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New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems
Fixed Point Theory and Applications volume 2014, Article number: 187 (2014)
Abstract
In this paper, we first introduce and investigate a bilevel split equilibrium problem (BSEP) which can be regarded as a new development in the field of equilibrium problems. We provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces.
MSC:47J25, 47H09, 65K10.
1 Introduction and preliminaries
Let K be a closed convex subset of a real Hilbert space H. Let and denote the inner product of H and the norm of H, respectively. For each point , there exists a unique nearest point in K, denoted by , such that
The mapping is called the metric projection from H onto K. It is well known that has the following properties:
-
(i)
for every .
-
(ii)
For and , for all .
-
(iii)
For and ,
(1.1)
Let and be two Hilbert spaces. Let and be two bounded linear operators. is called the adjoint operator (or adjoint) of A if
It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that if is an adjoint operator of A, then . The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively.
Example 1.1 ([1])
Let with the standard norm and with the norm for some . denotes the inner product of for some , and denotes the inner product of for some . Let for and for , then B is an adjoint operator of A.
Example 1.2 ([1])
Let with the norm for some and with the norm for some . Let and denote the inner product of and , respectively, where , . Let for and for . Obviously, B is an adjoint operator of A.
Let f be a bi-function from to ℝ. The classical equilibrium problem (EP, for short) is defined as follows.
(EP) Find such that , .
The set of such solutions is denoted by , that is, . In fact, equilibrium problem has an important relationship with variational inequality problem. For example, let be a nonlinear mapping satisfying for all . Then if and only if is a solution of the variational inequality for all . It is known that the EP is an important mathematical model for nonlinear analysis and applied sciences which is generalized to many new mathematical models and includes many important problems arising in physics, engineering, science optimization, economics, network, game theory, complementary problems, variational inequalities problems, saddle point problems, fixed point problems and others; for details, one can refer to [2–8] and references therein. Many authors have proposed some useful methods to solve the EP; see, for instance, [2–5, 9–17] and references therein.
Recent investigations and developments in equilibrium theory as well as optimization theory have been applied to connect fundamental sciences with the real world. According to our experience, useful methods of real world problems often need to be used to solve several problems arising in different spaces. In view of this, recent studies focus on split problems which are more closed in the real world applications; see, for instance, [1, 18–24] and the references therein. Recently, He [1] considered the following split equilibrium problem. Let and be two real Hilbert spaces. Let C be a closed convex subset of and K be a closed convex subset of . Let and be two bi-functions, and be a bounded linear operator. The split equilibrium problem (SEP, in short) is defined as follows:
(SEP) Find such that , , and satisfying , .
In [1], the author established weak convergence algorithms and strong convergence algorithms for SEP (see [1] for more details).
Motivated and inspired by the works mentioned above, in this paper we shall introduce and investigate the following new problem. Let , and be three real Hilbert spaces. Let C be a closed convex subset of , Q be a closed convex subset of and K be a closed convex subset of . Let , and be three bi-functions. Let and be two bounded linear operators with theirs adjoint operators and , respectively. The mathematical model about bilevel split equilibrium problem (BSEP, in short) is defined as follows:
(BSEP) Find and such that
-
(i)
and for all and ;
-
(ii)
;
-
(iii)
for all .
In fact, BSEP can be regarded as a new development in the field of equilibrium problems and contains several important problems as special cases. It was profoundly believed that BSEP will motivate and inspire further scientific activities in the fields of equilibrium problems, optimization problems, game problems, complementary problems, variational inequalities problems, fixed point problems and their applications.
Example A Let , and be three real Hilbert spaces. Let , and be three closed convex sets. Let , and be three convex functions. Let and be two bounded linear operators with their adjoint operators and , respectively. Let
and
Then BSEP reduces the bilevel convex optimization problem (BCOP):
(BCOP) Find and such that , , and for all , and .
Example B Let , and be three real Hilbert spaces. Let , and be three closed convex sets. Let , and be three nonlinear operators. Let and be two bounded linear operators with their adjoint operators and , respectively. If , and , then BSEP reduces to the bilevel split variational inequality problem (BSVI):
(BSVI) Find and such that satisfying , and for all , and .
Example C Let and be two real Hilbert spaces and be a bounded linear operator with its adjoint operator . Let , and be three closed convex sets. If and , then BSEP reduces to the following split equilibrium problem (1) (SEP):
() Find and such that satisfying , and for all , and .
Example D Let and be two real Hilbert spaces and be a bounded linear operator with its adjoint operator . Let , and be three closed convex sets with . If and (identity operator), then BSEP reduces to the following split equilibrium problem (2) (SEP):
() Find such that satisfying , and for each , and .
Especially, if for all , then (SEP) reduces to finding such that satisfying and for all and , which was studied in [1].
Example E Let and be two real Hilbert spaces and be a bounded linear operator with its adjoint operator . Let , and be three closed convex sets with . If and (identity operator), then BSEP reduces to the following split equilibrium problems (3) (SEP):
() Find such that satisfying , and for all , and .
Example F In Example A, if , and (identity operator), then BSEP reduces to the common solution of equilibrium problems (CEP):
(CEP) Find such that , and for each .
The paper is divided into four sections. In Sections 1 and 2, we first introduce and investigate a bilevel split equilibrium problem (BSEP) and then provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces. In Section 3, we give the proof of the main result Theorem 2.1 in detail. Finally, an example illustrating Theorem 2.1 is given in Section 4.
2 Feasible iterative algorithms for BSEP and their strong convergence theorems
In 1994, Blum and Oettli [2] established the following important existence theorem which plays a key role in solving equilibrium problems, variational inequality problems and optimization problems.
Lemma 2.1 (Blum and Oettli [2])
Let K be a nonempty closed convex subset of H and F be a bi-function of into ℝ satisfying the following conditions.
(A1) for all ;
(A2) F is monotone, that is, for all ;
(A3) for each ,
(A4) for each , is convex and lower semi-continuous.
Let and . Then there exists such that
In this paper, we first introduce a new iterative algorithm for BSEP and establish a strong convergence theorem for this iterative algorithm. Here, the space denotes the product space of two real Hilbert spaces and , which is endowed with the usual linear operation and norm, namely, for and ,
and
Theorem 2.1 Let , and be three real Hilbert spaces. Let C be a closed convex subset of , Q be a closed convex subset of and K be a closed convex subset of . Let , and be three bi-functions. and are two bounded linear operators with their adjoint operators and , respectively. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
The following conclusion is immediate from Theorem 2.1 by putting .
Corollary 2.1 Let and be two real Hilbert spaces. Let C and Q be two closed convex subsets of and K be a closed convex subset of . Let , and be three bi-functions. is a bounded linear operator with its adjoint operator . Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
If and , then Theorem 2.1 reduces to the following corollary.
Corollary 2.2 Let and be two real Hilbert spaces. Let and be three closed convex sets. Let , and be three bi-functions. is a bounded linear operator with its adjoint operator . Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
If and , then Theorem 2.1 reduces to the following corollary.
Corollary 2.3 Let and be two real Hilbert spaces. Let and be three closed convex sets. Let , and be three bi-functions. is a bounded linear operator with its adjoint operator . Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
Putting (identical operator), and , then we have the following result.
Corollary 2.4 Let H be a real Hilbert space. Let C be a closed convex subset of H. Let be three bi-functions. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , and be sequences generated by
where and with , is a projection operator from H into C. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
Remark 2.1 In Corollary 2.2, it is obvious that
implies
Hence, the problem studied in Corollary 2.4 is still the study of a common solution of three equilibrium problems in essence.
If C, Q, K are linear subspaces of a real Hilbert space, then we have the following corollaries from Theorem 2.1 and Corollary 2.1.
Corollary 2.5 Let , and be three real Hilbert spaces. Let , and be three linear subspaces. Let , and be three bi-functions. and are two bounded linear operators with their adjoint operators and , respectively. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
Corollary 2.6 Let and be two real Hilbert spaces. Let , and be three linear subspaces. Let , and be three bi-functions. is a bounded linear operator with its adjoint operator . Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let , , , , , , , and be sequences generated by
where and with , and are two projection operators from into C and from into Q, respectively. Suppose that
Then there exists such that
-
(a)
as ;
-
(b)
as ;
-
(c)
as .
Remark 2.2 In fact, the problem studied by Corollaries 2.1-2.3 and Corollary 2.6 is (SEP).
In order to prove Theorem 2.1, we need the following crucial known results.
Lemma 2.2 (see [10])
Let K be a nonempty closed convex subset of H, and let F be a bi-function of into ℝ satisfying (A1)-(A4). For , define a mapping as follows:
for all . Then the following hold:
-
(i)
is single-valued and for and is closed and convex;
-
(ii)
is firmly nonexpansive, that is, for any , .
Lemma 2.3 ([20])
Let be the same as in Lemma 2.2. If , then, for any and , .
Let the mapping be defined as in Lemma 2.2. Then, for and ,
In particular, for any and , that is, is nonexpansive for any .
Lemma 2.5 (see, e.g., [25])
Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and .
3 Proof of Theorem 2.1
Applying Lemmas 2.1 and 2.2, we know that , and are all well defined. It is also easy to verify that , , are closed convex sets for .
Now, we claim for all . Indeed, it suffices to prove that for all . Let . Then , and
Let be given. By Lemma 2.3, we have
From (2.1), (3.1) and Lemma 2.5, we obtain
and
By taking into account inequalities (3.1), (3.2) and (3.3), we obtain
which implies
Inequality (3.5) shows that . Hence and for all . For each , since , , we have
Similarly, since , we have
So, for any , we get
and
The last inequalities deduce that and are bounded and hence show that , , and are all bounded. For some with , from , and (1.1), we have
which yields that
Together with the boundedness of and , we know and exist. For some with , due to , and (1.1), we have
By (3.6), we have and . Hence and are all Cauchy sequences. Let and for some . We want to prove that . For any , since
from (2.1), we have
By taking the limit from both sides of (3.7), we obtain
Moreover, by (3.8), we get
Since , we have and as . Moreover, we obtain and as .
Now, we claim and . In fact, for , by Lemma 2.4, we have
and
So, and .
Finally, we prove . Setting
Then, for any , by (3.4) and (3.9), we have
Hence (3.10) implies
Since , and (3.11), we obtain and , where . On the other hand, for , by Lemma 2.4 again, we have
Hence , namely . Therefore, conclusions (a), (b) and (c) are all proved. The proof is completed.
4 An example of Theorem 2.1
In this section, we give an example illustrating Theorem 2.1.
Example 4.1 Let , and be three real Hilbert spaces with the standard norm and inner product. For each and , define
and
Then A is a bounded linear operator from into with , and is an adjoint operator of A with . For each and , let
and
Then B is a bounded linear operator from into with , and is an adjoint operator of B with . Put
and
For each , and , define
and
For each , let
For each , let
For each , let
It is not hard to verify that f, g and h satisfy conditions (A1)-(A4) with , , and
Let , , and for . Thus, for each and with , we have the following:
-
,
-
,
-
,
-
,
-
.
For and with , and , we obtain the following:
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
.
From , , we have , and . Since
and
we get the following:
-
,
-
,
-
,
-
.
Similarly, for with , we obtain
-
,
-
,
-
,
-
,
-
,
-
,
-
.
By mathematical induction, we know that , , , and all are decreasing sequences. Moreover, , , , and as . So, we have , , , and as .
5 Conclusion
In this paper, we first introduce and investigate BSEP which can be regarded as a new development in the field of equilibrium problems. We provide some new iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces. An example illustrating Theorem 2.1 is also given.
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Acknowledgements
The first author was supported by the Natural Science Foundation of Yunnan Province (201401CA00262) and the Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206); the second author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.
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Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
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He, Z., Du, WS. New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems. Fixed Point Theory Appl 2014, 187 (2014). https://doi.org/10.1186/1687-1812-2014-187
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DOI: https://doi.org/10.1186/1687-1812-2014-187