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Convergence analysis for the equilibrium problems with numerical results
Fixed Point Theory and Applications volume 2014, Article number: 167 (2014)
Abstract
In this paper, we propose an iterative scheme modified from the work of Ceng et al. (Nonlinear Anal. Hybrid Syst. 4:743-754, 2010) and Plubtieng and Punpaeng (J. Math. Anal. Appl. 336(1):455-469, 2007) to prove the strong convergence theorem for approximating a common element of the set of fixed points of nonspreading mappings and a finite family of the set of solutions of the equilibrium problem. Using this result, we obtain the strong convergence theorem for a finite family of nonspreading mappings and a finite family of the set of solutions of equilibrium problem. Moreover, in order to compare numerical results between the combination of the equilibrium problem and the classical equilibrium problem, some examples are given in one- and two-dimensional spaces of real numbers.
1 Introduction
Throughout this paper, let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . We denote weak convergence and strong convergence by the notations ‘⇀’ and ‘→’, respectively. We use ℝ to denote the set of real numbers and to represent the set of fixed points of , where is a mapping from C into itself.
In 2008, Kohsaka and Takahashi [1] introduced the nonspreading mapping in Hilbert space H as follows:
In 2009, it was shown by Iemoto and Takahashi [2] that (1.1) is equivalent to the following equation:
Many researchers proved the strong convergence theorem for a nonspreading mapping and some related mappings in Hilbert space; see for example [3–6].
Let . The variational inequality problem is to find a point satisfying the following inequality:
for all . Moreover, is used to denote the set of solutions of (1.2).
Let be a bifunction. The classical equilibrium problem for Φ is to find satisfying the following inequality:
We use to represent the set of solution of (1.3).
Let the bifunction Φ satisfy the following conditions for solving the equilibrium problem.
(A1) for all ;
(A2) Φ is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and lower semicontinuous.
In 1994, Blum and Oettli [7] showed that the classical equilibrium problem (1.3) covers monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, and certain fixed point problems.
Let be a finite family of bifunctions from to ℝ. The system of equilibrium problem for Ψ is to determine common equilibrium points for , that is, the set
The problem (1.4) extends (1.3) to a system of such problems covering various forms of feasibility problems [8]. Several iterative algorithms are proposed to solve the equilibrium problems and a finite family of equilibrium problems; see, for instance, [8–14].
Example 1.1 Let be a finite family of bifunctions from to ℝ, where the bifunctions are defined by
For each , it is obvious that the satisfy (A1)-(A4). Then we obtain
In 2010, Peng et al. [15] proposed the following iterative algorithm for solving a family of infinite nonexpansive mappings and a finite family of equilibrium problems in Hilbert space:
Under some appropriate conditions, they proved that , , and converge strongly to , where and f is a contractive mapping on H.
Over the past few years, many researchers have started working on the methods for finding a common solution of a finite family of equilibrium problems in Hilbert space; see, for instance, [16–18].
In 2013, Suwannaut and Kangtunyakarn [12] introduced the combination of equilibrium problem which is to find such that
where are bifunctions and with , for every . The set of solutions (1.6) is denoted by .
If , for all , then the combination of equilibrium problem (1.6) reduces to the classical equilibrium problem (1.3).
Moreover, they obtain Lemma 2.10 as shown in the next section.
Example 1.2 For every , let the bifunctions , be given by
For all , it is obvious that the satisfy (A1)-(A4). Let , and , thus we have
This implies that
Remark 1.3 For all , let the mapping be defined by for all . For each , if for all , and , then . Hence we have
After we have studied research related to equilibrium problems, we obtain the following question.
Question Is it possible to prove strong convergence theorem for a finite family of equilibrium problem using different method from the result of Peng et al. [15], Piri [17] and references therein?
Inspired and motivated by the work of Iemoto and Takahashi [2], Suwannaut and Kangtunyakarn [12] and related research, we propose an iterative scheme modified from the work of Plubtieng and Punpaeng [19] and Ceng et al. [11] to prove the strong convergence theorem for approximating a common element of the set of fixed points of a nonspreading mapping and a finite family of the set of solutions of equilibrium problems using Lemma 2.10 and a different method from the work of Peng et al. [15] and Piri [17] and references therein. Moreover, some examples are given in order to compare the numerical results between the combination of the equilibrium problem and the classical equilibrium problem.
2 Preliminaries
We now recall the following definition and well-known lemmas.
Definition 2.1
-
(i)
is strongly positive operator on H if there exists a constant such that
-
(ii)
is a nonexpansive mapping if
-
(iii)
For every , there is a unique nearest point in C such that
Such an operator is called the metric projection of H onto C.
Lemma 2.1 ([20])
For a given and ,
Furthermore, is a nonexpansive mapping.
Lemma 2.2 ([21])
Each Hilbert space H satisfies Opial’s condition, i.e., for any sequence with , the inequality
holds for every with .
Lemma 2.3 ([22])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
-
(1)
,
-
(2)
or .
Then .
Lemma 2.4 ([4])
Let H be a real Hilbert space. Then the following results hold:
-
(i)
For all and ,
-
(ii)
, for each .
Lemma 2.5 ([20])
Let H be a Hilbert space, let C be a nonempty closed convex subset of H and let be a mapping of C into H. Then, for ,
where is the metric projection of H onto C.
Lemma 2.6 ([23])
Assume is a strongly positive linear bounded operator on a Hilbert space H with coefficient and . Then .
Lemma 2.7 ([2])
Let C be a nonempty closed convex subset of H. Then a mapping is nonspreading if and only if
Remark 2.8 If is a nonexpansive mapping and , for every , then is a nonspreading mapping.
Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a nonspreading mapping with . Then we have the following statements:
-
(i)
;
-
(ii)
for every and ,
Proof To prove (i), let . Then . Since
we have , from which it follows that .
Next, we show .
Let . This implies that
Let . Then, by Lemma 2.7, we obtain
Observe that
From (2.1), (2.2), and (2.3), we get
which yields . Therefore .
To prove (ii), let and . Since is a nonspreading mapping and we have Lemma 2.7, we get
Thus we have
From (2.4) and (2.5), we obtain
From (i) and Lemma 2.5, we have
By the nonexpansiveness of , (2.6), and (2.7), we get
which implies that . □
Lemma 2.10 ([12])
Let C be a nonempty closed convex subset of a real Hilbert space H. For , let be bifunctions satisfying (A1)-(A4) with . Then
where for every and .
Lemma 2.11 ([7])
Let C be a nonempty closed convex subset of H and let Φ be a bifunction of into ℝ satisfying (A1)-(A4). Let and . Then there exists such that
Lemma 2.12 ([8])
Assume that satisfies (A1)-(A4). For , define a mapping as follows:
for all . Then the following hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for each ,
-
(iii)
;
-
(iv)
is closed and convex.
Remark 2.13 ([12])
From Lemma 2.10, it is easy to see that satisfies (A1)-(A4). By using Lemma 2.12, we obtain
where , for each , and .
3 Strong convergence theorem
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ℱ be an α-contractive mapping on H and let be a strongly positive linear bounded operator on H with coefficient and . Let be a nonspreading mapping. For every , let be a bifunction satisfying (A1)-(A4) with . Let , , and be sequences generated by and
where , , for all . Suppose the conditions (i)-(vi) hold.
-
(i)
and ;
-
(ii)
, for some ;
-
(iii)
;
-
(iv)
, for some ;
-
(v)
;
-
(vi)
, , , .
Then the sequences , , and converge strongly to .
Proof The proof of this theorem is divided into five steps.
Step 1. Claim that is a bounded sequence.
Since as , without loss of generality, we assume , for every . Since satisfies (A1)-(A4) and
by Lemma 2.12 and Remark 2.13, we have and .
From Lemma 2.5 and Lemma 2.9(i), we obtain
Let . By the nonexpansiveness of and , we have
From Lemma 2.6, Lemma 2.9(ii), and (3.2), we obtain
By induction, we obtain , . It shows that is bounded and so are and .
Step 2. Show that .
By the definition of and Lemma 2.6, we obtain
Using the same method as in [12] (Step 2 of Theorem 3.1), we have
Substitute (3.3) into (3.4) to get
where . From (3.5), the conditions (i), (iii), (v), and Lemma 2.3, we have
Step 3. Prove that .
To claim this, let . Since and is a firmly nonexpansive mapping, we have
from which it follows that
By the definition of , Lemma 2.6, Lemma 2.9(ii), and (3.7), we get
which implies that
From (3.6), the conditions (i) and (ii), this yields
By Lemma 2.6 and Lemma 2.9(ii), we get
from which it follows that
From (3.6), the conditions (i) and (ii), this implies that
Since
using (3.8) and (3.9), we have
Since
by (3.8) and (3.10), thus we obtain
Observe that
which implies by (3.6) and the condition (i) that
Since
by (3.11), (3.12), and the condition (iii), we obtain
Step 4. Show that , where .
First, take a subsequence of such that
Since is bounded, we can assume that as . By (3.8), it follows that as .
Assume . Since , we have . By the nonexpansiveness of , the condition (iii), (3.13), and Opial’s condition, we get
This is a contradiction. Then we have
By continuing the same argument as in [12] (Step 4 of Theorem 3.1), we obtain
From (3.14) and (3.15), we get . Since as , by Lemma 2.1 we can conclude that
Step 5. Finally, claim that the sequence converges strongly to .
By Lemma 2.4, Lemma 2.6, and Lemma 2.9(ii), we obtain
which implies that
From (3.16), the condition (i), and Lemma 2.3, we can conclude that converges strongly to . By (3.8) and (3.11), we see that and converge strongly to . This completes the proof. □
The following corollaries are direct results from Theorem 3.1.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ℱ be an α-contractive mapping on H and let be a strongly positive linear bounded operator on H with coefficient and . Let be a nonspreading mapping. Let be a bifunction satisfying (A1)-(A4) with . Let , , and be sequences generated by and
where . Suppose the conditions (i)-(vi) hold.
-
(i)
and ;
-
(ii)
, for some ;
-
(iii)
;
-
(iv)
, for some ;
-
(v)
, , , .
Then the sequences , , and converge strongly to .
Proof Put , for all . Using Theorem 3.1, the desired result is obtained. □
In 2007, Plubtieng and Punpaeng [19] introduced the general iterative method for an equilibrium problem and a nonexpansive mapping in Hilbert spaces. Let S be a nonexpansive mapping on H with . With an initial value , the sequences and are generated by
where and satisfy some appropriate conditions. Then and converge strongly to a point z, where .
Later, in 2010, Ceng et al. [11] studied the iterative scheme for equilibrium problem and an infinite family of nonexpansive mappings. Let . Let and be sequences in . Starting with , the sequences and are generated by the following iterative scheme:
where is a W-mapping generated by an infinite family of nonexpansive mappings and infinite real numbers. Then, under some suitable conditions, the sequences and converge strongly to , where .
Remark 3.3 For Corollary 3.2, we prove the strong convergence theorem for equilibrium problem and a nonspreading mapping. Motivated by the results of Ceng et al. [11] and Plubtieng and Punpaeng [19], we consider the following statements, different from this work.
-
(i)
We investigate the iterative algorithm for a nonspreading mapping instead of using a nonexpansive mapping.
-
(ii)
We study the general iterative method by using the sequence .
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ℱ be an α-contractive mapping on H and let be a strongly positive linear bounded operator with coefficient and . Let be a nonspreading mapping with . Let be the sequence generated by and
where . Suppose the conditions (i)-(vi) hold.
-
(i)
and ;
-
(ii)
;
-
(iii)
, .
Then the sequence converges strongly to .
Proof Take , for every . Then we have , for every . The result of Corollary 3.4 can be obtained by Theorem 3.1. □
4 Applications
By means of our main result, we obtain the strong convergence theorem for a finite family of nonspreading mappings and a finite family of equilibrium problems in the setting of Hilbert space. To prove this, the following definitions, remarks, and lemmas are needed.
Definition 4.1 A mapping is quasi-nonexpansive if
Remark 4.1 If is nonspreading with , then is quasi-nonexpansive.
Example 4.2 Let an inner product be defined by and a usual norm be given by , for all . Let and let be defined by
First, we show that is a nonspreading mapping.
For every , we obtain
and
This yields
Then is a nonspreading mapping and we observe that , where . For every and , from (4.1), we have
Therefore is a quasi-nonexpansive mapping.
The following example shows that the converse of Remark 4.1 does not hold.
Example 4.3 Let and let be defined by
First, show that is quasi-nonexpansive for all .
Observe that if and if , where and .
For any , we have
For every , we obtain
Therefore is a quasi-nonexpansive for all .
Choose and , we have
Thus we get
Hence we have
By changing from a nonspreading mapping to a quasi-nonexpansive mapping with , we obtain the same result as shown in Lemma 2.9.
Remark 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a quasi-nonexpansive mapping with . Then we have the following statement:
-
(i)
;
-
(ii)
for every and ,
Definition 4.2 ([24])
Let C be a nonempty convex subset of a real Banach space. Let be a finite family of (nonexpansive) mappings of C into itself. For each , let where and . Define the mapping as follows:
This mapping is called the S-mapping generated by and .
Lemma 4.5 ([25])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a finite family of nonspreading mappings of C into itself with and let where , , for all and , , for all . Let S be the S-mapping generated by and . Then and S is a quasi-nonexpansive mapping.
Theorem 4.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an α-contractive mapping, let be a strongly positive linear bounded operator with coefficient and . For , let be a bifunction satisfying (A1)-(A4). Let , for be a finite family of nonspreading mappings with . Let , , where , , for all and , , for all , and let S be the S-mapping generated by and . Let , , and be sequences generated by and
where , , for all . Suppose the conditions (i)-(vi) hold.
-
(i)
and ;
-
(ii)
, for some ;
-
(iii)
;
-
(iv)
, for some ;
-
(v)
;
-
(vi)
, , , .
Then the sequences , , and converge strongly to .
Proof Using Remark 4.4, Lemma 4.5, and the same method as in Theorem 3.1, we have the desired conclusion. □
Remark 4.7 Theorem 4.6 can be considered as an improvement of Theorem 3.1 in the result of Tian and Jin [26] in the sense that some conditions are not assumed.
-
(i)
, ,
-
(ii)
T is demi-closed on H,
where T is a quasi-nonexpansive mapping on H.
5 Examples for equilibrium problems and numerical results
In this section, the numerical examples are given for supporting Theorem 3.1. Using these examples, we see that our iteration for the combination of equilibrium problem converges faster than our iteration for the classical equilibrium problem.
Example 5.1 Let the mappings , , be defined by
For every , let and be defined by
Put , for every . Let , , , , and for every . Let the initial values be defined as in the following cases:
-
(i)
, , and ,
-
(ii)
and .
Then, for both cases, the sequences , , and converge strongly to 1.
Solution. It is obvious that is a nonspreading mapping and .
Since , we obtain
where . It is clear that satisfies all conditions in Theorem 3.1 and . Then we have
Observe that
Let . is a quadratic function of y with coefficients , , and . Determine the discriminant Δ of G as follows:
From (5.1), we have , for every . If has at most one solution in ℝ, thus we have . This implies that
where . Put , , , , . It is clear to see that the sequences , , , and satisfy all conditions in Theorem 3.1. For every , from (5.2), we rewrite (3.1) as follows:
From Theorem 3.1, we can conclude that the sequences , , and generated by (5.3) converge strongly to 1.
For case (i), with , we have . Then (5.2) becomes
Then we have
From Corollary 3.2, we can conclude that the sequences , , and generated by (5.5) converge strongly to 1.
Table 1 and Figure 1 show the values of sequences , , and in two cases.
Remark 5.2
-
(i)
From Table 1 and Figure 1, the sequences , , and converge to 1, where .
-
(ii)
For case (i), Corollary 3.2 guarantees the convergence of , , and .
-
(iii)
For case (ii), the convergence of , , and can be guaranteed by Theorem 3.1.
-
(iv)
The iteration (5.3) for the combination of equilibrium problem converges faster than the iteration (5.5) for the classical equilibrium problem.
Finally, we give the numerical example for our main theorem in two-dimensional space of real numbers.
Example 5.3 Let be the two-dimensional space of real numbers with an inner product defined by and a usual norm given by , for all . Let the mappings , be defined by
For every and , let and be defined by
where . Let , , , , and for every .
It is clear that is a nonspreading mapping and , where .
Put , for every . It is obvious that satisfies all conditions in Theorem 3.1 and , where . Then we have
Then, by Theorem 3.1, the sequences , , and converge strongly to .
Remark 5.4 From Example 5.3, putting , we obtain