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Viscosity approximation method with Meir-Keeler contractions for common zero of accretive operators in Banach spaces
Fixed Point Theory and Applications volume 2015, Article number: 9 (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.
1 Introduction
Let E be a real Banach space and let J be the normalized duality mapping from E into \(2^{E^{*}}\) given by
where \(E^{*}\) denotes the dual space of E and \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing. It is well known that if \(E^{*}\) is strictly convex then J is single-valued. In the sequel, we denote the single-valued normalized duality mapping by j. For an operator \(A: E\to2^{E}\), we define its domain, range, and graph as follows:
and
respectively. The inverse \(A^{-1}\) of A is defined by
An operator A is said to be accretive if, for each \(x,y\in D(A)\), there exists \(j(x-y)\in J(x-y)\) such that
for all \(u\in Ax\) and \(v\in Ay\). We denote by I the identity operator on E. An accretive operator A is said to be maximal accretive if there is no proper accretive extension of A and A is said to be m-accretive if \(R(I+\lambda A)=E\), for all \(\lambda>0\). If A is m-accretive, then it is maximal, but generally, the converse is not true. If A is accretive, then we can define, for each \(\lambda>0\), a nonexpansive single-valued mapping \(J_{\lambda}^{A}: R(I+\lambda A)\to D(A)\) by
It is called the resolvent of A which is denoted by \(J^{A}\) when \(\lambda=1\).
Let \(A: E\to2^{E}\) be an m-accretive operator. It is well known that many problems in nonlinear analysis and optimization can be formulated as the problem: Find \(x\in E\) such that
One popular method of solving the equation \(0\in A(x)\), where A is a maximal monotone operator in a Hilbert space H, is the proximal point algorithm. The proximal point algorithm generates, for any starting point \(x_{0}=x\in E\), a sequence \(\{x_{n}\}\) by the rule
for all \(n\in\mathbb{N}\), where \(\{r_{n}\}\) is a regularization sequence of positive real numbers, \(J_{r_{n}}^{A}=(I+r_{n}A)^{-1}\) is the resolvent of A, and ℕ is the set of all natural numbers. Some of them deal with the weak convergence theorem of the sequence \(\{x_{n}\}\) generated by (1.1) and others proved strong convergence theorems by imposing assumptions on A.
Note that algorithm (1.1) can be rewritten as
for all \(n\in\mathbb{N}\). This algorithm was first introduced by Martinet [1]. If \(\psi: H\to\mathbb{R}\cup\{\infty\}\) is proper lower semicontinuous convex function, then the algorithm reduces to
for all \(n\in\mathbb{N}\). Moreover, Rockafellar [2] has given a more practical method which is an inexact variant of the method:
for all \(n\in\mathbb{N}\), where \(\{e_{n}\}\) is regarded as an error sequence and \(\{r_{n}\}\) is a sequence of positive regularization parameters. Note that the algorithm (1.3) can be rewritten as
for all \(n\in\mathbb{N}\). This method is called inexact proximal point algorithm. It was shown in Rockafellar [2] that if \(e_{n}\to0\) quickly enough such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\), then \(x_{n}\rightharpoonup z\in H\) with \(0\in Az\).
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.
When A is maximal monotone in a Hilbert space H, Lehdili and Moudafi [5] obtained the convergence of the sequence \(\{x_{n}\}\) generated by the algorithm
where \(A_{n}=\mu_{n}I+A\), \(\mu>0\) is viewed as a Tikhonov regularization of A. Next, in 2006, Xu [6] and in 2009, Song and Yang [7] used the technique of nonexpansive mappings to get convergence theorems for \(\{x_{n}\}\) defined by the perturbed version of algorithm (1.4) in the form
Note that algorithm (1.6) can be rewritten as
In [8], Tuyen was studied an extension the results of Xu [6], when A is an m-accretive operator in a uniformly smooth Banach space E which has a weakly sequentially continuous normalized duality mapping j from E to \(E^{*}\) (cf. [9]). At that time, in [10], Sahu and Yao also extended the results of Xu [6] for the zero of an accretive operator in a Banach space which has a uniformly Gâteaux differentiable norm by combining the prox-Tikhonov method and the viscosity approximation method. They introduced the iterative method to define the sequence \(\{x_{n}\}\) as follows:
for all \(n\in\mathbb{N}\), where A is an accretive operator such that \(S=A^{-1}0\neq\emptyset\) and \(\overline{D(A)}\subset C\subset\bigcap_{t>0}R(I+tA)\), and f is a contractive mapping on C.
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]
Let E be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty, closed, and convex subset of E and \(A_{i}: K\to E \) be an m-accretive operator, for each \(i=1,2,\ldots,N\) with
For any \(u,x_{0}\in K\), let \(\{x_{n}\}\) be a sequence in K generated by the algorithm:
where \(S_{N}:=a_{0}I+a_{1}J^{A_{1}}+a_{2}J^{A_{2}}+\cdots+a_{N}J^{A_{N}}\) with \(J^{A_{i}}=(I+A_{i})^{-1}\) for \(0< a_{i}<1\), \(i=0,1,2,\ldots,N\), \(\sum_{i=0}^{N}a_{i}=1\), and \(\{\alpha_{n}\}\) is a real sequence which satisfies the following conditions:
-
(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 \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\).
If every nonempty, bounded, closed, and convex subset of E has the fixed point property for nonexpansive mapping, then \(\{x_{n}\}\) converges strongly to a common solution of the equations \(A_{i}(x)=0\) for \(i=1,2,\ldots,N\).
Motivated by Xu [6] and Zegeye and Shahzed [11], Tuyen [14] introduced an iterative algorithm as follows:
where \(S_{N}:=a_{0}I+a_{1}J^{A_{1}}+a_{2}J^{A_{2}}+\cdots+a_{N}J^{A_{N}}\) with \(a_{0},a_{1},\ldots,a_{N}\) in \((0,1)\) such that \(\sum_{i=0}^{N}a_{i}=1\) and \(\{\alpha_{n}\}\subset(0,1)\) is a real sequence of positive numbers. The result of Tuyen [14] is given by the following.
Theorem 1.2
[14]
Let E be a strictly convex and reflexive Banach space which has a weakly continuous duality mapping \(J_{\varphi}\) with gauge φ. Let C be a nonempty, closed, and convex subset of E and f be a contraction mapping of C into itself with the contractive coefficient \(c\in(0,1)\). Let \(A_{i}: C\rightarrow E\) be an m-accretive operator, for each \(i=1,2,\ldots,N\) with
Let \(J^{A_{i}}=(I+A_{i})^{-1}\) for \(i=1,2,\ldots,N\). For any \(x_{0}\in C\), let \(\{x_{n}\}\) be a sequence generated by algorithm (1.10). If the sequence \(\{\alpha_{n}\}\) satisfies the following conditions:
-
(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 \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\),
then \(\{x_{n}\}\) converges strongly to a common solution of the equations \(A_{i}(x)=0\) for \(i=1,2,\ldots,N\).
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 \(T: M\to M\).
Recall that a mapping \(\phi: (X,d)\to(X,d)\) from the metric space \((X,d)\) into itself is said to be a Meir-Keeler contraction, if, for every \(\varepsilon>0\), there exists \(\delta>0\) such that \(d(x,y)<\varepsilon+\delta\) implies
for all \(x,y\in X\). We know that if \((X,d)\) is a complete metric space, then ϕ has a unique fixed point [16]. In the sequel, we always use \(\Sigma_{M}\) to denote the collection of all Meir-Keeler contractions on M and \(S_{E}\) to denote the unit sphere \(S_{E}=\{x\in E: \|x\|=1\}\). A Banach space E is said to be strictly convex if \(x,y\in S_{E} \) with \(x\neq y\), and, for all \(t\in(0,1)\),
A Banach space E is said to be smooth provided the limit
exists for each x and y in \(S_{E}\). In this case, the norm of E is said to be Gâteaux differentiable. It is said to be uniformly Gâteaux differentiable if for each \(y\in S_{E}\), this limit is attained uniformly for \(x\in S_{E}\). It is well known that every uniformly smooth Banach space has a uniformly Gâteaux differentiable norm.
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 \(\lambda>0\).
Let f be a continuous linear functional on \(l_{\infty}\). We use \(f_{n}(x_{n+m})\) to denote
for \(m=0,1,2,\ldots \) . A continuous linear functional f on \(l_{\infty}\) is called a Banach limit if \(\|f\|=f(e)=1\) and \(f_{n}(x_{n})=f_{n}(x_{n+1})\) for each \(x=(x_{1},x_{2},\ldots)\) in \(l_{\infty}\). Fix any Banach limit and denote it by \(LIM\). Note that \(\|LIM\|=1\), and, for all \(\{x_{n}\} \in l_{\infty}\),
The following lemmas play crucial roles for the proof of main theorems in this paper.
Lemma 2.1
[17]
Let ϕ be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then for each \(\varepsilon>0\), there exists \(r\in (0,1)\) such that, for all \(x,y\in C\), \(\|x-y\|\geq\varepsilon\) implies
Remark 2.2
From Lemma 2.1, for each \(\varepsilon>0\), there exists \(r\in(0,1)\) such that
for all \(x,y\in C\).
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]
Let C be a convex subset of a smooth Banach space E, D a nonempty subset of C, and P a retraction from C onto D. Then the following statements are equivalent:
-
(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]
Let E be a Banach space with a uniformly Gâteaux differentiable norm, C a nonempty, closed, and convex subset of E and \(\{x_{n}\}\) a bounded sequence in E. Let \(LIM\) be a Banach limit and \(y\in C\) such that
Then \(LIM_{n}\langle x-y,j(x_{n}-y)\rangle\leq0\), for all \(x\in C\).
Lemma 2.6
[20]
Let \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{\sigma_{n}\}\) be sequences of positive numbers satisfying the inequality:
If \(\sum_{n=0}^{\infty}b_{n}=+\infty\) and \(\lim_{n\to\infty}\sigma_{n}/b_{n}=0\), then \(\lim_{n\rightarrow\infty}a_{n}=0\).
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
Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm and let C be a closed convex subset of E which has the fixed point property for nonexpansive mappings. Let T be a nonexpansive mapping on C. Then for each \(\phi\in\Sigma_{C}\) and every \(t\in(0,1)\), there exists a unique fixed point \(v_{t}\in C\) of the Meir-Keeler contraction \(C\ni v_{t}\mapsto t\phi v_{t} +(1-t)Tv_{t}\), such that \(\{v_{t}\}\) converges strongly to \(x^{*}\in F(T)\) as \(t\to0\) which solves the variational inequality:
for all \(x \in F(T)\).
Proof
By Lemma 2.3, the mapping \(C\ni v\mapsto t\phi v +(1-t)Tv\) is a Meir-Keeler contraction on C. So, there is a unique \(v_{t}\in C\) which satisfies
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.
Case 2. Let \(\|v_{t}-p\|\geq\varepsilon\). Then, by Lemma 2.4, there exists \(r\in(0,1)\) such that
So, we have
Therefore,
Hence, we conclude that \(\{v_{t}\}\) is bounded and \(\{\phi v_{t}\}\), \(\{ Tv_{t}\}\) are also bounded.
By the boundedness of \(\{v_{t}\}\), \(\{\phi v_{t}\}\), and \(\{Tv_{t}\}\), we have
Assume \(t_{n}\to0\). Set \(v_{n}:=v_{t_{n}}\) and define \(\varphi: C\to \mathbb{R}^{+}\) by
for all \(x\in C\) and let
Since E is reflexive, \(\varphi(x)\to\infty\) as \(\|x\|\to\infty\), and φ is a continuous convex function, from Barbu and Precupanu [22], we know that M is a nonempty subset of C. By Takahashi [23], we see that M is also closed, convex, and bounded.
For all \(x\in M\), from \(\|v_{n}-Tv_{n}\|\to\)0 as \(n\to\infty\), we have
So, M is invariant under T, i.e., \(T(M)\subset M\). By assumption, we have \(M\cap F(T)\neq\emptyset\). Let \(x^{*}\in M\cap F(T)\). By Lemma 2.7, we obtain
for all \(x\in C\). In particular,
Suppose that \(LIM_{n}\|v_{n}-x^{*}\|^{2}\geq\varepsilon>0\). By (2.1),
So, there exists a subsequence \(\{v_{n_{k}}\}\) of \(\{v_{n}\}\) such that, for all \(k\geq1\),
where \(\varepsilon_{0}\in(0,\sqrt{\varepsilon} )\). By Lemma 2.3, there is \(r_{0}\in(0,1)\) such that
From
for all \(k\geq1\), we have
which implies that
for all \(x\in C\). So, from (3.2), we get
for all \(x\in C\). In particular,
which is a contradiction. Hence, \(LIM_{n}\|v_{n}-x^{*}\|=0\) and there exists a subsequence \(\{v_{n_{k}}\}\) of \(\{v_{n}\}\) such that \(v_{n_{k}}\to x^{*}\) as \(k\to\infty\).
Assume that \(\{v_{n_{l}}\}\) is another subsequence of \(\{v_{n}\}\) such that \(v_{n_{l}}\to y^{*}\) with \(y^{*}\neq x^{*}\). It is easy to see that \(y^{*}\in F(T)\). By Lemma 2.3, there exists \(r_{1}\in(0,1)\) such that
Observe that
for all \(n\in\mathbb{N}\). Since \(v_{n_{k}}\to x^{*}\) and j is norm to weak* uniformly continuous, we obtain
Similarly, we have
Adding the above two inequalities yields
and combining with (3.4) implies that
which is a contradiction. Hence \(\{v_{t_{n}}\}\) converges strongly to \(x^{*}\).
Now, we prove that the net \(\{v_{t}\}\) converges strongly to \(x^{*}\) as \(t\to0\). We assume that there is another subsequence \(\{s_{n}\}\) with \(s_{n}\in(0,1)\), for all n and \(s_{n}\to0\) as \(n\to\infty\) such that \(v_{s_{n}}\to z^{*}\) as \(n\to\infty\). Then we have \(z^{*}\in F(T)\). For each t and \(z\in F(T)\), we have
So, we obtain
and similarly, we have
which implies that
and
Thus, we have \(x^{*}=z^{*}\). Therefore, \(\{v_{t}\}\) converges strongly to \(x^{*}\) and it is easy to see that \(x^{*}\) solves the variational inequality
for all \(x\in F(T)\). This completes the proof. □
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
Let C be a closed convex subset of a reflexive Banach space E with a uniformly Gâteaux differentiable norm and let T be a nonexpansive mapping on C with \(F(T)\neq\emptyset\). Assume \(\{x_{n}\}\) is a bounded sequence such that \(x_{n}-Tx_{n}\to0\) as \(n\to \infty\). Let \(x_{t}=t\phi x_{t}+(1-t)Tx_{t}\), for all \(t\in(0,1)\), where \(\phi\in\Sigma_{C}\). Assume that \(x^{*}=\lim_{t\rightarrow0}x_{t}\) exists. Then we have
Proof
Set \(M=\sup\{\|x_{n}-x_{t}\|: t\in(0,1), n\geq0\}\). Then we have
which implies that
Fix t and letting \(n\rightarrow\infty\) yields
This completes the proof. □
Now, let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm and C a closed convex subset of E which has the fixed point property for nonexpansive mappings. Let \(A_{i}: E\to2^{E}\) be an accretive operator, for each \(i=1,2,\ldots,N\) such that
and
for all \(i=1,2,\ldots,N\).
For each \(\phi\in\Sigma_{C}\), we study the strong convergence of the sequence \(\{z_{n}\}\) defined by
where \(S_{N}:=a_{0}I+a_{1}J^{A_{1}}+a_{2}J^{A_{2}}+\cdots+a_{N}J^{A_{N}}\) with \(a_{0},a_{1},\ldots,a_{N}\) are real numbers in \((0,1)\) such that \(\sum_{i=0}^{N}a_{i}=1\) and \(\{\alpha_{n}\}\subset(0,1)\) is a real sequence of positive numbers, under the conditions:
-
(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 \(\lim_{n\rightarrow\infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\).
Then we have the following theorem.
Theorem 3.4
If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{x_{n}\}\) generated by
converges strongly to Qu, where \(u\in C\) and Q is a sunny nonexpansive retraction from C onto S.
Proof
By Lemma 2.8, we have \(F(S_{N})=\bigcap_{i=1}^{N}A_{i}^{-1}0\neq\emptyset\). Now, for each \(p\in F(S_{N})\), we have
Hence \(\{x_{n}\}\) is bounded. Suppose that \(\max \{\sup\|x_{n}\|, \|u\| \}\le K\). It follows that
From (1.10), we get
where \(\beta_{n}=2K\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}\).
We consider two cases of condition (C2).
First, suppose that \(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty \). Then
where \(\sigma_{n}=2K|\alpha_{n}-\alpha_{n-1}|\). So, we have \(\sum_{n=1}^{\infty}\sigma_{n}<\infty\).
Second, suppose that \(\lim_{n\to\infty}\frac{|\alpha_{n}-\alpha _{n-1}|}{\alpha_{n}}=0\). Then
where \(\sigma_{n}=\alpha_{n}\beta_{n}\). So, we have \(\sigma_{n}=o(\alpha_{n})\).
For any case, we have \(\|x_{n+1}-x_{n}\|\to0\) as \(n\to\infty\), from Lemma 2.6. By (3.9) we obtain
Let \(y_{n}=\alpha_{n} u+(1-\alpha_{n})x_{n}\). Then we have
it follows that
For each \(t\in(0,1)\), let \(x_{t}=tu+(1-t)S_{N}x_{t}\). Apply Proposition 3.1 with \(\phi x =u\), for all \(x\in C\), we know that \(\{x_{t}\}\) converges strongly to \(x^{*}\in F(S_{N})\), satisfying \(Qu=x^{*}\). It follows from Proposition 3.3 that
Observe that
Hence, we have
Next, we have
From Lemma 2.6, we have the desired result. That is, the sequence \(\{ x_{n}\}\) converges strongly to \(Qu=x^{*}\). This completes the proof. □
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
Let \(x^{*}\) be a unique fixed point of Qϕ, that is, \(Q\phi x^{*}=x^{*}\). Let \(\{x_{n}\}\) be a sequence defined by
By Theorem 3.4, \(x_{n}\to Q\phi x^{*}=x^{*}\) as \(n\to\infty\).
Now, we prove that \(\|z_{n}-x_{n}\|\to0\) as \(n\to\infty\). Assume that
Then we choose ε with \(\varepsilon\in(0,\limsup_{n\to \infty}\|z_{n}-x_{n}\|)\). By Lemma 2.3, there exists \(r\in(0,1)\) satisfying (2.2). We also choose \(n_{1}\in\mathbb{N}\) such that
for all \(n\geq n_{1}\). We divide this into the following two cases:
-
(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 \(n\geq n_{1}\).
In the case of (i), we have
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 \(\varepsilon<\limsup_{n\rightarrow\infty}\|z_{n}-x_{n}\|\).
In the case of (ii), for each \(n\geq n_{1}\), we have
So, by Lemma 2.1, we get \(\lim_{n\rightarrow\infty}\|z_{n}-x_{n}\|=0\). This is a contradiction. Therefore \(\lim_{n\rightarrow\infty}\|z_{n}-x_{n}\|=0\). Thus we obtain
Hence \(\{z_{n}\}\) convergence strongly to \(Q\phi x^{*}=x^{*}\). This completes the proof. □
Corollary 3.6
Let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm and let C be a closed convex subset of E which has the fixed point property for nonexpansive mappings. Let \(A_{i}: E\to2^{E}\) be an m-accretive operator, for each \(i=1,2,\ldots,N\) such that
For each \(\phi\in\Sigma_{C}\), let \(\{z_{n}\}\) be a sequence generated by (3.6). If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{z_{n}\}\) converges strongly to \(x^{*}\in S\) which satisfies \(Q\phi x^{*}=x^{*}\), where Q is a sunny nonexpansive retraction from C onto S.
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
Let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm and let C be a closed convex subset of E which has the fixed point property for nonexpansive mappings. Let \(A: E\to2^{E}\) be an m-accretive operator such that \(S=A^{-1}0\neq\emptyset\). For each \(\phi\in\Sigma_{C}\), let \(\{z_{n}\}\) be a sequence defined by
for all \(n\geq0\). If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{z_{n}\}\) converges strongly to \(x^{*}\in S\) which satisfies \(Q\phi x^{*}=x^{*}\), where Q is a sunny nonexpansive retraction from C onto S.
Remark 3.10
Corollary 3.9 is a generalization of the results of Tuyen in [8].
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
Let H be a Hilbert space and let \(f: H\to (-\infty,\infty]\) be a proper lower semicontinuous convex function such that \((\partial f)^{-1}0\neq \emptyset\) for a subdifferential mapping ∂f of f. Let \(\{x_{n}\}\) be a sequence defined as follows:
for all \(n\geq0\), where \(\{\alpha_{n}\}\) is a sequence positive real numbers and \(\phi\in\Sigma_{H} \). If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{x_{n}\}\) converges strongly to \(x^{*}\) in \((\partial f)^{-1}0\).
Proof
By the Rockafellar theorem [25] (cf. [26]), the subdifferential mapping ∂f is maximal monotone in H. So,
is equivalent to \(\partial f(x_{n+1})+x_{n+1}\ni y_{n}\). Using Corollary 3.9, \(\{x_{n}\}\) converges strongly to an element \(x^{*}\) in \((\partial f)^{-1}0\). This completes the proof. □
We next apply Proposition 3.3 to the variational inequality problem. Let C be a nonempty, closed, and convex subset of a Hilbert space H and let \(A : C\to H\) be a single-valued monotone operator which is hemicontinuous. Then a point \(u \in C\) is said to be a solution of the variational inequality for A if
for all \(y\in C\). We denote by \(VI(C,A)\) the set of all solutions of the variational inequality (4.2) for A. We also denote by \(N_{C}(x)\) the normal cone for C at a point \(x \in C\), that is,
Theorem 4.2
Let C be a nonempty, closed, and convex subset of a Hilbert space H and let \(A : C\longrightarrow H\) be a single-valued monotone operator and hemicontinuous such that \(VI(C,A)\neq\emptyset\). Let \(\{x_{n}\}\) be a sequence defined as follows:
for all \(n\geq0\), where \(\{\alpha_{n}\}\) is a sequence of positive real numbers and \(\phi\in\Sigma_{H} \). If the sequence \(\{\alpha_{n}\}\) satisfies the conditions (C1)-(C2), then the sequence \(\{x_{n}\}\) converges strongly to \(x^{*}\) in \(VI(C,A)\).
Proof
Define a mapping \(T\subset H\times H\) by
By the Rockafellar theorem [27], we know that T is maximal monotone and \(T^{-1}0=VI(C,A)\).
Note that
if and only if
for all \(y\in C\), that is,
This implies that
Using Corollary 3.9, \(\{x_{n}\}\) converges strongly to an element \(x^{*}\) in \(VI(C,A)\). This completes the proof. □
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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).
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Kim, J.K., Tuyen, T.M. Viscosity approximation method with Meir-Keeler contractions for common zero of accretive operators in Banach spaces. Fixed Point Theory Appl 2015, 9 (2015). https://doi.org/10.1186/s13663-014-0256-3
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DOI: https://doi.org/10.1186/s13663-014-0256-3