We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
1. Introduction
Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see [1–8]. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.
Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. Let be a realvalued function and an equilibriumlike function, that is,
We consider the problem of finding and such that
which is called the generalized mixed equilibrium problem (for short, GMEP). If is a singlevalued mapping, then problem (1.2) is equivalent to finding such that
We denote for the set of solutions of GMEP (1.2). This class is a quite general and unifying one and includes several classes of equilibrium problems and variational inequalities as special cases. In recent years, several numerical techniques including projection, resolvent, and auxiliary principle have been developed and analyzed for solving variational inequalities. It is well known that projection and resolventtype methods cannot be extended for equilibrium problems. To overcome this drawback, one usually uses the auxiliary principle technique. Glowinski et al. [9] have used this technique to study the existence of a solution of mixed variational inequalities. The viscosity approximation method is one of the important methods for approximation fixed points of nonexpansive type mappings. It was first discussed by Moudafi [10]. Recently, Hirstoaga [11] and S. Takahashi and W. Takahashi [12] applied viscosity approximation technique for finding a common element of set of solutions of an equilibrium problem (EP) and set of fixed points of a nonexpansive mapping. Very recently, Yao et al. [13] introduced and studied an iteration process for finding a common element of the set of solutions of the EP and the set of common fixed points of infinitely many nonexpansive mappings in . Let be a sequence of nonexpansive mappings of into itself and let be a sequence of nonnegative numbers in . For any , define a mapping of into itself as follows:
Such a mapping is called the mapping generated by and , see [14].
The purpose of this paper is to develop an iterative algorithm for finding a common element of set of solutions of GMEP (1.2) and set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and extends the main result of S. Takahashi and W. Takahashi [12].
2. Preliminaries
Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that
is called metric projection of onto . It is well known that is nonexpansive. Furthermore, for and ,
We denote by the set of fixed points of a selfmapping on , that is, . It is well known that if is nonempty, bounded, closed, and convex and is nonexpansive, then is nonempty; see [15]. Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of selfmappings on by (1.4). Then we have the following lemmas which are important to prove our results.
Lemma 2.1 (see [14]).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, for every and the limit exists.
Using Lemma 2.1, one can define mapping of into itself as follows:
for every . Such a mapping is called the mapping generated by and Throughout this paper, we will assume that for every . Since is nonexpansive, is also nonexpansive.
Lemma 2.2 (see [14]).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, .
Let be a convex subset of a real Hilbert space and a Fréchet differential function. Then is said to be convex strongly convex if there exists a constant such that
If , then is said to be convex. In particular, if for all , then is said to be strongly convex.
Let be a nonempty subset of a real Hilbert space . A bifunction is said to be skewsymmetric if
If the skewsymmetric bifunction is linear in both arguments, then
We denote for weak convergence and for strong convergence. A function is called weakly sequentially continuous at , if as for each sequence in converging weakly to . The function is called weakly sequentially continuous on if it is weakly sequentially continuous at each point of .
Let denote the set of nonempty closed bounded subsets of . For , define the Hausdorff metric as follows:
Lemma 2.3 (see [16]).
Let and . Then for , there must exist a point such that .
Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. For , let . Let be a realvalued function satisfying the following:
is skew symmetric;
for each fixed , is convex and upper semicontinuous;
is weakly continuous on .
Let be a differentiable functional with Fréchet derivative at satisfying the following:
is sequentially continuous from the weak topology to the strong topology;
is Lipschitz continuous with Lipschitz constant .
Let be a function satisfying the following:
for all ;
is affine in the first coordinate variable;
for each fixed , is sequentially continuous from the weak topology to the weak topology.
Let us consider the equilibriumlike function which satisfies the following conditions with respect to the multivalued mapping :
for each fixed , is an upper semicontinuous function from to , that is, and imply ;
for each fixed , is a concave function;
for each fixed , is a convex function.
Let be a positive parameter. For a given element and , consider the following auxiliary problem for GMEP(1.2): find such that
It is easy to see that if , then is a solution of GMEP(1.2).
Lemma 2.4 (see [6]).
Let be a nonempty closed convex bounded subset of a real Hilbert space and a realvalued function satisfying the conditions . Let be a multivalued mapping and the equilibriumlike function satisfying the conditions . Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an strongly convex function with constant which satisfies the conditions and . For each , let . For , define a mapping by
Then one has the following:
(a) the auxiliary problem (2.8) has a unique solution;
(b) is single valued;
(c) if and for all and all , , it follows that is nonexpansive;
(d);
(e) is closed and convex.
We also need the following lemmas for our main results.
Lemma 2.5 (see [17]).
Let , and be three sequences of nonnegative numbers such that
If , , and , then exists.
Lemma 2.6.
Let and be sequences of nonnegative numbers such that
If and , then .
Proof.
It is easy to see that inequality (2.11) is equivalent to
where , and . It follows that
Note that Lemma 2.5 implies that exists. Suppose for some . It is obvious that and so inequality (2.12) implies that , which is a contradiction. Thus, . This completes the proof.
Lemma 2.7 (see [6]).
Let be a sequence in a normed space such that
where , and and are sequences satisfy the following conditions:
(i) for all and ;
(ii) for all and .
Then is a Cauchy sequence.
Lemma 2.8 (see [18]).
Let be a sequence of nonnegative real numbers such that
where , and are sequences of real numbers satisfying the following conditions:
(i), and ;
(ii);
(iii) for all and .
Then, .
3. Iterative Algorithm and Convergence Theorem
Let be a nonempty closed convex subset of a real Hilbert space , a multivalued mapping, a contraction mapping with constant , and an mapping generated by and , where sequence is nonexpansive. Let be a sequence in and a sequence in . We can develop Algorithm 3.1 for finding a common element of a set of fixed points of mapping and a set of solutions of GMEP(1.2).
Algorithm.
For given and , there exist sequences , in and in such that for all ,
We now prove the strong convergence of iterative sequence , , and generated by Algorithm 3.1.
Theorem 3.2.
Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a realvalued function satisfying the conditions and let be an equilibriumlike function satisfying conditions and :
for all and , where , and .
Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an strongly convex function with constant which satisfies conditions and with . Let be an mapping generated by and and , where sequence is nonexpansive. Let , and be sequences generated by Algorithm 3.1, where is a sequence in and in satisfying the following conditions:
, and ;
and ;
where .
Then the sequences and converge strongly to , and converges strongly to , where .
Proof.
It is easy to see from () that
for all and , where , , and . All the conclusions (a)–(e) of Lemma 2.4 hold.
Let . Then is a contraction of into itself. In fact,
Hence there exists a unique element such that . Noting that and , we get that .
Now, we prove that and as . Observe that
Noting that and , it follows from (3.1) that
Putting in (3.5) and in (3.6), respectively, we have
Adding up those inequalities, we obtain from (2.5), (), and () that
It follows that
since and are Lipschitz continuous wiht Lipschitz constants and , respectively. Noting that , without loss of generality, we assume that there exists a real number such that for all Thus,
which implies that
and hence
where . Set . Combining (3.4) and (3.12) yields
From conditions and ,
Set and
Then Lemmas 2.6 and 2.7 imply that and is a Cauchy sequence in . Hence from (3.12), we get
We know from that . It follows that
Thus,
Next, we prove that there exists , such that , , and as , where .
Let . Then
and so
By the convexity of , we have
It follows that
This implies that
Since is a Cauchy sequence in , there exists an element such that . Now implies that . From (3.1), we have
and for ,
Thus,
By (3.24) and (3.26), we have
It follows that is a Cauchy sequence in and so there exists an element in such that :
that is, . We conclude that as .
It follows that
and so , that is, . Since and , we know that . From (3.1) and (), we have
that is, . Thus, .
Since , we have for all . From , we have
and so
It follows from (3.19) that
Set
Then, , , and . It follows from Lemma 2.8 that and so . This completes the proof.
Remark 3.3.
Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi [12].
We now give some applications of Theorem 3.2. If the setvalued mapping in Theorem 3.2 is singlevalued, then we have the following corollary.
Corollary 3.4.
Let be a nonempty closed convex bounded subset of a real Hilbert space , a Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a realvalued function satisfying the conditions and let be an equilibriumlike function satisfying the conditions and :
for all and .
Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an strongly convex function with constant which satisfies the conditions and with . Let be an mapping generated by and and , where sequence is nonexpansive. Let , , and be sequences generated by
where is a sequence in and in satisfying conditions . Then the sequences and converge strongly to , where .
Corollary 3.5.
Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a realvalued function satisfying the conditions and let be an equilibriumlike function satisfying the conditions and . Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an strongly convex function with constant which satisfies the conditions and with . Let , , and be sequences generated by
where is a sequence in and in satisfying conditions and . Then the sequences and converge strongly to , and converges strongly to , where .
Proof.
Let in Theorem 3.2 for , where is an identity mapping. Then for Thus, the condition is satisfied. Now Corollary 3.5 follows from Theorem 3.2. This completes the proof.
Acknowledgments
The authors would like to thank the referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005).
References

Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student. 63(1–4), 123–145 (1994)

Huang, NJ, Lan, HY, Teo, KL: On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces. Journal of Inequalities and Applications. 2007, (2007)

Giannessi F, Maugeri A, Pardalos PM (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)

FloresBazán, F: Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case. SIAM Journal on Optimization. 11(3), 675–690 (2000)

Mosco, U: Implicit variational problems and quasi variational inequalities. Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Mathematics, pp. 83–156. Springer, Berlin, Germany (1976)

Sahu, DR, Wong, NC, Yao, JC: On convergence analysis of an iterative algorithm for finding common solution of generalized mixed equilibrium problems and fixed point problemes. to appear in Mathematical Inequalities & Applications

Peng, JW, Yao, JC: A new hybridextragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics. 12(6), 1401–1432 (2008)

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

Glowinski, R, Lions, JL, Tremolieres, R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications,p. xxix+776. NorthHolland, Amsterdam, The Netherlands (1981)

Moudafi, A: Viscosity approximation methods for fixedpoints problems. Journal of Mathematical Analysis and Applications. 241(1), 46–55 (2000). Publisher Full Text

Hirstoaga, SA: Iterative selection methods for common fixed point problems. Journal of Mathematical Analysis and Applications. 324(2), 1020–1035 (2006). Publisher Full Text

Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications. 331(1), 506–515 (2007). Publisher Full Text

Yao, Y, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory and Applications. 2007, (2007)

Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics. 5(2), 387–404 (2001)

Takahashi, W: Nonlinear Functional Analysis. Fixed Point Theory and Its Application,p. iv+276. Yokohama, Yokohama, Japan (2000)

Nadler, SB Jr..: Multivalued contraction mappings. Pacific Journal of Mathematics. 30, 475–488 (1969)

Osilike, MO, Aniagbosor, SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Mathematical and Computer Modelling. 32(10), 1181–1191 (2000). Publisher Full Text

Xu, HK: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. 66(1), 240–256 (2002). Publisher Full Text