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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 operator
MSC
- 47H06
- 47H09
- 47H10
- 47J25
1 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
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