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.
Keywords:hierarchical 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., .
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.
It is well known that the iterative methods for finding hierarchical fixed points of nonexpansive mappings can also be used to solve a convex minimization problem; see, for example, [13,14] and the references therein. In 2006, Marino and Xu  considered the following general iterative method:
where f is a contraction, T is a nonexpansive mapping, A is a bounded linear strongly positive operator: , , for some . And it is proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (2) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
In 2010, Tian  introduced the general steepest-descent method
where F is an L-Lipschitzian and η-strongly monotone operator. Under certain approximate conditions, the sequence generated by (3) converges strongly to a fixed point of T, which solves the variational inequality
Very recently, Ceng et al. investigated the following iterative method:
where U is a Lipschitzian (possibly non-self) mapping, and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (4) converges strongly to the unique solution of the variational inequality
On the other hand, in 2010, Yao et al. investigated an iterative method for a hierarchical fixed point problem by
Motivated and inspired by the above research work, we introduce the following modified iterative method for a hierarchical fixed point problem:
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 2.4 (Demiclosedness principle)
Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand letbe a nonexpansive mapping with. Ifis a sequence in Ω weakly converging toxandconverges strongly toy, then. In particular, if, then.
3 Main results
Theorem 3.1Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand letbe any given initial guess. Letbe nonexpansive mappings such that. Letbe anL-Lipschitzian andη-strongly monotone (possibly non-self) operator with coefficients. Letbe aγ-Lipschitzian (possibly non-self) mapping with a coefficient. Suppose the parameters satisfy, , where. And suppose the sequencessatisfy the following conditions:
Then the sequencegenerated by (7) converges strongly to a fixed pointofT, which is the unique solution of the variational inequality (5). In particular, if we take, , thendefined by (7) converges in norm to the minimum norm fixed pointof T, namely, the pointis the unique solution to the quadratic minimization problem.
Proof We divide the proof into six steps.
Together with (8) and (9), we have
where M is a constant such that
Substituting (10) into (11), we obtain
which implies that
(a) 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 .
(c) 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.
(d) 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 declare that they have no competing interests.
All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
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.. 340, 1088–1095 (2008). Publisher Full Text
Karapinar, E, Tas, K: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl.. 61(11), 3370–3380 (2011). Publisher Full Text
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, Article ID 651304 (2012)
Moudafi, A: Viscosity approximation methods for fixed point problems. J. Math. Anal. Appl.. 241, 46–55 (2000). Publisher Full Text
Matsushita, S, Takahashi, W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory. 134, 257–266 (2005). Publisher Full Text
Ceng, LC, Cubiotti, P, Yao, JC: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal.. 67, 1464–1473 (2007). Publisher Full Text
Zegeye, H, Ofoedu, EU, Shahzad, N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput.. 216, 3439–3449 (2010). Publisher Full Text
Takahashi, W, Wong, NC, Yao, JC: Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces. J. Fixed Point Theory Appl.. 11(1), 159–183 (2012). Publisher Full Text
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, Article ID 69 (2012)
Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl.. 116, 659–678 (2003). Publisher Full Text
Yamada, I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S (eds.) Inherently Parallel Algorithms and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001)
Marino, G, Xu, HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl.. 318, 43–52 (2006). Publisher Full Text
Tian, M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal.. 73, 689–694 (2010). Publisher Full Text
Ceng, LC, Anasri, QH, Yao, JC: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal.. 74, 5286–5302 (2011). Publisher Full Text
Yao, YH, Cho, YJ, Liou, YC: Iterative algorithms for hierarchical fixed point problems and variational inequalities. Math. Comput. Model.. 52, 1697–1705 (2010). Publisher Full Text
Suzuki, N: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl.. 325, 342–352 (2007). Publisher Full Text
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, Article ID 818970 (2012)