Strong Convergence of a General Iterative Algorithm for Mixed Equilibrium, Variational Inequality and Common Fixed Points Problems ()
1. Introduction
Let
be a real Hilbert space, whose inner product and norm are denoted by
and
respectively. Let
be a nonempty closed convex subset of H. A mapping
is called nonexpansive if
for all
We denote by
the set of fixed points of T. A linear bounded operator A is strongly positive if there is a constant
with the property
for all
A mapping
is said to be a contraction if there exists a coefficient
such that
for all
Let PC be the nearest point projection of
onto the convex subset
(i.e., for
, PC is the only point in C such that
It is known that projection operator PC is nonexpansive. It is also known that PC satisfies
for
The following characterizes the projection PC Given
and
Then
if and only if there holds the relations:
(1.1)
for all
(see [1]). Moreover,
is characterized by the properties:
and
for all
Let
be a nonlinear map. The classical variational inequality problem, denoted by
is to find
such that
(1.2)
for all
One can see that the variational inequality problem (1.2) is equivalent to the following fixed point problem: the element
is a solution of the variational inequality (1.2) if and only if
satisfies the relation
where
is a constant. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [2-6] and the references therein. A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space
:
(1.3)
where A is a linear bounded operator and b is a given point in H. In [5] (see also [6]), it is proved that the sequence
defined by the iterative method below, with the initial guess
chosen arbitrarily,
![](https://www.scirp.org/html/11-5300318\0e0e0009-a915-400a-9ecf-577bf42f4a13.jpg)
converges strongly to the unique solution of the minimization problem (1.3) provided the sequence
satisfies certain conditions. In 2006, Marino and Xu (see [3]) considered the following viscosity iterative method which was first introduced by Moudafi (see [7]):
(1.4)
They proved that the sequence
generated by iterative scheme (1.4) converges strongly to the unique solution of the variational inequality
,
which is the optimality condition for the minimization problem
![](https://www.scirp.org/html/11-5300318\c2756f03-f71c-420c-aca4-a7ffe2b366b4.jpg)
where h is a potential function for
(i.e.,
for
).
For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for
-cocoercive mapping, Takahashi and Toyoda (see [11]) introduced the following iterative process: ![](https://www.scirp.org/html/11-5300318\5798a32a-fbef-4fcd-a87f-0b9d1e1b4981.jpg)
(1.5)
where B is
-cocoercive,
and
. They showed that, if
is nonempty, then the sequence
generated by (1.5) converges weakly to some
In 2005, Iiduka and Takahashi (see [12]) introduced the following iterative process:
(1.6)
where
,
and
They proved that under certain appropriate conditions imposed on
and
the sequence
generated by (1.6) converges strongly to
In 2009, Qin, Kang and Shang, [13] introduced the following iterative algorithm given by ![](https://www.scirp.org/html/11-5300318\3e2f0739-353e-4330-adba-fdc54a1cd9cd.jpg)
(1.7)
where
,
a k-strict pseudo-contraction for some
,
defined by
A is a strongly positive linear bounded self-adjoint operator and f is a contraction. They proved that the sequence
generated by the iterative algorithm (1.7) converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A.
Let
be a proper extended realvalued function and F be a bifunction from
to
where
is the set of real numbers. Ceng and Yao [14] considered the following mixed equilibrium problem: Find
such that
(1.8)
for all
The set of solutions of (1.8) is denoted by
i.e.,
![](https://www.scirp.org/html/11-5300318\a9fd1aa7-09e6-4e5a-a43b-39339f949b8e.jpg)
It is easy to see that x is a solution of problem (1.8) implies that
Moreover, Ceng and Yao [14] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.8) and the set of common fixed points of a family of finitely nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. If
then the mixed equilibrium problem (1.8) becomes the following equilibrium problem:
(1.9)
for all
The set of solutions of (1.9) is denoted by
i.e.,
![](https://www.scirp.org/html/11-5300318\eb7107af-bcba-4430-bcef-f1a9b51c1075.jpg)
Given a mapping
let
and
for all
Then,
if and only if
for all
i.e., z is a solution of the variational inequality. Equilibrium problems have been studied extensively; see, for instance, [15,16]. The mixed equilibrium problem (1.8) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance, [14,16-19].
Combettes and Hirstoaga (see [15]) introduced an iterative scheme for finding the best approximation to the initial data when
is nonempty and proved a strong convergence theorem. In 2007, S. Takahashi and W. Takahashi (see [20]) introduced an iterative scheme using the viscosity approximation method for finding a common element of the set of solutions of equilibrium problem (1.9) and the set of fixed points of a nonexpansive nonself-mapping in a Hilbert space. The scheme is defined as follows: ![](https://www.scirp.org/html/11-5300318\8d0e8a3b-19f7-4350-90f8-8883e1c7db56.jpg)
(1.10)
They proved that under certain appropriate conditions imposed on
and
, the sequences
and
generated by (1.10) converge strongly to
, where
In the same year, Shang et al. (see [21]) introduced the following iterative scheme: ![](https://www.scirp.org/html/11-5300318\e70f3c11-4bb7-4fc5-96c0-e4f0a96452c8.jpg)
(1.11)
for finding a common element of the set of solutions of equilibrium problem (1.9) and the set of fixed points of a nonexpansive nonself-mapping in a Hilbert space. They proved that under some sufficient suitable conditions, the sequences
and
generated by (1.11) converge strongly to
![](https://www.scirp.org/html/11-5300318\14937301-0c91-4500-b14f-1c36e7f0bbb0.jpg)
where
![](https://www.scirp.org/html/11-5300318\80f7e94c-49de-4985-8f6c-fe018dd71eb5.jpg)
which is the unique solution of the variational inequality
![](https://www.scirp.org/html/11-5300318\b247a306-366c-4ad4-a431-33e19c87b26d.jpg)
for all ![](https://www.scirp.org/html/11-5300318\8e6706d2-9036-4eba-a141-719347017d3b.jpg)
Let
where
be a finite family of nonexpansive mappings. Finding an optimal point in the intersection
of the fixed points set of a finite family of nonexpansive mappings is a problem of interest in various branches of sciences; see [22-27] and also see [28] for solving the variational problems defined on the set of common fixed points of finitely many nonexpansive mappings. Atsushiba and Takahashi (see [29]), defined the mappings
(1.12)
where
Such a mapping
is called the W-mapping generated by
and
The concept of W-mappings was introduced in [30-33]. In 2008, Qin et al. (see [34]) introduced and studied the following iterative process: ![](https://www.scirp.org/html/11-5300318\519fa55e-52d9-4bba-ae7b-dc77c0a3daba.jpg)
(1.13)
where
is defined by (1.12),
is a strongly linear bounded operator and B is
-Lipschitzian, relaxed
-cocoercive mapping of C into H. They proved that the sequences
and
generated by the iterative scheme (1.13) converge strongly to
![](https://www.scirp.org/html/11-5300318\09194ef9-9b79-46ba-9c2d-25c908406f07.jpg)
where
![](https://www.scirp.org/html/11-5300318\b45efb0a-e113-411e-9ed7-a4d424e89c38.jpg)
which is the unique solution of the variational inequality
![](https://www.scirp.org/html/11-5300318\b701f96c-188e-4f49-a6ec-4781ed106233.jpg)
for all
.
In the same year, Colao et al. (see [35]) introduced a new iterative scheme: ![](https://www.scirp.org/html/11-5300318\f2abafb5-baea-4ba6-a0ff-6de8ee6b0497.jpg)
(1.14)
for approximating a common element of the set of solutions of equilibrium problem (1.9) and the set of common fixed points of a finite family of nonexpansive mappings and obtained a strong convergence theorem in a Hilbert space. In 2009, Yao et al. (see [36]) studied similar scheme as follows: ![](https://www.scirp.org/html/11-5300318\347a4ee7-479b-483a-bb22-b600b35877f4.jpg)
(1.15)
where
,
,
,
and
is the W-mapping defined by (1.12). They proved that under certain appropriate conditions imposed on
,
,
and
, the sequences
and
generated by (1.15) converge strongly to
![](https://www.scirp.org/html/11-5300318\5cebe377-e5f0-4734-b5db-7729c34c3f12.jpg)
where
![](https://www.scirp.org/html/11-5300318\1c778da4-d96a-4398-922f-fce0ab55e896.jpg)
which is the unique solution of the variational inequality
for all
.
If
for some
then (1.15) reduces to the iterative scheme (1.14). Very recently, Kangtunyakarn and Suantai (see [37]) defined the new mappings
(1.16)
where
Such a mapping Kn is called the K-mapping generated by
and
Nonexpansivity of each Ti ensures the nonexpansivity of Kn Also following they defined the new mappings
(1.17)
where
such that
for all
and
Such a mapping K is called the K-mapping generated by
and
In [37], Lemma 2.9 and Lemma 2.10, its shown that
![](https://www.scirp.org/html/11-5300318\43aa1417-5dfc-407d-a177-ed46429f9533.jpg)
and
for all
where Kn and K are the K-mappings defined by (1.16) and (1.17), respectively. Its important tool for the proof of the main results in this paper. Moreover, Kangtunyakarn and Suantai (see [37]) introduced a new iterative scheme:
and
,
(1.18)
where
,
,
,
and Kn is the K-mapping defined by (1.16). They proved that under certain appropriate conditions imposed on
,
and
, the sequences
and
generated by (1.18) converge strongly to
![](https://www.scirp.org/html/11-5300318\1acbdbbd-1ffb-4b8b-860d-fbc5a788f4a0.jpg)
where
![](https://www.scirp.org/html/11-5300318\63262a40-0502-4fe5-bbe2-19a490a195b9.jpg)
Motivated by the recent works, we introduce a more general iterative algorithm for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. The scheme is defined as follows:
and ![](https://www.scirp.org/html/11-5300318\dc187b6b-8d14-48d2-898a-f24702ad1c53.jpg)
(1.19)
where
,
,
,
,
,
is a
-Lipschitzian, relaxed
-cocoercive mapping, f is a contraction of H into itself with a coefficient
is a projection of H onto C, A is a strongly positive linear bounded operator on H, F is a mixed equilibrium bifunction,
is a proper lower semicontinuous and convex function and Kn is the K-mapping generated by
and
We prove that the sequences
and
generated by the iterative scheme (1.19) converge strongly to
![](https://www.scirp.org/html/11-5300318\04a2d206-e962-4639-b3e5-6e77bfcb842e.jpg)
where
![](https://www.scirp.org/html/11-5300318\572f2d4f-f2cd-4b05-a2bb-3997cf9c64f9.jpg)
which is the unique solution of the variational inequality for all ![](https://www.scirp.org/html/11-5300318\5b7566ba-fba2-43c0-8c65-fc561d551b30.jpg)
![](https://www.scirp.org/html/11-5300318\fedfa168-0d63-4227-9974-81797e8191f6.jpg)
and is also the optimality condition for the minimization problem
![](https://www.scirp.org/html/11-5300318\95bd3db4-f416-44e4-bf76-c93878131e13.jpg)
where h is a potential function for
(i.e.,
for
).
2. Preliminaries and Lemmas
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
A mapping B is called
-strongly monotone, if each
we have
![](https://www.scirp.org/html/11-5300318\c4abe569-a646-4529-8915-5ac083d9d214.jpg)
for a constant v > 0, which implies that
so that B is v-expansive and when v = 1, it is expansive. B is said to be v-cocoercive (see [8] and [9]), if for each
we have
![](https://www.scirp.org/html/11-5300318\db780ff3-1f89-420b-a304-56160a665ef3.jpg)
for a constant v > 0. Clearly, every v-cocoercive mapping B is
-Lipschitz continuous. B is called relaxed u-cocoercive, if there exists a constant u > 0 such that
![](https://www.scirp.org/html/11-5300318\4e71fc58-0f16-4a61-a48a-48ec0c75387a.jpg)
for all
B is said to be relaxed
-cocoercive, if there exist two constants u, v > 0 such that
![](https://www.scirp.org/html/11-5300318\f34e5766-c6f6-4dc1-9c5c-ac54fe02e94a.jpg)
for all
for
B is v-strongly monotone.
It is worth mentioning that the class of mappings which are relaxed
-cocoercive more general than the class of strongly monotone mappings. It is easy to see that if B is a v-strongly monotone mapping, then it is a relaxed
-cocoercive mapping (see [10]).
It is well known that for all
and
there holds
![](https://www.scirp.org/html/11-5300318\904530a4-fc98-446c-ade7-54ad0ec4eb58.jpg)
Recall that a space X is said to satisfy Opial’s condition (see [38]) if
weakly as
and
for all
then
![](https://www.scirp.org/html/11-5300318\eb2be521-85ee-4885-935c-ea8ab687f7bb.jpg)
A set-valued mapping
is called monotone if for all
,
,
and
imply ![](https://www.scirp.org/html/11-5300318\2255c57f-9ae5-4ed7-80b7-5133eb1a575e.jpg)
A monotone mapping
is maximal if graph
of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for
,
for every
implies
Let B be a monotone mapping of C into H and let
be normal cone to C at
i.e.,
![](https://www.scirp.org/html/11-5300318\76c1d149-ce4a-4acc-a265-a1ba7c055b75.jpg)
and define
![](https://www.scirp.org/html/11-5300318\8cc4613f-240a-4b5d-9436-7cb9176d935b.jpg)
Then T is a maximal monotone and
if and only if
; see [39].
In the sequel, the following lemmas are needed to prove our main results.
Lemma 2.1. (see [4,5]). Assume that
is a sequence of nonnegative real numbers such that
![](https://www.scirp.org/html/11-5300318\77d26d6f-8b8c-4547-b2b6-ecb51c4d0820.jpg)
where
is a sequence in
and
is a sequence such that 1) ![](https://www.scirp.org/html/11-5300318\5fe7698b-9c2f-4777-ac2f-61d3a6b0554b.jpg)
2)
Then ![](https://www.scirp.org/html/11-5300318\a836ca05-f1ed-4b71-b3cc-984d27d506be.jpg)
Lemma 2.2. (see [3]). Assume A is a strong positive linear bounded operator on a Hilbert space H with coefficient
and
. Then
.
Lemma 2.3. (see [40]). Let
and
be bounded sequences in a Banach space
and let
be a sequence in
with
![](https://www.scirp.org/html/11-5300318\b53a72af-9462-441b-8c74-d9d668786b91.jpg)
Suppose
for all integers n ≥ 0 and ![](https://www.scirp.org/html/11-5300318\899f203c-05ad-438b-88fc-ed1185e93200.jpg)
Then
![](https://www.scirp.org/html/11-5300318\4dcd00c0-e645-4d94-a6b4-5bc5a02be6e7.jpg)
Lemma 2.4. (see [37]). Let C be a nonempty closed convex set of a strictly convex Banach space. Let
be a finite family of nonexpansive mappings of C into itself with
and let
be real numbers such that
for every
and
Let K be the K-mapping generated by
and
Then
.
Lemma 2.5. (see [37]). Let C be a nonempty convex subset of a Banach space. Let
be a finite family of nonexpansive mappings of
into itself and ![](https://www.scirp.org/html/11-5300318\9c6cab82-a3be-492e-bccb-13f6afcf83e8.jpg)
be sequences in
such that
Moreover for every
let K and
be the Kmappings generated by
and ![](https://www.scirp.org/html/11-5300318\5b3d7853-f5f1-48a7-8d48-52687d20e344.jpg)
and
and
respectively. Then for every
it follows that
![](https://www.scirp.org/html/11-5300318\8cb75d50-44d0-42c1-a8f2-33a71b768984.jpg)
For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction
and the set C:
(A1)
for all ![](https://www.scirp.org/html/11-5300318\0ff8204d-fd22-4df1-bc5b-a8bf92045e55.jpg)
(A2)
is monotone, i.e.,
for all ![](https://www.scirp.org/html/11-5300318\67027e3b-6327-4142-bbdc-1334594d7ecb.jpg)
(A3) For each ![](https://www.scirp.org/html/11-5300318\c4e3fd28-1fc7-45d6-8b25-1af0550e3a83.jpg)
![](https://www.scirp.org/html/11-5300318\b284e0c1-bdb6-4bce-bdbf-120dbc75cff9.jpg)
(A4) For each
is convex and lower semicontinuous;
(B1) For each
and
there exists a bounded subset
and
such that for any ![](https://www.scirp.org/html/11-5300318\459bf277-1e81-4a1d-bfcf-ca0738041228.jpg)
![](https://www.scirp.org/html/11-5300318\28e95004-3469-4378-9a79-e8047c3edf07.jpg)
(B2) C is a bounded set.
By a similar argument as in the proof of Lemma 2.3 in [18], we have the following result.
Lemma 2.6. Let C be a nonempty closed convex subset of a Hilbert space H and let F be a mixed equilibrium bifunction of C × C into
satisfying conditions (A1)- (A4) and let
be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For
and
define a mapping
as follows:
![](https://www.scirp.org/html/11-5300318\22c9bf09-8b6b-4610-9f75-2c6e540fdbf3.jpg)
for all
Then
is well defined and the following hold:
1)
is single-valued;
2)
is firmly nonexpansive, i.e., for any ![](https://www.scirp.org/html/11-5300318\de96b9d0-1214-435a-94a1-e65e8d10f28a.jpg)
![](https://www.scirp.org/html/11-5300318\809aabf9-6c12-42dc-883c-7d8f8f31a652.jpg)
3)
;
4)
is closed and convex.
Remark 2.7. We remark that Lemma 1.6 is not a consequence of Lemma 3.1 in [14], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative
of the function
does not cover the case
![](https://www.scirp.org/html/11-5300318\29b5e046-46ab-4100-b485-d8c0e8ae7e92.jpg)
The following lemma is well known.
Lemma 2.8. In a real Hilbert space H, there holds the following inequality
![](https://www.scirp.org/html/11-5300318\13dc04d8-a48d-47ac-9abd-5b7d5ff1f828.jpg)
for all ![](https://www.scirp.org/html/11-5300318\6fb4bf9c-b611-439f-b496-3004fe6b906b.jpg)
3. Main Results
Theorem 3.1. Let H be a real Hilbert space, C a nonempty closed convex subset of H, B a
-Lipschitzian, relaxed
-cocoercive mapping of C into H, F a bifunction from C × C to
which satisfies (A1)-(A4),
a proper lower semicontinuous and convex function and
a finite family of nonexpansive mappings of C into H such that the common fixed points set
![](https://www.scirp.org/html/11-5300318\001e8b24-c006-47f5-8daf-3727aecdce45.jpg)
Let f be a contraction of H into itself with a coefficient
and A a strongly positive linear bounded operator on H with coefficient
such that ![](https://www.scirp.org/html/11-5300318\b237de64-22f6-45ae-b1de-9819334e3aba.jpg)
Assume that
and either (B1) or (B2) holds.
Let
be real numbers such that
for every
and
and
,
two real sequences in (0, 1) satisfying the following conditions:
(C1)
and ![](https://www.scirp.org/html/11-5300318\8ec55623-cce5-40e0-88a2-c3f51d31b9cd.jpg)
(C2) ![](https://www.scirp.org/html/11-5300318\a04c4de4-6586-4801-9c08-1be6748b2007.jpg)
(C3)
and
(this is weaker than the condition ); ![](https://www.scirp.org/html/11-5300318\36564956-1e4f-49d8-9fb6-de514a8f0071.jpg)
(C4) ![](https://www.scirp.org/html/11-5300318\0ed82c80-1371-48ac-927f-c76ff9bd03cf.jpg)
(C5)
for some a, b with
;
(C6) ![](https://www.scirp.org/html/11-5300318\6d7a1ece-137f-4f9d-adf2-dc7225fa7f15.jpg)
Then, the sequences
and
generated iteratively by (1.19) converge strongly to
![](https://www.scirp.org/html/11-5300318\25b4b4f5-bf41-4a38-9274-9eab2edfd5b6.jpg)
where
![](https://www.scirp.org/html/11-5300318\84286821-e51b-4716-8ce7-c6ed974fb8a4.jpg)
which solves the following variational inequality:
![](https://www.scirp.org/html/11-5300318\256db1a6-40e7-4c14-8a9d-b55220593edb.jpg)
for all
![](https://www.scirp.org/html/11-5300318\30d3f3fd-6cf0-4278-8bcf-5331ad20ce93.jpg)
Proof Since
as
by the condition (C1), we may assume, without loss of generality, that
![](https://www.scirp.org/html/11-5300318\466f11c0-d7b4-4e6b-8882-97650d7b0593.jpg)
for all n. We also have
for all n. By using Lemma 2.2, we have
![](https://www.scirp.org/html/11-5300318\71cb3522-4997-463c-8d17-e101d0bccc9e.jpg)
Since A is a strongly positive linear bounded operator on a Hilbert space H, we have
![](https://www.scirp.org/html/11-5300318\9aaa2aab-b48f-4ff3-a02b-86b8f93817bb.jpg)
and
![](https://www.scirp.org/html/11-5300318\cad84393-4fc2-43ed-a0bd-0c50e5f654cb.jpg)
Observe that
![](https://www.scirp.org/html/11-5300318\60d9e36d-3c65-4223-89d4-dd2b1bfbd0bd.jpg)
This shows that
is positive. It follows that
![](https://www.scirp.org/html/11-5300318\a042281d-9b8a-47d4-9d58-97ba619f8d7d.jpg)
Next, we will assume that
First, we show
is nonexpansive. Indeed, from the relaxed
-cocoercive and
-Lipschitzian definition on B and condition (C5), we have which implies the mapping
is nonexpansive.
![](https://www.scirp.org/html/11-5300318\5b51ed8d-7585-4b73-9379-15a339cd0bd4.jpg)
We shall divide our proof into 5 steps.
Step 1. We shall show that the sequence
is bounded. Let
![](https://www.scirp.org/html/11-5300318\fb0a944c-58b3-4378-9f1c-db27fce16b9d.jpg)
Since
we have
(3.1)
Putting
for all
we have
![](https://www.scirp.org/html/11-5300318\60de27c1-a4e3-418b-a8fe-ad182a335649.jpg)
Using (1.19), (3.1) and (3.2), we have
![](https://www.scirp.org/html/11-5300318\57155694-2984-463a-861c-3c2caa717e27.jpg)
which gives that
![](https://www.scirp.org/html/11-5300318\2bd4e1d3-94ad-42e9-ae11-052768061b94.jpg)
Hence
is bounded, so are
,
and ![](https://www.scirp.org/html/11-5300318\a5cc442a-422b-4941-a1f1-37277f3d6620.jpg)
Step 2. We will show that
![](https://www.scirp.org/html/11-5300318\99b7b1bc-ff8b-4a3e-bce2-bde4d5d835ae.jpg)
Observing that
and
we have
(3.3)
and
(3.4)
Putting
in (3.3) and
in (3.4), we have
![](https://www.scirp.org/html/11-5300318\2a6b7f92-1e1c-4c56-9781-ecb3f1911b9f.jpg)
and
![](https://www.scirp.org/html/11-5300318\8562edbb-ca14-46fa-9520-ccc14e383ff9.jpg)
Summing up the last two inequalities and using Lemma 2.6 (A2), we obtain
![](https://www.scirp.org/html/11-5300318\c3134119-b7ea-4290-87a7-42ffe4753beb.jpg)
That is,
![](https://www.scirp.org/html/11-5300318\49c51f77-4e2d-40de-96dd-410124c04484.jpg)
It then follows that
![](https://www.scirp.org/html/11-5300318\3eebac1d-f771-4c2a-bca5-9bd44f4c5989.jpg)
This implies that
![](https://www.scirp.org/html/11-5300318\35a782bd-b1f9-499d-a9cb-fa9b1c32d344.jpg)
where M1 is an appropriate constant such that
![](https://www.scirp.org/html/11-5300318\4c8a52c9-698b-41fd-af6f-a888b4d76ac1.jpg)
Since
is nonexpansive and
using (3.5), we also have
![](https://www.scirp.org/html/11-5300318\2a5f1efd-a029-4ad2-87c5-44a2500a7366.jpg)
where M2 is an appropriate constant such that
![](https://www.scirp.org/html/11-5300318\9ed2aeb6-d517-4bda-9253-c51d0b029cf5.jpg)
Define
![](https://www.scirp.org/html/11-5300318\18c511be-9491-4585-9470-fdaa9fc62a84.jpg)
for all
so that
![](https://www.scirp.org/html/11-5300318\8c0f34fc-6e7d-4f20-8237-0d711df16614.jpg)
It follows that
![](https://www.scirp.org/html/11-5300318\19bf4a4b-1078-4a09-b8c6-6f06d5965d08.jpg)
Observe that
from (3.6), we obtain
(3.7)
Next we estimate ![](https://www.scirp.org/html/11-5300318\0fcbccac-1e56-4690-90f0-d8fab9556453.jpg)
For
we have
(3.8)
and
(3.9)
where
![](https://www.scirp.org/html/11-5300318\156111ca-4b7a-4497-a198-59f5b956dd88.jpg)
Using (3.8) and (3.9), we have
(3.10)
Substitute (3.10) into (3.7) yields that
![](https://www.scirp.org/html/11-5300318\57620961-b991-4660-80ad-9a601821125e.jpg)
which implies that (noting that (C1), (C2), (C3), (C4) and (C6))
![](https://www.scirp.org/html/11-5300318\37fc73e3-4514-4574-bfce-c0951a487c84.jpg)
Hence by Lemma 2.3, we have
(3.11)
Using (3.11) and we have ![](https://www.scirp.org/html/11-5300318\1a65cc49-c10b-4c6a-b7da-ff8f1160f900.jpg)
(3.12)
Step 3. We shall show that
![](https://www.scirp.org/html/11-5300318\da93fa56-3610-471d-a4a3-cf95aed2263c.jpg)
where ![](https://www.scirp.org/html/11-5300318\2f4e99dc-3d5e-42e1-b8e5-e416fd2f4541.jpg)
Note that
![](https://www.scirp.org/html/11-5300318\89eca870-d26c-475c-8dab-2f77ea69df85.jpg)
This implies
![](https://www.scirp.org/html/11-5300318\c0a8de92-82e9-43c2-ab3a-157fd6cd10ec.jpg)
From condition (C1), (C4) and (3.12), we have
(3.13)
Next we prove that
![](https://www.scirp.org/html/11-5300318\9fca6359-0f61-4926-8092-2eaa16dff4bb.jpg)
as ![](https://www.scirp.org/html/11-5300318\4275fdab-a644-424e-9d52-df4e40083e47.jpg)
Indeed, picking
![](https://www.scirp.org/html/11-5300318\7dcc6d26-4c6e-4c55-baf6-745fdbdb759a.jpg)
Since
and Tr is firmly nonexpansive, we obtain and hence
(3.14)
Set
and let
be an appropriate constant such that
![](https://www.scirp.org/html/11-5300318\28eb8d05-3a4b-41ac-9bb3-9a2889737cf6.jpg)
Therefore, from the convexity of
using (3.2), (3.14) and Lemma 2.8 we have
![](https://www.scirp.org/html/11-5300318\7ee614f5-9e97-4e43-b5c6-04d86ac81103.jpg)
It follows that
![](https://www.scirp.org/html/11-5300318\3a060698-0a20-466b-bc4f-86262654606c.jpg)
By using condition (C1), (C4) and (3.12), we have
(3.15)
From (3.13) and (3.15), we obtain
(3.16)
From (3.11) and (3.13), we also obtain
(3.17)
Step 4. We shall show that
![](https://www.scirp.org/html/11-5300318\574251b5-bc40-4688-b3f3-7aa7515c8aa7.jpg)
where q is the unique solution of the variational inequality ![](https://www.scirp.org/html/11-5300318\72a78385-2b16-45cc-b6a4-94017354c913.jpg)
![](https://www.scirp.org/html/11-5300318\3117518d-b8fd-4b15-a44f-ac186d4e9003.jpg)
Let
Observe that
is a contraction. Indeed, for all
,
and
we have
![](https://www.scirp.org/html/11-5300318\0b027727-ff40-4408-8446-38c6b5edced8.jpg)
Banach’s Contraction Mapping Principle guarantees that
has a unique fixed point, say
That is,
![](https://www.scirp.org/html/11-5300318\b08fb31a-fde7-4574-b1ae-455c328cc2c6.jpg)
by (1.1) we obtain that
for all
![](https://www.scirp.org/html/11-5300318\06ed643e-c307-4cc0-b511-36f21cc89339.jpg)
Next, we show that
![](https://www.scirp.org/html/11-5300318\75cc6a64-31d9-43a2-9a9c-498b68a7db4a.jpg)
To see this, we choose a subsequence
of
such that
![](https://www.scirp.org/html/11-5300318\b6f4d0c4-2076-4389-a677-da2e81d7ebd0.jpg)
Since
is bounded, there exists
a subsequence of
which converges weakly to p. Without loss of generality, we can assume that
Claim that
![](https://www.scirp.org/html/11-5300318\7687fab9-895c-4ada-a003-66dca7575ed5.jpg)
First, we prove
.
Since
we have
![](https://www.scirp.org/html/11-5300318\d1a6bf7a-6d13-4425-827d-a9e37d626891.jpg)
for all
It follows from Lemma 2.6 (A2) that
![](https://www.scirp.org/html/11-5300318\bc800cfc-63b2-4269-8b70-cf7c4d2e114e.jpg)
and hence
![](https://www.scirp.org/html/11-5300318\ad2b5d7d-f06b-4b65-b304-8f8ec28bab5a.jpg)
Since
and
together with the lower semicontinuity of
and Lemma 2.6 (A4), we have
for all
For t with
and
let
Since
and
we have
and hence
So, from Lemma 2.6 (A1), (A4) and the convexity of
we have
![](https://www.scirp.org/html/11-5300318\86a215f1-9a8e-4cb2-aea9-3abd9e61150e.jpg)
Dividing by t, we get ![](https://www.scirp.org/html/11-5300318\ea5dde75-8f21-4d74-ba82-9e35553e5b39.jpg)
Letting
it follows from Lemma 2.6 (A3) and the lower semicontinuity of
that
for all
and hence
Next, we prove
To see this, we observe that we may assume (by passing to a further subsequence if necessary)
. Let K be the K-mapping generated by
and
Then by Lemma 2.5, we have, for every ![](https://www.scirp.org/html/11-5300318\552f0797-b523-4edf-8084-95848f7d26d9.jpg)
(3.18)
every
Moreover, from Lemma 2.4 it follows that
![](https://www.scirp.org/html/11-5300318\99405420-a82c-4e19-b4ea-f08922f60909.jpg)
Suppose for contradiction
. Then
. Since Hilbert space are Opial’s spaces and
![](https://www.scirp.org/html/11-5300318\e7f2cac9-8e1c-476e-86e0-307b0bac2dfd.jpg)
from (3.17) and (3.18), we have
![](https://www.scirp.org/html/11-5300318\6fff4dbd-c38e-4909-be6c-e932a757c248.jpg)
which derives a contradiction. Thus, we have
It follows from
![](https://www.scirp.org/html/11-5300318\2da5ba12-e1e0-4e7b-a30e-a32aa2861663.jpg)
that
![](https://www.scirp.org/html/11-5300318\8edd56bd-50a3-46f6-9aa7-4677760e41a3.jpg)
Next, we prove
Put
![](https://www.scirp.org/html/11-5300318\71456383-c9e7-40d4-a751-c3ddc14bbb64.jpg)
Since B is relaxed
-cocoercive and condition (C5), we have
![](https://www.scirp.org/html/11-5300318\b7331633-9dad-4206-961b-a8e6a4b11a3f.jpg)
which yields that B is monotone. Thus T is maximal monotone. Let
. Since
and
we have
![](https://www.scirp.org/html/11-5300318\27b92adf-f68f-4bca-8e25-ed2871e698bc.jpg)
On the other hand, from
and (1.1), we have
![](https://www.scirp.org/html/11-5300318\84bbfb31-8b95-45ac-8386-0a23d6e51873.jpg)
and hence
![](https://www.scirp.org/html/11-5300318\4ea391ea-5132-465c-8893-1e6a0ffeb25e.jpg)
It follows that
![](https://www.scirp.org/html/11-5300318\e3d9fd2f-8cba-486f-8c4f-919b0125dcb6.jpg)
which together with (3.16), (3.17) and B is Lipschitz continuous implies that
We have
and hence
That is,
![](https://www.scirp.org/html/11-5300318\0597fd9b-0d67-4d17-b143-c3448541609a.jpg)
It follows from the variational inequality
for all
![](https://www.scirp.org/html/11-5300318\e947df6b-df0a-4261-99d0-ab630ba5b6f8.jpg)
that
(3.19)
Using (3.16) and (3.19), we have
(3.20)
Moreover, from (3.15) and (3.19), we have
(3.21)
Step 5. Finally, we will show that the sequences
and
converge strongly to q.
Since
using (1.19), (3.1), (3.2) and Lemma 2.8, we have
![](https://www.scirp.org/html/11-5300318\1fa23176-3c9c-4b2e-babc-8b702d9b55e0.jpg)
which implies that
![](https://www.scirp.org/html/11-5300318\ae943058-7422-4cc9-ad3f-b359f75b6b72.jpg)
Since ![](https://www.scirp.org/html/11-5300318\a662200a-b345-4a7c-afd1-da1f59f232a9.jpg)
and
are bounded, we can take a constant
such that
![](https://www.scirp.org/html/11-5300318\35bdf0af-15f0-420c-a12f-a3160860de20.jpg)
for all
It then follows that
(3.22)
where
![](https://www.scirp.org/html/11-5300318\fafaeac8-2d69-42de-8b3d-45e3a5473612.jpg)
By using (3.20), (3.21) and condition (C1), we get
![](https://www.scirp.org/html/11-5300318\0e14ada8-46a6-4226-adc2-a9ac6b49d42c.jpg)
Now applying Lemma 2.1 to (3.22) concludes that
as
Finally, noticing
![](https://www.scirp.org/html/11-5300318\af9dbb35-c96a-49c7-b636-e13bd8f1c542.jpg)
we also conclude that
as
This completes the proof.
4. Applications
In this section, by Theorem 3.1, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows:
Let
for all
and setting
and
in Theorem 3.1, we obtain the following result.
Corollary 4.1. Let H be a real Hilbert space, C a nonempty closed convex subset of H, F a bifunction from
to
which satisfies (A1)-(A4),
a proper lower semicontinuous and convex function and
a finite family of nonexpansive mappings of C into H such that the common fixed points set
Assume that either (B1) or (B2) holds and
is an arbitrary point in C. Let
and
be sequences generated by
and ![](https://www.scirp.org/html/11-5300318\c2224ed9-2d93-46fd-ab27-cc164d562eb0.jpg)
![](https://www.scirp.org/html/11-5300318\ba8062fb-62f7-4734-84d5-2f7615454b7a.jpg)
where
,
,
,
satisfying the conditions (C1)-(C5) in Theorem 3.1. Then,
and
converge strongly to a point
![](https://www.scirp.org/html/11-5300318\bab5c03b-79a5-43a2-aae1-85b0f760285b.jpg)
where
![](https://www.scirp.org/html/11-5300318\196fb00a-c2ef-4dc7-8d09-814583d5cf81.jpg)
Setting
and
for all n in Theorem 3.1, we obtain the following result.
Corollary 4.2. Let H be a real Hilbert space, C a nonempty closed convex subset of H, F a bifunction from
to
which satisfies (A1)-(A4),
a proper lower semicontinuous and convex function and
a finite family of nonexpansive mappings of C into H such that the common fixed points set
Let Kn and K be the K-mappings defined by (1.16) and (1.17), respectively. Assume that either (B1) or (B2) holds and x is an arbitrary point in C. Let
and
be sequences generated by
and ![](https://www.scirp.org/html/11-5300318\c41fec05-8cfa-497b-a3ff-50d1a4f03211.jpg)
![](https://www.scirp.org/html/11-5300318\15080e05-fd61-4e2c-af53-cd34441024ef.jpg)
where
are real numbers such that
for every
and
and
,
, ![](https://www.scirp.org/html/11-5300318\759a8ec5-b088-467f-8618-b2afac7ed91e.jpg)
satisfying the conditions (C1), (C3), (C4) and (C6) in Theorem 3.1. Then,
and
converge strongly to a point
![](https://www.scirp.org/html/11-5300318\db177eb8-44de-4185-ae39-dbd809e94d5e.jpg)
where
![](https://www.scirp.org/html/11-5300318\cbef4c1a-75e2-49d7-8c20-0f65e28a3833.jpg)
Finally as applications, we will utilize the results presented in this paper to study the following optimization problem:
(4.1)
where C is a nonempty bounded closed convex subset of a Hilbert space and
is a proper lower semicontinuous and convex function. We denote by
the set of solutions in (4.1). Let
for all
in Corollary 4.1, then
![](https://www.scirp.org/html/11-5300318\ec299a07-34d8-441d-895a-e578f4cfae35.jpg)
It follows from Corollary 4.1 that the sequence
generated by
and
,
(4.2)
where
,
,
and
satisfying the conditions (C1)-(C5) in Theorem 3.1. Then the sequence
converges strongly to a point
![](https://www.scirp.org/html/11-5300318\9001d37f-954a-4522-821e-3436aff89699.jpg)
where
![](https://www.scirp.org/html/11-5300318\33820e17-83ba-4495-afd0-09d0b278b8a5.jpg)
Let
for all
and
for all
in Corollary 4.2, then
It follows from Corollary 4.2 that the iterative sequence
generated by
and
,
(4.3)
where
,
and
satisfying the conditions (C1), (C3) and (C4) in Theorem 3.1. Then the sequence
converges strongly to a point
where ![](https://www.scirp.org/html/11-5300318\fe03cc1c-0734-45e5-8e29-66189bd3a6a5.jpg)
Remark 4.3. The algorithms (4.2) and (4.3) are variants of the proximal method for optimization problems introduced and studied by Martinet [41], Rockafellar [42], Ferris [43] and many others.
5. Acknowledgements
This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The author is extremely grateful to the referees for useful suggestions that improved the contents of the paper.
NOTES