- Research
- Open access
- Published:
An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities
Fixed Point Theory and Applications volume 2013, Article number: 143 (2013)
Abstract
In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings and the set of solutions of new variational inequalities problems in Hilbert space. By using our main results, we obtain an interesting theorem involving a finite family of κ-strictly pseudo-contractive mappings and two sets of solutions of the variational inequalities problem.
1 Introduction
Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. A mapping is called nonexpansive if
for all .
A mapping S is called a κ-strictly pseudo-contractive mapping if there exists such that
for all .
It is easy to see that every noexpansive mapping is a κ-strictly pseudo-contractive mapping.
Let . The variational inequality problem is to find a point such that
for all . The set of solutions of (1.1) is denoted by .
Variational inequalities were initially studied by Kinderlehrer and Stampacchia [1] and Lions and Stampacchia [2]. Such a problem has been studied by many researchers, and it is connected with a wide range of applications in industry, finance, economics, social sciences, ecology, regional, pure and applied sciences; see, e.g., [3–9].
A mapping A of C into H is called α-inverse-strongly monotone, see [10], if there exists a positive real number α such that
for all .
Let be two mappings. In 2008, Ceng et al. [11] introduced a problem for finding such that
which is called a system of variational inequalities where . By a modification of (1.2), we consider the problem for finding such that
which is called a modification of system of variational inequalities, for every and . If , (1.3) reduce to (1.2).
In 2008, Ceng et al. [11] introduce and studied a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space as follows.
Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mappings be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let be a nonexpansive mapping such that , where Ω is the set of fixed points of the mapping , defined by , for all . Suppose that and is generated by
where , and , , are three sequences in such that
Then converges strongly to and is a solution of problem (1.2), where .
In the last decade, many author studied the problem for finding an element of the set of fixed points of a nonlinear mapping; see, for instance, [12–14].
From the motivation of [11] and the research in the same direction, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings and the set of solutions of a modified general system of variational inequalities problems. Moreover, in the last section, we prove an interesting theorem involving the set of a finite family of -strictly pseudo-contractive mappings and two sets of solutions of variational inequalities problems by using our main results.
2 Preliminaries
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Let C be a closed convex subset of a real Hilbert space H, let be the metric projection of H onto C, i.e., for , satisfies the property
It is well known that is a nonexpansive mapping and satisfies
Obviously, this immediately implies that
The following characterizes the projection .
Lemma 2.1 (See [15])
Given and . Then if and only if the following inequality holds:
Lemma 2.2 (See [16])
Let be a sequence of nonnegative real numbers satisfying
where , satisfy the conditions
Then .
Lemma 2.3 (See [17])
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose that
for all integer and
Then .
Definition 2.1 (See [18])
Let C be a nonempty convex subset of a real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself. For each , let , where and . Define the mapping as follows:
This mapping is called S-mapping generated by and .
Lemma 2.4 (See [18])
Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Then and S is a nonexpansive mapping.
Lemma 2.5 (See [19])
Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and let be a nonexpansive mapping. Then is demi-closed at zero.
Lemma 2.6 In a real Hilbert space H, the following inequality holds:
for all .
Lemma 2.7 Let C be a nonempty closed convex subset of a Hilbert space H and let be mappings. For every and , the following statements are equivalent:
-
(a)
is a solution of problem (1.3),
-
(b)
is a fixed point of the mapping , i.e., , defined by
where .
Proof (a) ⇒ (b) Let be a solution of problem (1.3). For every and , we have
From the properties of , we have
It implies that
Hence, we have , where .
(b) ⇒ (a) Let and . Then, we have
From the properties of , we have
Hence, we have is a solution of (1.3). □
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively. Define the mapping by for all , and . Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Suppose that and let be the sequence generated by
where , and , , are sequences in . Assume that the following conditions hold:
Then converges strongly to and is a solution of (1.3), where .
Proof First, we show that and are nonexpansive mappings for every , . Let . Since is -inverse strongly monotone and , we have
Thus is a nonexpansive mapping. By using the same method as (3.2), we have is a nonexpansive mapping. Hence, , are nonexpansive mappings. It is easy to see that the mapping G is a nonexpansive mapping. Let . Then we have and
Put and , we can rewrite (3.1) by
and .
From the definition of , we have
By induction we can conclude that for all . It implies that is bounded and so are and .
Next, we show that .
Let
where .
Since and (3.3), we have
It follows that
From conditions (ii) and (iii), we have
From Lemma 2.3 and (3.3) we have . Since , then we have
From the definition of , we have
From (3.4), we obtain
From the definition of , we have
From (3.4), conditions (ii) and (iii), we have
From the definition of , we have
From (3.4) and (3.7), we derive
From the nonexpansiveness of and , we have
It implies that
From (3.4), (3.9) conditions (ii) and (iii), we have
Put and . From the definition of , we have
which implies that
From (3.4), (3.11), conditions (ii) and (iii), we can conclude
Next, we show that
From the definition of , we have
It implies that
From the nonexpansiveness of and (3.14), we have
It follows that
From condition (ii), (3.4) and (3.10), we have
From the definition of , , , , we have
and
From the properties of , we have
From (3.6), (3.12) and (3.15), we have
Since
and (3.15), (3.16), then we have
From (3.6) and (3.17), we can conclude that
Next we show that
where . To show this inequality, take a subsequence of such that
Without loss of generality, we may assume that as , where . From (3.17), we have as . From Lemma 2.5 and (3.13), we have
From Lemma 2.4, we have . Then we obtain
From the nonexpansiveness of the mapping G and the definition of , we have
From (3.17), we have
From as , (3.19) and Lemma 2.5, we have
Hence, we can conclude that .
Since as and , we have
From the definition of and , we have
From condition (ii), (3.18) and Lemma 2.2, we can conclude that the sequence converges strongly to . This completes the proof. □
Remark 3.2 (1) If we take , then the iterative scheme (3.1) reduces to the following scheme:
which is an improvement to (1.4). From Theorem 3.1, we obtain that the sequence generated by (3.21) converges strongly to , where the mapping defined by for all and is a solution of (1.2) where .
(2) If we take , and , then the iterative scheme (3.1) reduces to the following scheme:
From Theorem 3.1, we obtain that the sequence generated by (3.22) converges strongly to , where the mapping defined by for all and is a solution of (1.3) where .
4 Applications
In this section we prove a strong convergence theorem involving variational inequalities problems by using our main result. We need the following lemmas to prove the desired results.
Lemma 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be nonexpansive mappings. Define a mapping by for every and . Then and is a nonexpansive mapping.
Proof It is easy to see that . Let and . By the definition of , we have
From (4.1), it implies that
Then we have , that is, . By the definition of , we have
It follows that . Then we have . Hence .
Next, we show that is a nonexpansive mapping. Let , since
Then we have is a nonexpansive mapping. □
Lemma 4.2 (See [15])
Let H be a real Hibert 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 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively, which . Define a mapping as in Lemma 2.7 for every , and . Then .
Proof First, we show that , are nonexpansive. Let . Since is -inverse strongly monotone and , we have
Thus is nonexpansive. By using the same method as (4.3), we have is a nonexpansive mapping. Hence , are nonexpansive mappings. From
for every and Lemma 4.1, we have
From Lemma 4.2, we have
□
Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively. Define the mapping by for all , and . Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Suppose that and let be a sequence generated by
where , and , , are sequences in . Assume that the following conditions hold:
Then converges strongly to .
Proof From Lemma 4.3 and Theorem 3.1 we can conclude the desired conclusion. □
References
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.
Lions JL, Stampacchia G: Variational inequalities. Commun. Pure Appl. Math. 1967, 20: 493–517.
Chang SS, Joseph Lee HW, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319.
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradientmethod for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006, 128: 191–201.
Yao JC, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Monotonicity. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York; 2005:501–558.
Yao Y, Yao JC: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 2007, 186(2):1551–1558.
Ceng LC, Yao JC: Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense. Acta Appl. Math. 2011, 115: 167–191.
Sahu DR, Wong NC, Yao JC: A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Control Optim. 2012, 50: 2335–2354.
Zeng LC, Ansari QH, Wong NC, Yao JC: An extragradient-like approximation method for variational inequalities and fixed point problems. Fixed Point Theory Appl. 2011., 2011: Article ID 22. doi:10.1186/1687–1812–2011–22
Iiduka H, Takahashi W: Weak convergence theorem by Ces‘aro means for nonexpansive mappings and inverse-strongly monotone mappings. J. Nonlinear Convex Anal. 2006, 7: 105–113.
Ceng LC, Wang CY, Yao JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008, 67: 375–390.
Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 2010, 11: 283–308.
Ceng LC, Petruşel A, Yao JC: Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense. Taiwan. J. Math. 2011, 15: 587–606.
Ceng LC, Shyu DS, Yao JC: Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings. Fixed Point Theory Appl. 2009., 2009: Article ID 402602. doi:10.1155/2009/402602
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.
Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116(3):659–678.
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: 227–239.
Kangtunyakarn A, Suantai S: Strong convergence of a new iterative scheme for a finite family of strict pseudo-contractions. Comput. Math. Appl. 2010, 60: 680–694.
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.
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. An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities. Fixed Point Theory Appl 2013, 143 (2013). https://doi.org/10.1186/1687-1812-2013-143
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-143