
Research Article

Open

 Published:
Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators
Journal of Inequalities and Applicationsvolume 2009, Article number: 896252 (2009)
Abstract
The purpose of this paper is to introduce and study two new hybrid proximalpoint algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximalpoint algorithms, respectively.
1. Introduction
Let be a real Banach space and its dual space. The normalized duality mapping is defined as
where denotes the generalized duality pairing. Recall that if is a smooth Banach space then is singlevalued. Throughout this paper, we will still denote by the singlevalued normalized duality mapping. Let be a nonempty closed convex subset of , a bifunction from to , and a nonlinear mapping. The generalized equilibrium problem is to find such that
The set of solutions of (1.2) is denoted by . Problem (1.2) and similar problems have been extensively studied; see, for example, [1–11]. Whenever , problem (1.2) reduces to the equilibrium problem of finding such that
The set of solutions of (1.3) is denoted by . Whenever , problem (1.2) reduces to the variational inequality problem of finding such that
Whenever a Hilbert space, problem (1.2) was very recently introduced and considered by Kamimura and Takahashi [12]. Problem (1.2) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for example, [13, 14]. A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . Iterative schemes for finding common elements of EP and fixed points set of nonexpansive mappings have been studied recently; see, for example, [12, 15–17] and the references therein.
On the other hand, a classical method of solving in a Hilbert space is the proximal point algorithm which generates, for any starting point , a sequence in by the iterative scheme
where is a sequence in , for each is the resolvent operator for , and is the identity operator on . This algorithm was first introduced by Martinet [14] and generally studied by Rockafellar [18] in the framework of a Hilbert space . Later many authors studied (1.5) and its variants in a Hilbert space or in a Banach space ; see, for example, [13, 19–23] and the references therein.
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the following conditions (A1)–(A4) which were imposed in [24]:
(A1) for all ;
(A2) is monotone, that is, , for all ;
(A3)for all ;
(A4)for all is convex and lower semicontinuous.
Let be a maximal monotone operator such that
(A5).
The purpose of this paper is to introduce and study two new iterative algorithms for finding a common element of the set of solutions for the generalized equilibrium problem (1.2) and the set for maximal monotone operators in a uniformly smooth and uniformly convex Banach space . First, motivated by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we introduce a sequence that, under some appropriate conditions, is strongly convergent to in Section 3. Second, inspired by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we define a sequence weakly convergent to an element , where in Section 4. Our results represent a generalization of known results in the literature, including Takahashi and Zembayashi [15], Kamimura and Takahashi [12], Li and Song [22], Ceng and Yao [25], and Ceng et al. [16]. In particular, compared with Theorems and in [16], our results (i.e., Theorems 3.2 and 4.2 in this paper) extend the problem of finding an element of to the one of finding an element of . Meantime, the algorithms in this paper are very different from those in [16] (because of considering the complexity involving the problem of finding an element of ).
2. Preliminaries
In the sequel, we denote the strong convergence, weak convergence and weak* convergence of a sequence to a point by , and , respectively.
A Banach space is said to be strictly convex, if for all with . is said to be uniformly convex if for each there exists such that for all with . Recall that each uniformly convex Banach space has the KadecKlee property, that is,
The proof of the main results of Sections 3 and 4 will be based on the following assumption.
Assumption A.
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the same conditions (A1)–(A4) as in Section 1. Let be two maximal monotone operators such that
(A5)′.
Recall that if is a nonempty closed convex subset of a Hilbert space , then the metric projection of onto is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined as in [26] by
It is clear that in a Hilbert space , (2.2) reduces to .
The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem
The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [27]). In a Hilbert space, . From [26], in a smooth strictly convex and reflexive Banach space , we have
Moreover, by the property of subdifferential of convex functions, we easily get the following inequality:
Let be a mapping from into itself. A point in is called an asymptotically fixed point of if contains a sequence which converges weakly to such that [28]. The set of asymptotically fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive if and , for all and [15].
Observe that, if is a reflexive strictly convex and smooth Banach space, then for any if and only if . To this end, it is sufficient to show that if then . Actually, from (2.4), we have which implies that . From the definition of , we have and therefore, ; see [29] for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (Kamimura and Takahashi [12]).
Let be a smooth and uniformly convex Banach space and let and be two sequences of . If and either or is bounded, then .
Lemma 2.2 (Alber [26], Kamimura and Takahashi [12]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space . Let and let . Then
Lemma 2.3 (Alber [26], Kamimura and Takahashi [12]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space . Then
Lemma 2.4 (Rockafellar [18]).
Let be a reflexive strictly convex and smooth Banach space and let be a multivalued operator. Then there hold the following hold:
(i) is closed and convex if is maximal monotone such that ;
(ii) is maximal monotone if and only if is monotone with for all .
Lemma 2.5 (Xu [30]).
Let be a uniformly convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function such that and
for all and , where .
Lemma 2.6 (Kamimura and Takahashi [12]).
Let be a smooth and uniformly convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function such that and
The following result is due to Blum and Oettli [24].
Lemma 2.7 (Blum and Oettli [24]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and . Then, there exists such that
Motivated by Combettes and Hirstoaga [31] in a Hilbert space, Takahashi and Zembayashi [15] established the following lemma.
Lemma 2.8 (Takahashi and Zembayashi [15]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space , and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:
for all . Then, the following hold:
(i) is singlevalued;
(ii) is a firmly nonexpansivetype mapping, that is, for all ,
(iii);
(iv) is closed and convex.
Using Lemma 2.8, one has the following result.
Lemma 2.9 (Takahashi and Zembayashi [15]).
Let be a nonempty closed convex subset of a smooth strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let . Then, for and ,
Utilizing Lemmas 2.7, 2.8 and 2.9 as previously mentioned, Zhang [17] derived the following result.
Proposition 2.10 (Zhang [21, Lemma ]).
Let be a smooth strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Let be an inversestrongly monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let . Then the following hold:
for , there exists such that
if is additionally uniformly smooth and is defined as
then the mapping has the following properties:
(i) is singlevalued,
(ii) is a firmly nonexpansivetype mapping, that is,
(iii),
(iv) is a closed convex subset of ,
(v) for all
Let be two maximal monotone operators in a smooth Banach space . We denote the resolvent operators of and by and for each , respectively. Then and are two singlevalued mappings. Also, and for each , where and are the sets of fixed points of and , respectively. For each , the Yosida approximations of and are defined by and , respectively.It is known that
Lemma 2.11 (Kohsaka and Takahashi [13]).
Let be a reflexive strictly convex and smooth Banach space and let be a maximal monotone operator with . Then
Lemma 2.12 (Tan and Xu [32]).
Let and be two sequences of nonnegative real numbers satisfying the inequality: for all . If , then exists.
3. Strong Convergence Theorem
In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .
Lemma 3.1.
Let be a reflexive strictly convex and smooth Banach space and let be a maximal monotone operator. Then for each , the following holds:
where and is the duality mapping on . In particular, whenever a real Hilbert space, is a nonexpansive mapping on .
Proof.
Since for each
we have that
Thus, from the monotonicity of it follows that
and hence
Theorem 3.2.
Suppose that Assumption A is fulfilled and let be chosen arbitrarily. Consider the sequence
where
is defined by (2.15), satisfy
and satisfies . Then, the sequence converges strongly to provided for any sequence with , where is the generalized projection of onto .
Remark 3.3.
In Theorem 3.2, if a real Hilbert space, then is a sequence of nonexpansive mappings on . This implies that as ,
In this case, we can remove the requirement that for any sequence with .
Proof of Theorem 3.2.
For the sake of simplicity, we define
so that
We divide the proof into several steps.
Step 1.
We claim that is closed and convex for each .
Indeed, it is obvious that is closed and is closed and convex for each . Let us show that is convex. For and , put . It is sufficient to show that . We first write for each . Next, we prove that
is equivalent to
Indeed, from (2.4) we deduce that the following equations hold:
which combined with (3.12) yield that (3.12) is equivalent to (3.13). Thus we have
This implies that . Therefore, is closed and convex.
Step 2.
We claim that for each and that is well defined.
Indeed, take arbitrarily. Note that is equivalent to
Then from Lemma 2.11 we obtain
Moreover, we have
where . So for all . Now, let us show that
We prove this by induction. For , we have . Assume that . Since is the projection of onto , by Lemma 2.2 we have
As by the induction assumption, the last inequality holds, in particular, for all . This, together with the definition of implies that . Hence (3.20) holds for all . So, for all . This implies that the sequence is well defined.
Step 3.
We claim that is bounded and that as .
Indeed,it follows from the definition of that . Since and , so for all ; that is, is nondecreasing. It follows from and Lemma 2.3 that
for each for each . Therefore, is bounded which implies that the limit of exists. Since
so is bounded. From Lemma 2.3, we have
for each . This implies that
Step 4.
We claim that , , and .
Indeed, from , we have
for all . Therefore, from and , it follows that . Since and is uniformly convex and smooth, we have from Lemma 2.1 that
and therefore, . Since is uniformly normtonorm continuous on bounded subsets of and , then .
Let us set . Then, according to Lemma 2.4 and Proposition 2.10, we know that is a nonempty closed convex subset of such that . Fix arbitrarily. As in the proof of Step 2 we can show that , and . Hence it follows from the boundedness of that , and are also bounded. Let . Since is a uniformly smooth Banach space, we know that is a uniformly convex Banach space. Therefore, by Lemma 2.5 there exists a continuous, strictly increasing, and convex function with such that
for and . So, we have that
and hence
for all . Consequently we have
Since and is uniformly normtonorm continuous on bounded subsets of , we obtain . From , and we have
Therefore, from the properties of we get
recalling that is uniformly normtonorm continuous on bounded sunsets of . Next let us show that
Observe first that
Since , and is bounded, so it follows that . Also, observe that
Since and the sequences are bounded, so it follows that . Meantime, observe that
and hence
Since and , it follows from the boundedness of that . Thus, in terms of Lemma 2.1, we have that and so . Furthermore, since , from the uniform normtonorm continuity of on bounded subsets of , we obtain
Observe that
Thus, from (3.35) it follows that . Since is uniformly normtonorm continuous on bounded subsets of , it follows that .
Step 5.
We claim that , where
Indeed, since is bounded and is reflexive, we know that . Take arbitrarily. Then there exists a subsequence of such that . Hence it follows from , and that , and converge weakly to the same point . On the other hand, from (3.35), (3.36), and we obtain that
If and , then it follows from (2.17) and the monotonicity of the operators that for all
Letting , we have that and . Then the maximality of the operators implies that and . Next, let us show that . Since we have by (3.32)
from and Proposition 2.10 it follows that
Also, since
so we get
Thus from (3.47), , and , we have . Since is uniformly convex and smooth, we conclude from Lemma 2.1 that
From , and (3.51), we have and . Since is uniformly normtonorm continuous on bounded subsets of , from (3.51) we derive
From , it follows that
By the definition of , we have
where
Replacing by , we have from (A2) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.53) and (A4) we have
For with and , let . Since and , and hence . So, from (A1) we have
Dividing by , we have
Letting , from (A3) it follows that
Thus . Therefore, we obtain that by the arbitrariness of .
Step 6.
We claim that converges strongly to .
Indeed, from and , It follows that
Since the norm is weakly lower semicontinuous, we have
From the definition of , we have . Hence and
which implies that . Since has the KadecKlee property, then . Therefore, converges strongly to .
Remark 3.4.
In Theorem 3.2, put , and . Then, for all and , we have that
Moreover, the following hold:
and hence
In this case, the previous Theorem 3.2 reduces to [20, Theorem ].
4. Weak Convergence Theorem
In this section, we present the following algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .
Let be chosen arbitrarily and consider the sequence generated by
where , and , is defined by (2.15).
Before proving a weak convergence theorem, we need the following proposition.
Proposition 4.1.
Suppose that Assumption A is fulfilled and let be a sequence defined by (4.1), where satisfy the following conditions:
Then, converges strongly to , where is the generalized projection of onto .
Proof.
We set and
so that
Then, in terms of Lemma 2.4 and Proposition 2.10, is a nonempty closed convex subset of such that . We first prove that is bounded. Fix . Note that by the first and third of (4.3), , and
Here, each is relatively nonexpansive. Then from Proposition 2.10 we obtain
and hence by Proposition 2.10, we have
Consequently, the last two inequalities yield that
for all . So, from , and Lemma 2.12, we deduce that exists. This implies that is bounded. Thus, is bounded and so are , and .
Define for all . Let us show that is bounded. Indeed, observe that
for each . This, together with the boundedness of , implies that is bounded and so is . Furthermore, from and (4.10) we have
Since is the generalized projection, then, from Lemma 2.3 we obtain
Hence, from (4.12), it follows that .
Note that , and is bounded, so that . Therefore, is a convergent sequence. On the other hand, from (4.10) we derive, for all ,
In particular, we have
Consequently, from and Lemma 2.3, we have
and hence
Let . From Lemma 2.6, there exists a continuous, strictly increasing, and convex function with such that
So, we have
Since is a convergent sequence, is bounded and is convergent, from the property of we have that is a Cauchy sequence. Since is closed, converges strongly to . This completes the proof.
Now, we are in a position to prove the following theorem.
Theorem 4.2.
Suppose that Assumption A is fulfilled and let be a sequence defined by (4.1), where satisfy the following conditions:
and satisfies . If is weakly sequentially continuous, then converges weakly to , where .
Proof.
We consider the notations (4.3). As in the proof of Proposition 4.1, we have that , and are bounded sequences. Let
From Lemma 2.5 and as in the proof of Theorem 3.2, there exists a continuous, strictly increasing, and convex function with such that
for and . Observe that for ,
Hence,
Consequently, the last two inequalities yield that
Thus, we have
By the proof of Proposition 4.1, it is known that is convergent; since , and , then we have
Taking into account the properties of , as in the proof of Theorem 3.2, we have
since is uniformly normtonorm continuous on bounded subsets of . Note that . Hence, from the uniform normtonorm continuity of on bounded subsets of we obtain . Also, observe that
From it follows that . Since is uniformly normtonorm continuous on bounded subsets of , we have
Now let us show that
Indeed, from (4.10) we get
which, together with , yields that
From (4.9) it follows that
Note that
Since and , we obtain , which, together with , yields that
We have from (4.8) that
which, together with , yields that
Also from (4.7) it follows that
which, together with , yields that
Similarly from (4.6) it follows that
which, together with , yields that
On the other hand, let us show that
Indeed, let . From Lemma 2.6, there exists a continuous, strictly increasing, and convex function with such that
Since and , we deduce from Proposition 2.10 that for ,
This implies that
Since is uniformly normtonorm continuous on bounded subsets of , from the properties of we obtain
Note that
Since , it follows from (4.31) and (4.35) that and . Also, observe that
and hence
Thus, from , and , it follows that . In terms of Lemma 2.1, we derive .
Next, let us show that , where .
Indeed, since is bounded, there exists a subsequence of such that . Hence it follows from (4.31), (4.35), and that both and converge weakly to the same point . Furthermore, from and (4.31) we have that
If and , then it follows from (2.17) and the monotonicity of the operators that for all
Letting , we obtain that
Then the maximality of the operators implies that .
Now, by the definition of , we have
where . Replacing by , we have from (A2) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (4.35) and (A4) we have
For , with , and , let . Since and , then and hence . So, from (A1) we have
Dividing by , we get . Letting , from (A3) it follows that . So, . Therefore, . Let . From Lemma 2.2 and , we get
From Proposition 4.1, we also know that . Note that . Since is weakly sequentially continuous, then as . In addition, taking into account the monotonicity of , we conclude that . Hence
From the strict convexity of , it follows that . Therefore, , where . This completes the proof.
Remark 4.3.
In Theorem 4.2, put , , and . Then, for all and , we have that
Moreover, the following hold:
In this case, Algorithm (4.1) reduces to the following one:
Corollary 4.4.
Suppose that conditions (A1)–(A5) are fulfilled and let be a sequence defined by (4.65), where is defined in Lemma 2.8, satisfy the conditions and , and satisfies . If is weakly sequentially continuous, then converges weakly to , where .
References
 1.
Zeng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006,10(5):1293–1303.
 2.
Schaible S, Yao JC, Zeng LC: A proximal method for pseudomonotone type variationallike inequalities. Taiwanese Journal of Mathematics 2006,10(2):497–513.
 3.
Zeng LC, Lin LJ, Yao JC: Auxiliary problem method for mixed variationallike inequalities. Taiwanese Journal of Mathematics 2006,10(2):515–529.
 4.
Peng JW, Yao JC: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudocontraction mapping. to appear in Journal of Global Optimization to appear in Journal of Global Optimization
 5.
Zeng LC, Wu SY, Yao JC: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese Journal of Mathematics 2006,10(6):1497–1514.
 6.
Peng JW, Yao JC: Some new extragradientlike methods for generalized equilibrium problems, fixed points problems and variational inequality problems. to appear in Optimization Methods & Software to appear in Optimization Methods & Software
 7.
Ceng LC, Lee C, Yao JC: Strong weak convergence theorems of implicit hybrid steepestdescent methods for variational inequalities. Taiwanese Journal of Mathematics 2008,12(1):227–244.
 8.
Peng JW, Yao JC: A new hybridextragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
 9.
Peng JW, Yao JC: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Mathematical and Computer Modelling 2009,49(9–10):1816–1828. 10.1016/j.mcm.2008.11.014
 10.
Ceng LC, Ansari QH, Yao JC: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. Journal of Global Optimization 2009,43(4):487–502. 10.1007/s1089800893426
 11.
Peng JW, Yao JC: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudocontractions and monotone mappings. to appear in Taiwanese Journal of Mathematics to appear in Taiwanese Journal of Mathematics
 12.
Kamimura S, Takahashi W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM Journal on Optimization 2003,13(3):938–945.
 13.
Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstract and Applied Analysis 2004,2004(3):239–249. 10.1155/S1085337504309036
 14.
Martinet B: Régularisation d'inéquations variationnelles par approximations successives. Revue Française d'Informatique et de Recherche Opérationnelle 1970, 4: 154–158.
 15.
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):45–57. 10.1016/j.na.2007.11.031
 16.
Ceng LC, Mastroeni G, Yao JC: Hybrid proximalpoint methods for common solutions of equilibrium problems and zeros of maximal monotone operators. Journal of Optimization Theory and Applications 2009,142(3):431–449. 10.1007/s109570099538z
 17.
Zhang SS: Shrinking projection method for solving generalized equilibrium problem, variational inequality and common fixed point in Banach spaces with applications. to appear in Science in China Series A to appear in Science in China Series A
 18.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056
 19.
Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022
 20.
Zeng LC, Yao JC: An inexact proximaltype algorithm in Banach spaces. Journal of Optimization Theory and Applications 2007,135(1):145–161. 10.1007/s1095700792616
 21.
Kamimura S, Kohsaka F, Takahashi W: Weak and strong convergence theorems for maximal monotone operators in a Banach space. SetValued Analysis 2004,12(4):417–429. 10.1007/s1122800481964
 22.
Li L, Song W: Modified proximalpoint algorithm for maximal monotone operators in Banach spaces. Journal of Optimization Theory and Applications 2008,138(1):45–64. 10.1007/s109570089370x
 23.
Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization 1991,29(2):403–419. 10.1137/0329022
 24.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
 25.
Ceng LC, Yao JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Applied Mathematics and Computation 2008,198(2):729–741. 10.1016/j.amc.2007.09.011
 26.
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:15–50.
 27.
Alber YI, GuerreDelabriere S: On the projection methods for fixed point problems. Analysis 2001,21(1):17–39.
 28.
Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:313–318.
 29.
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic, Dordrecht, The Netherlands; 1990:xiv+260.
 30.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1127–1138. 10.1016/0362546X(91)90200K
 31.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.
 32.
Tan KK, 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
Acknowledgments
The first author (http://zenglc@hotmail.com) was partially supported by the National Science Foundation of China (10771141), Ph. D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author (http://yaojc@math.nsysu.edu.tw) was partially supported by the Grant NSF 972115M110001.
Author information
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 Hilbert Space
 Banach Space
 Variational Inequality
 Nonexpansive Mapping
 Lower Semicontinuous