Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method

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.

MSC: 47H10, 47H09.

Let C be a nonempty closed convex subset of a Hilbert space H. A subset is called proximal if for each there exists an element such that

We denote by and the collection of all nonempty closed bounded subsets and nonempty proximal bounded subsets of C, respectively. The Hausdorff metric H on is defined by

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 .

Definition 1.1 A multivalued mapping is called

(i) nonexpansive if

(ii) quasi-nonexpansive if and for all and all .

Recently, J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki [1] introduced a new condition on singlevalued mappings, called condition (E), which is weaker than nonexpansiveness.

Definition 1.2 A mapping is said to satisfy the condition provided that

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 [2]):

Definition 1.3 A multivalued mapping is said to satisfy the condition provided that

for some .

It is obvious that every nonexpansive multivalued mapping satisfies the condition . The theory of multivalued mappings has applications in control theory, convex optimization, differential equations and economics. Theory of nonexpansive multivalued mappings is harder than the corresponding theory of nonexpansive single valued mappings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings (see [3-8]). Let Φ be a bifunction from into , where is the set of real numbers. The equilibrium problem for is to find such that

The set of solutions is denoted by . It is well known that this problem is closely related to minimax inequalities (see [9] and [10]). 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 [15] and many others.

Let us recall the following definitions and results which will be used in the sequel.

Lemma 2.1 ([6])

LetHbe a real Hilbert space. Then forwe have

for allandwith.

Let C be a closed convex subset of H. For every point , there exists a unique nearest point in C, denoted by such that

is called the metric projection of H onto C. It is well known that is a nonexpansive mapping.

Lemma 2.2 ([16])

LetCbe a closed convex subset ofH. Givenand a point. Thenif and only if

Lemma 2.3 ([30])

LetCbe a closed convex subset ofH. Then for allandwe have

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 ,

(A3) Φ is upper-hemicontinuous, i.e., for each ,

(A4) is convex and lower semicontinuous for each .

The following lemma was proved in [11].

Lemma 2.4LetCbe a nonempty closed convex subset ofHand let Φ be a bifunction ofintosatisfying (A1)-(A4). Letand. Then, there existssuch that

The following lemma was given in [14].

Lemma 2.5Assume thatsatisfies (A1)-(A4). Forand, define a mappingas follows:

Then, the following hold:

(i) is single valued;

(ii) is firmly nonexpansive, i.e., for any,

(iii) ;

(iv) is closed and convex.

The following lemma was proved in [31] 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 2.6LetCbe a closed convex subset of a real Hilbert spaceH. Letbe a quasi-nonexpansive multivalued mapping such thatfor all. Thenis closed and convex.

Now, following Shahzad and Zegeye [3], we remove the restriction for all . Let be a multivalued mapping and

We use a similar argument as in the proof of Lemma 3.1 in [31] to obtain the following lemma.

Lemma 2.7LetCbe a closed convex subset of a real Hilbert spaceH. 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 [3] for details), so that the assumption on T is not artificial.

Theorem 3.1LetCbe a nonempty closed convex subset of a real Hilbert spaceH, andbe two bifunctions ofintosatisfying (A1)-(A4). Let, (), be a finite family of quasi-nonexpansive multivalued mappings, each satisfying the condition (P). Assume further thatand, (), for each. For, letandbe sequences generated by the following algorithm:

where, for. Assume that, , and, satisfy the following conditions:

(i) (),

(ii) , andand.

Then, the sequencesandconverge strongly to.

Proof First, we show that for all . Fix . We set

Hence we have and . By Lemma 2.5, we have

and

which implies that

Since is quasi-nonexpansive, for , we have

and also

Therefore , which implies that

We observe that is closed and convex (see [32]). Now we show that exists. By Lemma 2.6 we have is closed and convex. Put . From and we have

Also from and for all , we get that

It follows that the sequence is bounded and nondecreasing. Hence the limit exists. We show that . For we have . Now by applying Lemma 2.3 we have

Since exists, it follows that is a Cauchy sequence, and hence there exists such that . Putting , in the above inequality we have

From , we have

so that . This implies that . Take . By Lemma 2.1, for each , we have

and also

So we have that

which implies that

Hence

As above so that

and hence

And also by we have

and hence

Now we use (2) and (3) to obtain

It follows from (1) and the last inequality that

So we have

and

Since , we obtain that

which implies that

Since , for , we obtain that

We observe that . Indeed,

which implies that . Let us show that . From and , we have as . Since we obtain

From (A2), we have

and hence

Since

and , from (A4) we have

For and , let . Since , and C is convex we have and hence . So, from (A1) and (A4), we have

which gives . From (A3) we have , and hence . Similarly, we have . Now we show that . Since , by Lemma 2.2 we have

Since we get

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.

Theorem 3.2LetCbe a nonempty closed convex subset of a real Hilbert spaceH, andbe two bifunctions ofintosatisfying (A1)-(A4). Let, (), be a finite family of multivalued mappings such that eachis quasi-nonexpansive and satisfies the condition (P). Assume further that. For, letandbe sequences generated by the following algorithm:

where, for. Assume that, , and, satisfy the following conditions:

(i) (),

(ii) , andand.

Then, the sequencesandconverge strongly to.

As a result, for single valued mappings we obtain the following theorem.

Theorem 3.3LetCbe a nonempty closed convex subset of a real Hilbert spaceHandandbe two bifunctions ofintosatisfying (A1)-(A4). Let (), be a finite family of quasi-nonexpansive mappings, each satisfying the condition (P). Assume further that. For, letandbe sequences generated by the following algorithm:

Assume that, , and, satisfy the following conditions:

(i) , , (),

(ii) , andand.

Then, the sequencesandconverge strongly to.

Theorem 3.4LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let (), be a finite family of multivalued mappings such thatis quasi-nonexpansive and satisfies the condition (P). Assume further that. For, letbe the sequence generated by the following algorithm:

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.

Example 3.5 We consider the nonempty closed convex subset of the Hilbert space . Define two mappings and on C as follows:

We note that and are quasi-nonexpansive mappings satisfying the condition (P), (for details, see [2]). Also we define two bifunctions and as follows:

It is easy to see that and satisfy the conditions (A1)-(A4). If we put and , then and (for details, see [26]). Put for and . For any arbitrary we have

Since , we obtain that

By continuing this process we obtain

and hence

Now, we have the following algorithm:

Putting and we get that

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 [15], of a nonexpansive single valued mapping to a finite family of generalized nonexpansive multivalued mappings.

The authors declare that they have no competing interests.

All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.

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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