Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method
© Abkar and Eslamian; licensee Springer 2012
Received: 18 December 2011
Accepted: 6 July 2012
Published: 23 July 2012
In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of the set of common fixed points of a finite family of generalized nonexpansive multivalued mappings and the solution set of two equilibrium problems in a Hilbert space is proved. Our results extend some important recent results.
for all .
Let be a multivalued mapping. An element is said to be a fixed point of T, if . The set of fixed points of T will be denoted by .
- (i)nonexpansive if
quasi-nonexpansive if and for all and all .
Recently, J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki  introduced a new condition on singlevalued mappings, called condition (E), which is weaker than nonexpansiveness.
We say that T satisfies the condition (E) whenever T satisfies for some .
Now we modify this condition for multivalued mappings as follows (see also ):
for some .
The set of solutions is denoted by . It is well known that this problem is closely related to minimax inequalities (see  and ). The equilibrium problem includes fixed point problems, optimization problems and variational inequality problems as special cases. Some methods have been proposed to solve the equilibrium problem, see, for example, [11–14].
Recently, many authors have studied the problems of finding a common element of the set of fixed points of nonexpansive single valued mappings and the set of solutions of an equilibrium problem in the framework of Hilbert spaces: see, for instance, [15–29] and the references therein. In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of a set of common fixed points of a finite family of multivalued mappings satisfying the condition (P) and the solution set of two equilibrium problems in a Hilbert space is proved. Our results generalize some results of Tada, Takahashi  and many others.
Let us recall the following definitions and results which will be used in the sequel.
Lemma 2.1 ()
is called the metric projection of H onto C. It is well known that is a nonexpansive mapping.
Lemma 2.2 ()
Lemma 2.3 ()
For solving the equilibrium problem, we assume that the bifunction Φ satisfies the following conditions:
(A1) for any ,
(A2) Φ is monotone, i.e., for any ,
(A4) is convex and lower semicontinuous for each .
The following lemma was proved in .
The following lemma was given in .
is single valued;
- (ii)is firmly nonexpansive, i.e., for any ,
is closed and convex.
The following lemma was proved in  for nonexpansive multivalued mappings. The statement is true for quasi-nonexpansive multivalued mappings as well. To avoid repetition, we omit the details of the proof.
Lemma 2.6 Let C be a closed convex subset of a real Hilbert space H. Letbe a quasi-nonexpansive multivalued mapping such thatfor all. Thenis closed and convex.
We use a similar argument as in the proof of Lemma 3.1 in  to obtain the following lemma.
Lemma 2.7 Let C be a closed convex subset of a real Hilbert space H. Letbe a multivalued mapping such thatis quasi-nonexpansive. Thenis closed and convex.
Note that for all , . We remark that there exist some examples of multivalued mappings for which is nonexpansive (see  for details), so that the assumption on T is not artificial.
3 The main result
, and and .
Then, the sequencesandconverge strongly to.
Now by Lemma 2.2 we obtain that . □
By substituting by and using a similar argument as in Theorem 3.1, we obtain the following result.
, and and .
Then, the sequencesandconverge strongly to.
As a result, for single valued mappings we obtain the following theorem.
, , (),
, and and .
Then, the sequencesandconverge strongly to.
where, for. Assume that (), Then, the sequenceconverges strongly to.
Proof Putting for all and in Theorem 3.2, we have and hence . Now, the desired conclusion follows directly from Theorem 3.2. □
Now, we supply an example to illustrate the main result of this paper.
We observe that for an arbitrary , is convergent to zero. We note that .
Remark 3.6 Since every nonexpansive mapping is quasi-nonexpansive and satisfies the condition (P), our results hold for nonexpansive mappings.
Remark 3.7 Our results generalize the results of Tada and Takahashi , of a nonexpansive single valued mapping to a finite family of generalized nonexpansive multivalued mappings.
Research of the first author was supported in part by a grant from Imam Khomeini International University, under the grant number 751164-91.
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