Abstract
The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others.
MSC:90C25, 90C30, 47J25, 47H09.
Similar content being viewed by others
1 Introduction
Let H be a real Hilbert space with inner product and induced norm , let be a bifunction. Then we consider the following equilibrium problem (EP): find such that
The set of the EP is denoted by Ω, i.e.,
The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, the Nash equilibrium problems and others, see, for instance, [1–3]. Some methods have been proposed to solve the EP, see, e.g., [4–6] and [7, 8].
The split feasibility problem (SFP) was proposed by Censer and Elfving in [9]. It can be formulated as the problem of finding a point x satisfying the property:
where A is a given real matrix, and C and Q are nonempty, closed and convex subsets in and , respectively.
Due to its extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems have been received great attention (see, for instance [10–13] and also [14–18]).
We assume the SFP (1.2) is consistent, and let Γ be the solution set, i.e.,
It is not hard to see that Γ is closed convex and if and only if it solves the fixed-point equation
where and are the orthogonal projection onto C and Q, respectively, is any positive constant and denotes the adjoint of A.
Recently, for the purpose of generality, the SFP (1.2) has been studied in a more general setting. For instance, see [16, 19]. However, the algorithms in these references have only weak convergence in the setting of infinite-dimensional Hilbert spaces. Very recently, He and Zhao [20] introduce a new relaxed CQ algorithm (1.4) such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces:
Motivated and inspired by the research going on in the sections of equilibrium problems and split feasibility problems, the purpose of this article is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of He et al. [20] and some others.
2 Preliminaries and lemmas
Throughout this paper, we assume that H, or is a real Hilbert space, A is a bounded linear operator from to , and I is the identity operator on H, or . If is a differentiable function, then we denote by ∇f the gradient of the function f. We will also use the notations: → to denote strong convergence, ⇀ to denote weak convergence and to denote by
the weak ω-limit set of .
Recall that a mapping is said to be nonexpansive if
is said to be firmly nonexpansive if
A mapping is said to be demi-closed at origin, if for any sequence with and , then .
It is easy to prove that if is a firmly nonexpansive mapping, then T is demi-closed at the origin.
A function is called convex if
Lemma 2.1 Let be a firmly nonexpansive mapping such that is a convex function from to . Let be a bounded linear operator and
Then
-
(i)
, .
-
(ii)
∇f is -Lipschitz, i.e., , .
Proof (i) From the definition of f, we know that f is convex. First we prove that the limit
exists in and satisfies
If fact, if , then
Since f is convex and , it follows that
and hence that
This shows that this difference quotient is increasing, therefore it has a limit in as :
This implies that f is differential. Taking , (2.1) implies that
Next we prove that
In fact, since
and
Substituting (2.3) into (2.2), simplifying and then letting and taking the limit we have
It follows from (2.1) that
(ii) From (i) we have
□
Lemma 2.2 (See, for example, [21])
Let be an operator. The following statements are equivalent.
-
(i)
T is firmly nonexpansive.
-
(ii)
, .
-
(iii)
is firmly nonexpansive.
Proof (i) ⇒ (ii): Since T is firmly nonexpansive, for all we have
hence
(ii) ⇒ (iii): From (ii), we know that for all
This implies that is firmly nonexpansive.
(iii) ⇒ (i): From (iii) we immediately know that T is firmly nonexpansive.
Let C be a nonempty closed convex subset of H. Recall that for every point , there exists a unique nearest point of C, denoted by , such that for all . Such a is called the metric projection from H onto C. We know that is a firmly nonexpansive mapping from H onto C, i.e.,
Further, for any and , if and only if
Throughout this paper, let us assume that a bifunction satisfies the following conditions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3) , ;
(A4) for each , is convex and lower semicontinuous.
□
Let H be a Hilbert space and let satisfy (A1), (A2), (A3), and (A4). Then, for any and , there exists such that
Furthermore, if
then the following hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive;
-
(3)
;
-
(4)
Ω is closed and convex.
The following results play an important role in this paper.
Lemma 2.4 ([22])
Let X be a real Hilbert space, then we have
Lemma 2.5 ([23])
Let and be bounded sequences in a Banach space X. Let be a sequence in satisfying . Suppose that
for all integer and
Then .
Lemma 2.6 ([24])
Let be a sequence of nonnegative real numbers such that
where is a sequence in , and is a sequence in ℝ such that
-
(i)
;
-
(ii)
, or .
Then .
3 Main results
We are now in a position to prove the following theorem.
Theorem 3.1 Let , be two real Hilbert spaces, be a bifunction satisfying (A1), (A2), (A3), and (A4). Let be a bounded linear operator, be a firmly nonexpansive mapping, and let be a firmly nonexpansive mapping such that is a convex function from to ℝ. Assume that and . Let and be the sequence generated by
where
If the solution set Γ of SPF (1.2) is not empty, and the sequences , satisfy the following conditions:
-
(i)
, ;
-
(ii)
;
-
(iii)
and ,
then the sequence converges strongly to .
Proof Since the solution set Ω of EP and the solution set of SPF (1.2) are both closed and convex, Γ (≠∅) is closed and convex. Thus, the metric projection is well defined.
Letting , it follows from Lemma 2.3 that and
Observing that S is firmly nonexpansive, we have
Since , . Observe that is firmly nonexpansive, from Lemma 2.2(ii) we have
This implies that
Substituting (3.5) into (3.3), we get
Thus, from (3.2) and (3.6) we have
It turns out that
By induction, we have
This implies that the sequence is bounded. From (3.2) and (3.6) we know that and both are bounded.
From Lemma 2.4 and (3.5), we have
Therefore, from Lemma 2.6 and (3.2), (3.7) we have
On the other hand, without loss of generality, we may assume that there is a constant such that
Setting , we get the following inequality:
Now, we prove by employing the technique studied by Maingé [25]. For the purpose we consider two cases.
Case 1: is eventually decreasing, i.e., there exists a sufficient large positive integer such that holds for all . In this case, must be convergent, and from (3.9) it follows that
where M is a constant such that for all . Using the condition (i) and (3.10), we have
Moreover, it follows from Lemma 2.1(ii) that for all
This implies that is bounded. From (3.11) it yields , namely
Furthermore, we have
For any , and if is a subsequence of such that , then
On the other hand, from (3.12), we have
Since T is demi-closed at origin, from (3.14) and (3.15) we have , i.e., .
In order to prove , we need to prove and . In fact, from (3.1) we have
Taking , we get
Similarly, we also have
Adding up the above two inequalities, we get
From (A2), we have
Multiplying the above inequality by and simplifying, we have
Hence we have
and hence
By (3.1) we have
where
This implies that
It follows that
In view of condition (iii) and (3.17) we get
By Lemma 2.5, we obtain
Consequently
Since S is firmly nonexpansive, it follows from (3.1) that
where
Therefore we have
This together with (3.8) shows that
Then we obtain
Therefore, we get
By virtue of (3.18), we have
Now, we turn to a proof that . For this purpose, we denote
In view of condition (i) and (3.13) we have
Since S is firmly nonexpansive (and so it is also nonexpansive), it follows from Lemma 2.4 that
Thus, we have
It follows from (3.19), (3.20), and (3.22) that . In view of and that S is demi-closed at origin, we get .
On the other hand, from and (3.18), we obtain . From (3.1), for any , we have
From (A2), we have
Replacing n by , we have
Since and , from (A4) we have
Put for all and . Then we get . So, from (3.24) we have
From (A4), we have
and hence . Letting , we have
This implies . Consequently, , and hence . Furthermore, in view of (3.20) we have
On the other hand, from (3.9), we have
Applying Lemma 2.6 to (3.25), from the condition (i) we obtain , that is, .
Case 2: is not eventually decreasing, that is, we can find a positive integer such that . Now we define
It easy to see that is nonempty and satisfies . Let
It is clear that as (otherwise, is eventually decreasing). It is also clear that for all . Moreover, we prove that
In fact, if , then the inequality (3.26) is trivial; if , from the definition of , there exists some such that , we deduce that
and the inequality (3.26) holds again. Since for all , it follows from (3.10) that
Noting that is bounded, we get . By the same argument to the proof in case 1, we have . From (3.19) we have
Furthermore, in view of (3.20), we can deduce that
Since , it follows from (3.9) that
Combining (3.28) and (3.29) we have
and hence , which together with (3.27) implies that
Noting the inequality (3.26), this shows that , that is, . This completes the proof of Theorem 3.1. □
References
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 2003, 117(2):245–266. 10.1023/A:1023627606067
Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 2004, 121: 571–596.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042
Reich S, Sabach S: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces, optimization theory and related topics. Contemp. Math. 2012, 568: 225–240.
Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002
Censor Y, Elfving T: A multiprojection algorithm using Bregman projection in product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692
Aleyner A, Reich S: Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach. J. Math. Anal. Appl. 2008, 343(1):427–435. 10.1016/j.jmaa.2008.01.087
Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38(3):367–426. 10.1137/S0036144593251710
Moudafi A: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. 2013, 79: 117–121.
Masad E, Reich S: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 2007, 8: 367–371.
Yao Y, Chen R, Marino G, Liou YC: Applications of fixed point and optimization methods to the multiple-sets split feasibility problem. J. Appl. Math. 2012., 2012: Article ID 927530
Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007
Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26(10): Article ID 105018
Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014
Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017
López G, Martín-Márquez V, Wang FH, Xu HK: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 2012., 28: Article ID 085004 10.1088/0266-5611/28/8/085004
He S, Zhao Z: Strong convergence of a relaxed CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2013., 2013: Article ID 197 10.1186/1029-242X-2013-197
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York; 1984.
Chang SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Maingé PE: New approach to solving a system of variational inequalities and hierarchical problems. J. Optim. Theory Appl. 2008, 138: 459–477. 10.1007/s10957-008-9433-z
Acknowledgements
This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199) and the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and by the National Natural Science Foundation of China (Grant No. 11361070).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly to this research work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tang, J., Chang, Ss. & Yuan, F. A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces. Fixed Point Theory Appl 2014, 36 (2014). https://doi.org/10.1186/1687-1812-2014-36
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-36