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Hybrid iterative method for systems of generalized equilibria with constraints of variational inclusion and fixed point problems
Fixed Point Theory and Applications volume 2014, Article number: 164 (2014)
Abstract
In this paper, we introduce and analyze an iterative algorithm by the hybrid iterative method for finding a solution of the system of generalized equilibrium problems with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inclusions, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Weak convergence result under mild assumptions will be established.
MSC: 49J30, 47H09, 47J20, 49M05.
1 Introduction and formulations
Let H be a real Hilbert space with the inner product and the norm , let C be a nonempty closed convex subset of H and be the metric projection of H onto C. Let be a nonlinear mapping on C. We denote by the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant such that
A mapping is called L-Lipschitz continuous if there exists a constant such that
In particular, S is called a nonexpansive mapping if and A is called a contraction if .
Let be a real-valued function, be a nonlinear mapping and be a bifunction. Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that
We denote the set of solutions of GMEP (1.2) by . GMEP (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others.
Throughout this paper, we assume as in [1] that is a bifunction satisfying conditions (H1)-(H4) and is a lower semicontinuous and convex function with restriction (H5), where
-
(H1)
for all ;
-
(H2)
Θ is monotone, i.e., for any ;
-
(H3)
Θ is upper-hemicontinuous, i.e., for each ,
-
(H4)
is convex and lower semicontinuous for each ;
-
(H5)
for each and , there exists a bounded subset and such that for any ,
Given a positive number , let be a solution set of the auxiliary mixed equilibrium problem, that is, for each ,
In particular, whenever , , is rewritten as .
Let be two bifunctions and be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that
where and are two constants. It is introduced and studied in [2]. When , the SGEP reduces to a system of variational inequalities, which is considered and studied in [3]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.
In 2010, Ceng and Yao [2] transformed the SGEP into a fixed point problem in the following way.
Proposition 1.1 (see [2])
Let be two bifunctions satisfying conditions (H1)-(H4), and let be -inverse-strongly monotone for . Let for . Then is a solution of SGEP (1.3) if and only if is a fixed point of the mapping defined by , where . Here, we denote the fixed point set of G by .
Let be an infinite family of nonexpansive mappings on H and be a sequence of nonnegative numbers in . For any , define a mapping on H as follows:
Such a mapping is called the W-mapping generated by and .
In 2011, for the case where , Yao et al. [4] proposed the following hybrid iterative algorithm:
where is a contraction, is differentiable and strongly convex, and are given, for finding a common element of the set and the fixed point set of an infinite family of nonexpansive mappings on H. They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (1.5) to a point under some appropriate conditions. This point also solves the following optimization problem:
where is the potential function of γf.
Let be a contraction and V be a strongly positive bounded linear operator on H. Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)-(H4), and that are inverse-strongly monotone. Let the mapping G be defined as in Proposition 1.1. Very recently, Ceng et al. [5] introduced the following hybrid extragradient-like iterative algorithm:
for finding a common solution of GMEP (1.2), SGEP (1.3) and the fixed point problem of an infinite family of nonexpansive mappings on H, where , , , , and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (1.6) to a point under some suitable conditions. This point also solves the following optimization problem:
where is the potential function of γf.
On the other hand, let B be a single-valued mapping of C into H and R be a set-valued mapping with domain . Consider the following variational inclusion [6]: find a point such that
We denote by the solution set of the variational inclusion (1.7). It is known that problem (1.7) provides a convenient framework for the unified study of optimal solutions in many optimization-related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with R and λ as follows:
where λ is a positive number.
In 2011, for the case where , Yao et al. [7] introduced and analyzed the following iterative algorithms for finding an element of the intersection of the solution set of GMEP (1.2), the solution set of the variational inclusion (1.7) and the fixed point set of a countable family of nonexpansive mappings: for arbitrarily given , let the sequence be generated by
where , are two sequences in and is the W-mapping defined by (1.4). It is proven that under appropriate conditions the sequence converges strongly to , where is a unique solution of the VIP:
Next, we recall some concepts. Let C be a nonempty subset of a normed space X. A mapping is called uniformly Lipschitzian if there exists a constant such that
Recently, Kim and Xu [8] introduced the concept of asymptotically k-strict pseudocontractive mappings in a Hilbert space as follows.
Definition 1.1 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically k-strict pseudocontractive mapping with sequence if there exists a constant and a sequence in with such that
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically k-strict pseudocontractive mapping with sequence is a uniformly ℒ-Lipschitzian mapping with . Subsequently, Sahu et al. [9] considered the concept of asymptotically k-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.
Definition 1.2 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that
Put . Then (), () and we get the relation
Whenever for all in (1.10), then S is an asymptotically k-strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [9] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence . More precisely, they first established one weak convergence theorem for the following iterative scheme:
where , and ; and then obtained another strong convergence theorem for the following iterative scheme:
where , and .
Motivated and inspired by the above results and the method in [10], we introduce and analyze an iterative algorithm by the hybrid iterative method for finding a solution of the system of generalized equilibrium problems with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inclusions, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. A weak convergence theorem for the iterative algorithm will be established under mild conditions.
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. We use the notations and to indicate the weak convergence of to x and the strong convergence of to x, respectively. Moreover, we use to denote the weak ω-limit set of , i.e.,
Definition 2.1 A mapping is called
-
(i)
monotone if
-
(ii)
η-strongly monotone if there exists a constant such that
-
(iii)
ζ-inverse-strongly monotone if there exists a constant such that
It is easy to see that the projection is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.
Definition 2.2 A differentiable function is called:
-
(i)
convex if
where is the Frechet derivative of K at x;
-
(ii)
strongly convex if there exists a constant such that
It is easy to see that if is a differentiable strongly convex function with constant , then is strongly monotone with constant .
The metric projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
Some important properties of projections are listed in the following proposition.
Proposition 2.1 ([[11], p.17])
For given and ,
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, . (This implies that is nonexpansive and monotone.)
By using the technique of [12], we can readily obtain the following elementary result where is the solution set of the mixed equilibrium problem [5].
Proposition 2.2 (see [[5], Lemma 1 and Proposition 1])
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a lower semicontinuous and convex function. Let be a bifunction satisfying conditions (H1)-(H4). Assume that
-
(i)
is strongly convex with constant and the function is weakly upper semicontinuous for each ;
-
(ii)
for each and , there exists a bounded subset and such that for any ,
Then the following hold:
-
(a)
for each , ;
-
(b)
is single-valued;
-
(c)
is nonexpansive if is Lipschitz continuous with constant and
where for ;
-
(d)
for all and ,
-
(e)
;
-
(f)
is closed and convex.
Remark 2.1 In Proposition 2.2, whenever is a bifunction satisfying conditions (H1)-(H4) and , , we have, for any ,
( is firmly nonexpansive) and
If, in addition, , then is rewritten as ; see [[2], Lemma 2.1] for more details.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1 Let X be a real inner product space. Then the following inequality holds:
Lemma 2.2 ([[11], p.20])
Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and with ;
-
(c)
If is a sequence in H such that , it follows that
We have the following crucial lemmas concerning the W-mappings defined by (1.4).
Lemma 2.3 (see [[13], Lemma 3.3])
Let be a sequence of nonexpansive self-mappings on H such that , and let be a sequence in for some . Then .
Lemma 2.4 (see [[14], Demiclosedness principle])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then is demiclosed. That is, whenever is a sequence in C weakly converging to some and the sequence strongly converges to some y, it follows that . Here I is the identity operator of H.
Lemma 2.5 ([[9], Lemma 2.6])
Let C be a nonempty subset of a Hilbert space H and be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence . Then
for all and .
Lemma 2.6 (Demiclosedness principle [[9], Proposition 3.1])
Let C be a nonempty closed convex subset of a Hilbert space H and be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in C such that and , then .
Recall that a Banach space X is said to satisfy the Opial condition [4] if for any given sequence which converges weakly to an element , the following inequality holds:
It is well known in [4] that every Hilbert space H satisfies the Opial condition.
Lemma 2.7 (see [[15], Proposition 3.1])
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a sequence in H. Suppose that
where and are sequences of nonnegative real numbers such that and . Then converges strongly in C.
3 Weak convergence theorem
In this section, we will prove weak convergence of another iterative algorithm by the hybrid Mann-type viscosity method for finding a solution of the system of generalized equilibrium problems with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inclusions, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. This iterative algorithm is based on the extragradient method, viscosity approximation method and Mann-type iterative method.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ, , be three bifunctions from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping, and let and be ζ-inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively, where and . Let be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive mappings on H and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Let be the W-mapping defined by (1.4). Assume that is nonempty, where G is defined as in Proposition 1.1. Let be a sequence in and , and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:
Assume that the following conditions are satisfied:
-
(i)
is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
, and , .
If is firmly nonexpansive, then converges weakly to .
Proof First, let us show that exists for any . Put
for all , , and , where I is the identity mapping on H. Then we get . Take arbitrarily. Repeating the same arguments as in the proof of [[16], Theorem 3.1], we can obtain that
We observe that
Set . Then by Lemma 2.1 we deduce from (3.2)-(3.5) and (3.8) and that
which hence yields
where (due to and ). Since , and , by Lemma 2.7 we have that exists. Thus is bounded and so are the sequences , and .
Also, utilizing Lemmas 2.1 and 2.2(b), we obtain from (3.3)-(3.5) and (3.8) that
which leads to
Since , and , it follows from the existence of and condition (iii) that
Note that
which yields
So, from (3.11) and , we get
In the meantime, we conclude from (3.3), (3.4), (3.8) and (3.10) that
which together with implies that
Consequently, from , , and the existence of , we get
Since , from (3.13) we have
Combining (3.3), (3.4), (3.8) and (3.10), we have
which implies
Since , and , from condition (iii) and the existence of we get
Repeating the same arguments as those of (3.17) in the proof of [[16], Theorem 3.1], we can get
Combining (3.8), (3.10) and (3.16), we have
which implies
Since , and , from (3.15) and the existence of we obtain
Combining (3.6), (3.8) and (3.10), we have
where , which implies
Since , and , from , and the existence of we obtain
Combining (3.7), (3.8) and (3.10), we get
which implies
Since , and , from (3.18) and the existence of we obtain
By (3.19), we have
From (3.17) and (3.20), we have
By (3.14) and (3.21), we obtain
which together with (3.12) and (3.22) implies that
On the other hand, we observe that
By (3.21) and (3.23), we have
We note that
From (3.13), (3.24), Lemma 2.5 and the uniform continuity of S, we obtain
On the other hand, for simplicity, we write , and for all . Then
We now show that , i.e., . As a matter of fact, utilizing the arguments similar to those of (3.29) in the proof of [[16], Theorem 3.1], we deduce from (3.1)-(3.5) and (3.8) that for ,
which immediately leads to
Since , , and , we conclude from (3.23) and condition (iv) that
Utilizing the arguments similar to those of (3.31) and (3.32) in the proof of [[16], Theorem 3.1], we can obtain
and
Consequently, from (3.3), (3.4), (3.8), (3.26) and (3.28) it follows that
which hence leads to
Since , , and , from (3.23) and (3.27) we have
Furthermore, from (3.3), (3.4), (3.8), (3.26) and (3.29) it follows that
which hence yields
Since , , and , from (3.23) and (3.27) we have
Note that
Hence from (3.30) and (3.31) we get
which together with (3.11) and (3.32) implies that
Also, observe that
From (3.33), [[17], Remark 2.3] and the boundedness of we immediately obtain
Since is bounded, there exists a subsequence of which converges weakly to w. From (3.21) and (3.22), we have that and . From (3.17), (3.19), (3.21), we have that , , and , where . Since S is uniformly continuous, by (3.25) we get for any . Hence from Lemma 2.4 we obtain . In the meantime, utilizing Lemma 2.4, we deduce from , (3.32) and (3.34) that and (due to Lemma 2.3). Utilizing similar arguments to those in the proof of [[16], Theorem 3.1], we can derive . Consequently, . This shows that .
Next let us show that is a single-point set. As a matter of fact, let be another subsequence of such that . Then we get . If , from the Opial condition, we have
This attains a contradiction. So we have . Put . Since , we have . By Lemma 2.7, we have that converges strongly to some . Since converges weakly to w, we have
Therefore we obtain . This completes the proof. □
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ, , be three bifunctions from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping, and let and be ζ-inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, for and . Let be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive mappings on H and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Let be the W-mapping defined by (1.4). Assume that is nonempty, where G is defined as in Proposition 1.1. Let be a sequence in and , and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:
Assume that the following conditions are satisfied:
-
(i)
is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
and for and .
If is firmly nonexpansive, then converges weakly to .
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ, , be three bifunctions from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping, and let and be ζ-inverse strongly monotone, -inverse strongly monotone and η-inverse strongly monotone, respectively, for . Let be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty, where G is defined as in Proposition 1.1. Let be a sequence in and , and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:
Assume that the following conditions are satisfied:
-
(i)
is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
and for .
If is firmly nonexpansive, then converges weakly to .
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant No. (30-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The authors thank the referees for their appreciation with valuable comments.
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Latif, A., Al-Mazrooei, A.E., Alofi, A.S. et al. Hybrid iterative method for systems of generalized equilibria with constraints of variational inclusion and fixed point problems. Fixed Point Theory Appl 2014, 164 (2014). https://doi.org/10.1186/1687-1812-2014-164
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DOI: https://doi.org/10.1186/1687-1812-2014-164