- Research Article
- Open access
- Published:
Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators
Journal of Inequalities and Applications volume 2009, Article number: 896252 (2009)
Abstract
The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively.
1. Introduction
Let be a real Banach space and
its dual space. The normalized duality mapping
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ1_HTML.gif)
where denotes the generalized duality pairing. Recall that if
is a smooth Banach space then
is singlevalued. Throughout this paper, we will still denote by
the single-valued normalized duality mapping. Let
be a nonempty closed convex subset of
,
a bifunction from
to
, and
a nonlinear mapping. The generalized equilibrium problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ2_HTML.gif)
The set of solutions of (1.2) is denoted by . Problem (1.2) and similar problems have been extensively studied; see, for example, [1–11]. Whenever
, problem (1.2) reduces to the equilibrium problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ3_HTML.gif)
The set of solutions of (1.3) is denoted by . Whenever
, problem (1.2) reduces to the variational inequality problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ4_HTML.gif)
Whenever a Hilbert space, problem (1.2) was very recently introduced and considered by Kamimura and Takahashi [12]. Problem (1.2) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for example, [13, 14]. A mapping
is called nonexpansive if
for all
. Denote by
the set of fixed points of
, that is,
. Iterative schemes for finding common elements of EP and fixed points set of nonexpansive mappings have been studied recently; see, for example, [12, 15–17] and the references therein.
On the other hand, a classical method of solving in a Hilbert space
is the proximal point algorithm which generates, for any starting point
, a sequence
in
by the iterative scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ5_HTML.gif)
where is a sequence in
,
for each
is the resolvent operator for
, and
is the identity operator on
. This algorithm was first introduced by Martinet [14] and generally studied by Rockafellar [18] in the framework of a Hilbert space
. Later many authors studied (1.5) and its variants in a Hilbert space
or in a Banach space
; see, for example, [13, 19–23] and the references therein.
Let be a uniformly smooth and uniformly convex Banach space and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying the following conditions (A1)–(A4) which were imposed in [24]:
(A1) for all
;
(A2) is monotone, that is,
, for all
;
(A3)for all ;
(A4)for all is convex and lower semicontinuous.
Let be a maximal monotone operator such that
(A5).
The purpose of this paper is to introduce and study two new iterative algorithms for finding a common element of the set of solutions for the generalized equilibrium problem (1.2) and the set
for maximal monotone operators
in a uniformly smooth and uniformly convex Banach space
. First, motivated by Kamimura and Takahashi [12, Theorem
], Ceng et al. [16, Theorem
], and Zhang [17, Theorem
], we introduce a sequence
that, under some appropriate conditions, is strongly convergent to
in Section 3. Second, inspired by Kamimura and Takahashi [12, Theorem
], Ceng et al. [16, Theorem
], and Zhang [17, Theorem
], we define a sequence weakly convergent to an element
, where
in Section 4. Our results represent a generalization of known results in the literature, including Takahashi and Zembayashi [15], Kamimura and Takahashi [12], Li and Song [22], Ceng and Yao [25], and Ceng et al. [16]. In particular, compared with Theorems
and
in [16], our results (i.e., Theorems 3.2 and 4.2 in this paper) extend the problem of finding an element of
to the one of finding an element of
. Meantime, the algorithms in this paper are very different from those in [16] (because of considering the complexity involving the problem of finding an element of
).
2. Preliminaries
In the sequel, we denote the strong convergence, weak convergence and weak* convergence of a sequence to a point
by
,
and
, respectively.
A Banach space is said to be strictly convex, if
for all
with
.
is said to be uniformly convex if for each
there exists
such that
for all
with
. Recall that each uniformly convex Banach space has the Kadec-Klee property, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ6_HTML.gif)
The proof of the main results of Sections 3 and 4 will be based on the following assumption.
Assumption A.
Let be a uniformly smooth and uniformly convex Banach space and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying the same conditions (A1)–(A4) as in Section 1. Let
be two maximal monotone operators such that
(A5)′.
Recall that if is a nonempty closed convex subset of a Hilbert space
, then the metric projection
of
onto
is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined as in [26] by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ7_HTML.gif)
It is clear that in a Hilbert space , (2.2) reduces to
.
The generalized projection is a mapping that assigns to an arbitrary point
the minimum point of the functional
; that is,
, where
is the solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ8_HTML.gif)
The existence and uniqueness of the operator follows from the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [27]). In a Hilbert space,
. From [26], in a smooth strictly convex and reflexive Banach space
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ9_HTML.gif)
Moreover, by the property of subdifferential of convex functions, we easily get the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ10_HTML.gif)
Let be a mapping from
into itself. A point
in
is called an asymptotically fixed point of
if
contains a sequence
which converges weakly to
such that
[28]. The set of asymptotically fixed points of
will be denoted by
. A mapping
from
into itself is called relatively nonexpansive if
and
, for all
and
[15].
Observe that, if is a reflexive strictly convex and smooth Banach space, then for any
if and only if
. To this end, it is sufficient to show that if
then
. Actually, from (2.4), we have
which implies that
. From the definition of
, we have
and therefore,
; see [29] for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (Kamimura and Takahashi [12]).
Let be a smooth and uniformly convex Banach space and let
and
be two sequences of
. If
and either
or
is bounded, then
.
Lemma 2.2 (Alber [26], Kamimura and Takahashi [12]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space
. Let
and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ11_HTML.gif)
Lemma 2.3 (Alber [26], Kamimura and Takahashi [12]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ12_HTML.gif)
Lemma 2.4 (Rockafellar [18]).
Let be a reflexive strictly convex and smooth Banach space and let
be a multivalued operator. Then there hold the following hold:
(i) is closed and convex if
is maximal monotone such that
;
(ii) is maximal monotone if and only if
is monotone with
for all
.
Lemma 2.5 (Xu [30]).
Let be a uniformly convex Banach space and let
. Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ13_HTML.gif)
for all and
, where
.
Lemma 2.6 (Kamimura and Takahashi [12]).
Let be a smooth and uniformly convex Banach space and let
. Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ14_HTML.gif)
The following result is due to Blum and Oettli [24].
Lemma 2.7 (Blum and Oettli [24]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ15_HTML.gif)
Motivated by Combettes and Hirstoaga [31] in a Hilbert space, Takahashi and Zembayashi [15] established the following lemma.
Lemma 2.8 (Takahashi and Zembayashi [15]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space
, and let
be a bifunction from
to
satisfying (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ16_HTML.gif)
for all . Then, the following hold:
(i) is singlevalued;
(ii) is a firmly nonexpansive-type mapping, that is, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ17_HTML.gif)
(iii);
(iv) is closed and convex.
Using Lemma 2.8, one has the following result.
Lemma 2.9 (Takahashi and Zembayashi [15]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
. Then, for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ18_HTML.gif)
Utilizing Lemmas 2.7, 2.8 and 2.9 as previously mentioned, Zhang [17] derived the following result.
Proposition 2.10 (Zhang [21, Lemma ]).
Let be a smooth strictly convex and reflexive Banach space and let
be a nonempty closed convex subset of
. Let
be an
-inverse-strongly monotone mapping, let
be a bifunction from
to
satisfying (A1)–(A4), and let
. Then the following hold:
for , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ19_HTML.gif)
if is additionally uniformly smooth and
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ20_HTML.gif)
then the mapping has the following properties:
(i) is singlevalued,
(ii) is a firmly nonexpansive-type mapping, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ21_HTML.gif)
(iii),
(iv) is a closed convex subset of
,
(v) for all  
Let be two maximal monotone operators in a smooth Banach space
. We denote the resolvent operators of
and
by
and
for each
, respectively. Then
and
are two single-valued mappings. Also,
and
for each
, where
and
are the sets of fixed points of
and
, respectively. For each
, the Yosida approximations of
and
are defined by
and
, respectively.It is known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ22_HTML.gif)
Lemma 2.11 (Kohsaka and Takahashi [13]).
Let be a reflexive strictly convex and smooth Banach space and let
be a maximal monotone operator with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ23_HTML.gif)
Lemma 2.12 (Tan and Xu [32]).
Let and
be two sequences of nonnegative real numbers satisfying the inequality:
for all
. If
, then
exists.
3. Strong Convergence Theorem
In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators
and
.
Lemma 3.1.
Let be a reflexive strictly convex and smooth Banach space and let
be a maximal monotone operator. Then for each
, the following holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ24_HTML.gif)
where and
is the duality mapping on
. In particular, whenever
a real Hilbert space,
is a nonexpansive mapping on
.
Proof.
Since for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ25_HTML.gif)
we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ26_HTML.gif)
Thus, from the monotonicity of it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ27_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ28_HTML.gif)
Theorem 3.2.
Suppose that Assumption A is fulfilled and let be chosen arbitrarily. Consider the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ29_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_IEq276_HTML.gif)
is defined by (2.15), satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ31_HTML.gif)
and satisfies
. Then, the sequence
converges strongly to
provided
for any sequence
with
, where
is the generalized projection of
onto
.
Remark 3.3.
In Theorem 3.2, if a real Hilbert space, then
is a sequence of nonexpansive mappings on
. This implies that as
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ32_HTML.gif)
In this case, we can remove the requirement that for any sequence
with
.
Proof of Theorem 3.2.
For the sake of simplicity, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ33_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ34_HTML.gif)
We divide the proof into several steps.
Step 1.
We claim that is closed and convex for each
.
Indeed, it is obvious that is closed and
is closed and convex for each
. Let us show that
is convex. For
and
, put
. It is sufficient to show that
. We first write
for each
. Next, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ35_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ36_HTML.gif)
Indeed, from (2.4) we deduce that the following equations hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ37_HTML.gif)
which combined with (3.12) yield that (3.12) is equivalent to (3.13). Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ38_HTML.gif)
This implies that . Therefore,
is closed and convex.
Step 2.
We claim that for each
and that
is well defined.
Indeed, take arbitrarily. Note that
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ39_HTML.gif)
Then from Lemma 2.11 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ41_HTML.gif)
Moreover, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ42_HTML.gif)
where . So
for all
. Now, let us show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ43_HTML.gif)
We prove this by induction. For , we have
. Assume that
. Since
is the projection of
onto
, by Lemma 2.2 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ44_HTML.gif)
As by the induction assumption, the last inequality holds, in particular, for all
. This, together with the definition of
implies that
. Hence (3.20) holds for all
. So,
for all
. This implies that the sequence
is well defined.
Step 3.
We claim that is bounded and that
as
.
Indeed,it follows from the definition of that
. Since
and
, so
for all
; that is,
is nondecreasing. It follows from
and Lemma 2.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ45_HTML.gif)
for each for each
. Therefore,
is bounded which implies that the limit of
exists. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ46_HTML.gif)
so is bounded. From Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ47_HTML.gif)
for each . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ48_HTML.gif)
Step 4.
We claim that ,
, and
.
Indeed, from , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ49_HTML.gif)
for all . Therefore, from
and
, it follows that
. Since
and
is uniformly convex and smooth, we have from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ50_HTML.gif)
and therefore, . Since
is uniformly norm-to-norm continuous on bounded subsets of
and
, then
.
Let us set . Then, according to Lemma 2.4 and Proposition 2.10, we know that
is a nonempty closed convex subset of
such that
. Fix
arbitrarily. As in the proof of Step 2 we can show that
, and
. Hence it follows from the boundedness of
that
, and
are also bounded. Let
. Since
is a uniformly smooth Banach space, we know that
is a uniformly convex Banach space. Therefore, by Lemma 2.5 there exists a continuous, strictly increasing, and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ51_HTML.gif)
for and
. So, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ52_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ53_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ55_HTML.gif)
for all . Consequently we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ56_HTML.gif)
Since and
is uniformly norm-to-norm continuous on bounded subsets of
, we obtain
. From
, and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ57_HTML.gif)
Therefore, from the properties of we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ58_HTML.gif)
recalling that is uniformly norm-to-norm continuous on bounded sunsets of
. Next let us show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ59_HTML.gif)
Observe first that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ60_HTML.gif)
Since , and
is bounded, so it follows that
. Also, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ61_HTML.gif)
Since and the sequences
are bounded, so it follows that
. Meantime, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ62_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ63_HTML.gif)
Since and
, it follows from the boundedness of
that
. Thus, in terms of Lemma 2.1, we have that
and so
. Furthermore, since
, from the uniform norm-to-norm continuity of
on bounded subsets of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ64_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ65_HTML.gif)
Thus, from (3.35) it follows that . Since
is uniformly norm-to-norm continuous on bounded subsets of
, it follows that
.
Step 5.
We claim that , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ66_HTML.gif)
Indeed, since is bounded and
is reflexive, we know that
. Take
arbitrarily. Then there exists a subsequence
of
such that
. Hence it follows from
, and
that
, and
converge weakly to the same point
. On the other hand, from (3.35), (3.36), and
we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ67_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ68_HTML.gif)
If and
, then it follows from (2.17) and the monotonicity of the operators
that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ69_HTML.gif)
Letting , we have that
and
. Then the maximality of the operators
implies that
and
. Next, let us show that
. Since we have by (3.32)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ70_HTML.gif)
from and Proposition 2.10 it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ71_HTML.gif)
Also, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ72_HTML.gif)
so we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ73_HTML.gif)
Thus from (3.47), , and
, we have
. Since
is uniformly convex and smooth, we conclude from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ74_HTML.gif)
From , and (3.51), we have
and
. Since
is uniformly norm-to-norm continuous on bounded subsets of
, from (3.51) we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ75_HTML.gif)
From , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ76_HTML.gif)
By the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ77_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ78_HTML.gif)
Replacing by
, we have from (A2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ79_HTML.gif)
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting
in the last inequality, from (3.53) and (A4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ80_HTML.gif)
For with
and
, let
. Since
and
,
and hence
. So, from (A1) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ81_HTML.gif)
Dividing by , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ82_HTML.gif)
Letting , from (A3) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ83_HTML.gif)
Thus . Therefore, we obtain that
by the arbitrariness of
.
Step 6.
We claim that converges strongly to
.
Indeed, from and
, It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ84_HTML.gif)
Since the norm is weakly lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ85_HTML.gif)
From the definition of , we have
. Hence
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ86_HTML.gif)
which implies that . Since
has the Kadec-Klee property, then
. Therefore,
converges strongly to
.
Remark 3.4.
In Theorem 3.2, put , and
. Then, for all
and
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ87_HTML.gif)
Moreover, the following hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ88_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ89_HTML.gif)
In this case, the previous Theorem 3.2 reduces to [20, Theorem ].
4. Weak Convergence Theorem
In this section, we present the following algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators
and
.
Let be chosen arbitrarily and consider the sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ90_HTML.gif)
where , and
, is defined by (2.15).
Before proving a weak convergence theorem, we need the following proposition.
Proposition 4.1.
Suppose that Assumption A is fulfilled and let be a sequence defined by (4.1), where
satisfy the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ91_HTML.gif)
Then, converges strongly to
, where
is the generalized projection of
onto
.
Proof.
We set and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ92_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ93_HTML.gif)
Then, in terms of Lemma 2.4 and Proposition 2.10, is a nonempty closed convex subset of
such that
. We first prove that
is bounded. Fix
. Note that by the first and third of (4.3),
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ94_HTML.gif)
Here, each is relatively nonexpansive. Then from Proposition 2.10 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ95_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ96_HTML.gif)
and hence by Proposition 2.10, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ97_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ98_HTML.gif)
Consequently, the last two inequalities yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ99_HTML.gif)
for all . So, from
, and Lemma 2.12, we deduce that
exists. This implies that
is bounded. Thus,
is bounded and so are
, and
.
Define for all
. Let us show that
is bounded. Indeed, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ100_HTML.gif)
for each . This, together with the boundedness of
, implies that
is bounded and so is
. Furthermore, from
and (4.10) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ101_HTML.gif)
Since is the generalized projection, then, from Lemma 2.3 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ102_HTML.gif)
Hence, from (4.12), it follows that .
Note that , and
is bounded, so that
. Therefore,
is a convergent sequence. On the other hand, from (4.10) we derive, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ103_HTML.gif)
In particular, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ104_HTML.gif)
Consequently, from and Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ105_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ106_HTML.gif)
Let . From Lemma 2.6, there exists a continuous, strictly increasing, and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ107_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ108_HTML.gif)
Since is a convergent sequence,
is bounded and
is convergent, from the property of
we have that
is a Cauchy sequence. Since
is closed,
converges strongly to
. This completes the proof.
Now, we are in a position to prove the following theorem.
Theorem 4.2.
Suppose that Assumption A is fulfilled and let be a sequence defined by (4.1), where
satisfy the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ109_HTML.gif)
and satisfies
. If
is weakly sequentially continuous, then
converges weakly to
, where
.
Proof.
We consider the notations (4.3). As in the proof of Proposition 4.1, we have that , and
are bounded sequences. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ110_HTML.gif)
From Lemma 2.5 and as in the proof of Theorem 3.2, there exists a continuous, strictly increasing, and convex function with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ111_HTML.gif)
for and
. Observe that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ112_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ113_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ114_HTML.gif)
Consequently, the last two inequalities yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ115_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ116_HTML.gif)
By the proof of Proposition 4.1, it is known that is convergent; since
, and
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ117_HTML.gif)
Taking into account the properties of , as in the proof of Theorem 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ118_HTML.gif)
since is uniformly norm-to-norm continuous on bounded subsets of
. Note that
. Hence, from the uniform norm-to-norm continuity of
on bounded subsets of
we obtain
. Also, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ119_HTML.gif)
From it follows that
. Since
is uniformly norm-to-norm continuous on bounded subsets of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ120_HTML.gif)
Now let us show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ121_HTML.gif)
Indeed, from (4.10) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ122_HTML.gif)
which, together with , yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ123_HTML.gif)
From (4.9) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ124_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ125_HTML.gif)
Since and
, we obtain
, which, together with
, yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ126_HTML.gif)
We have from (4.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ127_HTML.gif)
which, together with , yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ128_HTML.gif)
Also from (4.7) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ129_HTML.gif)
which, together with , yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ130_HTML.gif)
Similarly from (4.6) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ131_HTML.gif)
which, together with , yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ132_HTML.gif)
On the other hand, let us show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ133_HTML.gif)
Indeed, let . From Lemma 2.6, there exists a continuous, strictly increasing, and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ134_HTML.gif)
Since and
, we deduce from Proposition 2.10 that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ135_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ136_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded subsets of
, from the properties of
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ137_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ138_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ139_HTML.gif)
Since , it follows from (4.31) and (4.35) that
and
. Also, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ140_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ141_HTML.gif)
Thus, from , and
, it follows that
. In terms of Lemma 2.1, we derive
.
Next, let us show that , where
.
Indeed, since is bounded, there exists a subsequence
of
such that
. Hence it follows from (4.31), (4.35), and
that both
and
converge weakly to the same point
. Furthermore, from
and (4.31) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ142_HTML.gif)
If and
, then it follows from (2.17) and the monotonicity of the operators
that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ143_HTML.gif)
Letting , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ144_HTML.gif)
Then the maximality of the operators implies that
.
Now, by the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ145_HTML.gif)
where . Replacing
by
, we have from (A2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ146_HTML.gif)
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting
in the last inequality, from (4.35) and (A4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ147_HTML.gif)
For , with
, and
, let
. Since
and
, then
and hence
. So, from (A1) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ148_HTML.gif)
Dividing by , we get
. Letting
, from (A3) it follows that
. So,
. Therefore,
. Let
. From Lemma 2.2 and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ149_HTML.gif)
From Proposition 4.1, we also know that . Note that
. Since
is weakly sequentially continuous, then
as
. In addition, taking into account the monotonicity of
, we conclude that
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ150_HTML.gif)
From the strict convexity of , it follows that
. Therefore,
, where
. This completes the proof.
Remark 4.3.
In Theorem 4.2, put ,
, and
. Then, for all
and
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ151_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ152_HTML.gif)
Moreover, the following hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ153_HTML.gif)
In this case, Algorithm (4.1) reduces to the following one:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F896252/MediaObjects/13660_2009_Article_2026_Equ154_HTML.gif)
Corollary 4.4.
Suppose that conditions (A1)–(A5) are fulfilled and let be a sequence defined by (4.65), where
is defined in Lemma 2.8,
satisfy the conditions
and
, and
satisfies
. If
is weakly sequentially continuous, then
converges weakly to
, where
.
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Acknowledgments
The first author (http://zenglc@hotmail.com) was partially supported by the National Science Foundation of China (10771141), Ph. D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author (http://yaojc@math.nsysu.edu.tw) was partially supported by the Grant NSF 97-2115-M-110-001.
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Zeng, L.C., Lin, Y.C. & Yao, J.C. Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators. J Inequal Appl 2009, 896252 (2009). https://doi.org/10.1155/2009/896252
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DOI: https://doi.org/10.1155/2009/896252