- Research Article
- Open Access
Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems
© J. F. Tan and S. S. Chang. 2011
- Received: 30 October 2010
- Accepted: 9 November 2010
- Published: 25 November 2010
The main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.
- Equilibrium Problem
- Nonexpansive Mapping
- Multivalued Mapping
- Maximal Monotone
- Real Hilbert Space
Throughout this paper, we assume that is a real Hilbert space with inner product and norm , is a nonempty closed convex subset of , and is the set of fixed points of mapping .
The set of solutions to quasivariational inclusion problem (1.2) is denoted by .
This problem is called the Hartman-Stampacchia variational inequality (see ). The set of solutions to variational inequality (1.5) is denoted by .
The set of solutions to (1.7) is denoted by EP.
where is the fixed point set of a nonexpansive mapping on .
Under suitable conditions, they proved the sequence generated by (1.9) converges strongly to the fixed point , which solves the quadratic minimization problem (1.8).
Motivated and inspired by the researches going on in this direction, especially inspired by Zhang et al. , the purpose of this paper is to introduce an explicit iterative algorithm to studying the existence problem and the approximation problem of the solution to the quadratic minimization problem (1.8) and prove some strong convergence theorems for a family of nonexpansive mappings in the setting of Hilbert spaces.
Such a mapping from onto is called the metric projection. It is well-known that the metric projection is nonexpansive.
In the sequel, we use and to denote the weak convergence and the strong convergence of the sequence , respectively.
Proposition 2.2 (see ).
Let be an -inverse strongly monotone mapping. Then, the following statements hold:
(i) is an -Lipschitz continuous and monotone mapping;
(ii)if is any constant in , then the mapping is nonexpansive, where is the identity mapping on .
Lemma 2.3 (see ).
is called the resolvent operator associated with , where is any positive number and is the identity mapping.
A single-valued mapping is said to be hemicontinuous if for any , function is continuous at 0.
It is well-known that every continuous mapping must be hemicontinuous.
Lemma 2.7 (see ).
Lemma 2.8 (see ).
Let be a real Banach space, be the dual space of , be a maximal monotone mapping, and be a hemicontinuous bound monotone mapping with . Then, the mapping is a maximal monotone mapping.
Lemma 2.9 (see ).
Throughout this paper, we assume that the bifunction satisfies the following conditions:
for all ;
for each , is convex and lower semi-continuous.
Lemma 2.10 (see ).
then the following results hold:
(ii)EP is closed and convex, and .
- (i)(see ) is a solution of variational inclusion (1.2) if and only if
- (ii)(see ) is a solution of generalized equilibrium problem (1.6) if and only if
(iii) (see ) Let be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. If and , then is a closed convex subset in and GEP is a closed convex subset in .
Lemma 2.12 (see ).
where is a sequence in and is a sequence such that:
(ii) or .
We divide the proof of Theorem 3.1 into four steps.
Step 1 (The sequence is bounded).
where . This shows that is bounded. Hence, it follows from (3.9) that the sequence and are also bounded.
This shows that is bounded.
Step 3 (sequence converges strongly to ).
Because is bounded, without loss of generality, we can assume that . In view of (3.12), it yields that and . From Lemma 2.9 and (3.30), we know that .
Next, we prove that .
This shows that .
Step 4 (now, we prove that ).
Since is maximal monotone, , that is, .
This completes the proof of Theorem 3.1.
In Theorem 3.1, if , then the following corollary can be obtained immediately.
In Theorem 3.1, if , where is the indicator function of , then the variational inclusion problem (1.2) is equivalent to variational inequality (1.5), that is, to find such that , for all . Since . Consequently, we have the following corollary.
- Noor MA, Noor KI: Sensitivity analysis for quasi-variational inclusions. Journal of Mathematical Analysis and Applications 1999,236(2):290–299. 10.1006/jmaa.1999.6424MATHMathSciNetView ArticleGoogle Scholar
- Chang SS: Set-valued variational inclusions in Banach spaces. Journal of Mathematical Analysis and Applications 2000,248(2):438–454. 10.1006/jmaa.2000.6919MATHMathSciNetView ArticleGoogle Scholar
- Chang S-S: Existence and approximation of solutions for set-valued variational inclusions in Banach space. Nonlinear Analysis. Theory, Methods & Applications 2001,47(1):583–594. 10.1016/S0362-546X(01)00203-6MATHMathSciNetView ArticleGoogle Scholar
- Noor MA: Generalized set-valued variational inclusions and resolvent equations. Journal of Mathematical Analysis and Applications 1998,228(1):206–220. 10.1006/jmaa.1998.6127MATHMathSciNetView ArticleGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MATHMathSciNetGoogle Scholar
- Tang F: Strong convergence theorem for a generalized equilibrium problems and a family of infinitely relatively nonexpansive mappings in a Banach space. Acta Analysis Functionalis Applicata 2010,12(3):259–265.MATHMathSciNetView ArticleGoogle Scholar
- Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022MATHMathSciNetView ArticleGoogle Scholar
- Li S, Li L, Su Y: General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space. Nonlinear Analysis. Theory, Methods & Applications 2009,70(9):3065–3071. 10.1016/j.na.2008.04.007MATHMathSciNetView ArticleGoogle Scholar
- Colao V, Marino G, Xu H-K: An iterative method for finding common solutions of equilibrium and fixed point problems. Journal of Mathematical Analysis and Applications 2008,344(1):340–352. 10.1016/j.jmaa.2008.02.041MATHMathSciNetView ArticleGoogle Scholar
- Zhang S-S, Lee H-W, Chan C-K: Quadratic minimization for equilibrium problem variational inclusion and fixed point problem. Applied Mathematics and Mechanics 2010,31(7):917–928. 10.1007/s10483-010-1326-6MATHMathSciNetView ArticleGoogle Scholar
- Zhang S-S, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Applied Mathematics and Mechanics 2008,29(5):571–581. 10.1007/s10483-008-0502-yMATHMathSciNetView ArticleGoogle Scholar
- Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MATHMathSciNetView ArticleGoogle Scholar
- Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory and Applications 2005, (1):103–123.Google Scholar
- Pascali D: Nonlinear Mappings of Monotone Type. Sijthoff and Noordhoff International Publishers, The Netherlands; 1978.MATHView ArticleGoogle Scholar
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.MATHView ArticleGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MATHMathSciNetGoogle Scholar
- Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MATHMathSciNetView ArticleGoogle Scholar
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