- Research
- Open access
- Published:
Convergence theorems for a system of equilibrium problems and fixed point problems of a strongly nonexpansive sequence
Fixed Point Theory and Applications volume 2013, Article number: 193 (2013)
Abstract
The purpose of this paper is to prove a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence. Moreover, in the last section, by using our main result, we obtain a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of a finite family of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence in a Hilbert space, and we also give some examples to support our main result.
1 Introduction
Throughout this paper, we assume that H is a real Hilbert space with the inner product and the norm . A mapping T of C into itself is called nonexpansive if for all . The set of fixed points of T is denoted by , i.e., . It is known that is closed and convex if T is nonexpansive. Let be a metric projection of H onto C, i.e., for , satisfies the property
We use and to denote weak and strong convergence, respectively. Let be a sequence of mappings of C into H. The set of common fixed points of is denoted by . Recall the main concepts as follows:
-
(1)
A sequence in C is said to be an approximate fixed point sequence of if . The set of all bounded approximate fixed point sequences of is denoted by ; see [1]. It is clear that if has a common fixed point, then is nonempty.
-
(2)
A sequence is said to be a strongly nonexpansive sequence if each is nonexpansive and
whenever and are sequences in C such that is bounded and .
-
(3)
A sequence having a common fixed point is said to satisfy the condition (Z) if every weak cluster point of is a common fixed point whenever .
-
(4)
A sequence of nonexpansive mappings of C into H is said to satisfy the condition (R) if
for every nonempty bounded subset D of C; see [2].
Example 1.1 Let ℝ be a set of real numbers. For every , the mapping is defined by for all .
Then is a nonexpansive sequence, but it is not a strongly nonexpansive sequence.
Example 1.2 For every , the mapping is defined by for all .
Then is a strongly nonexpansive sequence.
Solution It is easy to see that is a nonexpansive mapping for all .
Let and be sequences in with being bounded and as .
Since , for all , then we have
Then is a strongly nonexpansive sequence.
Let be a bifunction. The equilibrium problem for G is to determine its equilibrium points, i.e., the set
It is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, etc.; see [3–5]. Several iterative methods have been proposed to solve the equilibrium problem; see, for instance, [6–8]. In 2005, Combettes and Hirstoaga [4] introduced some iterative schemes of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.
Also in [4], Combettes and Hiratoaga, following [3], defined by
They proved that under suitable hypotheses is single-valued and firmly nonexpansive with .
In 2007, Takahashi and Takahashi [9] proved the following theorem.
Theorem 1.3 Let C be a nonempty closed convex subset of H. Let G be a bifunction from to ℝ satisfying
-
(A1)
, ;
-
(A2)
G is monotone, i.e., , ;
-
(A3)
, ;
-
(A4)
, is convex and lower semicontinuous;
and let S be a nonexpansive mapping of C into H such that . Let f be a contraction of H into itself, and let and be sequences generated by and
for all , where and satisfy
-
(C1)
;
-
(C2)
;
-
(C3)
;
and and .
Then and converge strongly to , where .
Very recently, in 2011, Aoyama and Kimura [10] proved a strong convergence theorem of the iterative scheme of associated to a strongly nonexpansive sequence as follows.
Theorem 1.4 Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let and be sequences of nonexpansive self-mappings of C. Suppose that is nonempty, both and satisfy the conditions (R) and (Z), and or is a strongly nonexpansive sequence. Let and be sequences in [0, 1] such that
Let and let be a sequence in C defined by and
for all . Then converges strongly to .
For , let , , and be the sequences defined by
where is a contractive mapping with and , are sequences of nonexpansive mappings, one of them is a strongly nonexpansive sequence.
In this paper, inspired and motivated by [10] and [9], we prove that a strong convergence theorem of the iterative scheme defined by (1.1) converges strongly to , where , under the conditions (R) and (Z) and suitable conditions of , , , and .
2 Preliminaries
In this section, we need the following lemmas to prove our main result in the next section.
Lemma 2.1 (See [11])
Given and . Then if and only if the following inequality holds:
Lemma 2.2 (See [12])
Let be a sequence of nonnegative real numbers satisfying
where , satisfy the conditions
-
(1)
, ;
-
(2)
or .
Then .
Lemma 2.3 (See [13])
Let and be bounded sequences in a Banach space X, and let be a sequence in with . Suppose that
for all integers and
Then .
Lemma 2.4 (See [14])
Let C be a closed convex subset of a strictly convex Banach space E. Let be a sequence of nonexpansive mappings on C. Suppose that is nonempty. Let be a sequence of positive numbers with . Then a mapping S on C defined by
for all is well defined, nonexpansive and holds.
Lemma 2.5 (See [4])
Let C be a nonempty closed convex subset of a Hilbert space H, and let satisfy
-
(A1)
, ;
-
(A2)
G is monotone, i.e., , ;
-
(A3)
, ;
-
(A4)
, is convex and lower semicontinuous.
For and , define a mapping as follows:
Then is well defined and the following hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e., , ;
-
(3)
;
-
(4)
is closed and convex.
Lemma 2.6 (See [11]) (Demiclosedness principle)
Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then is demiclosed. That is, whenever is a sequence in C weakly converging to some and the sequence converges strongly to some y, it follows that . Here, I is the identity mapping of H.
Lemma 2.7 Let H be a real Hilbert space. Then, for all ,
Lemma 2.8 (See [10])
Let H be a Hilbert space, let C be a nonempty subset of H, and let and be the sequences of nonexpansive self-mappings of C. Suppose that and satisfy the condition (R) and that is bounded for any bounded subset D of C. Then satisfies the condition (R).
Lemma 2.9 (See [1])
Let H be a Hilbert space, let C be a nonempty subset of H, and let and be the sequences of nonexpansive self-mappings of C. Suppose that or is a strongly nonexpansive sequence and that is nonempty. Then .
3 Main result
Theorem 3.1 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let and be two bifunctions from into ℝ satisfying (A1)-(A4), respectively, and let and be sequences of nonexpansive self-mappings of C with . Let or be a sequence of strongly nonexpansive mappings, and let be a contractive mapping with . Let , , be sequences generated by and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
;
-
(v)
and satisfy the conditions R and Z.
Then the sequences , , , converge strongly to .
Proof Let . From the definition of , we have
From Lemma 2.5 and (3.1), we have , , and . By and the nonexpansiveness of and , we have
Substituting (3.3) into (3.2), we have
By induction we can conclude that is bounded and so are , , . Next, we show that and , where .
Let be a bounded sequence in C. From the nonexpansiveness of , we have
From (3.4) and as , we have
Let , then we have
From (3.5), we have
which implies that . It follows that
Let , then we have
From (3.5), we have
which implies that . It follows that
From (3.6) and (3.7), we have
Let be a bounded sequence in C, then we have is bounded and so is . Since
and as , we have
Let , then we have
From (3.9), we have
which implies that
It follows that
Let , then we have
From (3.9), we have
which implies that
It follows that
From (3.10) and (3.11), we have
Next, we show that
Since is nonempty, from (3.8), (3.12), we have
and
Suppose that is a strongly nonexpansive sequence. From (3.14) and Lemma 2.9, we have
On the other hand, suppose that is a strongly nonexpansive sequence. From (3.13) and Lemma 2.9, we have
From (3.16) and (3.15), we have . Next, we show that and satisfy the condition (R). It is easy to see that is a nonexpansive mapping for every and that is bounded, where D is a bounded subset of C. Let , then we have
From the condition (i), we have
It follows that satisfies the condition (R). From Lemma 2.8, we have that satisfies the condition (R). From the nonexpansiveness of and , we have is bounded for any bounded subset D of C. From Lemma 2.8, we have that satisfies the condition (R).
Next, we show that
Put
where . From the definition of , we have
where D is a bounded subset of C. Besides, we have
From (3.1) and Lemma 2.5, we have . This implies that
and
Putting in (3.21) and in (3.22), we have
and
Summing up the last two inequalities and using (A2), we obtain
This implies that
Hence,
Then we have
From (3.1) and Lemma 2.5, we have . This implies that
By using the same method as (3.25), we have
Substituting (3.25) and (3.26) into (3.20), we have
where . Substituting (3.27) into (3.19), we have
From (3.28), the conditions (iii), (iv) and satisfying the condition (R), we have
From Lemma 2.3 and the definition of , we have
From the definition of , we have
From (3.30), (3.31) and the condition (ii), we have
Next, we show that
From the definition of , we have
Next, we show that
Let . From the definition of , we have
From the firm nonexpansiveness of and , we have
It implies that
Since is a firmly nonexpansive mapping and , by using the same method as (3.34), we have
Substituting (3.34), (3.35) into (3.33), we have
From (3.36), we have
From the conditions (i), (ii), (iv) and (3.17), we have
By using the method as (3.37), we have
From (3.32), (3.37) and (3.38), we have
Next, we show that
Since
from (3.30) and (3.39), we have
Since is bounded, we have
Since and (3.41), we have (3.40).
Next, we show that
where . From the definition of , we have
From (3.39), (3.40), (3.30) and the condition (i), we have
Next, we show that
where . Since is bounded, there exists a subsequence of converging weakly to v, that is, as . From (3.40), and satisfying the condition (Z), we have .
Define the mapping by
where . From the nonexpansiveness of , and Lemma 2.4, we have
From the definitions of and Q, we have
From (3.39), (3.42) and the condition (iv), we have
From (3.39) and as , we have as . By (3.43), as and Lemma 2.6, we have
Hence,
By (3.40), (3.44) and the condition (i), we have
Finally, we show that the sequence converges strongly to . From the definition of , we have
Since , we have
This implies that
Substituting (3.46) into (3.45), we have
Applying (3.47), the conditions (i), (ii) and Lemma 2.2, we have converges strongly to . From (3.39), (3.37) and (3.38), it is easy to see that , , converge strongly to . This completes the proof. □
4 Applications
In this section, we give three examples for a strongly nonexpansive sequence and prove a strong convergence theorem associated to the variational inequality problem.
Before we give three examples, we need the following definition and lemmas.
Definition 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping is called an α-inverse strongly monotone mapping if there exists an such that
for all .
A mapping is called α-strongly monotone if there exists such that
for all .
A mapping is called a κ-strictly pseudo-contractive mapping if there is such that
for all .
Then (4.1) is equivalent to
for all .
The set of solutions of the variational inequality problem of the mapping is denoted by , that is,
Let be two mappings. In 2013, Kangtunyakarn [15] modified as follows:
Remark 4.1 If is a κ-strictly pseudo-contractive mapping with , then is a -inverse strongly monotone mapping and .
Lemma 4.2 (See [16])
Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let A be a mapping of C into H. Let . Then, for ,
where is the metric projection of H onto C.
Lemma 4.3 (See [15])
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be α and β-inverse strongly monotone mappings, respectively, with and . Then
Furthermore, if , where , we have is a nonexpansive mapping.
Example 4.4 Let be a κ-strictly pseudo-contractive mapping with . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by . Then is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).
Proof Since T is a κ-strictly pseudo-contractive mapping, then is -inverse strongly monotone. From Example 4.3 in [10], we have is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □
Example 4.5 Let be -inverse strongly monotone mappings, respectively, with and . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by , where for all . Then is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).
Proof Let , then we have
Then D is a -inverse strongly monotone mapping. From Example 4.3 in [10], we have that is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □
Example 4.6 Let be an α-strongly monotone and L-Lipschitzian mapping with . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by . Then is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).
Proof Let , then we have
Then A is an -inverse strongly monotone mapping. From Example 4.3 in [10], we have that is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □
Example 4.7 (See [10])
Let be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let be a sequence in . For each , a W-mapping [17] generated by and is defined as follows:
If and , for all and , then satisfies the conditions (R) and (Z).
By using our main result and these three examples, we obtain the following results.
Theorem 4.8 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let and be two bifunctions from into ℝ satisfying (A1)-(A4), respectively. Let be a κ-strictly pseudo-contractive mapping with . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by . Let be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let be a sequence in . For each , is a W-mapping generated by and . Assume that . Let be a contractive mapping with . Let , , be sequences generated by and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
Then the sequences , , , converge strongly to .
Proof From Example 4.4, we have is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemma 4.2, we have for all . It implies that . From [18], we have . It follows that . From Example 4.7, we have is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □
Theorem 4.9 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let and be two bifunctions from into ℝ satisfying (A1)-(A4), respectively. Let be -inverse strongly monotone mappings, respectively, with and . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by , where for all . Let be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let be a sequence in . For each , is a W-mapping generated by and . Assume that . Let be a contractive mapping with . Let , , be sequences generated by and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
Then the sequences , , , converge strongly to .
Proof From Example 4.5, we have is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemmas 4.2 and 4.3, we have for all . It implies that . From [18], we have . It follows that . From Example 4.7, we have is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □
Theorem 4.10 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let and be two bifunctions from into ℝ satisfying (A1)-(A4), respectively. Let be an α-strongly monotone and L-Lipschitzian mapping with . Let be a sequence of positive real numbers such that
and let be a sequence of mappings defined by . Let be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let be a sequence in . For each , is a W-mapping generated by and . Assume that . Let be a contractive mapping with . Let , , be sequences generated by and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
Then the sequences , , , converge strongly to .
Proof From Example 4.6, we have is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemma 4.2, we have for all . It implies that . From [18], we have . It follows that . From Example 4.7, we have is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □
Theorem 4.11 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let be a bifunction from into ℝ satisfying (A1)-(A4), and let and be sequences of nonexpansive self-mappings of C with . Let or be a sequence of strongly nonexpansive mappings, and let be a contractive mapping with . Let , be sequences generated by and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
and satisfy the conditions R and Z.
Then the sequences , converge strongly to .
Proof Put , and . From Theorem 3.1, we can conclude the desired conclusion. □
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
Theorem 4.12 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let be bifunctions from into ℝ, for every , satisfying (A1)-(A4), and let and be sequences of nonexpansive self-mappings of C with . Let or be a sequence of strongly nonexpansive mappings, and let be a contractive mapping with . Let , , be sequences generated by , for every , and
where , . Assume that the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
;
-
(v)
, ;
-
(vi)
and satisfy the conditions R and Z.
Then the sequences , and , for every , converge strongly to .
References
Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.
Aoyama K: An iterative method for fixed point problems for sequences of nonexpansive mappings. In Fixed Point Theory and Applications. Yokohama Publ., Yokohama; 2010:1–7.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63(1–4):123–145.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6(1):117–136.
Kangtunyakarn A: Iterative methods for finding common solution of generalized equilibrium problems and variational inequality problems and fixed point problems of a finite family of nonexpansive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 836714 10.1155/2010/836714
Kangtunyakarn A: Hybrid algorithm for finding common elements of the set of generalized equilibrium problems and the set of fixed point problems of strictly pseudocontractive mapping. Fixed Point Theory Appl. 2011., 2011: Article ID 274820 10.1155/2011/274820
Cholamjiak W, Suantai S: A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 349158 10.1155/2010/349158
Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 528476 10.1155/2008/528476
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331(1):506–515. 10.1016/j.jmaa.2006.08.036
Aoyama K, Kimura Y: Strong convergence theorems for strongly nonexpansive sequences. Appl. Math. Comput. 2011, 217: 7537–7545. 10.1016/j.amc.2011.01.092
Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach space. Arch. Ration. Mech. Anal. 1967, 24: 82–89.
Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116(3):659–678. 10.1023/A:1023073621589
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017
Bruck RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.
Kangtunyakarn A: Convergence theorem of κ -strictly pseudocontractive mapping and a modification of generalized equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 89
Takahashi W: Nonlinear Functional Analysis. Yokohama Publ., Yokohama; 2000.
Takahashi W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Proceedings of Workshop on Fixed Point Theory Kazimierz Dolny 1997, 277–292.
Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 1999, 41: 435–453.
Acknowledgements
This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kangtunyakarn, A. Convergence theorems for a system of equilibrium problems and fixed point problems of a strongly nonexpansive sequence. Fixed Point Theory Appl 2013, 193 (2013). https://doi.org/10.1186/1687-1812-2013-193
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-193