Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems
© L. Yang and Y. Su. 2009
Received: 21 October 2008
Accepted: 5 March 2009
Published: 17 March 2009
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung (2008) and some others in some respects.
1. Introduction and Preliminaries
where denotes the metric projection from onto a closed convex subset of . The iterative method (1.3) is said to be hybrid method or method. In recent years, the hybrid method (1.3) has been modified by many authors for other nonlinear operators [2, 5–10].
In 2008, Nilsrakoo and Saejung  used the hybrid method to obtain a strong convergence theorem for countable Lipschitzian mappings as follows.
as . Let for any bounded subset of , and let be a mapping of into itself defined by , for all and suppose that , then converges strongly to .
Nilsrakoo and Saejung also apply the aforementioned result to obtain an applied result for equilibrium problems.
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the Nilsrakoo and Saejung results in some respects.
The following lemma is well known.
Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence of -Lipschitzian mappings from into itself with . is said to satisfy the (SU) condition, if the following conditions hold:
(1)for any strong convergence sequence , the sequence is also strong convergent;
(2)the common fixed points set is nonempty;
(3) , where is defined by , for all , denotes the fixed points set of .
In Section 3 of this paper, we will give an important example of sequence of -Lipschitzian mappings which satisfies the (SU) condition.
Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence of -Lipschitzian mappings from into itself with . If satisfies the (SU) condition. Then
(1) is bounded implies is uniformly continuous on the ;
(2) implies is -Lipschitzian;
(3) implies is nonexpansive.
The results (1)–(3) are easy to prove.
2. Main Results
Then converges strongly to , where is the metric projection from onto closed convex subset .
for all . It follows from the definition of that . By using the principle of induction, we claim that , for all . Therefore, we have , for all . Now the sequence is well defined.
for all . Therefore, is nondecreasing and bounded. So that exists.
From this inequality, we know that is a Cauchy sequence in , so that there exists a point such that .
as . Then .
It follows that implies that This is a contradiction, hence . This completes the proof.
The following theorem directly follows from Theorem 2.1.
Then converges strongly to . Where is the metric projection from onto closed convex subset .
3. Application for Equilibrium and Optimization
The set of solutions of (3.1) is denoted by . Given a mapping , let , for all . Then, if and only if , for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (3.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [12–16].
For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1) , for all ;
(A2) is monotone, that is, , for all ;
(A4)for each , is convex and lower semicontinuous.
We need the following lemmas for the proof of our main results.
for all . Then, the following hold:
(1) is single-valued;
(4) is closed and convex.
is also nonexpansive, for all .
Now, we prove the following lemma which is very important for the main results of this section.
Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let be a positive real sequence such that . Then the sequence satisfies the (SU) condition.
- (1)Let be a convergent sequence in . Let , for all , then(3.6)
- (2)By using Lemma 3.2, we know that(3.15)
- (3)From (1) we know that, exists, for all . So, we can define a mapping from into itself by(3.16)
and hence . From the aforementioned two respects, we know that . This completes the proof.
Assume and . Then converges strongly to , where is the metric projection from onto .
By using Lemma 3.4, we know that satisfies the (SU) condition. Then . By using Theorem 2.2, we obtain the result of Theorem 3.5
Now, we study a kind of optimization problem by using the aforementioned results of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
It is obvious that , where denotes the set of solutions of equilibrium Problem (3.25). In addition, it is easy to see that satisfies the conditions (A1)–(A4) in Section 2. Therefore, from Theorem 3.5, we can obtain the following theorem.
Assume and . Then converges strongly to .
Remark 3.7 ..
It is easy to see that this paper hassome new methods and results than the results of Nilsrakoo and Saejung :
(1)proposed a modified hybrid iterative scheme, so that the new simple method of proof has been used;
(2)removed the bounded restriction for closed convex set ;
(3)relax the conditions of sequence ;
(4)give an application for optimization problem;
so that to raise the convergence rate of is possible.
This project is supported by the National Natural Science Foundation of China under grant(10771050).
- Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953,4(3):506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(6):1958–1965. 10.1016/j.na.2006.08.021MathSciNetView ArticleMATHGoogle Scholar
- Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
- Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1140–1152. 10.1016/j.na.2005.05.059MathSciNetView ArticleMATHGoogle Scholar
- Nakajo K, Shimoji K, Takahashi W: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese Journal of Mathematics 2006,10(2):339–360.MathSciNetMATHGoogle Scholar
- Su Y, Qin X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3657–3664. 10.1016/j.na.2007.04.008MathSciNetView ArticleMATHGoogle Scholar
- Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, Article ID 284613, 2008:-8.Google Scholar
- Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, Article ID 528476, 2008:-11.Google Scholar
- Cheng Y, Tian M: Strong convergence theorem by monotone hybrid algorithm for equilibrium problems, hemirelatively nonexpansive mappings, and maximal monotone operators. Fixed Point Theory and Applications 2008, Article ID 617248, 2008:-12.Google Scholar
- Nilsrakoo W, Saejung S: Weak and strong convergence theorems for countable Lipschitzian mappings and its applications. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2695–2708. 10.1016/j.na.2007.08.044MathSciNetView ArticleMATHGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MathSciNetMATHGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MathSciNetMATHGoogle Scholar
- Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar
- Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar
- Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.