An Iterative Algorithm for Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities
Fixed Point Theory and Applications volume 2010, Article number: 564361 (2010)
We present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithms strongly converge to which solves some variational inequality.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonlinear mapping, let be a function, and let be a bifunction of into . Now we consider the following mixed equilibrium problem:
The set of solution of problem (1.1) is denoted by EP.
If , then the mixed equilibrium problem (1.1) becomes the following mixed equilibrium problem:
which was considered by Ceng and Yao . If , then the mixed equilibrium problem (1.1) becomes the following equilibrium problem:
which was studied by S. Takahashi and W. Takahashi . If and , then the mixed equilibrium problem (1.1) becomes the following equilibrium problem:
If for all , the mixed equilibrium problem (1.1) becomes the following variational inequality problem:
The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, [3–8]. Some methods have been proposed to solve the mixed equilibrium problem and the equilibrium problem. In 1997, Flaim and Antipen  introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi  introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.2) and the set of fixed point points of a nonexpansive mapping. Furthermore, Yao et al.  introduced some new iterative schemes for finding a common element of the set of solutions of the equilibrium problem (1.2) and the set of common fixed points of finitely (infinitely) nonexpansive mappings. Very recently, Ceng and Yao  considered a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Peng and Yao  developed a CQ method. They obtained some strong convergence results for finding a common element of the set of solutions of the mixed equilibrium problem (1.1) and the set of the variational inequality and the set of fixed points of a nonexpansive mapping. Their results extend and improve the corresponding results in [1, 9, 12].
Recall that a mapping is said to be -inverse strongly monotone if there exists a constant such that for all . A mapping is strongly positive on if there exists a constant such that .
Let be a single-valued nonlinear mapping and let be a set-valued mapping. Now we concern the following variational inclusion, which is to find a point such that
where is the zero vector in . The set of solutions of problem (1.6) is denoted by . If , then problem (1.6) becomes the generalized equation introduced by Robinson . If , then problem (1.6) becomes the inclusion problem introduced by Rockafellar . It is known that (1.6) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized. Recently, Zhang et al.  introduced a new iterative scheme for finding a common element of the set of solutions to the problem (1.6) and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al.  introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. For some related works, please see [1, 2, 9–11, 13–34] and the references therein.
Inspired and motivated by the works in the literature, in this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithms strongly converge to which solves some variational inequality.
Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that
Such a is called the metric projection of onto . We know that is nonexpansive. Further, for and ,
A set-valued mapping is called monotone if, for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies .
Let the set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows:
where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone and that a solution of problem (1.6) is a fixed point of the operator for all , see, for instance, .
Throughout this paper, we assume that a bifunction and a convex function satisfy the following conditions:
(H1) for all ;
(H2) is monotone, that is, for all ;
(H3)for each , is weakly upper semicontinuous;
(H4)for each , is convex and lower semicontinuous;
(H5)for each and , there exists a bounded subset and such that for any ,
Lemma (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction and let be a proper lower semicontinuous and convex function. For and , define a mapping as follows:
for all . Assume that the conditions (H1)–(H5) hold. Then one has the following results:
(1)for each , and is single-valued;
(2) is firmly nonexpansive, that is, for any ,
(4) is closed and convex.
Lemma (see ).
Let be a maximal monotone mapping and let be a Lipschitz-continuous mapping. Then the mapping is maximal monotone.
Lemma (see ).
Assume taht is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that
(2) or .
3. Main Results
In this section, we will prove our main result. First, we give some assumptions on the operators and the parameters. Subsequently, we introduce our iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion. Finally, we will show that the proposed algorithm has strong convergence.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function and let be a bifunction satisfying conditions (H1)–(H5). Let be a strongly positive bounded linear operator with coefficient and let be a maximal monotone mapping. Let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Let and be two constants such that and .
Now we introduce the following iteration algorithm.
For given arbitrarily, compute the sequences and as follows:
where is a real sequence in .
Now we study the strong convergence of the algorithm (3.1).
Suppose that . Assume the following conditions are satisfied:
Then the sequence generated by (3.1) converges strongly to which solves the following variational inequality:
Take . It is clear that
We divide our proofs into the following five steps:
(1)the sequences and are bounded;
Since is -inverse strongly monotone and is -inverse strongly monotone, we have
It is clear that if and , then and are all nonexpansive. Set . It follows that
By Lemma 2.1, we have for all . Then, we have
Hence, we have
Since is linear bounded self-adjoint operator on , then
that is to say is positive. It follows that
From (3.1), we deduce that
Therefore, is bounded. Hence, , , and are all bounded.
From (3.1), we have
Substituting (3.14) into (3.13), we get
Notice that This together with the last inequality and Lemma 2.3 implies that
From (3.5) and (3.7), we get
By (3.1), we obtain
where is some constant satisfying . From (3.17) and (3.18), we have
which imply that
Since is firmly nonexpansive, we have
Hence, we have
Since is 1-inverse strongly monotone, we have
which implies that
Thus, by (3.23) and (3.25), we obtain
Substitute (3.26) into (3.18) to get
Then we derive
So, we have
We note that is a contraction. As a matter of fact,
for all . Hence has a unique fixed point, say . That is, . This implies that for all . Next, we prove that
First, we note that there exists a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .
We next show that . By , we know that
It follows from (H2) that
For and , let . From (3.35) we have
Since , we have . Further, from the inverse strongly monotonicity of , we have . So, from (H4) and the weakly lower semicontinuity of , and weakly, we have
From (H1), (H4), and (3.37), we also have
Letting , we have, for each ,
This implies that .
Next, we show that . In fact, since is -inverse strongly monotone, is Lipschitz continuous monotone mapping. It follows from Lemma 2.2 that is maximal monotone. Let , that is, . Again since , we have , that is, . By virtue of the maximal monotonicity of , we have
It follows from , and that
It follows from the maximal monotonicity of that , that is, . Therefore, . It follows that
First, we note that ; then for all , we have .
From (3.1), we have
where and . It is easy to see that and . Hence, by Lemma 2.3, we conclude that the sequence converges strongly to . This completes the proof.
Ceng L-C, Yao J-C: 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
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.
Zeng L-C, Wu S-Y, Yao J-C: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese Journal of Mathematics 2006,10(6):1497–1514.
Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. Journal of Optimization Theory and Applications 2003,117(2):245–266. 10.1023/A:1023627606067
Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with application to noncoercive hemivariational inequalities. Journal of Optimization Theory and Applications 2004,121(3):571–596.
Konnov IV, Schaible S, Yao JC: Combined relaxation method for mixed equilibrium problems. Journal of Optimization Theory and Applications 2005,126(2):309–322. 10.1007/s10957-005-4716-0
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036
Yao Y, Liou Y-C, Yao J-C: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory and Applications 2007, 2007:-12.
Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.
Robinson SM: Generalized equations and their solutions. I. Basic theory. Mathematical Programming Study 1979, (10):128–141.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056
Zhang S-S, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Applied Mathematics and Mechanics. English Edition 2008,29(5):571–581. 10.1007/s10483-008-0502-y
Peng J-W, Wang Y, Shyu DS, Yao J-C: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. Journal of Inequalities and Applications 2008, 2008:-15.
Ding XP, Lin YC, Yao JC: Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems. Applied Mathematics and Mechanics 2006,27(9):1157–1164. 10.1007/s10483-006-0901-1
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,336(1):455–469. 10.1016/j.jmaa.2007.02.044
Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In Nonlinear Analysis and Convex Analysis. Edited by: Takahashi W, Tanaka T. Yokohama Publishers, Yokohama, Japan; 2007:609–617.
Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.
Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007,186(2):1551–1558. 10.1016/j.amc.2006.08.062
Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
Adly S: Perturbed algorithms and sensitivity analysis for a general class of variational inclusions. Journal of Mathematical Analysis and Applications 1996,201(2):609–630. 10.1006/jmaa.1996.0277
Brézis H: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, no. 5. North-Holland, Amsterdam, The Netherlands; 1973:vi+183.
Lemaire B: Which fixed point does the iteration method select? In Recent Advances in Optimization (Trier, 1996), Lecture Notes in Economics and Mathematical Systems. Volume 452. Springer, Berlin, Germany; 1997:154–167.
Yao Y, Liou Y-C: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Problems 2008,24(1):-8.
Yao Y, Liou Y-C, Yao J-C: An iterative algorithm for approximating convex minimization problem. Applied Mathematics and Computation 2007,188(1):648–656. 10.1016/j.amc.2006.10.039
Yao Y, Chen R, Yao J-C: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1687–1693. 10.1016/j.na.2007.01.009
Ding XP: Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions. Applied Mathematics and Computation 2003,141(2–3):359–373. 10.1016/S0096-3003(02)00261-8
Huang N-J: Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions. Computers & Mathematics with Applications 1998,35(10):1–7. 10.1016/S0898-1221(98)00066-2
Fang Y-P, Huang N-J: -monotone operator and resolvent operator technique for variational inclusions. Applied Mathematics and Computation 2003,145(2–3):795–803. 10.1016/S0096-3003(03)00275-3
Lin L-J: Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems. Journal of Global Optimization 2007,39(4):509–527. 10.1007/s10898-007-9153-1
Verma RU: General system of -monotone variational inclusion problems based on generalized hybrid iterative algorithm. Nonlinear Analysis: Hybrid Systems 2007,1(3):326–335. 10.1016/j.nahs.2006.07.002
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
The author was supported in part by NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.
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Liou, YC. An Iterative Algorithm for Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities. Fixed Point Theory Appl 2010, 564361 (2010). https://doi.org/10.1155/2010/564361