A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces
© S. Plubtieng and W. Sriprad. 2009
Received: 12 February 2009
Accepted: 18 May 2009
Published: 16 June 2009
We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al. (2008) and many others.
We denote by the set of fixed points of .
where is a potential function for
then the variational inclusion problem (1.8) is equivalent to variational inequality problem (1.2). It is known that (1.8) 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, game theory. Also various types of variational inclusions problems have been extended and generalized (see  and the references therein.)
for all , where , and . They proved that under certain appropriate conditions imposed on and , the sequences , , and generated by (1.10) converge strongly to , where .
for all , where , , and let ; be a strongly bounded linear operator on , and is a sequence of nonexpansive mappings on . Under suitable conditions, some strong convergence theorems for approximating to this common elements are proved. Our results extend and improve some corresponding results in [3, 9] and the references therein.
This section collects some lemmas which will be used in the proofs for the main results in the next section.
Let be a real Hilbert space with inner product and norm , respectively.
For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1) for all
(A2) is monotone, that is,
(A4)for each is convex and lower semicontinuous.
for all . Then, the following hold:
(1) is single-valued;
(4) is closed and convex.
We also need the following lemmas for proving our main result.
Lemma 2.2 (See ).
Let be a Hilbert space, a nonempty closed convex subset of , a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Then :
(1)if , then
(2)if , then .
Lemma (See ).
where is a sequence in and is a sequence in such that
Let be the identity mapping on . It is well known that if is -inverse-strongly monotone, then is -Lipschitz continuous and monotone mapping. In addition, if , then is a nonexpansive mapping.
A set-valued is called monotone if for all and imply . A monotone mapping is maximal if its graph of 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 .
where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive and -inverse-strongly monotone, see for example  and that a solution of problem (1.8) is a fixed point of the operator for all , see for instance .
Lemma 2.4 (See ).
Let be a maximal monotone mapping and be a Lipschitz-continuous mapping. Then the mapping is a maximal monotone mapping.
Remark (See ).
Lemma 2.4 implies that is closed and convex if is a maximal monotone mapping and be an inverse strongly monotone mapping.
Lemma 2.6 (See ).
3. Main Results
We begin this section by proving a strong convergence theorem of an implicit iterative sequence obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.
Equivalently, we have
Since is bounded and it follows that as
Put . From (3.10), it follows by the nonexpansive of and the inverse strongly monotonicity of that
Since , as , we have as . It follows from the inequality that as . Moreover, we have as .
Put . Since both and are nonexpansive, we have is a nonexpansive mapping on and then we have for all . It follows by Theorem 3.1 of Plubtieng and Punpaeng  that converges strongly to , where and , for all . We will show that . Since converges strongly to , we also have . Let us show Assume Since and , we have Since it follows by the Opial's condition that
Since is maximal monotone, this implies that that is, . Hence, . Since , we have . It implies that is the unique solution of the variational inequality (3.4).
Now we prove the following theorem which is the main result of this paper.
Suppose that for any bounded subset of . Let be a mapping of into itself defined by and suppose that . Then, , and converges strongly to , where is a unique solution of the variational inequalities (3.4).
Since is bounded, it follows that . Hence, by Lemma 2.3, we have as . From (3.34) and , we have . Moreover, we have from (3.27) that .
We note from (3.23) that . Then, we have
Since is bounded, and , it follows that as
Put . It follows from (3.38), the nonexpansive of and the inverse strongly monotonicity of that
where and . It easily verified that , and . Hence, by Lemma 2.1, the sequence converges strongly to .
As in [10, Theorem 4.1], we can generate a sequence of nonexpansive mappings satisfying condition . for any bounded of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point.
Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping, be a maximal monotone mapping. Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let be a family of nonnegative numbers with indices with such that
If and in Theorem 3.2, we obtain the following corollary.
Corollary 3.4 (see Peng et al. ).
Then, , and converges strongly to , where .
If and in Theorem 3.2, we obtain the following corollary.
Corollary 3.5 (see S. Plubtieng and R. Punpaeng ).
The first author thank the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the "National Centre of Excellence in Mathematics", PERDO, under the Commission on Higher Education, Ministry of Education, Thailand.
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