Non-convex hybrid algorithm for a family of countable quasi-Lipschitz mappings and application
- Jinyu Guan^{1},
- Yanxia Tang^{1},
- Pengcheng Ma^{1},
- Yongchun Xu^{1}Email author and
- Yongfu Su^{2}
https://doi.org/10.1186/s13663-015-0457-4
© Guan et al. 2015
Received: 28 July 2015
Accepted: 5 November 2015
Published: 21 November 2015
Abstract
The purpose of this article is to establish a kind of non-convex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Meanwhile, the main result is applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. It is worth pointing out that a non-convex hybrid iteration algorithm is first presented in this article, a new technique is applied in our process of proof. Finally, an example is given which is a uniformly closed asymptotically family of countable quasi-Lipschitz mappings. The results presented in this article are interesting extensions of some current results.
Keywords
MSC
1 Introduction
Construction of fixed points of nonexpansive mappings (and asymptotically nonexpansive mappings) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas. Recently, a great deal of literature on iteration algorithms for approximating fixed points of nonexpansive mappings has been published since one has a variety of applications in inverse problem, image recovery, and signal processing; see [1–8]. Mann’s iteration process [1] is often used to approximate a fixed point of the operators, but it has only weak convergence (see [3] for an example). However, strong convergence is often much more desirable than weak convergence in many problems that arise in infinite dimensional spaces (see [7] and references therein). So, attempts have been made to modify Mann’s iteration process so that strong convergence is guaranteed (see [9–24] and references therein).
In 2003, Nakajo and Takahashi [25] proposed a modification of Mann iteration method for a single nonexpansive mapping in a Hilbert space. In 2006, Kim and Xu [26] proposed a modification of Mann iteration method for asymptotically nonexpansive mapping T in a Hilbert space. They also proposed a modification of the Mann iteration method for asymptotically nonexpansive semigroup in a Hilbert space. In 2006, Martinez-Yanes and Xu [27] proposed a modification of the Ishikawa iteration method for nonexpansive mapping in a Hilbert space. Martinez-Yanes and Xu [27] proposed also a modification of the Halpern iteration method for nonexpansive mapping in a Hilbert space. In 2008, Su and Qin [28] proposed first a monotone hybrid iteration method for nonexpansive mapping in a Hilbert space. In 2015, Dong and Lu [29] proposed a new iteration method for nonexpansive mapping in a Hilbert space. In 2015, Liu et al. [30] proposed a new iteration method for a finite family of quasi-asymptotically pseudocontractive mappings in a Hilbert spaces.
- (1)
the fixed point set \(F(T)\) is nonempty;
- (2)
\(\|Tx-p\|\leq L\|x-p\|\) for all \(x \in C\), \(p \in F(T)\),
Recall that a mapping \(T:C\rightarrow C\) is said to be closed if \(x_{n}\rightarrow x\) and \(\|Tx_{n}-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\) implies \(Tx=x\). A mapping \(T:C\rightarrow C\) is said to be weak closed if \(x_{n}\rightharpoonup x\) and \(\|Tx_{n}-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\) implies \(Tx=x\). It is obvious that a weak closed mapping must be a closed mapping, the inverse is not true.
Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let \(\{T_{n}\}\) be sequence of mappings from C into itself with a nonempty common fixed point set F. \(\{T_{n}\}\) is said to be uniformly closed if for any convergent sequence \(\{z_{n}\} \subset C\) such that \(\|T_{n}z_{n}-z_{n}\|\rightarrow0\) as \(n\rightarrow\infty\), the limit of \(\{z_{n}\}\) belongs to F.
The purpose of this article is to establish a kind of non-convex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Meanwhile, the main result was applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. It is worth pointing out that a non-convex hybrid iteration algorithm was first presented in this article, a new technique has been applied in our process of proof. Finally, an example has been given which is a uniformly closed asymptotically family of countable quasi-Lipschitz mappings. The results presented in this article are interesting extensions of some current results.
2 Main results
The following lemma is well known and is useful for our conclusions.
Lemma 2.1
Definition 2.2
Let H be a Hilbert space, let C be a closed convex subset of E, and let \(\{T_{n}\}\) be a family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself, \(\{T_{n}\}\) is said to be asymptotically, if \(\lim_{n\rightarrow\infty}L_{n}=1\).
Lemma 2.3
Let H be a Hilbert space, let C be a closed convex subset of E, and let \(\{T_{n}\}\) be a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself. Then the common fixed point set F is closed and convex.
Proof
The following conclusion is well known.
Lemma 2.4
Theorem 2.5
Proof
We split the proof into seven steps.
Next, we give an example of \(C_{n}\) not involving a convex subset.
Example 2.6
Corollary 2.7
Proof
Take \(T_{n}\equiv T\), \(L_{n}\equiv1\) in Theorem 2.5, in this case, \(C_{n}\) is closed and convex, for all \(n\geq0\), by using Theorem 2.5, we obtain Corollary 2.7. □
Since a nonexpansive mapping must be a closed quasi-nonexpansive mapping, from Corollary 2.7, we obtain the following result.
Corollary 2.8
3 Application to family of quasi-asymptotically nonexpansive mappings
Theorem 3.1
Proof
It is sufficient to prove the following two conclusions.
Conclusion 1
\(\{T_{i(n)}^{j(n)}\}_{n=0}^{\infty}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself.
Conclusion 2
\(F=\bigcap_{n=0}^{N}F(T_{n})=\bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), where \(F(T)\) denotes the fixed point set of the mapping T.
Proof of Conclusion 1
Proof of Conclusion 2
By using Theorem 2.5, the iterative sequence \(\{x_{n}\}\) converges strongly to \(P_{\bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})}x_{0}=P_{F}x_{0}\). This completes the proof of Theorem 3.1. □
Corollary 3.2
Since a nonexpansive mapping must be a Lipschitz asymptotically quasi-nonexpansive mapping, from Corollary 3.2, we can obtain Corollary 2.8.
4 Example
Conclusion 4.1
Proof
Remark
In the result of Liu et al. [30], the boundedness of C was assumed and the hybrid iterative process was complex. In our hybrid iterative process, \(C_{n}\) was constructed as a non-convex set can makes it more simple, meanwhile, the boundedness of C can be removed. Of course, a new technique has been applied in our process of proof.
Declarations
Acknowledgements
This project is supported by major project of Hebei North University under grant (No. ZD201304).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953) MATHView ArticleGoogle Scholar
- Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967) MATHView ArticleGoogle Scholar
- Genel, A, Lindenstrass, J: An example concerning fixed points. Isr. J. Math. 22, 81-86 (1975) MATHView ArticleGoogle Scholar
- Youla, D: Mathematical theory of image restoration by the method of convex projection. In: Stark, H (ed.) Image Recovery: Theory and Applications, pp. 29-77. Academic Press, Orlando (1987) Google Scholar
- Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000) MATHMathSciNetView ArticleGoogle Scholar
- Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279-291 (2004) MATHMathSciNetView ArticleGoogle Scholar
- Bauschke, HH, Combettes, PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26(2), 248-264 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Podilchuk, CI, Mammone, RJ: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. 7(3), 517-521 (1990) View ArticleGoogle Scholar
- Matsushita, S-Y, Takahashi, W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257-266 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Dekker, New York (1996) Google Scholar
- Alber, YI, Reich, S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panam. Math. J. 4, 39-54 (1994) MATHMathSciNetGoogle Scholar
- Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002) MathSciNetView ArticleGoogle Scholar
- Cioranescu, I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990) MATHView ArticleGoogle Scholar
- Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
- Butnariu, D, Reich, S, Zaslavski, AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151-174 (2001) MATHMathSciNetGoogle Scholar
- Butnariu, D, Reich, S, Zaslavski, AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 24, 489-508 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Censor, Y, Reich, S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323-339 (1996) MATHMathSciNetView ArticleGoogle Scholar
- Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970) MATHMathSciNetView ArticleGoogle Scholar
- Takahashi, W: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama (2000) (in Japanese) Google Scholar
- Kohsaka, F, Takahashi, W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 2004, 239-249 (2004) MATHMathSciNetView ArticleGoogle Scholar
- Ohsawa, S, Takahashi, W: Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces. Arch. Math. 81, 439-445 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Reich, S: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis (Proceedings of the Third International Conference, University of Texas, Arlington, TX, 1978), pp. 335-345. Academic Press, New York (1979) Google Scholar
- Reich, S: A weak convergence theorem for the alternating method with Bregman distance. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 313-318. Dekker, New York (1996) Google Scholar
- Solodov, MV, Svaiter, BF: Forcing strong convergence of proximal point iterations in Hilbert space. Math. Program. 87, 189-202 (2000) MATHMathSciNetGoogle Scholar
- Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Kim, T-H, Xu, H-K: Strong convergence of modified Mann iterations for asymptotically mappings and semigroups. Nonlinear Anal. 64, 1140-1152 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Martinez-Yanes, C, Xu, H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400-2411 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Su, Y, Qin, X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal. 68, 3657-3664 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Dong, Q, Lu, Y: A new hybrid algorithm for a nonexpansive mapping. Fixed Point Theory Appl. 2015, 37 (2015) MathSciNetView ArticleGoogle Scholar
- Liu, Y, Zheng, L, Wang, P, Zhou, H: Three kinds of new hybrid projection methods for a finite family of quasi-asymptotically pseudocontractive mappings in Hilbert spaces. Fixed Point Theory Appl. 2015, 118 (2015) MathSciNetView ArticleGoogle Scholar