An Iterative Algorithm for Generalized Mixed Equilibrium Problems and Fixed Points of Nonexpansive Semigroups ()
1. Introduction
As you know, there are many problems that are reduced to find solutions of equilibrium problems which cover variational inequalities, fixed point problems, saddle point problems, complementarity problems as special cases. Equilibrium problem which was first introduced by Blum and Oettli [1] has been extensively studied as effective and powerful tools for a wide class of real world problems, which arises in economics, finance, image reconstruction, ecology, transportation network and related optimization problems.
From now on, we assume that
is a real Hilbert space with inner product
and norm
, and
is a nonempty closed convex subset of
.
is denoted by the set of real numbers. Let
be a bifunction. Blum and Oettli [1] consider the equilibrium problem of finding
such that
(1.1)
The solution set of problem (1.1) is denoted by
, i.e.,
![](//html.scirp.org/file/74148x12.png)
Recently the so-called generalized mixed equilibrium problem has been investigated by many authors [2] [3]. The generalized mixed equilibrium problem is to find
such that
(1.2)
where
is a mapping and
is a real valued function. We use
to denote the solution set of generalized mixed equilibrium problem i.e.,
![](//html.scirp.org/file/74148x18.png)
The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequality problem, minimax problems, the Nash equilibrium problems in noncooperative games and others (see [4] [5] [6] [7] [8] [9] [10] [11] [12]).
Special Cases: The following problems are the special cases of problem (1.2).
1) If
then (1.2) is equivalent to finding
such that
(1.3)
is called mixed equilibrium problems.
2) If
then (1.2) is equivalent to finding
such that
(1.4)
is called mixed variational inequality of Browder type [13].
3) If
then (1.2) is equivalent to find
such that
(1.5)
is called generalized equilibrium problems (shortly, (GEP)). We denote GEP(G,A) the solution set of problem (GEP).
4) If
and
then (1.2) is equivalent to (1.1).
5) Let
for all
. Then we see that (1.1) is reduces to the following classical variational inequalities for finding
such that
(1.6)
It is known that
is a solution to (1.6) if and only if
is a fixed point of the mapping
, where
is a constant and I is an identity mapping.
Let
be a mapping from
into itself. Let denote
the set of fixed points of the mapping T. A mapping
is said to be nonexpansive if
![]()
A mapping
is said to be contractive if there exists a constant
such that
![]()
A mapping
is called
-inverse strongly monotone if there exists a constant
such that
![]()
Remark 1.1 Every
-inverse strongly monotone mapping is monotone and
-Lipschitz continuous.
In 1967, Halpern [14] introduced the following iterative method for a nonexpansive mapping
in a real Hilbert space, for finding
and
(1.7)
where
and
is fixed.
Moudafi [15] introduced the viscosity approximation method for a nonexpansive mapping
as follows: For finding
and
(1.8)
where
and
is a contraction mapping.
A viscosity approximation method with Meir-Keeler contraction was first studied by Suzuki [16]. Very recently Petrusel and Yao [17] studied the following viscosity approximation method with a generalized contraction: for finding
and
![]()
where
and
is a family of nonexpansive mappings on
.
Takahashi and Takahashi [18] introduced the following iterative scheme for solving a generalized equilibrium problems and a fixed point problems of a nonexpansive mapping
in a Hilbert spaces
: Finding
and
(1.9)
where
and A is an
-inverse strongly monotone mapping. They proved that the sequence
generated by (1.9) strongly converges to an element in
under suitable conditions.
In this paper, from the recent works [19] [20] [21] [22] [23] [24] [25] [26], we introduced an iterative scheme by the modified viscosity approximation method associated with Meir-Keeler contraction (see [27]) for solving the generalized mixed equilibrium problems and fixed point problem of a nonexpansive semigroup in Hilbert spaces, and also we discussed a convergence theorem. Finally we apply our main results for commutative nonexpansive mappings and semigroup of strongly continuous mappings.
2. Preliminaries
Let
be a semigroup and
be the Banach space of all bounded real valued functionals on
with superimum norm. For each
, we define the left and right translation operators
and
on
by
and
for each
and
respectively. Let
be a subspace of
containing 1. An element
in the dual space
of
is said to be a mean on
if
We denote the value of
at the function
by
. According to the time and circumstances, we write the value
by
or
. It is well known that
is a mean of
if and only if for each ![]()
![]()
Let X be a translation invariant subspace of
(i.e.,
and
for each
) containing 1. Then a mean μ on X is said to be left invariant (resp. right invariant) if
(resp.
) for each
and
. A mean μ on X is said to be invariant if μ is both left and right invariant [28] [29]. S is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. S is amenable if S is left and right amenable [30]. In this case
also has an invariant mean. It is known that
is amenable when S is commutative semigroup or solvable group. However the free group or semigroup of two generators is not left or right amenable (see [31]). A net
of mean on X is said to be left regular if
![]()
for each
where
is the adjoint operator of ![]()
Let
be a nonempty closed convex subset of
. A family
is called a nonexpansive semigroup on
if for each
, the mapping
is nonexpansive and
for each
(see [30] [30]). We denote by
the set of common fixed point of
, i.e.,
![]()
Assume that
is a open ball of radius
centered at 0 and
is a closed convex hull of
. For
and a mapping
, the set of
-approximate fixed points of
will be denoted by
, i.e.,
![]()
Lemma 2.1 [32] Let
be a function of a semigroup
into a Banach space E such that the weak closure of
is weakly compact and
a subspace of
containing all the function
with
Then for any
there exists a unique element
in
such that for all
,
![]()
Moreover if
is a mean on
then
![]()
We can write
by ![]()
Lemma 2.2 [32] Let
be a closed convex subset of a Hilbert space H.Let
be a nonexpansive semigroup from
into itself such that
,
be a subspace of
containing 1, the mapping
be an element of
for each
and
and
be a mean on
. If we write
instead of
then the following state- ments hold:
1)
is a nonexpansive mapping from
into
,
2)
for each ![]()
3)
, for each
;
4) if
is left invariant then
is a nonexpansive retraction from
into ![]()
Let
be a nonempty closed convex subset of a real Hilbert space
. Then for any
there exists a unique nearest point in
, denoted by
such that for all
,
![]()
where
is the metric projection of
onto
. We also know that for
and
if and only if for all
,
![]()
A mapping
is said to be an
-function if
for each
and for every
there exists
such that
for all
. As a consequence, every
-function
satisfies
for each
.
Definition 2.3 Let
be a metric space. A mapping
is said to be a
1)
-contraction if
is an
-function and
![]()
for all
with ![]()
2) Meir-Keeler type mapping if for each
there exists
such that for each
with
we have
(see [33] [34]).
Theorem 2.4 [34] Let
be a complete metric space and
is a Meir-Keeler type mapping. Then
has a unique fixed point.
Theorem 2.5 [35] Let
be a complete metric space and
is a mapping. Then the following statements are equivalent.
1)
is a Meir-Keeler type mapping;
2) there exists an
-function
such that
is a
-con- traction.
Theorem 2.6 [16] Let
be a convex subset of a Banach space
and let
be a Meir-Keeler type mapping. Then for each
there exists
such that for each
with
we have
![]()
Proposition 2.7 [31] Let
be a convex subset of a Banach space
,
be a nonexpansive mapping on
and
be a Meir-Keeler type mapping. Then the following statements hold:
1)
is a Meir-Keeler type mapping on
.
2) For each
, the mapping
is a Meir- Keeler type mapping on
.
Lemma 2.8 [36] Assume that
is a sequence of nonnegative real number such that
![]()
where
is a sequence in
and
is a sequence in
satisfying
1) ![]()
2)
or ![]()
Then ![]()
Lemma 2.9 [37] Let
and
be bounded sequences in a Banach space
such that
![]()
where
is a real sequence in
with
![]()
If
![]()
then
![]()
Lemma 2.10 [38] Let
for all
Suppose that
and
are sequences in
such that
![]()
and
![]()
for some
. Then we have
![]()
Lemma 2.11 [39] Let
be a nonempty closed convex subset of a real Hilbert space
and
be a nonexpansive mapping with
Then
is demiclosed at zero, that is, for all sequence
with
and
it follows that ![]()
For solving the equilibrium problem we assume that bifunction
satisfies the following conditions:
(A1) ![]()
(A2)
is monotone, i.e., ![]()
(A3) for each ![]()
(A4) for each
,
is convex and lower semicontinuous.
Lemma 2.12 [1] Let
be a nonempty closed convex subset of a real Hilbert space H and G be a bifunction from
to
satisfying (A1)-(A4). Then for any
and
, there exists
such that
![]()
Further, if
![]()
then we have the followings:
1)
is single-valued;
2)
is firmly nonexpansive, i.e., for any ![]()
![]()
3) ![]()
4)
is closed and convex.
Lemma 2.13 [18] Let
and
be as in Lemma 2.12. Then we have
![]()
for all
and ![]()
3. Main Results
Theorem 3.1 Let K be a nonempty closed convex subset of a Hilbert space
. Let
be a semigroup,
be a nonexpansive semigroup on
be a bifunction satisfying (A1)-(A4) and
be an
- inverse strongly monotone mapping with
![]()
Let
be a proper lower semicontinuous and convex function, X be a left invariant subspace of
such that
and the function
be an element of X for each
Let
be a left regular sequence of means on X such that
as
and
be a Meir-Keeler contraction. Let
be the sequence generated by
and
![]()
where
is bounded sequence in
,
and
are real number sequences in
and
satisfying the conditions:
(C1) ![]()
(C2) ![]()
(C3) ![]()
(C4) ![]()
Then the sequence
strongly converges to
which is also solves the following variational inequality problem:
(3.1)
Proof. We give the several steps for the proof.
Step 1: First we show that
is bounded. Put
and
for all
Then for
, we have
(3.2)
Set
, then
is nonexpansive and
. Hence we have
![]()
By induction, we can prove that
![]()
Hence the sequence
is bounded. So
and
are all bounded.
Step 2: We next show that
![]()
Observe that
(3.3)
Indeed
![]()
Since
is bounded and
(3.3) holds. Since
and
, we have
(3.4)
From
and
, we have
![]()
it follows that
(3.5)
We see that
(3.6)
Combining (3.4) and (3.5) with (3.6), we obtain
![]()
Using Lemma 2.13, (3.3),(C1) and (C4), then we have
![]()
From this inequality and (C3), it follows from Lemma 2.9 that
(3.7)
It implies that
(3.8)
Step 3: Next we prove that for all
,
![]()
Put
![]()
Set
It is easily seen that D is a nonempty bounded closed convex subset of K. Further
and
are in D. To complete our proof, we follows that proof line as in [30]. From [40], for every
there exists
such that for all
,
(3.9)
From Corollary 1.1 in [40], there exists a natural number
such that for all ![]()
(3.10)
Since
is left regular, for
there exists
such that
![]()
for all
Therefore, we have for all ![]()
(3.11)
We observe from Lemma 2.2 (iii) that
(3.12)
Combining (3.10), (3.12) and (3.12), we have for all ![]()
(3.13)
Let
and
. Then there exists
which satisfies (3.9). From (C3) there exist
such that
. From (3.7) there exists
such that
and
for all
So from (3.9) and (3.13), we have
![]()
Hence
Since
is arbitrary,
![]()
Step 4: We next show that
(3.14)
Using inequality (3.2), we obtain
(3.15)
which implies that
![]()
From (C1)-(C4) and (3.8), we obtain
(3.16)
Since
is firmly nonexpansive,
![]()
Therefore
![]()
Then we have
![]()
which yields
![]()
Hence, from (C2), (C3) and (3.16) we obtain
(3.17)
Since
we have
and hence
(3.18)
On the other hand, by Proposition 2.7 (i), we know that
is a Meir-Keeler contraction. From Theorem 2.4, there exists a unique element
such that
which is equivalent to
![]()
Step 5: We next show that
![]()
To see this, we chose a subsequence
of
such that
![]()
Since
is a bounded, K is closed and H is reflexive, there exists a point
such that
. From (3.17) and (3.18) there exists a corresponding subsequence
of
(resp.
of
) such that
(resp.
). We next show that
Since
We can write
![]()
From (A2), we have
![]()
Then
(3.19)
Put
for
and
. Since
and
,
. So from (3.19) we have
![]()
From (A4), we have
(3.20)
From (A1)-(A4) and (3.20), we have
![]()
It follows that
![]()
letting
by (A3), we have
![]()
Hence
It is easily seen that
Indeed, since
and
for all
we conclude from Lemma 2.1 that
. Consequently, we have
and hence
(3.21)
Step 6: Now we are in a position to show that
is a fixed point of
.
Let
. Then we have
![]()
We note that
![]()
and
![]()
It follows from Lemma 2.10 that
(3.22)
On the other hand, we have
![]()
It follows from (3.17) and (3.22) that
(3.23)
Therefore
Let
be an another subsequence of
converging to
with
. Similarly, we can find
. Hence we have
![]()
This is a contradiction. Hence we have ![]()
Step 7: We finally show that
as
.
Suppose that
does not strongly converge to
. Then there exists
and a subsequence
of
such that
for all
By Proposition 2.7, for this
there exists
such that
![]()
So we have
![]()
This implies that
![]()
Hence
![]()
Using (3.21) and (C2), we can conclude by Lemma 2.8 that
as
. This is a contradiction and hence the sequence
converges to
. Thus we completes the proof. ![]()
Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea(2015R1D1A1A09058177).