Viscosity approximation method with Meir-Keeler contractions for common zero of accretive operators in Banach spaces
- Jong Kyu Kim^{1}Email author and
- Truong Minh Tuyen^{2}
https://doi.org/10.1186/s13663-014-0256-3
© Kim and Tuyen; licensee Springer 2015
Received: 6 October 2014
Accepted: 30 December 2014
Published: 1 February 2015
Abstract
The purpose of this paper is to introduce a new iteration by the combination of the viscosity approximation with Meir-Keeler contractions and proximal point algorithm for finding common zeros of a finite family of accretive operators in a Banach space with a uniformly Gâteaux differentiable norm. The results of this paper improve and extend corresponding well-known results by many others.
Keywords
accretive operators prox-Tikhonov method Meir-Keeler contraction common zero of accretive operatorMSC
47H06 47H09 47H10 47J251 Introduction
Further, Rockafellar [2] posed the open question of whether the sequence generated by (1.1) converges strongly or not. In 1991, Güler [3] gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly.
An example of the authors, Bauschke et al. [4] also showed that the proximal algorithm only converges weakly but not strongly.
Zegeye and Shahzed [11] studied the convergence problem of finding a common zero of a finite family of m-accretive operators (cf. [12, 13]). More precisely, they proved the following result.
Theorem 1.1
[11]
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty }\alpha_{n}=\infty\),
- (ii)
\(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty\) or \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\).
Theorem 1.2
[14]
- (i)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\),
- (ii)
\(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty\) or \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\),
In this paper, we combine the proximal point method [9] and the viscosity approximation method [15] with Meir-Keeler contractions to get strong convergence theorems for the problem of finding a common zero of a finite family of accretive operators in Banach spaces. We also give some applications of our results for the convex minimization problem and the variational inequality problem in Hilbert spaces.
2 Preliminaries
Let E be a real Banach space and \(M\subseteq E\). We denote by \(F(T)\) the set of all fixed points of the mapping \(T: M\to M\).
A closed convex subset C of a Banach space E is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a nonempty, closed, and convex subset M of C into itself has a fixed point in M.
A subset C of a Banach space E is called a retract of E if there is a continuous mapping P from E onto C such that \(Px=x\), for all \(x\in C\). We call such P a retraction of E onto C. It follows that if P is a retraction, then \(Py=y\), for all y in the range of P. A retraction P is said to be sunny if \(P(Px+t(x-Px))=Px\), for all \(x\in E\) and \(t\geq 0\). If a sunny retraction P is also nonexpansive, then C is said to be a sunny nonexpansive retract of E.
An accretive operator A defined on a Banach space E is said to satisfy the range condition if \(\overline{D(A)}\subset R(I+\lambda A)\), for all \(\lambda>0\), where \(\overline{D(A)}\) denotes the closure of the domain of A. We know that for an accretive operator A which satisfies the range condition, \(A^{-1}0=F(J_{\lambda}^{A})\), for all \(\lambda>0\).
The following lemmas play crucial roles for the proof of main theorems in this paper.
Lemma 2.1
[17]
Remark 2.2
Lemma 2.3
[17]
Let C be a convex subset of a Banach space E. Let T be a nonexpansive mapping on C and ϕ be a Meir-Keeler contraction on C. Then, for each \(t\in(0,1)\), a mapping \(x\mapsto(1-t)Tx+t\phi x\) is also a Meir-Keeler contraction on C.
Lemma 2.4
[18]
- (i)
P is sunny nonexpansive.
- (ii)
\(\langle x-Px,j(z-Px)\rangle\leq0\), for all \(x\in C\), \(z\in D\).
- (iii)
\(\langle x-y,j(Px-Py)\rangle\geq\|Px-Py\|^{2}\), for all \(x,y\in C\).
We can easily prove the following lemma from Lemma 1 in [19].
Lemma 2.5
[19]
Lemma 2.6
[20]
Lemma 2.7
[21]
Let E be a Banach space with a uniformly Gâteaux differentiable norm and let C be a nonempty, closed, and convex subset of E with fixed point property for nonexpansive self-mappings. Let \(A: D(A)\subset E\to2^{E}\) be an accretive operator such that \(A^{-1}0\neq\emptyset\) and \(\overline{D(A)}\subset \bigcap_{t>0}R(I+tA)\). Then \(A^{-1}0\) is a sunny nonexpansive retract of C.
Lemma 2.8
[11]
Let C be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let \(A_{i}:C\to E \) be an m-accretive operator for each \(i=1,2,\ldots,N\) with \(\bigcap_{i=1}^{N} N(A_{i})\neq\emptyset\). Let \(a_{0},a_{1},\ldots,a_{N}\) be real numbers in \((0,1)\) such that \(\sum_{i=0}^{N} a_{i}=1\) and let \(S_{N}:=a_{0}I+a_{1}J^{A_{1}}+a_{2}J^{A_{2}}+\cdots+a_{N}J^{A_{N}}\), where \(J^{A_{i}}:=(I+A_{i})^{-1}\). Then \(S_{N}\) is a nonexpansive mapping and \(F(S_{N})=\bigcap_{i=1}^{N} N(A_{i})\).
3 Main results
Now, we are in a position to introduce and prove the main theorems.
Propositon 3.1
Proof
Now we show that \(\{v_{t}\}\) is bounded. Indeed, take a \(p\in F(T)\) and a number \(\varepsilon>0\).
Case 1. Let \(\|v_{t}-p\|\leq\varepsilon\). Then we can see easily that \(\{v_{t}\}\) is bounded.
Remark 3.2
Let Q be a sunny nonexpansive retraction from C onto \(F(T)\). By the uniqueness of Q, inequality (3.1) and Lemma 2.4, we obtain \(Q\phi x^{*} =x^{*}\).
Proposition 3.3
Proof
- (C1)
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty }\alpha_{n}=\infty\),
- (C2)
\(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty\) or \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\).
Theorem 3.4
Proof
We consider two cases of condition (C2).
The following is a strong convergence theorem for the sequence \(\{z_{n}\} \) in (3.6).
Theorem 3.5
If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{z_{n}\}\) generated by (3.6) converges strongly to \(x^{*}\in S\), which satisfies \(Q\phi x^{*}=x^{*}\), where Q is a sunny nonexpansive retraction from C onto S.
Proof
- (i)
There exists \(n_{2}\in\mathbb{N}\) satisfying \(n_{2}\geq n_{1}\) and \(\|z_{n_{2}}-x_{n_{2}}\|\leq\varepsilon\).
- (ii)
\(\|z_{n}-x_{n}\| >\varepsilon\), for all \(n\geq n_{1}\).
By induction, we can show that \(\|z_{n}-x_{n}\|\leq\varepsilon\), for all \(n\geq n_{2}\). This is a contradiction to the fact that \(\varepsilon<\limsup_{n\rightarrow\infty}\|z_{n}-x_{n}\|\).
Corollary 3.6
Proof
Since for each \(i=1,2,\ldots,N\), \(A_{i}\) is an m-accretive operator, the condition \(\overline{D(A_{i})}\subset C\subset\bigcap_{r>0}R(I+rA_{i})\) is satisfied, for all \(i=1,2,\ldots,N\). By the assumption and Theorem 3.5, we have \(z_{n}\to x^{*}\) as \(n\to\infty \) which satisfies \(Q\phi x^{*}=x^{*}\). This completes the proof. □
Remark 3.7
Corollary 3.6 is a generalization of the results of Tuyen [14], Zegeye and Shahzad [11] and Jung [24].
Remark 3.8
If we take \(r=1\), then we may take \(S_{1}:=J^{A}=(I+A)^{-1}\) and strict convexity of E and the real constants \(a_{i}\), \(i=0,1\), may not be needed.
Corollary 3.9
4 Applications
In this section, we give some applications in the framework of Hilbert spaces. We first apply Corollary 3.9 to the convex minimization problem.
Theorem 4.1
Proof
Theorem 4.2
Proof
Declarations
Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation Grant funded by Ministry of Education of the republic of Korea (2014046293).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Martinet, B: Regularisation d’inequations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154-158 (1970) MATHMathSciNetGoogle Scholar
- Rockaffelar, RT: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 887-897 (1976) MathSciNetGoogle Scholar
- Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403-419 (1991) View ArticleMATHMathSciNetGoogle Scholar
- Bauschke, HH, Matoušková, E, Reich, S: Projection and proximal point methods convergence results and counterexamples. Nonlinear Anal. TMA 56, 715-738 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Lehdili, N, Moudafi, A: Combining the proximal algorithm and Tikhonov regularization. Optimization 37, 239-252 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Xu, H-K: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115-125 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Song, Y, Yang, C: A note on a paper: A regularization method for the proximal point algorithm. J. Glob. Optim. 43, 171-174 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Tuyen, TM: A regularization proximal point algorithm for zeros of accretive operators in Banach spaces. Afr. Diaspora J. Math. 13, 62-73 (2012) MATHMathSciNetGoogle Scholar
- Kim, JK, Tuyen, TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2011, 52 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Sahu, DR, Yao, JC: The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces. J. Glob. Optim. 51, 641-655 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Zegeye, H, Shahzad, N: Strong convergence theorems for a common zero of a finite family of m-accretive mappings. Nonlinear Anal. TMA 66, 1161-1169 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Yao, Y, Liou, YC, Wong, MM, Yao, JC: Hierarchical convergence to the zero point of maximal monotone operators. Fixed Point Theory 13, 293-306 (2012) MATHMathSciNetGoogle Scholar
- Ceng, LC, Ansari, QH, Schaible, S, Yao, JC: Hybrid viscosity approximation method for zeros of m-accretive operators in Banach spaces. Numer. Funct. Anal. Optim. 32(11), 1127-1150 (2011) View ArticleMathSciNetGoogle Scholar
- Tuyen, TM: Strong convergence theorem for a common zero of m-accretive mappings in Banach spaces by viscosity approximation methods. Nonlinear Funct. Anal. Appl. 17, 187-197 (2012) MATHGoogle Scholar
- Witthayarat, U, Kim, JK, Kumam, P: A viscosity hybrid steepest-descent methods for a system of equilibrium problems and fixed point for an infinite family of strictly pseudo-contractive mappings. J. Inequal. Appl. 2012, 224 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Meir, A, Keeler, R: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326-329 (1969) View ArticleMATHMathSciNetGoogle Scholar
- Suzuki, T: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 325, 342-352 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker, New York (1984) MATHGoogle Scholar
- Ha, KS, Jung, JS: Strong convergence theorems for accretive operators in Banach space. J. Math. Anal. Appl. 147, 330-339 (1990) View ArticleMATHMathSciNetGoogle Scholar
- Xu, H-K: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631-643 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Wong, NC, Sahu, DR, Yao, JC: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal. TMA 69, 4732-4753 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Barbu, V, Precupanu, T: Convexity and Optimization in Banach Spaces. Editura Academiei R.S.R., Bucharest (1978) MATHGoogle Scholar
- Takahashi, W: Nonlinear Functional Analysis. Fixed Point Theory and Applications. Yokohama Publishers, Yokohama (2009) Google Scholar
- Jung, JS: Strong convergence of an iterative method for finding common zeros of a finite family of accretive operators. Commun. Korean Math. Soc. 24(3), 381-393 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Rockafellar, RT: Characterization of the subdifferentials of convex functions. Pac. J. Math. 17, 497-510 (1966) View ArticleMATHMathSciNetGoogle Scholar
- Jung, JS: Some results on Rockafellar-type iterative algorithms for zeros of accretive operators. J. Inequal. Appl. 2013, 255 (2013). doi:10.1186/1029-242X-2013-255 View ArticleMATHMathSciNetGoogle Scholar
- Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970) View ArticleMATHMathSciNetGoogle Scholar