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Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings
Fixed Point Theory and Applications volume 2011, Article number: 973028 (2011)
Abstract
The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set of solutions of a generalized equilibrium problem, the set
of fixed points of a relatively nonexpansive mapping
, and the set
of zeros of a maximal monotone operator
in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in
. These new results represent the improvement, generalization, and development of the previously known ones in the literature.
1. Introduction
Let be a real Banach space with the dual
and
be a nonempty closed convex subset of
. We denote by
and
the sets of positive integers and real numbers, respectively. Also, we denote by
the normalized duality mapping from
to
defined by

where denotes the generalized duality pairing. Recall that if
is smooth, then
is single valued and if
is uniformly smooth, then
is uniformly norm-to-norm continuous on bounded subsets of
. We will still denote by
the single valued duality mapping. Let
be a bifunction and
be a nonlinear mapping. We consider the following generalized equilibrium problem:

The set of such is denoted by
, that is,

Whenever a Hilbert space, problem (1.2) was introduced and studied by S. Takahashi and W. Takahashi [1]. Similar problems have been studied extensively recently. See, for example, [2–11]. In the case of
is denoted by
. In the case of
,
is also denoted by
. The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for example, [12–14]. A mapping
is called nonexpansive if
for all
. Denote by
the set of fixed points of
, that is,
. A mapping
is called α-inverse-strongly monotone, if there exists an
such that

It is easy to see that if is an α-inverse-strongly monotone mapping, then it is
-Lipschitzian.
Let be a real Banach space with the dual
. A multivalued operator
with domain
is called monotone if
for each
and
,
. A monotone operator
is called maximal if its graph
is not properly contained in the graph of any other monotone operator. A method for solving the inclusion
is the proximal point algorithm. Denote by
the identity operator on
a Hilbert space. The proximal point algorithm generates, for any initial point
, a sequence
in
, by the iterative scheme

where is a sequence in the interval
. Note that this iteration is equivalent to

This algorithm was first introduced by Martinet [12] and generally studied by Rockafellar [15] in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance, [16–21] and the references therein.
Let be a reflexive, strictly convex, and smooth Banach space with the dual
and
be a nonempty closed convex subset of
. Let
be a maximal monotone operator with domain
and
be a relatively nonexpansive mapping. Let
be an α-inverse-strongly monotone mapping and
be a bifunction satisfying (A1)–(A4): (A1)
,
; (A2)
is monotone, that is,
,
; (A3)
,
; (A4) the function
is convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms 1.1 and 1.2 for finding an element of
.
Algorithm 1.1.

where is a sequence in
and
,
are sequences in
.
Algorithm 1.2.

where is a sequence in
and
is a sequence in
.
In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence generated by Algorithm 1.1 and the sequence
generated by Algorithm 1.2, converge strongly to the same point
. These new results represent the improvement, generalization and development of the previously known ones in the literature including Solodov and Svaiter [22], Kamimura and Takahashi [23], Qin and Su [24], and Ceng et al. [25].
Throughout this paper the symbol ⇀ stands for weak convergence and → stands for strong convergence.
2. Preliminaries
Let be a real Banach space with the dual
. We denote by
the normalized duality mapping from
to
defined by

where denotes the generalized duality pairing. A Banach space
is called strictly convex if
for all
with
and
. It is said to be uniformly convex if
for any two sequences
such that
and
. Let
be a unit sphere of
, then the Banach space
is called smooth if

exists for each . If
is smooth, then
is single valued. We still denote the single valued duality mapping by
.
It is also said to be uniformly smooth if the limit is attained uniformly for . Recall also that if
is uniformly smooth, then
is uniformly norm-to-norm continuous on bounded subsets of
. A Banach space
is said to have the Kadec-Klee property if for any sequence
, whenever
and
, we have
. It is known that if
is uniformly convex, then
has the Kadec-Klee property; see [26, 27] for more details.
Let be a nonempty closed convex subset of a real Hilbert space
and
be the metric projection of
onto C, then
is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber [28] recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that is a smooth Banach space. Consider the functional defined as in [28, 29] by

It is clear that in a Hilbert space , (2.3) reduces to
, for all
.
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

The existence and uniqueness of the operator follows from the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [30]). In a Hilbert space
,
. From [28], in uniformly smooth and uniformly convex Banach spaces, we have

Let be a nonempty closed convex subset of
, and let
be a mapping from
into itself. A point
is called an asymptotically fixed point of
[31] if
contains a sequence
which converges weakly to
such that
. The set of asymptotical fixed points of
will be denoted by
. A mapping
from
into itself is called relatively nonexpansive [32–34] if
and
for all
and
.
We remark that if is a reflexive, strictly convex and smooth Banach space, then for any
if and only if
. It is sufficient to show that if
then
. From (2.5), we have
. This implies that
. From the definition of
, we have
. Therefore, we have
; see [26, 27] for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (see [23]).
Let be a smooth and uniformly convex Banach space and let
and
be two sequences of
. If
and either
or
is bounded, then
.
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, let
and let
, then

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, then

Lemma 2.4 (see [35]).
Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space
, and let
be a relatively nonexpansive mapping, then
is closed and convex.
The following result is according to Blum and Oettli [36].
Lemma 2.5 (see [36]).
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

Motivated by Combettes and Hirstoaga [37] in a Hilbert space, Takahashi and Zembayashi [38] established the following lemma.
Lemma 2.6 (see [38]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
, and let
be a bifunction from
to
satisfying (A1)–(A4). For
and
, define a mapping
as follows:

for all , then, the following hold:
(i) is single valued;
(ii) is a firmly nonexpansive-type mapping, that is, for all
,

(iii);
(iv) is closed and convex.
Using Lemma 2.6, one has the following result.
Lemma 2.7 (see [38]).
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
,

Utilizing Lemmas 2.5, 2.6 and 2.7 as above, Chang [39] derived the following result.
Proposition 2.8 (see [39, Lemma 2.5]).
Let be a smooth, strictly convex and reflexive Banach space and
be a nonempty closed convex subset of
. Let
be an α-inverse-strongly monotone mapping, let
be a bifunction from
to
satisfying (A1)–(A4), and let
, then there hold the following:
(I)for , there exists
such that

(II)if is additionally uniformly smooth and
is defined as

then the mapping has the following properties:
(i) is single valued,
(ii) is a firmly nonexpansive-type mapping, that is,

(iii),
(iv) is a closed convex subset of
,
(v), for all
.
Proof.
Define a bifunction as follows:

Then it is easy to verify that satisfies the conditions (A1)–(A4). Therefore, The conclusions (I) and (II) of Proposition 2.8 follow immediately from Lemmas 2.5, 2.6 and 2.7.
Let be a reflexive, strictly convex and smooth Banach space, and let
be a maximal monotone operator with
, then,

3. Main Results
Throughout this section, unless otherwise stated, we assume that is a maximal monotone operator with domain
,
is a relatively nonexpansive mapping,
is an α-inverse-strongly monotone mapping and
is a bifunction satisfying (A1)–(A4), where
is a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space
. In this section, we study the following algorithm.
Algorithm 3.1.

where is a sequence in
and
,
are sequences in
.
First we investigate the condition under which the Algorithm 3.1 is well defined. Rockafellar [40] proved the following result.
Lemma 3.2 (Rockafellar [40]).
Let be a reflexive, strictly convex, and smooth Banach space and let
be a multivalued operator, then there hold the following:
(i) is closed and convex if
is maximal monotone such that
;
(ii) is maximal monotone if and only if
is monotone with
for all
.
Utilizing this result, we can show the following lemma.
Lemma 3.3.
Let be a reflexive, strictly convex, and smooth Banach space. If
, then the sequence
generated by Algorithm 3.1 is well defined.
Proof.
For each , define two sets
and
as follows:

It is obvious that is closed and
are closed convex sets for each
. Let us show that
is convex. For
and
, put
. It is sufficient to show that
. Indeed, observe that

is equivalent to

Note that there hold the following:

Thus we have

This implies that . Therefore,
is convex and hence
is closed and convex.
On the other hand, let be arbitrarily chosen, then
and
. From Algorithm 3.1, it follows that

So for all
. Now, from Lemma 3.2 it follows that there exists
such that
and
. Since
is monotone, it follows that
, which implies that
and hence
. Furthermore, it is clear that
, then
, and therefore
is well defined. Suppose that
and
is well defined for some
. Again by Lemma 3.2, we deduce that
such that
and
, then from the monotonicity of
we conclude that
, which implies that
and hence
. It follows from Lemma 2.4 that

which implies that . Consequently,
and so
. Therefore
is well defined, then, by induction, the sequence
generated by Algorithm 3.1, is well defined for each integer
.
Remark 3.4.
From the above proof, we obtain that

for each integer .
We are now in a position to prove the main theorem.
Theorem 3.5.
Let be a uniformly smooth and uniformly convex Banach space. Let
be a sequence in
and
be sequences in
such that

Let . If
is uniformly continuous, then the sequence
generated by Algorithm 3.1 converges strongly to
.
Proof.
First of all, if follows from the definition of that
. Since
, we have

Thus is nondecreasing. Also from
and Lemma 2.3, we have that

for each and for each
. Consequently,
is bounded. Moreover, according to the inequality

we conclude that is bounded. Thus, we have that
exists. From Lemma 2.3, we derive the following:

for all . This implies that
. So it follows from Lemma 2.1 that
. Since
, from the definition of
, we also have

Observe that

At the same time,

Since and
, it follows that

and hence that . Further, from
, we have
, which yields

Then it follows from that
. Hence it follows from Lemma 2.1 that
. Since from (3.15) we derive

we have

Thus, from ,
, and
, we know that
. Consequently from (3.16),
, and
it follows that

So it follows from (3.15), , and
that
. Utilizing Lemma 2.1 we deduce that

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

Since is uniformly norm-to-norm continuous on bounded subsets of
, it follows from (3.23) that
and
, which hence yield
. Utilizing Lemma 2.1, we get
. Observe that

due to (3.23). Since is uniformly norm-to-norm continuous on bounded subsets of
, we have that

On the other hand, we have

Noticing that

we have

From (3.26) and , we obtain

Since is also uniformly norm-to-norm continuous on bounded subsets of
, we obtain

Observe that

Since is uniformly continuous, it follows from (3.27), (3.31) and
that
.
Now let us show that , where

Indeed, since is bounded and
is reflexive, we know that
. Take
arbitrarily, then there exists a subsequence
of
such that
. Hence
. Let us show that
. Since
, we have that
. Moreover, since
is uniformly norm-to-norm continuous on bounded subsets of
and
, we obtain

It follows from and the monotonicity of
that

for all and
. This implies that

for all and
. Thus from the maximality of
, we infer that
. Therefore,
. Further, let us show that
. Since
and
, from
we obtain that
and
.
Since is uniformly norm-to-norm continuous on bounded subsets of
, from
we derive

From , it follows that

By the definition of , we have

where

Replacing by
, we have from (A2) that

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting
in the last inequality, from (3.38) and (A4) we have

For , with
, and
, let
. Since
and
, then
and hence
. So, from (A1) we have

Dividing by , we have

Letting , from (A3) it follows that

So, . Therefore, we obtain that
by the arbitrariness of
.
Next, let us show that and
.
Indeed, put . From
and
, we have
. Now from weakly lower semicontinuity of the norm, we derive for each

It follows from the definition of that
and hence

So we have . Utilizing the Kadec-Klee property of
, we conclude that
converges strongly to
. Since
is an arbitrary weakly convergent subsequence of
, we know that
converges strongly to
. This completes the proof.
Theorem 3.5 covers [25, Theorem 3.1] by taking and
. Also Theorem 3.5 covers [24, Theorem 2.1] by taking
,
and
.
Theorem 3.6.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a maximal monotone operator with domain
,
be a relatively nonexpansive mapping,
be an α-inverse-strongly monotone mapping and
be a bifunction satisfying (A1)–(A4). Assume that
is a sequence in
satisfying
and that
is a sequences in
satisfying
.
Define a sequence .
Algorithm 3.7.

where is the single valued duality mapping on
. Let
. If
is uniformly continuous, then
converges strongly to
.
Proof.
For each , define two sets
and
as follows:

It is obvious that is closed and
are closed convex sets for each
. Let us show that
is convex and so
is closed and convex. Similarly to the proof of Lemma 3.3, since

is equivalent to

we know that is convex and so is
. Next, let us show that
for each
. Indeed, utilizing Proposition 2.8, we have, for each
,

So for all
and
. As in the proof of Lemma 3.3, we can obtain
and hence
. It follows from Lemma 2.4 that

which implies that . Consequently,
and so
for all
. Therefore, the sequence
generated by Algorithm 3.7 is well defined. As in the proof of Theorem 3.5, we can obtain
. Since
, from the definition of
we also have

As in the proof of Theorem 3.5, we can deduce not only from that
but also from
,
and
that

Since , from the definition of
, we also have

It follows from (3.55) and that

Utilizing Lemma 2.1 we have

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

Since is uniformly norm-to-norm continuous on bounded subsets of
, it follows from (3.58) that
and
, which together with
, yield
. Utilizing Lemma 2.1, we get
. Observe that

due to (3.58). Since is uniformly norm-to-norm continuous on bounded subsets of
, we have

Note that

Therefore, from we get

Since is also uniformly norm-to-norm continuous on bounded subsets of
, we obtain

It follows that

Since is uniformly continuous, it follows from (3.58) and (3.64) that
.
Finally, we prove that . Indeed, for
arbitrarily fixed, there exists a subsequence
of
such that
, then
. Now let us show that
. Since
, we have that
. Moreover, since
is uniformly norm-to-norm continuous on bounded subsets of
, and
, we obtain that
. It follows from
and the monotonicity of
that
for all
and
. This implies that
for all
and
. Thus from the maximality of
, we infer that
. Further, let us show that
. Since
and
, from
we obtain that
and
.
Since is uniformly norm-to-norm continuous on bounded subsets of
, from
we derive
. From
it follows that

By the definition of , we have

where . Replacing
by
, we have from (A2) that

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting
in the last inequality, from (3.66) and (A4) we have
, for all
. For
, with
, and
, let
. Since
and
, then
and hence
. So, from (A1) we have

Dividing by , we have
, for all
. Letting
, from (A3) it follows that
, for all
. So,
. Therefore, we obtain that
by the arbitrariness of
.
Next, let us show that and
.
Indeed, put . From
and
, we have
. Now from weakly lower semicontinuity of the norm, we derive for each

It follows from the definition of that
and hence
. So we have
. Utilizing the Kadec-Klee property of
, we know that
. Since
is an arbitrary weakly convergent subsequence of
, we know that
. This completes the proof.
Theorem 3.6 covers [25, Theorem 3.2] by taking and
. Also Theorem 3.6 covers [24, Theorem 2.2] by taking
and
.
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Acknowledgments
This research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (no. DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (no. 09ZZ133), National Science Foundation of China (no. 11071169), Ph.D. Program Foundation of Ministry of Education of China (no. 20070270004), Science and Technology Commission of Shanghai Municipality Grant (no. 075105118), and Shanghai Leading Academic Discipline Project (no. S30405). Al-Homidan is grateful to KFUPM for providing research facilities.
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Zeng, LC., Ansari, Q. & Al-Homidan, S. Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings. Fixed Point Theory Appl 2011, 973028 (2011). https://doi.org/10.1155/2011/973028
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DOI: https://doi.org/10.1155/2011/973028