if
for
and
. It is also said to be uniformly convex if for each
there exists
such that
for
and
.
is said to have the Kadec-Klee property, that is, for any sequence
, if
and
, then
.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ8_HTML.gif)
for and
. A norm of
is said to be
differentiable if
has a limit for each
. In this case,
is said to be smooth. A norm of
is said to be Fréchet differentiable if
is attained uniformly for
for each
. It is known that
has a Fréchet differentiable norm if and only if
is strictly convex and reflexive, and has the Kadec-Klee property. We know that if
is smooth, strictly convex, and reflexive, then the duality mapping
is single valued, one to one, and onto. In this case, the inverse mapping
coincides with the duality mapping
on
. See [8] for more details.
Remark 2.1.
If is a reflexive and strictly convex Banach space, then
is hemicontinuous, that is,
is norm-weak continuous.
Let be a smooth, strictly convex and reflexive Banach space and let
be a closed convex subset of
. Throughout this paper, we denote by
the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ9_HTML.gif)
Let be a sequence of nonempty closed convex subset of a reflexive Banach space
. We define two subsets
and
as follows:
if and only if there exists
such that
converges strongly to
and that
for all
. Similarly,
if and only if there exists a subsequence
of
and a sequence
such that
converges weakly to
and
for all
. We define the Mosco convergence [9] of
as follows: if
satisfies that
, then it is said that
converges to
in the sense of Mosco and we write
. For more details, see [10].
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
. Then, for arbitrarily fixed
, a function
has a unique minimizer
. Using such a point, we define the metric projection
by
for every
. In a similar fashion, we can see that a function
has a unique minimizer
. The generalized projection
of
onto
is defined by
for every
; see [11].
The generalized projection from
onto
is well defined and single valued and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ10_HTML.gif)
If is a Hilbert space, then
and
is the metric projection
of
onto
.
It is well-known that the following conclusions hold.
Let be a nonempty closed convex subsets of a smooth, strictly convex and reflexive Banach spaces. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ11_HTML.gif)
Lemma 2.3.
Let be a nonempty closed convex subsets of a smooth, strictly convex and reflexive Banach spaces
, let
and let
. Then the following conclusions hold:
(a)
(b)For if and only if
.
The following theorem proved by Tsukada [13] plays an important role in our results.
Theorem 2.4.
Let be a smooth, reflexive, and strictly convex Banach space having the Kadec-Klee property. Let
be a sequence of nonempty closed convex subset of
. If
exists and is nonempty, then
converges strongly to
for each
.
Theorem 2.4 is still valid if we replace the metric projections with the generalized projections as follows:
Theorem 2.5.
Let be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let
be a sequence of nonempty closed convex subsets of
. If
exists and is nonempty, then
converges strongly to
for each
.
Let be a nonempty closed convex subsets of
, and let
be a mapping from
into itself. We denoted by
the set of fixed points of
.
is said to be
-asymptotically nonexpansive, if there exists some real sequence
with
and
such that
for all
and
.
is said to be quasi-
-asymptotically nonexpansive [14], if there exists some real sequence
with
and
and
such that
for all
,
and
.
is said to be uniformly Lipschitzian continuous if there exists some
such that
for all
and
. A point
is said to be an asymptotic fixed point of
[15, 16] if there exists
in
which converges weakly to
and
. We denote the set of all asymptotic fixed point of
by
. Following Matsushita and Takahashi [17], a mapping
is said to be relatively nonexpansive if the following conditions are satisfied:
(1) is nonempty,
(2), for all
,
(3).
A mapping is said to be quasi-
-nonexpansive, if
.
We remark that a quasi--nonexpansive mapping with a nonempty fixed point set
is a quasi-
-asymptotically nonexpansive mapping, but the converse may be not true.
A mapping is said to be closed, if for any sequence
with
and
,
.
Lemma 2.6.
Let be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm,
be a nonempty closed convex subset of
, and let
be a uniformly Lipschitzian continuous and quasi-
-asymptotically nonexpansive mapping from
into itself. Then
is closed and convex.
Proof.
We first show that is closed. To see this, let
be a sequence in
with
as
; we shall prove that
. In fact, from the definition of
, we have
. Therefore we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ12_HTML.gif)
that is, . We next show that
is convex. To end this, for arbitrary
, by setting
, it is sufficient to show that
. Indeed, by using (2.3) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ13_HTML.gif)
which implies that as
. From (2.4) we have
. Consequently
. This implies that
is bounded in
. Since
is reflexive, so is
, we can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ14_HTML.gif)
In view of the reflexive of , we see that
. Hence there exists
such that
. By virtue of the weak lower semicontinuity of norm
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ15_HTML.gif)
that is, . This implies that
. Thus from (2.8) we have
. Since
and
has the Kadec-Klee property, we have
. Note that
is hemicontinuous, it yields that
. Again since
, by using the Kadec-Klee property of
, we have
. Hence
as
. Since
is uniformly Lipschitzian continuous, we have
. This completes the proof.
For solving the equilibrium problem for bifunction , let us assume that
satisfies the following conditions:
for all
,
is monotone, that is,
for all
,
for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ16_HTML.gif)
for each is a convex and lower semicontinuous.
If an equilibrium bifunction satisfies conditions
, then we have the following two important results.
Lemma 2.7 (see[18]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, let
be an equilibrium bifunction from
to
satisfying conditions
, let
and let
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ17_HTML.gif)
Lemma 2.8 (see[5]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
, and let
be an equilibrium bifunction satisfying conditions
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ18_HTML.gif)
for all . Then, the following hold:
(1) is single-valued,
(2) is a firmly nonexpansive-type mapping, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ19_HTML.gif)
(3),
(4) is a closed and convex set.
Lemma 2.9 (see[5]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, and let
be an equilibrium bifunction satisfying conditions
. For
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ20_HTML.gif)
3. The Main Results
Lemma 3.1.
Let be a strictly convex reflexive Banach space having a Fréchet differentiable norm,
a nonempty closed convex subset of
and
a sequence of mappings of
into itself. Let
be a strongly convergent sequence in
with a limit
and
a sequence in
defined by
for each
, where
is a convergent sequence in
with a limit
. Suppose that
for all
and that
converges weakly to
, where
. Then
converges strongly to
0.
Moreover, if
has the Kadec-Klee property, then
converges strongly to
.
Proof.
Since for
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ21_HTML.gif)
and hence . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ22_HTML.gif)
for , we have that
and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ23_HTML.gif)
Using weak lower semicontinuity of the norm, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ24_HTML.gif)
Therefore, we have that and hence
. Thus we have that
converges weakly to
. It also holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ25_HTML.gif)
Since has a Fréchet differentiable norm, it follows that
has the Kadec-Klee property, and thus we have that
converges strongly to
. Then, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ26_HTML.gif)
for . Using norm-to-norm continuity of
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ27_HTML.gif)
and since , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ28_HTML.gif)
We also have that converges strongly to
and hence we obtain that
converges strongly to 0. Further, let us suppose that
has the Kadec-Klee property. Then, the norm of
is Fréchet differentiable and, therefore,
is norm-to-norm continuous. Hence we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ29_HTML.gif)
which is the desired result.
Theorem 3.2.
Let be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm,
a nonempty closed convex subset of
a continuous and monotone mapping,
a lower semicontinuous and convex function
a bifunction from
which satisfies the conditions
and
a family of uniformly Lipschitzian continuous and quasi-
-asymptotically nonexpansive mappings such that
. Assume that
. Let
be the sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ30_HTML.gif)
where is the duality mapping on
and
for all
, where
. Let
be a sequence in
such that
and
for some
, then
converge strongly to
, where
is the metric projection of
onto
.
Proof.
We define a bifunction by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ31_HTML.gif)
Next, we prove that the bifunction satisfies conditions
as follows
for all
,
since for all
.
is monotone, that is,
for all
.
Since is a continuous and monotone operator, hence from the definition of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ32_HTML.gif)
for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ33_HTML.gif)
Since is continuous and
is lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ34_HTML.gif)
For each is a convex and lower semicontinuous.
For each and
, since
satisfies
and
is convex, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ35_HTML.gif)
So, is convex.
Similarly, we can prove that is lower semi-continuous.
Therefore, the generalized mixed equilibrium problem (1.1) is equivalent to the following equilibrium problem: find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ36_HTML.gif)
then, . We have
. So, (3.10) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ37_HTML.gif)
Since the bifunction satisfies conditions
, from Lemma 2.8, for given
and
, we can define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ38_HTML.gif)
Moreover, satisfies the conclusions in Lemma 2.8.
Putting for all
, we have from Lemma 2.8 and Lemma 2.9 that
is relatively nonexpansive.
We divide the proof of Theorem 3.2 into five steps.
Step 1.
We first show that is closed and convex for every
. From the definition of
, we may show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ39_HTML.gif)
and thus is closed and convex for every
.
Step 2.
Next we show that for each
and
.
for any , since
is relatively nonexpansive and
is quasi-
-asymptotically nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ40_HTML.gif)
Hence, we have , that is,
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ41_HTML.gif)
Step 3.
Now we prove that the limit exists.Since
is nonempty,
is a nonempty closed convex subset of
, and, thus,
exists for every
, hence
is well defined. Also, since
is a decreasing sequence of closed convex subsets of
such that
is nonempty, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ42_HTML.gif)
By Theorem 2.4, converges strongly to
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ43_HTML.gif)
Step 4.
Next we prove that .
(a) First, we prove that .
Since for every
, it follows that
for every
. Fix
arbitrarily. From the assumption that
, we may take subsequences
of
and
of
such that
with
and
converges weakly to a point
. Then, by Lemma 3.1, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ44_HTML.gif)
From (3.23) and (3.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ45_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ46_HTML.gif)
By using (3.24),(3.25), and (3.26), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ47_HTML.gif)
Since is uniformly Lipschitzian continuous, from (3.24) and (3.27), we have
, that is,
.
(b) Next we prove that .
() In fact, since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ48_HTML.gif)
In view of , from the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ49_HTML.gif)
From (3.28) and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ50_HTML.gif)
From (2.4), it yields . Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ51_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ52_HTML.gif)
This implies that is bounded in
. Since
is reflexive, and so is
, we can assume that
. In view of the reflexive of
, we see that
. Hence there exists
such that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ53_HTML.gif)
taking on the both sides of equality above and in view of the weak lower semi-continuity of norm
, it yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ54_HTML.gif)
that is, . This implies that
, and so
. It follows from (3.32) and the Kadec-Klee property of
that
. Note that
is hemicontinuous, it yields that
. It follows from (3.31) and the Kadec-Klee property of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ55_HTML.gif)
From (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ56_HTML.gif)
Since is norm-to-norm continuous, hence we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ57_HTML.gif)
() Next we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ58_HTML.gif)
From (3.36) and (3.37), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ59_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ60_HTML.gif)
hence it follows from (3.38) and (3.40) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ61_HTML.gif)
From (2.3) and (3.41) it yields . Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ62_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ63_HTML.gif)
This implies that is bounded in
. Since
is reflexive, and so is
, we can assume that
. In view of the reflexive of
, we see that
. Hence there exists
such that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ64_HTML.gif)
Taking on the both sides of equality above and in view of the weak lower semi-continuity of norm
, it yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ65_HTML.gif)
that is, . This implies that
, and so
. It follows from (3.43) and the Kadec-Klee property of
that
. Note that
is hemicontinuous; it yields that
. It follows from (3.42) and the Kadec-Klee property of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ66_HTML.gif)
Since , from (3.46), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ67_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, from (3.47), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ68_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ69_HTML.gif)
By , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ70_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ71_HTML.gif)
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. So, letting
, we have from (3.51) and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ72_HTML.gif)
For any with
and
, let
. Since
and, hence,
, from conditions
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ73_HTML.gif)
This implies that . Hence from condition
, we have
for all
, and hence
.
Step 5.
Finally we prove that .
Since and
is a nonempty closed convex subset of
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ74_HTML.gif)
This completes the proof of Theorem 3.2.
The proof of Theorem 3.2 shows that the properties of projections used in the iterative scheme do not interact with the properties of mappings . Therefore, we may prove similar results as follows by replacing Theorem 2.4 with Theorem 2.5 in the proof.
Theorem 3.3.
Let be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm,
a nonempty closed convex subset of
,
a continuous and monotone mapping,
a lower semicontinuous and convex function,
a bifunction from
which satisfies the conditions
and let
a family of uniformly Lipschitzian continuous and quasi-
- asymptotically nonexpansive mappings such that
. Assume that
. Let
be the sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F157278/MediaObjects/13663_2009_Article_1207_Equ75_HTML.gif)
where is the duality mapping on
and
for all
, where
. Let
be a sequence in
such that
and
for some
, then
converge strongly to
, where
is the generalized projection of
onto
.
References
Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Journal of Optimization Theory and Applications 2007,133(3):359–370. 10.1007/s10957-007-9187-z
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036
Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-11.
Kimura Y, Takahashi W: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. Journal of Mathematical Analysis and Applications 2009,357(2):356–363. 10.1016/j.jmaa.2009.03.052
Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker, New York, NY, USA; 1984:ix+170.
Mosco U: Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics 1969, 3: 510–585. 10.1016/0001-8708(69)90009-7
Beer G: Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications. Volume 268. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+340.
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsators AG. Marcel Dekker, New York, NY, USA; 1996:15–50.
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002,13(3):938–945. 10.1137/S105262340139611X
Tsukada M: Convergence of best approximations in a smooth Banach space. Journal of Approximation Theory 1984,40(4):301–309. 10.1016/0021-9045(84)90003-0
Su Y, Qin X: Strong convergence of modified Ishikawa iterations for nonlinear mappings. Proceedings of the Indian Academy of Sciences. Mathematical Sciences 2007,117(1):97–107. 10.1007/s12044-007-0008-y
Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996,37(4):323–339. 10.1080/02331939608844225
Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsators AG. Marcel Dekker, New York, NY, USA; 1996:313–318.
Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2004,2004(1):37–47. 10.1155/S1687182004310089
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
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Liu, M., Chang, Ss. & Zuo, P. On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces.
Fixed Point Theory Appl 2010, 157278 (2010). https://doi.org/10.1155/2010/157278
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DOI: https://doi.org/10.1155/2010/157278