The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces
- Ping Luo^{1},
- Gang Cai^{1}Email author and
- Yekini Shehu^{2}
https://doi.org/10.1186/s13660-017-1426-8
© The Author(s) 2017
Received: 10 October 2016
Accepted: 22 May 2017
Published: 28 June 2017
Abstract
The aim of this paper is to introduce a viscosity iterative algorithm for the implicit midpoint rule of nonexpansive mappings in uniformly smooth spaces. Under some appropriate conditions on the parameters, we prove some strong convergence theorems. As applications, we apply our main results to solving fixed point problems of strict pseudocontractive mappings, variational inequality problems in Banach spaces and equilibrium problems in Hilbert spaces. Finally, we give some numerical examples for supporting our main results.
Keywords
strong convergence nonexpansive mapping implicit midpoint rule uniformly smooth Banach spaceMSC
49H09 47H10 47J201 Introduction
Question 1. Can we extend and improve the main results of Xu et al. [11] from Hilbert space to general Banach space? For example we might consider a uniformly smooth Banach space.
Question 2. We note that the proof of step 6 in Theorem 3.1 of [11] is very complicated. Can we simplify it?
In this paper, we give the affirmative answers to the above two questions. More precisely, we investigate the viscosity iterative algorithm (1.4) for the implicit midpoint rule of a nonexpansive mapping in a real uniformly smooth space. Under some suitable conditions on the parameters, we prove some strong convergence theorems. We also apply our main results to solve fixed point problems for strict pseudocontractive mappings, variational inequality problems in Banach spaces and equilibrium problems in Hilbert spaces.
2 Preliminaries
The following lemmas are fundamental in the proof of our main results of this section.
Lemma 2.1
[1]
- (i)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\), and
- (ii)
either \(\limsup_{n\rightarrow\infty}\frac{\delta_{n}}{\alpha _{n}}\leq0\) or \(\sum_{n=1}^{\infty} \vert \delta_{n} \vert <\infty\).
Then \(\lim_{n\rightarrow\infty}a_{n}=0\).
Lemma 2.2
[1]
Lemma 2.3
[3]
3 Main results
Theorem 3.1
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\),
- (ii)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\),
- (iii)
either \(\sum_{n=0}^{\infty} \vert \alpha_{n+1}-\alpha _{n} \vert <\infty\) or \(\lim_{n\rightarrow\infty}\frac{\alpha _{n+1}}{\alpha_{n}}=1\).
Proof
It is well known that Hilbert space is uniformly smooth, then we obtain the main results of [11].
Corollary 3.2
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\),
- (ii)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\),
- (iii)
either \(\sum_{n=0}^{\infty} \vert \alpha_{n+1}-\alpha _{n} \vert <\infty\) or \(\lim_{n\rightarrow\infty}\frac{\alpha _{n+1}}{\alpha_{n}}=1\).
4 Applications
(I) Application to fixed point problems for strict pseudocontractive mappings.
Now we give a relationship between strict pseudocontractive mapping and nonexpansive mapping.
Lemma 4.1
[12]
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E and \(T:C\rightarrow C\) be a λ-strict pseudocontractive mapping. For \(\alpha\in(0,1)\), we define \(T_{\alpha}x:=(1-\alpha)x+\alpha Tx\). Then, as \(\alpha\in(0,\frac{\lambda }{K^{2}}]\), where K is the 2-uniformly smooth constant. Then \(T_{\alpha}: C\rightarrow C\) is nonexpansive such that \(F(T_{\alpha})=F(T)\).
Using Theorem 3.1 and Lemma 4.1, we obtain the following results.
Theorem 4.1
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\),
- (ii)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\),
- (iii)
either \(\sum_{n=0}^{\infty} \vert \alpha_{n+1}-\alpha _{n} \vert <\infty\) or \(\lim_{n\rightarrow\infty}\frac{\alpha _{n+1}}{\alpha_{n}}=1\).
(II) Application to variational inequality problems in Banach spaces.
The following lemmas are very important for proving our main results.
Lemma 4.2
[14]
Lemma 4.3
[14]
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let \(Q_{C}\) be the sunny nonexpansive retraction from E onto C. Let \(A,B:C\rightarrow E\) be two nonlinear mappings. For given \(x^{*},y^{*}\in C\), \((x^{*},y^{*})\) is a solution of problem (4.6) if and only if \(x^{*}=Q_{C}(y^{*}-\lambda Ay^{*})\) where \(y^{*}=Q_{C}(x^{*}-\mu Bx^{*})\), that is, \(x^{*}=Gx^{*}\), where G is defined by Lemma 4.2.
Theorem 4.2
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\),
- (ii)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\),
- (iii)
either \(\sum_{n=0}^{\infty} \vert \alpha_{n+1}-\alpha _{n} \vert <\infty\) or \(\lim_{n\rightarrow\infty}\frac{\alpha _{n+1}}{\alpha_{n}}=1\).
Proof
By Lemma 4.2, we see that G is nonexpansive. So we obtain the desired results by Theorem 3.1 immediately. □
(III) Application to equilibrium problems in Hilbert spaces.
- (A1)
\(\phi(x,x)=0\) for all \(x\in C\);
- (A2)
ϕ is monotone, that is, \(\phi(x,y)+\phi(y,x)\leq0\) for all \(x,y\in C\);
- (A3)ϕ is upper-hemicontinuous, i.e., for any \(x,y,z\in C\)$$\limsup_{t\rightarrow0^{+}}\phi\bigl(tz+(1-t)x,y\bigr)\leq\phi(x,y); $$
- (A4)
\(\phi(x,\cdot)\) is convex and weakly lower semicontinuous for each \(x\in C\).
In order to prove our main results, we need the following lemmas.
Lemma 4.4
[15]
Lemma 4.5
[16]
- (1)
\(T_{r}\) is single-valued.
- (2)
\(T_{r}\) is firmly nonexpansive, i.e., for any \(x,y\in H\), \(\Vert T_{r}x-T_{r}y \Vert ^{2}\leq \langle T_{r}x-T_{r}y,x-y \rangle\).
- (3)
\(F(T_{r})=EP(\phi)\), \(\forall r>0\).
- (4)
\(EP(\phi)\) is a closed and convex set.
Lemma 4.6
[17]
Suppose that E is strictly convex, \(T_{1}\) an attracting nonexpansive and \(T_{2}\) a nonexpansive mapping which have a common fixed point. Then we have \(F(T_{1}T_{2})=F(T_{2}T_{1})=F(T_{1})\cap F(T_{2})\).
Theorem 4.3
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\),
- (ii)
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\),
- (iii)
either \(\sum_{n=0}^{\infty} \vert \alpha_{n+1}-\alpha _{n} \vert <\infty\) or \(\lim_{n\rightarrow\infty}\frac{\alpha _{n+1}}{\alpha_{n}}=1\).
Proof
5 Numerical examples
In the last section, we give two numerical examples where our main results may be applied.
Example 5.1
Remark 5.2
By Figure 1, we know that \(\{y_{n} \}\) converges to 0 more quickly than \(\{x_{n} \}\). So the rate of convergence of viscosity implicit midpoint rule (3.1) is better than viscosity iterative algorithm (1.2).
Example 5.3
Declarations
Acknowledgements
This work was supported by the Training Programs of Famous Teachers in Chongqing Normal University (NO.02030307-00047) and the Key Project of Teaching Reforms for Postgraduates in Chongqing (NO.yjg20162006).
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
- Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279-291 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Moudafi, A: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241, 46-55 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Song, Y, Chen, R, Zhou, H: Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Anal. 66, 1016-1024 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Jung, JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 302, 509-520 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Ceng, LC, Xu, HK, Yao, JC: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Anal. 69, 1402-1412 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Zegeye, H, Shahzad, N: Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings. Nonlinear Anal. 68, 2005-2012 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Sunthrayuth, P, Kumam, P: Viscosity approximation methods base on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces. Math. Comput. Model. 58, 1814-1828 (2013) View ArticleMATHGoogle Scholar
- Bader, G, Deuflhard, P: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 41, 373-398 (1983) MathSciNetView ArticleMATHGoogle Scholar
- Deuflhard, P: Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev. 27(4), 505-535 (1985) MathSciNetView ArticleMATHGoogle Scholar
- Somalia, S: Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 79(3), 327-332 (2002) MathSciNetView ArticleGoogle Scholar
- Xu, HK, Aoghamdi, MA, Shahzad, N: The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2015, 41 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Zhou, H: Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal. 69, 3160-3173 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Ceng, LC, Wang, C, Yao, JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375-390 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Cai, G, Bu, S: Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces. Math. Comput. Model. 55, 538-546 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) MathSciNetMATHGoogle Scholar
- Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 6, 117-136 (2005) MathSciNetMATHGoogle Scholar
- Chancelier, J-P: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 353, 141-153 (2009) MathSciNetView ArticleMATHGoogle Scholar