- Research Article
- Open access
- Published:
Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space
Fixed Point Theory and Applications volume 2010, Article number: 794503 (2010)
Abstract
Let be a real Banach space with the dual space
. Let
be a proper functional and let
be a bifunction. In this paper, a new concept of
-proximal mapping of
with respect to
is introduced. The existence and Lipschitz continuity of the
-proximal mapping of
with respect to
are proved. By using properties of the
-proximal mapping of
with respect to
, a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space
. An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space
, and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in
a Hilbert space.
1. Introduction
Let be a real Banach space with norm
and let
be its dual space. The value of
at
will be denoted by
. The normalized duality mapping
from
into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual space
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ1_HTML.gif)
It is known that the norm of is said to be Gateaux differentiable (and
is said to be smooth) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ2_HTML.gif)
exists for every in
the unit sphere of
. It is said to be uniformly Gateaux differentiable if for each
, this limit is attained uniformly for
. The norm of
is said to be uniformly Frechet differentiable (and
is said to be uniformly smooth) if the limit in (1.2) is attained uniformly for
. Every uniformly smooth Banach space
is reflexive and has a uniformly Gateaux differentiable norm.
Recall also that if is smooth, then
is single-valued and continuous from the norm topology of
to the weak star topology of
, that is, norm-to-weak
continuous. It is also well known that if
has a uniformly Gateaux differentiable norm, then
is uniformly continuous on bounded subsets of
from the strong topology of
to the weak star topology of
, that is, uniformly norm-to-weak* continuous on any bounded subset of
. Moreover, if
is uniformly smooth, then
is uniformly continuous on bounded subsets of
from the strong topology of
to the strong topology of
, that is, uniformly norm-to-norm continuous on any bounded subset of
. See [1] for more details.
It is well known that the variational inequality theory has played an important and powerful role in the studying of a wide class of linear and nonlinear problems arising in many diverse fields of pure and applied sciences, such as mathematical programming, optimization theory, engineering, elasticity theory and equilibrium problems of mathematical economics, and game theory; see, for instance, [1–6] and the references therein.
One of the most interesting and important problems in the theory of variational inequalities is the development of an efficient iterative algorithm to compute approximate solutions. In the setting of Hilbert spaces, one of the most efficient numerical techniques is the projection method and its variant forms; see [4, 6–15]. Since the standard projection method strictly depends on the inner product property of Hilbert spaces, it can no longer be applied for general mixed type variational inequalities in Banach spaces. This fact motivates us to develop alterative methods to study the existence and iterative algorithms of solutions for general mixed variational inequalities in Banach spaces. Recently, [16–18] extended the auxiliary principle technique to study the existence of solutions and to suggest the iterative algorithms for solving various mixed type variational inequalities in Banach spaces. For some related work, we refer to [19–21] and the references therein.
Very recently, inspired by the research work going on in this field, Xia and Huang [14] first introduced a new concept of -proximal mapping for a proper subdifferentiable functional on a Banach space. They proved an existence theorem and Lipschitz continuity of the
-proximal mapping. Using the properties of the
-proximal mapping, they proved an existence theorem of solutions for a new class of general mixed variational inequalities in a Banach space and suggested an iterative algorithm for computing approximate solutions. Moreover, they gave the strong convergence criteria of the iterative sequence generated by this algorithm. Their results include some known results in [8, 9, 11, 12, 16–18] as special cases.
Let be a real Banach space with the dual space
. Let
be a proper functional and let
be a bifunction. In this paper, motivated by Xia and Huang [14], we first introduce a new concept of
-proximal mapping of
with respect to
. We prove an existence theorem and Lipschitz continuity of the
-proximal mapping of
with respect to
. Utilizing the properties of the
-proximal mapping of
with respect to
, we introduce and consider a generalized mixed equilibrium problem with perturbation (for short, GMEPP) which includes as a special case the general mixed variational inequality studied by Xia and Huang [14]. We show an existence theorem of solutions for this problem under some appropriate conditions. In order to compute approximate solutions of the GMEPP, we propose a new iterative algorithm which includes as a special case the iterative algorithm considered by Xia and Huang [14]. Finally, we establish the strong convergence criteria of the iterative sequence generated by the new algorithm in a uniformly smooth Banach space
, and also derive the weak convergence criteria of the iterative sequence generated by this new algorithm in
a Hilbert space. Our results are new and represent the improvement, extension, and development of Xia and Huang's results in [14].
2. Preliminaries
Let be a real Banach space with the topological dual space
and let
be the pairing between
and
. We write
to indicate that the sequence
converges weakly to
.
implies that
converges strongly to
. Let
and
denote the family of all subsets of
and the family of all nonempty closed bounded subsets of
, respectively. Let
be a bifunction, let
and
be single-valued mappings, and let
be a proper lower semicontinuous functional. We consider the following generalized mixed equilibrium problem with perturbation (for short, GMEPP): find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ3_HTML.gif)
Some special cases of problem (2.1) are the following.
(1)If a real Hilbert space,
an identity mapping on
, and
is a lower semicontinuous and convex functional, then GMEPP (2.1) reduces to the generalized mixed equilibrium problem with perturbation considered by Ceng et al. [22].
(2)If a real Hilbert space,
an identity mapping on
,
, and
is a lower semicontinuous and convex functional, then GMEPP (2.1) reduces to the generalized mixed equilibrium problem considered by Peng and Yao [23].
(3)If , then GMEPP (2.1) reduces to the general mixed variational inequality problem (for short, GMVIP) considered by Xia and Huang [14].
(4)If a real Hilbert space,
, and
is a proper convex lower semicontinuous functional, then GMEPP (2.1) was studied by many authors (see, e.g., [1, 17–19]).
We first recall the following definitions and some known results.
Definition 2.1.
Let be a set-valued mapping, and let
and
be two single-valued mappings. We say that
(i) is
-strongly monotone with constant
if, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ4_HTML.gif)
(ii) is
-strongly monotone if, for any
, and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ5_HTML.gif)
(iii) is
-Lipschitz continuous with constant
if, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ6_HTML.gif)
where is the Hausdorff metric on
;
(iv) is
-strongly accretive (where
) if, for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ7_HTML.gif)
where is the normalized duality mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ8_HTML.gif)
Definition 2.2.
Let be a Banach space with the dual space
, let
be a bifunction, and let
be a proper functional. If, for
, there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ9_HTML.gif)
then is said to be
-subdifferentiable at
. We denote by
the set of such elements
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ10_HTML.gif)
The set is said to be the
-subdifferential of
at
. If there exists the
-subdifferential
at each
, then
is said to be
-subdifferentiable. The mapping
is said to be the
-subdifferential of
.
Definition 2.3.
Let be a Banach space with the dual space
, and let
be a proper subdifferentiable functional. If for any given
and any constant
, there is a unique
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ11_HTML.gif)
then the mapping , denoted by
, is said to be an
-proximal mapping of
with respect to
.
Remark 2.4.
From Definitions 2.2 and 2.3 it follows that is
-subdifferentiable at each
the range of
. If
is additionally
-subdifferentiable at each
, then there exists the
-subdifferential
at each
; that is,
is
-subdifferentiable. Observe that
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ12_HTML.gif)
In particular, whenever , then the concept of
-proximal mapping of
with respect to
reduces to the one of
-proximal mapping of
by Xia and Huang [14, Definition
]. In this case,
is rewritten as
. By the definition of the subdifferential, we know that
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ13_HTML.gif)
Lemma 2.5 (see [24]).
Let be a nonempty convex subset of a topological vector space and let
be such that
(i)for each is lower semicontinuous on each nonempty compact subset of
;
(ii)for each nonempty finite set and for each
with
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ14_HTML.gif)
(iii)there exist a nonempty compact convex subset of
and a nonempty compact subset
of
such that for each
, there is an
with
.
Then there exists such that
for all
.
Recall now that satisfies Opial's property [25] provided that, for each sequence
in
, the condition
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ15_HTML.gif)
It is known [25] that each enjoys this property, while
does not unless
. It is known [26] that any separable Banach space can be equivalently renormed so that it satisfies Opial's property.
Furthermore, recall that in a Hilbert space, there holds the following equality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ16_HTML.gif)
for all and
.
In order to investigate the generalized mixed equilibrium problem (2.1) with perturbation, we will need the following conditions on mapping and bifunction
in the sequel:
(H1);
(H2) is monotone, that is,
;
(H3) for each is convex and lower semicontinuous;
(H4) the following equilibrium problem (for short, EP) or generalized equilibrium problem (for short, GEP) has a solution in
:
(EP) Find such that
for all
, or
(GEP) Find such that
for all
.
Now we give some sufficient conditions which guarantee the existence and Lipschitz continuity of an -proximal mapping of
with respect to
in a reflexive Banach space
.
Theorem 2.6.
Let be a reflexive Banach space with the dual space
, let
be a bifunction satisfying conditions (H1)–(H4), let
be an
-strongly monotone and continuous mapping, and let
be a lower semicontinuous subdifferentiable proper functional. Then for any given
and any
, there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ17_HTML.gif)
that is, and the
-proximal mapping of
with respect to
is well defined. If
is additionally
-subdifferentiable at each
, then
is
-subdifferentiable and
.
Proof.
For any given and any
, define a functional
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ18_HTML.gif)
By the continuity of and the lower semicontinuity of
and
, we know that the function
is lower semicontinuous on
for each fixed
.
Now, let us show that satisfies condition (ii) of Lemma 2.5. If it is false, then there exist a finite set
and
with
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ19_HTML.gif)
Since is subdifferentiable at
, there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ20_HTML.gif)
From (H2) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ21_HTML.gif)
Utilizing the convexity of , we get from (H1)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ22_HTML.gif)
which leads to a contradiction. Therefore, satisfies condition (ii) of Lemma 2.5.
From (H4) we know that the following equilibrium problem (for short, EP) or generalized equilibrium problem (for short, GEP) has a solution :
(EP), for all
or
(GEP) for all
Since is subdifferentiable at
, there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ23_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ24_HTML.gif)
Next, we discuss two cases.
Case 1.
If EP has solution , then from (2.22) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ25_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ26_HTML.gif)
Then and
are both weakly compact convex subset of
. For each
, there exists a point
such that
and so all conditions of Lemma 2.5 are satisfied. By Lemma 2.5, there exists a point
such that
for all
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ27_HTML.gif)
Case 2.
If GEP has solution , then from (2.22) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ28_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ29_HTML.gif)
Then and
are both weakly compact convex subset of
. For each
, there exists a point
such that
and so all conditions of Lemma 2.5 are satisfied. By Lemma 2.5, there exists a point
such that
for all
, that is, (2.25) holds.
Now let us show that is a unique solution of auxiliary equilibrium problem (2.15). Suppose that
are arbitrary two solutions of auxiliary equilibrium problem (2.15). Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ31_HTML.gif)
Taking in (2.28) and
in (2.29) and adding these inequalities we have from the monotonicity of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ32_HTML.gif)
This together with the -strong monotonicity of
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ33_HTML.gif)
and so . Therefore,
and the
-proximal mapping of
with respect to
is well defined. In the meantime, it is known that
is
-subdifferentiable at each
. If
is additionally
-subdifferentiable at each
, then by Remark 2.4 we obtain that
is
-subdifferentiable and
. This completes the proof.
Corollary 2.7 (see [14, Theorem ]).
Let be a reflexive Banach space with the dual space
, let
be a lower semicontinuous subdifferentiable proper functional, and let
be an
-strongly monotone and continuous mapping. Then for any given
and any
, there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ34_HTML.gif)
that is, and the
-proximal mapping of
is well defined.
Proof.
Putting in Theorem 2.6, we obtain the desired result.
Remark 2.8.
Theorem 2.6 shows that if is a bifunction satisfying conditions (H1)–(H4) and
is a lower semicontinuous subdifferentiable proper functional, then for any strongly monotone and continuous mapping
, the
-proximal mapping
of
with respect to
is well defined. Furthermore, if
is additionally
-subdifferentiable at each
, then for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ35_HTML.gif)
is the unique solution of auxiliary equilibrium problem (2.15).
Theorem 2.9.
Let be a reflexive Banach space with the dual space
, let
be an
-strongly monotone and continuous mapping, let
be a bifunction satisfying conditions (H1)–(H4), and let
be a lower semicontinuous subdifferentiable proper functional. If
is
-subdifferentiable at each
, then
is
-subdifferentiable and the
-proximal mapping
of
with respect to
is
-Lipschitz continuous. Furthermore, if additionally the
-subdifferential
for
is
-strongly monotone, then the
-proximal mapping
of
with respect to
is
-Lipschitz continuous.
Proof.
First, utilizing Theorem 2.6 we know that whenever is
-subdifferentiable at each
,
is
-subdifferentiable. For any
, let
and
. Then
and
. By the definition of the
-subdifferential of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ37_HTML.gif)
Taking in (2.34) and
in (2.35) and adding these inequalities, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ38_HTML.gif)
Utilizing condition (H2) we get and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ39_HTML.gif)
Since is
-strongly monotone,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ40_HTML.gif)
which implies that is
-Lipschitz continuous.
Now we suppose that the -subdifferential
for
is
-strongly monotone. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ41_HTML.gif)
Since is
-strongly monotone,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ42_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ43_HTML.gif)
Thus, is
-Lipschitz continuous.
Corollary 2.10 (see [14, Theorem ]).
Let be a reflexive Banach space with the dual space
, let
be an
-strongly monotone and continuous mapping, and let
be a lower semicontinuous subdifferentiable proper functional. Then the
-proximal mapping
is
-Lipschitz continuous. Furthermore, if the subdifferential
for
is
-strongly monotone, then the
-proximal mapping
is
-Lipschitz continuous.
Proof.
Putting in Theorem 2.9, we derive the desired result.
3. Existence and Algorithm
We first transfer GMEPP (2.1) into a fixed point problem.
Theorem 3.1.
Let be a reflexive Banach space with the dual space
, let
be an
-strongly monotone and continuous mapping, let
be a bifunction satisfying conditions (H1)–(H4), and let
be a lower semicontinuous subdifferentiable proper functional. If
is
-subdifferentiable at each
, then
is a solution of the GMEPP (2.1) if and only if
satisfies the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ44_HTML.gif)
where is the
-proximal mapping of
with respect to
and
is a constant.
Proof.
Since is
-subdifferentiable at each
, in terms of Theorem 2.6,
is
-subdifferentiable.
Assume that satisfies relation (3.1). Noting
, we know that relation (3.1) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ45_HTML.gif)
By the definition of the -subdifferential of
with respect to
, the above relation holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ46_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ47_HTML.gif)
Thus, is a solution of GMEPP (2.1) if and only if
satisfies (3.1). This completes the proof.
Remark 3.2.
Relation (3.1) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ48_HTML.gif)
where .
Remark 3.3.
By Theorem 2.6, we can choose a strongly monotone and Lipschitz continuous mapping such that it is easy to compute the values of the
-proximal mapping
of
with respect to
. Theorem 3.1 shows that, by using the mapping
, GMEPP (2.1) can be transferred into a fixed point problem (3.5). Based on these observations, we can suggest the following new and general iterative algorithms for computing the approximate solutions of GMEPP (2.1) in reflexive Banach spaces.
Lemma 3.4 (see [27]).
Let be a real Banach space and let
be the normalized duality mapping. Then for any
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ49_HTML.gif)
We now use Theorem 3.1 to construct the following algorithms for solving the GMEPP (2.1) in Banach spaces.
Algorithm 3.5.
Let be two single-valued mappings, let
be a single-valued mapping with
let
be an
-strongly monotone and
-Lipschitz continuous mapping, let
be a bifunction satisfying conditions (H1)–(H4), and let
be a lower semicontinuous subdifferentiable proper functional. For any given
, an iterative sequence
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ50_HTML.gif)
where and
for all
. Algorithm 3.5 is called the Mann-type iterative algorithm.
Algorithm 3.6.
Let and
be the same as in Algorithm 3.5. For any given
, the iterative sequences
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ51_HTML.gif)
where for all
. Algorithm 3.6 is called the Ishikawa-type iterative algorithm.
Remark 3.7.
If for all
, then Algorithm 3.6 reduces to Algorithm 3.5. Whenever
, Algorithms 3.5 and 3.6 reduce to Algorithms 3.1 and 3.2 by Xia and Huang [14], respectively.
Now we prove an existence theorem of solutions for GMEPP (2.1).
Theorem 3.8.
Let be a reflexive Banach space with the dual space
let
be two continuous mappings, and let
be a continuous mapping. Let
be
-strongly monotone and continuous, let
be a bifunction satisfying conditions (H1)–(H4), and let
be a lower semicontinuous subdifferentiable proper functional. If
is
-subdifferentiable at each
, and the ranges
and
are totally bounded, then there exists
which is a solution of GMEPP (2.1).
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ52_HTML.gif)
By Theorem 2.9, the mapping is Lipschitz continuous. Since the ranges
and
are totally bounded, we know that the range
is also totally bounded in
; that is,
is totally bounded in
. Thus,
is a compact subset of
. Since
and
are continuous, so does
. By Schauder fixed point theorem,
has a fixed point
. It follows from Theorem 3.1 that
is a solution of GMEPP (2.1). This completes the proof.
Remark 3.9.
From the proof of Theorem 3.8, we know that in [14, Theorem ], the assumption of the boundedness of the ranges
and
cannot guarantee that all the conditions of Schauder fixed point theorem are satisfied. Thus, in [14, Theorem
], the assumption
the ranges
and
are bounded
should be replaced by the stronger condition
the ranges
and
are totally bounded
. Here the well-known Schauder fixed point theorem is stated as follows.
Let be a Banach space and let
be a nonempty closed convex subset of
. Assume that
is a continuous mapping such that the closure
is a compact subset in
. Then
has a fixed point
in
, that is,
.
In order to give some sufficient conditions, which guarantee the convergence of the iterative sequences generated by Algorithm 3.6, we will need the following lemma in the sequel.
Lemma 3.10 (see [28]).
Let be a sequence of nonnegative real numbers satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ53_HTML.gif)
where and
are sequences of real numbers such that
(i) and
, or equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ54_HTML.gif)
(ii), or
is convergent.
Then, .
Theorem 3.11.
Let be a uniformly smooth Banach space with the dual space
let
be
-Lipschitz continuous, let
be
-Lipschitz continuous, and let
be
-strongly accretive and
-Lipschitz continuous. Suppose that
is
-strongly monotone and
-Lipschitz continuous,
is a bifunction satisfying conditions (H1)–(H4), and
is a lower semicontinuous subdifferentiable proper functional which is
-subdifferentiable at each
. Let
and
be two sequences in
with
, and
. Assume that the
-subdifferential
of
is
-strongly monotone, the ranges
, and
are totally bounded, and
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ55_HTML.gif)
Then for any given , the sequence
defined by Algorithm 3.6 converges strongly to a solution
of GMEPP (2.1).
Proof.
By Theorem 3.8 and the assumptions in Theorem 3.11, we know that the solution set of GMEPP (2.1) is nonempty. Let be a solution of GMEPP (2.1). Since
and
, we can choose a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ56_HTML.gif)
By Algorithm 3.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ57_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ58_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ59_HTML.gif)
It follows from Theorem 2.9 that is Lipschitz continuous. Since the ranges
, and
are totally bounded, we know that the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ60_HTML.gif)
is bounded. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ61_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ62_HTML.gif)
Since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ63_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ64_HTML.gif)
By induction we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ65_HTML.gif)
On the other hand, by Lemma 3.4,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ66_HTML.gif)
Now we consider the third term on the right-hand side of (3.23). From (3.19) and (3.22) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ67_HTML.gif)
Since is a uniformly smooth Banach space, the normalized duality mapping
is uniformly norm-to-norm continuous on any bounded subset of
. Hence it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ68_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ69_HTML.gif)
Since is bounded,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ70_HTML.gif)
Next we consider the second term on the right-hand side of (3.23). Since is a solution of GMEPP (2.1), by Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ71_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ72_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ73_HTML.gif)
Substituting (3.27) and (3.29) into (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ74_HTML.gif)
Next we make an estimation for . Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ75_HTML.gif)
Substituting (3.32) into (3.31) and simplifying it, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ76_HTML.gif)
From condition (3.13), we get . Since
, and
, we deduce that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ77_HTML.gif)
Thus, utilizing Lemma 3.10 we conclude that as
. This completes the proof.
Remark 3.12.
By the careful analysis of the proof of Theorem 3.11, we can see readily that the following conditions are used to derive the following conclusion:
(i)the normalized duality mapping is uniformly norm-to-norm continuous on any bounded subset of
;
(ii)the constant in Algorithm 3.6 satisfies the inequality
.
In addition, we apply Lemma 3.4 to derive the strong convergence of the iterative sequence generated by Algorithm 3.6. Moreover, we simplify the original proof by Xia and Huang [14, Theorem 3.11] to a great extent. Therefore, Theorem 3.11 is a generalization and modification of Xia and Huang's Theorem [14].
Remark 3.13.
We would like to point out that, in Theorem 3.11, the functional may not be convex, the mappings
and
may not have any monotonicity, and their domains and ranges are reflexive Banach space
and the dual space
of
, respectively. Hence Theorem 3.11 improves and generalizes some known results in [8, 9, 11, 16, 17]. Furthermore, the argument methods presented in this paper are quite different from those in [8–11, 13, 14, 16, 21].
Finally, we give a weak convergence theorem for the iterative sequence generated by Algorithm 3.6 in a real Hilbert space. However, we first recall the following lemmas.
Lemma 3.14 (see [29, page 303]).
Let and
be sequences of nonnegative real numbers satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ78_HTML.gif)
If converges, then
exists.
Lemma 3.15 (see [30]).
Demiclosedness Principle. Assume that is a nonexpansive self-mapping of a nonempty closed convex subset
of a Hilbert space
. If
has a fixed point, then
is demiclosed. That is, whenever
is a sequence in
weakly converging to some
and the sequence
strongly converges to some
, it follows that
. Here
is the identity operator of
.
Theorem 3.16.
Let be a real Hilbert space, let
be
-Lipschitz continuous, let
be
-Lipschitz continuous, and let
be
-strongly monotone and
-Lipschitz continuous. Suppose that
is
-strongly monotone and
-Lipschitz continuous,
is a bifunction satisfying conditions (H1)–(H4), and
is a lower semicontinuous subdifferentiable proper functional which is
-subdifferentiable at each
. Let
and
be two sequences in
such that
. Assume that the
-subdifferential
of
is
-strongly monotone, the ranges
, and
are totally bounded, and
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ79_HTML.gif)
Then for any given , the sequence
defined by Algorithm 3.6 converges weakly to a solution of GMEPP (2.1).
Proof.
By Theorem 3.8 and the assumptions in Theorem 3.16, we know that the solution set of GMEPP (2.1) is nonempty. Let be a solution of GMEPP (2.1). It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ80_HTML.gif)
Define a mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ81_HTML.gif)
where . Since the
-subdifferential
of
is
-strongly monotone, by Theorem 2.9 we conclude that
is
-Lipschitz continuous.
Observe that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ82_HTML.gif)
where and
. This implies that
is nonexpansive on
. It is easy to see that the set of GMEPP (2.1) coincides with
.
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ83_HTML.gif)
This together with Lemma 3.14 implies that exists. And also, from (3.40) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ84_HTML.gif)
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ85_HTML.gif)
Now, let us show that . Indeed, let
. Then there exists a subsequence
of
such that
. Since
, by Lemma 3.15 we know that
.
Further, let us show that is a singleton. Indeed, let
be another subsequence of
such that
. Then
is also a fixed point of
. If
, by Opial's property of
, we reach the following contradiction:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F794503/MediaObjects/13663_2010_Article_1344_Equ86_HTML.gif)
This implies that is a singleton. Consequently,
converges weakly to a fixed point of
, that is, a solution of GMEPP (2.1). This completes the proof.
References
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities, Applications to Free Boundary Problems. John Wiley & Sons, New York, NY, USA; 1984:ix+452.
Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York, NY, USA; 2003.
Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313.
Xiu N, Zhang J: Some recent advances in projection-type methods for variational inequalities. Journal of Computational and Applied Mathematics 2003,152(1–2):559–585. 10.1016/S0377-0427(02)00730-6
Cho YJ, Kim JK, Verma RU: A class of nonlinear variational inequalities involving partially relaxed monotone mappings and general auxiliary problem principle. Dynamic Systems and Applications 2002,11(3):333–337.
Cohen G: Auxiliary problem principle extended to variational inequalities. Journal of Optimization Theory and Applications 1988,59(2):325–333.
Schaible S, Yao JC, Zeng L-C: Iterative method for set-valued mixed quasivariational inequalities in a Banach space. Journal of Optimization Theory and Applications 2006,129(3):425–436. 10.1007/s10957-006-9077-9
Zeng L-C, Guu S-M, Yao J-C: An iterative method for generalized nonlinear set-valued mixed quasi-variational inequalities with -monotone mappings. Computers & Mathematics with Applications 2007,54(4):476–483. 10.1016/j.camwa.2007.01.025
Zeng L-C: Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities. Journal of Mathematical Analysis and Applications 1994,187(2):352–360. 10.1006/jmaa.1994.1361
Zeng L-C: Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities. Journal of Mathematical Analysis and Applications 1996,201(1):180–194. 10.1006/jmaa.1996.0249
Huang N-J, Deng C-X: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Mathematical Analysis and Applications 2001,256(2):345–359. 10.1006/jmaa.2000.6988
Xia F-Q, Huang N-J: Algorithm for solving a new class of general mixed variational inequalities in Banach spaces. Journal of Computational and Applied Mathematics 2008,220(1–2):632–642. 10.1016/j.cam.2007.09.011
Zeng L-C, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-z
Ding XP: General algorithm of solutions for nonlinear variational inequalities in Banach space. Computers & Mathematics with Applications 1997,34(9):131–137. 10.1016/S0898-1221(97)00194-6
Ding XP: General algorithm for nonlinear variational-like inequalities in reflexive Banach spaces. Indian Journal of Pure and Applied Mathematics 1998,29(2):109–120.
Ding XP, Yao J-C, Zeng L-C: Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces. Computers & Mathematics with Applications 2008,55(4):669–679. 10.1016/j.camwa.2007.06.004
Chang SS: Set-valued variational inclusions in Banach spaces. Journal of Mathematical Analysis and Applications 2000,248(2):438–454. 10.1006/jmaa.2000.6919
Huang N-J, Fang Y-P: Iterative processes with errors for nonlinear set-valued variational inclusions involving accretive type mappings. Computers & Mathematics with Applications 2004,47(4–5):727–738. 10.1016/S0898-1221(04)90060-0
Huang N-J, Yuan GX-Z: Approximating solution of nonlinear variational inclusions by Ishikawa iterative process with errors in Banach spaces. Journal of Inequalities and Applications 2001,6(5):547–561. 10.1155/S1025583401000339
Zeng LC, Hu HY, Wong MM: Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed point problem of infinitely many nonexpansive mappings. to appear in Taiwanese Journal of Mathematics
Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
Ding XP, Tan K-K: A minimax inequality with applications to existence of equilibrium point and fixed point theorems. Colloquium Mathematicum 1992,63(2):233–247.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
van Dulst D: Equivalent norms and the fixed point property for nonexpansive mappings. The Journal of the London Mathematical Society 1982,25(1):139–144. 10.1112/jlms/s2-25.1.139
Petryshyn WV: A characterization of strict convexity of Banach spaces and other uses of duality mappings. Journal of Functional Analysis 1970, 6: 282–291. 10.1016/0022-1236(70)90061-3
Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. Journal of Optimization Theory and Applications 2003,119(1):185–201.
Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.
Acknowledgments
This research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph D Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405) (to Lu-Chuan Ceng). It was supported by Basic Science Research Program through KOSEF 2009-0077742 (to Sangho Kum) and was partially supported by the Grant NSC 98-2115-M-110-001 (to Jen-Chih Yao).
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Ceng, LC., Kum, S. & Yao, JC. Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space. Fixed Point Theory Appl 2010, 794503 (2010). https://doi.org/10.1155/2010/794503
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DOI: https://doi.org/10.1155/2010/794503