Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities
© Wang and Xu; licensee Springer 2013
Received: 24 November 2012
Accepted: 23 April 2013
Published: 7 May 2013
This article aims to deal with a new modified iterative projection method for solving a hierarchical fixed point problem. It is shown that under certain approximate assumptions of the operators and parameters, the modified iterative sequence converges strongly to a fixed point of T, also the solution of a variational inequality. As a special case, this projection method solves some quadratic minimization problem. The results here improve and extend some recent corresponding results by other authors.
MSC:47H10, 47J20, 47H09, 47H05.
Keywordshierarchical fixed point nonexpansive mapping Lipschitzian and strongly monotone mapping quadratic minimization modified iterative projection algorithm
Let Ω be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . Recall that a mapping is called L-Lipschitzian if there exits a constant L such that , . In particular, if , then T is said to be a contraction; if , then T is called a nonexpansive mapping. We denote by the set of the fixed points of T, i.e., .
In particular, if , then F is said to be monotone.
Recently many authors investigated the fixed point problem of nonexpansive mappings, generalized nonexpansive mappings with C-conditions, a family of finite or infinite nonexpansive mappings and pseudo-contractions and obtained many useful results; see, for example, [1–12] and the references therein.
Now, we focus on the following problem.
which is equivalent to the following fixed point problem: to find an that satisfies . We know that is closed and convex, so the metric projection is well defined.
where h is a potential function for γf, i.e., , .
where S, T are nonexpansive mappings with , U is a γ-Lipschitzian (possibly non-self) mapping, F is an L-Lipschitzian and η-strongly monotone operator. We prove that the sequence generated by (7) converges strongly to the unique solution of the variational inequality (5) if the operators and parameters satisfy some approximate conditions. As a special case, this projection method also solves the quadratic minimization problem .
This section contains some lemmas which will be used in the proofs of our main results in the following section.
Lemma 2.1 
- (1)is nonexpansive and if and only if the following relation holds:
- (2)if and only if the following relation holds:
Lemma 2.2 
Lemma 2.3 
That is to say, the operator is -strongly monotone.
Lemma 2.4  (Demiclosedness principle)
Let Ω be a nonempty closed convex subset of a real Hilbert space H and let be a nonexpansive mapping with . If is a sequence in Ω weakly converging to x and converges strongly to y, then . In particular, if , then .
Lemma 2.5 
Lemma 2.6 
3 Main results
Then the sequence generated by (7) converges strongly to a fixed point of T, which is the unique solution of the variational inequality (5). In particular, if we take , , then defined by (7) converges in norm to the minimum norm fixed point of T, namely, the point is the unique solution to the quadratic minimization problem .
Proof We divide the proof into six steps.
therefore the operator is -strongly monotone, and we get the uniqueness of the solution of the variational inequality (5) and denote it by .
We get the sequence is bounded, and so are , , , .
Notice the conditions (i) and (iii), by Lemma 2.6, we have as .
Notice that , , and are bounded, and we have as .
According to Lemma 2.6, we have .
So, , we deduce , i.e., the point is the unique solution to the quadratic minimization problem . This completes the proof. □
Remark 3.1 Prototypes for the iteration parameters in Theorem 3.1 are, for example, , (with ). It is not difficult to prove that the conditions (i)-(iii) are satisfied.
Some self-mappings in other papers (see [15, 16, 19]) are extended to the cases of non-self-mappings. Such as the self-contraction mapping in [15, 16, 19] is extended to the case of a Lipschitzian (possibly non-self-)mapping on a nonempty closed convex subset C of H. The Lipschitzian and strongly monotone (self-)mapping in  is extended to the case of a Lipschitzian and strongly monotone (possibly non-self-)mapping .
The Mann-type iterative format in [15–17, 19] has been extended to the Ishikawa-type iterative format (7) in our Theorem 3.1. So, their iterative formats (2), (3), (4) and (6) are some special cases of our iterative format (7), and some of their main results have been included in our Theorem 3.1, respectively.
The iterative approximating fixed point of T in Theorem 3.1 is also the unique solution of the variational inequality (5). In fact, (5) is a hierarchical fixed point problem which closely relates to a convex minimization problem. In hierarchical fixed point problem (1), if , then we can get the variational inequality (5). In (5), if , then we get the variational inequality , , which just is the variational inequality studied by Suzuki . If the Lipschitzian mapping , , , we get the variational inequality , , which is the variational inequality studied by Yao et al. . So, the results of Theorem 3.1 in this paper have many useful applications such as the quadratic minimization problem .
The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the article. This study was supported by the National Natural Science Foundations of China (Grant Nos. 11271330, 11071169 ), the Natural Science Foundations of Zhejiang Province of China (Grant No. Y6100696).
- Suzuki T: Fixed point theorems and convergence theorems for some generalized non expansive mappings. J. Math. Anal. Appl. 2008, 340: 1088–1095. 10.1016/j.jmaa.2007.09.023MathSciNetView ArticleGoogle Scholar
- Karapinar E, Tas K: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl. 2011, 61(11):3370–3380. 10.1016/j.camwa.2011.04.035MathSciNetView ArticleGoogle Scholar
- Karapinar E, Yuce IS: Fixed point theory for cyclic generalized weak ϕ -contraction on partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 491542Google Scholar
- Wang YH: Strong convergence theorems for asymptotically weak G -pseudo- ψ -contractive nonself mappings with the generalized projection in Banach spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 651304Google Scholar
- Moudafi A: Viscosity approximation methods for fixed point problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleGoogle Scholar
- Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 2003, 119: 185–201.MathSciNetView ArticleGoogle Scholar
- Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleGoogle Scholar
- Ceng LC, Cubiotti P, Yao JC: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal. 2007, 67: 1464–1473. 10.1016/j.na.2006.06.055MathSciNetView ArticleGoogle Scholar
- Zegeye H, Ofoedu EU, Shahzad N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 2010, 216: 3439–3449. 10.1016/j.amc.2010.02.054MathSciNetView ArticleGoogle Scholar
- Wang YH, Xia YH: Strong convergence for asymptotically pseudo-contractions with the demiclosedness principle in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 45Google Scholar
- Takahashi W, Wong NC, Yao JC: Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces. J. Fixed Point Theory Appl. 2012, 11(1):159–183. 10.1007/s11784-012-0074-3MathSciNetView ArticleGoogle Scholar
- Chang SS, Wang L, Tang YK, Zhao YH, Ma ZL: Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi- ϕ -asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 69Google Scholar
- Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
- Yamada I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam; 2001:473–504.Google Scholar
- Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleGoogle Scholar
- Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058MathSciNetView ArticleGoogle Scholar
- Ceng LC, Anasri QH, Yao JC: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 2011, 74: 5286–5302. 10.1016/j.na.2011.05.005MathSciNetView ArticleGoogle Scholar
- Yao YH, Cho YJ, Liou YC: Iterative algorithms for hierarchical fixed point problems and variational inequalities. Math. Comput. Model. 2010, 52: 1697–1705. 10.1016/j.mcm.2010.06.038MathSciNetView ArticleGoogle Scholar
- Suzuki N: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 2007, 325: 342–352. 10.1016/j.jmaa.2006.01.080MathSciNetView ArticleGoogle Scholar
- Wang YH, Yang L: Modified relaxed extragradient method for a general system of variational inequalities and nonexpansive mappings in Banach spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 818970Google Scholar
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