A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem
© Filomena Cianciaruso et al. 2010
Received: 3 June 2009
Accepted: 16 September 2009
Published: 1 November 2009
We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.
In this paper, we define an iterative method to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings generated by two sequences of firmly nonexpansive mappings and two nonlinear mappings. Let us recall from  that the -strict pseudocontractions in Hilbert spaces were introduced by Browder and Petryshyn in .
The iterative approximation problems for nonexpansive mappings, asymptotically nonexpansive mappings, and asymptotically pseudocontractive mappings were studied extensively by Browder , Goebel and Kirk , Kirk , Liu , Schu , and Xu [8, 9] in the setting of Hilbert spaces or uniformly convex Banach spaces. Although nonexpansive mappings are 0-strict pseudocontractions, iterative methods for -strict pseudocontractions are far less developed than those for nonexpansive mappings. The reason, probably, is that the second term appearing in the previous definition impedes the convergence analysis for iterative algorithms used to find a fixed point of the -strict pseudocontraction . However, -strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems. In the recent years the study of iterative methods like Mann's like methods and CQ-methods has been extensively studied by many authors [1, 10–13] and the references therein.
The equilibrium problems, in its various forms, found application in optimization problems, fixed point problems, convex minimization problems; in other words, equilibrium problems are a unified model for problems arising in physics, engineering, economics, and so on (see ).
As in the case of nonexpansive mappings, also in the case of -strict pseudocontraction mappings, in the recent years many papers concern the convergence of iterative methods to a solutions of variational inequality problems or equilibrium problems; see example for, [10, 14–18].
Here we prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.
If we say that is firmly nonexpansive. Note that every -inverse strongly monotone operator is also Lipschitz continuous (see ).
(see ). Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudocontractive mapping. Then with is a nonexpansive mapping with .
3. Main Theorem
Moreover suppose that
Now we divide the proof in more steps.
So the claim immediately follows by induction.
We will use three times the Opial's Lemma 2.1.
which is a contradiction.
Let us remember that the metric projection on a convex closed set is a firmly nonexpansive mapping (see ) so we claim that have the following proposition.
Moreover, (iii) follows directly by (2.2).
We notice that for and the problem is the well-known equilibrium problem [23–25]. If and is an -inverse strongly monotone operator we have the equilibrium problems studied firstly in  and then in [18, 22, 27]. If and we refound the mixed equilibrium problem studied in [16, 28, 29].
Moreover let us suppose that
that is absurd.
To prove (4), it is enough to follow (iii) and (iv) in [25, Lemma 2.12].
We note that if , our lemma reduces to [25, Lemma 2.12]. The coercivity condition (H) is fulfilled.
Moreover our lemma is more general than [16, Lemma 2.2]. In fact
(iii)the coercivity condition (H) by the equivalence of (3.36) and (3.37) is the same.
and thus the claim holds.
Let us suppose that and are two bi-functions satisfying the hypotheses of Lemma 3.5. Let be the resolvent of and . Let be an -inverse strongly monotone operator. Let us suppose that is such that . Then realize (ii) and (iii) in Theorem 3.1.
In next theorem our purpose is to prove a strong convergence theorem to approximate a fixed point of that is also a solution of a mixed equilibrium problem and a solution of a variational inequality . One can note that we relax the hypotheses on the convergence of the sequences and .
By condition , for fixed, the function is lower semicontinuos and convex, and thus weakly lower semicontinuous .
Since also Step 9 can be followed as in Theorem 3.1, we obtain the claim.
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