In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an
-inverse-strongly monotone mapping in real Hilbert spaces.
Theorem 3.1.
Let
be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
into
satisfying (a1)–(a4),
an inverse-strongly monotone mapping with constant
,
an inverse-strongly monotone mapping with constant
,
a contraction mapping with constant
. Let
be a
-mapping generated by
and
and
, where sequence
is nonexpansive and
is a sequence in
for some
. For
, suppose that
,
and
are generated by
for all
, where
,
, and
are three sequences in
,
is a sequence in
for some
and
for some
satisfying
(i)
;
(ii)
and
;
(iii)
;
(iv)
and
;
(v)
and
.
Then
,
, and
converge strongly to the point
, where
.
Proof.
For any
and
, we have
which implies that
is nonexpansive. Remark 2.6 implies that the sequences
and
are well defined. In view of the iterative sequence (3.1), we have
It follows from Lemma 2.5 that
for all
. Let
. For each
, we have
. By Lemma 2.5,
and so (3.2) implies that
For
, we have
from (2.4). Since
is a nonexpansive mapping and
is an inverse-strongly monotone mapping with constant
, by (3.1), we have
Thus, (3.5) and (3.6) imply that
and so
This implies that
is bounded. Therefore,
,
,
,
and
are also bounded.
From
and
, we have
Putting
in (3.9) and
in (3.10), we get
Adding the above two inequalities, the monotonicity of
implies that
and so
It follows from (3.2) that
and hence
From (3.1),
Putting
we have
Obviously, we get
From (2.11) and (3.16), we have
for some constant
. Combining (3.15), (3.19), and (3.20), we deduce
It is easy to check that
and so
Thus, by Lemma 2.7, we obtain
. It then follows that
By (3.15) and (3.16), we have
Since
, we get
On the other hand, for
,
It follows that
It is easy to see that
and hence
.
From (3.5) and (3.6), we obtain
Since
, without loss of generality, we may assume that there exists a real number
such that
for all
. Therefore, we have
Since
,
and
is bounded, (3.30) implies that
as
. From (2.2), we have
and so
It follows that
which implies that
Since
,
,
, and the sequences
,
and
are bounded, it follows from (3.34) that
. On the other hand, from (3.5), we have
The same as in (3.30), we have
as
. Likewise, using (3.5), we find
The same as in (3.34), we have
. Since
we get
as
. From (2.10) and
we get
.
Next, we show
, where
. To show this inequality, we choose a subsequence
of
such that
Since
is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
. From
, we obtain
. We now show that
. Indeed, we observe that
and
From (a2), we deduce that
and hence
Form
, we get
. Put
for all
and
. Consequently, we get
. From (3.42), it follows that
From the Lipschitz continuous of
and
, we obtain
. Since
is monotone, we know that
. Further,
. It follows from (a4) that
Owing to (a1) and (a4), we get that
and hence
Letting
, we have
This implies that
.
Furthermore, we prove that
. Assume
, since
, we have
. From Opial's theorem [27], we get
This is a contradiction. Hence,
.
Now, we will show that
. Let
Then
is a maximal monotone [23]. Let
, since
and
, we have
. From
, we have
This is,
Therefore, we obtain
Noting that
and
is Lipschitz continuous, we obtain
Since
is maximal monotone, we have
and so
. Thus,
. The property of the metric projection implies that
From (3.1) we obtain
which implies that
Setting
we have
Applying Lemma 2.8 to (3.56), we conclude that
converges strongly to
. Consequently,
and
converge strongly to
. This completes the proof.
As direct consequences of Theorem 3.1, we have the following two corollaries.
Corollary 3.2.
Let
be a nonempty closed convex subset of a real Hilbert space
. Let
be a contraction mapping with Lipschitz constant
and let
be an inverse-strongly monotone mapping with constant
. Suppose
and
generated by
for all
, where
,
,
are three sequences in
,
is a sequence in
for some
satisfying conditions (i)–(iv). Then
converges strongly to the point
, where
.
Proof.
Let
and
for all
and
in Theorem 3.1. Then
for
Letting
(the identity mapping) for all
, then
for
It is easy to see that all conditions of Theorem 3.1 hold. Therefore, we know that the sequence
generated by (3.59) converges strongly to
. This completes the proof.
Remark 3.3.
From Corollary 3.2, we can get an iterative scheme for finding the solution of the variational inequality involving the
-inverse-strongly monotone mapping
.
Corollary 3.4 (see [17, Theorem 3.5]).
Let
be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
into
satisfying (a1)–(a4),
a contraction mapping with constant
. Let
be an
-mapping generated by
and
and
, where sequence
is nonexpansive and
is a sequence in
for some
. Suppose
and
,
are generated by
for all
, where
,
,
are three sequences in
, and
is a sequence in
satisfying conditions (i)–(iii) and (v). Then, the sequences
and
converge strongly to the point
, where
.
Proof.
Let
for
and
and
for all
in Theorem 3.1. Since
, we get that
. It follows from Theorem 3.1 that the sequences
and
converge strongly to the point
. This completes the proof.
Remark 3.5.
The main result of Yao et al. [17, Corollary 3.2] improved and extended the corresponding theorems in Combettes and Hirstoaga [3] and S. Takahashi and W. Takahashi [14]. Our Theorem 3.1 improves and extends Theorem 3.5 of Yao et al. [17] in the following aspects:
(1)the equilibrium problem is extended to the generalized equilibrium problem;
(2)our iterative process (3.1) is different from Yao et al. iterative process (3.60) because there are a project operator and an
-inverse-strongly monotone mapping;
(3)our iterative process (3.1) is more general than Yao et al. iterative process (3.60) because it can be applied to solving the problem of finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for
-inverse-strongly monotone mapping.