The split common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces
- Lin Wang^{1}Email author and
- Zhaoli Ma^{2}
https://doi.org/10.1186/s13663-016-0588-2
© Wang and Ma 2016
Received: 30 January 2016
Accepted: 22 September 2016
Published: 30 September 2016
Abstract
In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results.
Keywords
MSC
1 Introduction
The class of nonexpansive mappings is one of the most important classes of mappings in nonlinear science. The class of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings, which was introduced by Goebel and Kirk [1] in 1972. They proved that if C is a nonempty closed convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping, then T has a fixed point.
Example 1.1
([2])
- (i)
\(\| Sx - Sy\| \le2 \|x-y\|\), for all \(x, y \in C\);
- (ii)
\(\| S^{n} x - S^{n} y\| \le2 \prod_{j=2}^{n} a_{j} \|x-y\|\), for all \(x, y \in C, \forall n \ge2\).
Definition 1.2
([3])
- (i)
\(T(0)x=x\), for all \(x\in E\);
- (ii)
\(T(s+t)=T(s)T(t)\), for all \(s, t \geq0\);
- (iii)
for each \(x\in E\), the mapping \(t\mapsto T(t)x\) is continuous;
- (iv)for each \(t>0\), there exists a bounded measurable function \(L(t):[0,\infty)\rightarrow[0,\infty)\) such that$$ \bigl\| T(t)x-T(t)y\bigr\| \leq L(t)\|x-y\|, \quad \mbox{for all } x,y\in E. $$(1.4)
Example 1.3
([4] (Example of asymptotically nonexpansive semigroup))
In 1994, in finite dimensional Hilbert spaces, Censor and Elfving [5] introduced the split feasibility problem for modeling inverse problems which arise from phase retrievals and in medical imagine reconstruction [6]. It has been found that split feasibility problems can be used in various disciplines, such as imagine restoration, computer tomograph and radiation therapy treatment planning [7–9].
When C and Q in (1.7) are the sets of fixed points of two nonlinear mappings, and C and Q are nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively, then the split feasibility problem (1.7) is also said to be split common fixed point problem [10]. It is well known that each nonempty closed convex subset of a Hilbert space is the set of fixed points of its projection, therefore, the split common fixed point problem may be considered as a generalization of split feasibility problem.
In the setting of two Hilbert spaces, for demicontractive mappings, Moudafi [10] proposed an iteration scheme and obtained a weak convergence theorem of the split common fixed point problem. Since then, the split common fixed point problems of other nonlinear mappings in the setting of two Hilbert spaces have been studied by some authors; see, for instance, [2, 11–14]. Especially, Cholamjiak et al. [15] obtained a strong convergence theorem of split common fixed point problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically strict pseudo-contractive mapping in Hilbert spaces.
In 2015, in the setting of one Hilbert space and one Banach space, Takahashi [16] investigated the split feasibility problem and split common null point problem, and obtained some strong and weak convergence theorems under some mild control conditions.
- (1)
\(E_{1}\) is a real uniformly convex and 2-uniformly smooth Banach space having the Opial property and the best smoothness constant k satisfying \(0< k<\frac{1}{\sqrt{2}}\).
- (2)
\(E_{2}\) is a real Banach space.
- (3)
\(A: E_{1} \to E_{2}\) be a bounded linear operator and \(A^{*}\) is the adjoint of A.
- (4)
\(S : E_{1}\rightarrow E_{1}\) is an \(\{l_{n}\}\)-asymptotically nonexpansive mapping with \(\{l_{n}\}\subset(1,\infty)\) and \(l_{n}\rightarrow1\). \(T: E_{2}\rightarrow E_{2}\) is a τ-quasi-strict pseudo-contractive mapping with \(F(S)\neq\emptyset\) and \(F(T)\neq\emptyset\), and T is demiclosed at zero.
Theorem 1.4
([17])
- (I)
If \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\phi\), then the sequence \(\{ x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma\).
- (II)
In addition, if \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\phi\) and S is semi-compact, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in \Gamma\).
This naturally brings about the following question:
Question
Can we obtain the convergence results of split common fixed point problem for asymptotically nonexpansive semigroups in the setting of two Banach spaces?
In this paper, motivated and inspired by the recent research going on in the direction of split feasibility problems and split common fixed point problems, we construct an iteration scheme to approximate a split common fixed point of two asymptotically nonexpansive semigroups in the setting of two Banach spaces. Under some suitable conditions on parameters, the iteration scheme proposed is shown to converge strongly and weakly to a split common fixed point of asymptotically nonexpansive semigroups in two Banach spaces.
2 Preliminaries
We now recall some definitions and elementary facts which will be used in the proofs of our main results.
Lemma 2.1
([18])
Let \(T:C\rightarrow C\) be a mapping with \(F(T)\neq\emptyset\). Then T is said to be demiclosed at zero if for any \(\{x_{n}\}\subset C\) with \(x_{n}\rightharpoonup x\) and \(\| x_{n}-Tx_{n}\|\rightarrow0\), \(x = Tx\).
A mapping \(T:C\rightarrow C\) is said to be semi-compact, if for any sequence \(\{x_{n}\}\) in C such that \(\|x_{n}-Tx_{n}\|\rightarrow0\) (\(n\rightarrow\infty\)), there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(\{x_{x_{j}}\}\) converges strongly to \(x^{*}\in C\).
Lemma 2.2
([19])
Let E be a real uniformly convex Banach space, C be a nonempty closed subset of E, and let \(T : C\rightarrow C\) be an asymptotically nonexpansive mapping. Then \(I-T\) is demiclosed at zero, that is, if \(\{x_{n}\}\subset C\) converges weakly to a point \(p\in C\) and \(\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\| =0\), then \(p=Tp\).
Lemma 2.3
([20])
- (1)
\(\lim_{n\rightarrow\infty}a_{n}\) exists;
- (2)
if \(\liminf_{n\rightarrow\infty}a_{n}=0\), then \(\lim_{n\rightarrow\infty} a_{n}=0\).
Lemma 2.4
([18])
3 Main results
Theorem 3.1
- (1)
\(t_{n}>0\) and \(\lim_{n\rightarrow\infty}t_{n}=\infty\);
- (2)
\(L(t)=\max\{L^{(1)}(t), L^{(2)}(t)\}\) and \(\sum_{n=1}^{\infty}(L^{2}(t_{n})-1)<\infty\);
- (3)
\(M=\sup_{n}L^{2}(t_{n})\), \(\liminf_{n\rightarrow\infty}\alpha _{n}(1-\alpha_{n})>0\) and \(0<\gamma<\min\{\frac{1-2k^{2}}{\|A\|^{2}M},\frac{1}{\| A\|^{2}}\}\).
- (I)
If \(\Gamma=\{p\in C:Ap\in Q\} \neq\emptyset\), then the sequence \(\{ x_{n}\}\) converges weakly to a split common fixed point \(x^{*} \in\Gamma\).
- (II)
In addition, if \(\Gamma=\{p\in C:Ap\in Q\} \neq\emptyset\) and there exists at least one \(S(t)\in\{S(t):t\geq0\}\) that is semi-compact, then \(\{x_{n}\}\) converges strongly to a split common fixed point \(x^{*} \in\Gamma\).
Proof
Now we prove the conclusion (I).
We shall divide the proof into four steps.
Step 1. We first show that the limit \(\lim_{n\rightarrow\infty}\| x_{n}-p\|\) exists for any \(p\in\Gamma\).
Since \(\lim_{t_{n}\rightarrow\infty}L(t_{n})=1\), \(\sum_{n=1}^{\infty}(L^{2}(t_{n})-1)<\infty\), \(0< k<\frac{1}{\sqrt{2}}\), \(0<\gamma<\frac {1-2k^{2}}{\|A\|^{2}M}\), so \(0<\gamma\|A\|^{2}M+2k^{2}<1\), and from (3.3) and Lemma 2.3 we see that the \(\lim_{n\rightarrow\infty}\|x_{n}-p\|\) exists. This implies that \(\{x_{n}\}\) is bounded. Further, it follows from (3.2) that \(\{z_{n}\}\) is bounded, too.
Step 2. We prove that \(\lim_{n\rightarrow\infty}\|x_{n+1}-x_{n}\|=0\) and \(\lim_{n\rightarrow\infty}\|z_{n+1}-z_{n}\|=0\).
Step 3. We prove that \(\lim_{n\rightarrow\infty}\|z_{n}-S(t)z_{n}\|=0\) and \(\lim_{n\rightarrow\infty}\|(T(t)-I)Az_{n}\|=0\) for all \(t\geq 0\).
Step 4. We prove that \(\{x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma\).
By the reflexivity of Banach space \(E_{1}\) and boundedness of \(\{x_{n}\}\), there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) converging weakly to \(x^{*}\). By using (3.15) this implies that \(\{z_{n_{i}}\}\) of \(\{z_{n}\}\) converges weakly to \(x^{*}\), too. Since \(S(t)\) is asymptotically nonexpansive for all \(t\geq0\), it is demiclosed at zero, we know from Lemma 2.2 that \(x^{*}\in F(S(t))\).
On the other hand, since A is a bounded linear operator, we know that \(\{Ax_{n_{i}}\}\) converges weakly to \(Ax^{*}\). It follows from (3.20) that \(\lim_{n_{i}\rightarrow\infty}\|(T(t)-I)Ax_{n_{i}}\|=0\). Since \(T(t)\) is demiclosed at zero for all \(t\geq0\), we have \(Ax^{*}\in F(T(t))\). This means that \(x^{*}\in\Gamma\).
We now prove that \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\).
Next, we prove conclusion (II).
Since there exists at least one \(S(t)\in\{S(t):t\geq0\}\) that is semi-compact and \(\lim_{n\rightarrow\infty}\|z_{n} - S(t)z_{n}\|=0\) for all \(t\geq0\), there exists subsequence \(\{z_{n_{j}}\}\) of \(\{z_{n}\}\) such that \(\{z_{n_{j}}\}\) converges strongly to \(\mu^{*}\in E_{1}\). By using (3.15) again, we know that the subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) converges strongly to \(\mu^{*}\). Due to \(\{x_{n}\}\) converging weakly to \(x^{*}\), we obtain \(\mu^{*}=x^{*}\). Since \(\lim_{n\rightarrow\infty}\|x_{n}-x^{*}\|\) exists and \(\lim_{n\rightarrow\infty}\|x_{n_{j}}-x^{*}\|=0\), we know that \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Gamma\). This completes the proof of conclusion (II). □
Corollary 3.2
- (I)
If \(\Gamma=\{p\in C:Ap\in Q\} \neq\emptyset\), then the sequence \(\{ x_{n}\}\) converges weakly to a split common fixed point \(x^{*} \in\Gamma\).
- (II)
In addition, if \(\Gamma=\{p\in C:Ap\in Q\} \neq\emptyset\) and there exists at least one \(S(t)\in\{S(t):t\geq0\}\) that is semi-compact, then \(\{x_{n}\}\) converges strongly to a split common fixed point \(x^{*} \in\Gamma\).
4 Application to hierarchical variational inequality problem in Banach spaces
Let E be a strictly convex and real reflexive Banach space and K be a nonempty closed and convex subset of E. Then, for any \(x\in E\), there exists a unique element \(z \in K\) such that \(\|x-z\|\leq\|x- y\|\) for all \(y \in K\). Putting \(z = P_{K}x\), we call \(P_{K}\) the metric projection of E onto K.
Lemma 4.1
([21])
- (i)
\(z = P_{K}x\);
- (ii)
\(\langle z-y, J(x-z)\rangle\geq0, \forall y \in K\), where J is the normalized duality mapping on E.
Definition 4.2
([17])
In Theorem 3.1, we take \(E_{1}=E_{2}=E\), \(A=I\), \(T(t)=P_{F(S(t))}V(t)\), \(J_{1}=J_{2}=J\) (where J is the normalized duality mapping on E), the following conclusion can be obtained from Theorem 3.1 immediately.
Theorem 4.3
- (1)
\(t_{n}>0\) and \(\lim_{n\rightarrow\infty}t_{n}=\infty\);
- (2)
\(L(t)=\max\{L^{(1)}(t), L^{(2)}(t)\}\) and \(\sum_{n=1}^{\infty}(L^{2}(t_{n})-1)<\infty\);
- (3)
\(M=\sup_{n}L^{2}(t_{n})\), \(\liminf_{n\rightarrow\infty}\alpha _{n}(1-\alpha_{n})>0\), and \(0<\gamma<\min\{\frac{1-2k^{2}}{\|A\|^{2}M},\frac {1}{\|A\|^{2}}\}\).
- (I)
If \(\Gamma_{1}\neq\emptyset\) (the set of solutions of hierarchical variational inequality problem (4.1) is nonempty), then the sequence \(\{x_{n}\}\) converges weakly to a split common fixed point \(x^{*} \in\Gamma_{1}\).
- (II)
In addition, if \(\Gamma_{1}=\{p\in C:Ap\in Q\} \neq\emptyset\) and there exists at least one \(S(t)\in\{S(t):t\geq0\}\) that is semi-compact, then \(\{x_{n}\}\) converges strongly to a split common fixed point \(x^{*} \in\Gamma_{1}\).
Declarations
Acknowledgements
The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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
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