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A new iterative scheme with nonexpansive mappings for equilibrium problems
Journal of Inequalities and Applications volume 2012, Article number: 116 (2012)
Abstract
In this paper, we suggest a new iteration scheme for finding a common of the solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of a nonexpansive mapping. The scheme is based on both hybrid method and extragradient-type method. We obtain a strong convergence theorem for the sequences generated by these processes in a real Hilbert space. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.
AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C25, 90 C33.
1 Introduction
Let be a real Hilbert space with inner product 〈·,·〉 and norm || · ||. Let C be a nonempty closed convex subset of a real Hilbert space . A mapping S : C → C is a contraction with a constant δ ∈ (0, 1), if
If δ = 1, then S is called nonexpansive on C. Fix(S) is denoted by the set of fixed points of S. Let be a bifunction such that f(x, x) = 0 for all x ∈ C. We consider the equilibrium problem in the sense of Blum and Oettli (see [1]) which is presented as follows:
The set of solutions of EP(f, C) is denoted by Sol(f, C). The bifunction f is called strongly monotone on C with ß > 0, if
monotone on C, if
pseudomonotone on C, if
Lipschitz-type continuous on C with constants c1 > 0 and c2 > 0 (see [2]), if
It is well-known that Problem EP(f, C) includes, as particular cases, the optimization problem, the variational inequality problem, the Nash equilibrium problem in noncooperative games, the fixed point problem, the nonlinear complementarity problem and the vector minimization problem (see [2–6]).
In recent years, the problem to find a common point of the solution set of problem (EP) and the set of fixed points of a nonexpansive mapping becomes an attractive field for many researchers (see [7–15]). An important special case of equilibrium problems is the variational inequalities (shortly (VIP)), where F : C → and f(x, y) = 〈F(x), y - x〉. Various methods have been developed for finding a common point of the solution set of problem (VIP) and the set of fixed points of a nonexpansive mapping when F is monotone (see [16–18]).
Motivated by fixed point techniques of Takahashi and Takahashi in [19] and an improvement set of extragradient-type iteration methods in [20], we introduce a new iteration algorithm for finding a common of the solution set of equilibrium problems with a monotone and Lipschitz-type continuous bifunction and the set of fixed points of a nonexpansive mapping. We show that all of the iterative sequences generated by this algorithm convergence strongly to the common element in a real Hilbert space.
2 Preliminaries
Let C be a nonempty closed convex subset of a Hilbert space . We write xn ⇀ x to indicate that the sequence {xn} converges weakly to x as n → ∞, xn → x implies that {xn} converges strongly to x. For any x ∈ , there exists a nearest point in C, denoted by Pr C (x), such that
Pr C is called the metric projection of to C. It is well known that Pr C satisfies the following properties:
Let us assume that a bifunction and a nonexpansive mapping S : C → C satisfy the following conditions:
A1. f is Lipschitz-type continuous on C;
A2. f is monotone on C;
A3. for each x ∈ C, f (x, ·) is subdifferentiable and convex on C;
A4. Fix(S) ∩ Sol(f, C) ≠∅.
Recently, Takahashi and Takahashi in [19] first introduced an iterative scheme by the viscosity approximation method. The sequence {xk} is defined by:
where C is a nonempty closed convex subset of and g is a contractive mapping of into itself. The authors showed that under certain conditions over {α k } and {r k }, sequences {x k } and {u k } converge strongly to z = PrSol(f,C)∩Fix(S)(g(x0)). Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping have further developed by many authors. These methods require to solve approximation auxilary equilibrium problems.
In this paper, we introduce a new iteration method for finding a common point of the set of fixed points of a nonexpansive mapping S and the set of solutions of problem EP(f, C). At each our iteration, the main steps are to solve two strongly convex problems
and compute the next iteration point by Mann-type fixed points
where g : C → C is a δ-contraction with .
To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.
Lemma 2.1 (see [21]) Let {a n } be a sequence of nonnegative real numbers such that:
where {α n }, and {ß n } satisfy the conditions:
(i) α n ⊂ (0, 1) and ;
(ii)
Then
Lemma 2.2 ([22]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space . If Fix(S) ≠Ø, then I - S is demiclosed; that is, whenever {xk} is a sequence in C weakly converging to some and the sequence{(I - S)(xk)} strongly converges to some , it follows that . Here I is the identity operator of .
Lemma 2.3 (see [20], Lemma 3.1) Let C be a nonempty closed convex subset of a real Hilbert space . Let be a pseudomonotone, Lipschitz-type continuous bifunction with constants c1 > 0 and c2 > 0. For each × ∈ C, let f(x, ·) be convex and subdifferentiable on C. Suppose that the sequences {xk}, {yk}, {tk} generated by Scheme (2.4) and x* ∈ Sol(f, C). Then
3 Main results
Now, we prove the main convergence theorem.
Theorem 3.1 Suppose that Assumptions A1-A4 are satisfied, x0 ∈ C and two positive sequences {λ k }, {a k } satisfy the following restrictions:
Then the sequences {xk}, {yk} and {tk} generated by (2.4) and (2.5) converge strongly to the same point x*, where
The proof of this theorem is divided into several steps.
Step 1. Claim that
Proof of Step 1. For each x* ∈ Fix(S) ∩ Sol(f, C), it follows from xk+1= a k g(xk) + (1 - a k )S(tk), Lemma 2.3 and that
Then, we have
and
By the similar way, also
Combining this, (3.1) and the inequality ||xk - tk|| = ||xk - yk || + || yk - tk ||, we have
Step 2. Claim that
Proof of Step 2. It is easy to see that if and only if
where N C (x) is the (outward) normal cone of C at x ∈ C. This means that , where w ∈ ∂2f(yk, tk) and . By the definition of the normal cone N C we have, from this relation that
Substituting t = tk+1into this inequality, we get
Since f(x, ·) is convex on C for all x ∈ C, we have
Using this and (3.3), we have
By the similar way, we also have
Using (3.4), (3.5) and f is Lipschitz-type continuous and monotone, we get
Hence
Since (3.6), ak+1- a k → 0 as k →∞, g is contractive on C, Lemma 2.3, Step 2 and the definition of xk+1that xk+1= a k g(xk) + a k S(tk), we have
where δ is contractive constant of the mapping g, M = sup{||g(xk - 1) - S(tk - 1)||: k = 0, 1, ...} and , since and , in view of Lemma 2.1, we have .
Step 3. Claim that
Proof of Step 3. From xk+1= a k g(xk) + (1 - a k )S(tk), we have
and hence
Using this, , Step 1 and Step 2, we have
Step 4. Claim that
Proof of Step 4. By Step 1, {tk} is bounded, there exists a subsequence of {tk} such that
Since the sequence is bounded, there exists a subsequence of which converges weakly to . Without loss of generality we suppose that the sequence converges weakly to such that
Since Lemma 2.2 and Step 3, we have
Now we show that . By Step 1, we also have
Since yk is the unique solution of the strongly convex problem
we have
This follows that
where w ∈ ∂2f (xk, yk) and wk ∈ N C (yk). By the definition of the normal cone N C , we have
On the other hand, since f(xk, ·) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists w ∈ ∂2f(xk, yk) such that
Combining this with (3.9), we have
Hence
Then, using and the continuity of f , we have
Combining this and (3.8), we obtain
By (3.7) and the definition of x*, we have
Using this and Step 3, we get
Step 5. Claim that the sequences {xk}, {yk} and {tk} converge strongly to x*.
Proof of Step 5. Using xk+1= α k g(xk) + (1 - a k )S(tk) and Lemma 2.3, we have
where A k and B k are defined by
Since , Step 4, we have and hence
By Lemma 2.1, we obtain that the sequence {xk} converges strongly to x*. It follows from Step 1 that the sequences {yk} and {tk} also converge strongly to the same solution x*= PrFix(S)∩Sol(f,C)g(x*).
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4 Applications
Let C be a nonempty closed convex subset of a real Hilbert space and F be a function from C into . In this section, we consider the variational inequality problem which is presented as follows:
Let be defined by f(x, y) = 〈F(x), y - x〉. Then Problem EP(f, C) can be written in VI(F, C). The set of solutions of VI(F, C) is denoted by Sol(F, C). Recall that the function F is called strongly monotone on C with ß > 0 if
monotone on C if
pseudomonotone on C if
Lipschitz continuous on C with constants L > 0 if
Since
(2.4), (2.5) and Theorem 3.1, we obtain that the following convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping S and the solution set of problem VI(F, C).
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space , F be a function from C to such that F is monotone and L-Lipschitz continuous on C, g : C → C is contractive with constant S: C→ C be nonexpansive and positive sequences {a k } and {λ k } satisfy the following restrictions
Then sequences {xk}, {yk} and {tk} generated by
converge strongly to the same point x* ∈ PrFix(S)∩Sol(F,C)g(x*).
Thus, this scheme and its convergence become results proposed by Nadezhkina and Takahashi in [23]. As direct consequences of Theorem 3.1, we obtain the following corollary.
Corollary 4.2 Suppose that Assumptions A1-A3 are satisfied, Sol(f, C) ≠Ø, x0 ∈ C and two positive sequences {λ k }, {a k } satisfy the following restrictions:
Then, the sequences {xk}, {yk} and {tk} generated by
where g : C → C is a δ-contraction with , converge strongly to the same point x*=PrSol(f,C)g(x*).
References
Blum E, Oettli W: From optimization and variational inequality to equilibrium problems. The Math Stud 1994, 63: 127–149.
Mastroeni G: On auxiliary principle for equilibrium problems. In Nonconvex Optimization and its Applications. Edited by: Daniele P, Giannessi F, Maugeri A. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003.
Anh PN: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math Vietnamica 2009, 34: 183–200.
Anh PN: An LQP regularization method for equilibrium problems on polyhedral. Vietnam J Math 2008, 36: 209–228.
Anh PN, Kim JK: Outer approximation algorithms for pseudomonotone equilibrium problems. Comp Math Appl 2011, 61: 2588–2595. 10.1016/j.camwa.2011.02.052
Quoc TD, Anh PN, Muu LD: Dual extragradient algorithms to equilibrium Problems. J Glob Optim 2012, 52: 139–159. 10.1007/s10898-011-9693-2
Anh PN: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J Optim Theory Appl 2012. DOI 10.1007/s10957–012–0005-x
Anh PN, Kim JK, Nam JM: Strong convergence of an extragradient method for equilibrium problems and fixed point problems. J Korean Math Soc 2012, 49: 187–200. 10.4134/JKMS.2012.49.1.187
Anh PN, Son DX: A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions. J Appl Math Inform 2011, 29: 1179–1191.
Ceng LC, Schaible S, Yao JC: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J Optim Theory Appl 2008, 139: 403–418. 10.1007/s10957-008-9361-y
Chen R, Shen X, Cui S: Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontractions mappings in Hilbert space. J Ineq Appl 2010. DOI:10.1155/2010/474813
Wang S, Cho YJ, Qin X: A new iterative method for solving equilibrium problems and fixed point problems for an infinite family of nonexpansive mappings. Fix Point Theory Appl 2010. DOI: 10.1155/2010/165098
Wangkeeree R: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fix Point Theory Appl 2008. DOI:10.1155/2008/134148
Yao Y, Liou YC, Jao JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fix Point Theory Appl 2007. DOI:10.1155/2007/64363
Yao Y, Liou YC, Wu YJ: An extragradient method for mixed equilibrium problems and fixed point problems. Fix Point Theory Appl 2009. DOI: 10.1155/2009/632819
Ceng LC, Petrusel A, Lee C, Wong MM: Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions. Taiwanese J Math 2009, 13: 607–632.
Takahashi S, Toyoda M: Weakly convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2003, 118: 417–428. 10.1023/A:1025407607560
Zeng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math 2010, 10: 1293–1303.
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036
Anh PN: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 2012. DOI:10.1080/02331934.2011.607497
Xu HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059
Goebel K, Kirk WA: Topics on metric fixed point theory. Cambridge University Press, Cambridge, England; 1990.
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2006, 128: 191–201. 10.1007/s10957-005-7564-z
Acknowledgements
We are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.
The work was supported by National Foundation for Science and Technology Development of Vietnam (NAFOSTED).
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The main idea of this paper is proposed by P.N. Anh. The revision is made by DDT. PNA and DDT prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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PN, A., DD, T. A new iterative scheme with nonexpansive mappings for equilibrium problems. J Inequal Appl 2012, 116 (2012). https://doi.org/10.1186/1029-242X-2012-116
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DOI: https://doi.org/10.1186/1029-242X-2012-116