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On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 305 (2015)
Abstract
In this paper, we prove a weak convergence theorem and a strong convergence theorem for split common fixed point problem involving a quasistrict pseudo contractive mapping and an asymptotical nonexpansive mapping in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces.
1 Introduction
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, C and Q be nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively, and \(A:H_{1}\to H_{2}\) be a bounded linear operator. The split feasibility problem (SFP) is formulated as to find a point \(q\in H_{1}\) such that
It is easy to see that \(q\in C\) solves equation (1.1) if and only if it solves the following fixed point equation:
where \(P_{C}\) (resp. \(P_{Q}\)) is the (orthogonal) projection from \(H_{1}\) (resp. \(H_{2}\)) onto C (resp. Q), \(\gamma>0\), and \(A^{*}\) is the adjoint of A.
Let \(S: C \to C \) and \(T: Q \to Q\) be two mappings. The socalled split common fixed point problem (SCFP) for mappings S and T is to find a point \(q\in C\) such that
where \(F(S)\) and \(F(T)\) denote the sets of fixed points of S and T, respectively. We use Î“ to denote the set of solutions of SCFP for mappings S and T, that is,
The SFP in finitedimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The split common fixed point problems in Hilbert spaces were introduced by Moudafi [3] in 2010. Since then, various algorithms have been invented to solve SFP and SCFP [4â€“16]. In 2014, Cui and Wang [17] investigated the split common fixed point problems of Ï„quasistrict pseudocontractive mappings in the setting of two Hilbert spaces.
In 2015, Takahashi [18], Takahashi and Yao [19] first attempted to introduce and consider the split feasibility problem and split common null point problem in the setting of a Banach space. By using hybrid methods and Halperntype methods and under suitable conditions, some strong and weak convergence theorems for such kinds of problems are obtained. The results presented in [18] and [19] seem to be the first outside Hilbert spaces.
Motivated and inspired by the research going on in the direction of split feasibility problems and split common fixed point problems, we have the purpose in this article to consider and study the split common fixed point problem for a Ï„quasistrict pseudocontractive mapping and asymptotical nonexpansive mappings in the setting of two Banach spaces. We construct an iterative scheme to approximate a solution for such kind of split common fixed point problem in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces on this problem.
2 Preliminaries
Throughout this paper, we assume that E is a real Banach space with the dual \(E^{*}\) and C is a nonempty closed convex subset of E. Let T be a mapping. We denote by \(F(T)\) the set of fixed points of T. We denote by â€˜â†’â€™ and â€˜â‡€â€™ strong convergence and weak convergence, respectively.
In the sequel, we denote by \(J : E \to2^{E^{*}}\) the normalized duality mapping defined by
where \(\langle\cdot,\cdot\rangle\) is the generalized duality pairing between E and \(E^{*}\).
A Banach space E is said to be strictly convex if \(\frac{\x+y\}{2} < 1\) for all \(x,y\in U=\{z\in E:\z\=1\}\) with \(x\neq y\). The modulus of convexity of E is defined by
for all \(\epsilon\in[0,2]\). E is said to be uniformly convex if \(\delta_{E}(0)=0\) and \(\delta_{E}(\epsilon)>0\) for all \(0<\epsilon\le2\).
Let \(\rho_{E}:[0,\infty)\to[0,\infty)\) be the modulus of smoothness of E defined by
A Banach space E is said to be uniformly smooth if \(\frac{\rho_{E}(t)}{t}\to0\) as \(t\to0\). Let \(q > 1\) be a fixed number. Then a Banach space E is said to be quniformly smooth if there exists a constant \(c>0\) such that \(\rho_{E}(t)\leq ct^{q}\) for all \(t>0\). It is well known that every quniformly smooth Banach space is uniformly smooth.
Lemma 2.1
Let E be a uniformly convex Banach space. Then, for any given number \(r>0\), there exists a continuous strictly increasing function \(g:[0,\infty)\to[0,\infty)\), \(g(0)=0\) such that
for all \(x,y\in E\) with \(\x\\leq r\) and \(\y\\leq r\), \(t\in[0,1]\).
Lemma 2.2
If E is a 2uniformly smooth Banach space with the best smoothness constants \(k>0\), then the following inequality holds:
Definition 2.3
Let \(T:C\to C\) be an operator. Then

(i)
T is said to be \(\{k_{n}\}\)asymptotically nonexpansive if there exists a sequence \(\{k_{n}\} \subset[1, \infty)\) with \(k_{n} \to1\) such that
$$\bigl\Vert T^{n} xT^{n} y\bigr\Vert \leq k_{n} \xy\,\quad \forall n \ge1, x,y\in C; $$ 
(ii)
T is said to be Ï„strict pseudocontractive if there exists a constant \(\tau\in[0,1)\) such that
$$ \TxTy\^{2}\leq\xy\^{2}+\tau\bigl\Vert (IT)x(IT)y \bigr\Vert ^{2}, \quad \forall x,y\in C; $$(2.1) 
(iii)
T is said to be Ï„quasistrict pseudocontractive if \(F(T)\neq\emptyset\) and there exists a constant \(\tau\in[0,1)\) such that
$$ \Txp\^{2}\leq\xp\^{2}+\tau\bigl\Vert (IT)x\bigr\Vert ^{2},\quad \forall p\in F(T),x\in C. $$(2.2)
Example of \(\{k_{n}\}\)asymptotically nonexpansive mapping
Let C be a unit ball in a real Hilbert \(l^{2}\), and let \(T:C\rightarrow C\) be a mapping defined by
where \(\{a_{i}\}\) is a sequence in \((0,1)\) such that \(\prod_{i=2}^{\infty}\alpha_{i}=\frac{1}{2}\).
It is proved in Goebel and Kirk [20] that

(i)
\(\TxTy\\leq2\xy\\), \(\forall x,y \in C\);

(ii)
\(\T^{n}xT^{n} y\\leq2\prod_{i=2}^{n}a_{j}\xy\\), \(\forall n\geq2\) and \(x,y \in C\).
Denote by \(k_{1}^{\frac{1}{2}}=2\), \(k_{n}^{\frac{1}{2}}=2\prod_{i=2}^{n}a_{i}\), \(n\geq2\), then
Therefore \(\forall x, y \in C\), \(n \ge1\), we have
This implies that T is a \(\{k_{n}\}\)asymptotically nonexpansive mapping.
Example of Ï„strict pseudocontractive mapping
Now, we give an example of a Ï„strict pseudocontractive mapping.
Let \(X=l^{2}\) with the norm \(\\cdot\\) defined by
and \(C=\{x=(x_{1},x_{2},\ldots,x_{n},\ldots) \in l^{2}: x_{i}\in R^{1},i=1,2,\ldots\}\) be an orthogonal subspace of X (i.e., we assume that the set C satisfies the condition that for any \(x, y \in C\), we have \(\langle x,y\rangle=0\)). It is obvious that C is a nonempty closed convex subset of X. For each \(x =(x_{1}, x_{2},\ldots, x_{n},\ldots)\in C\), we define a mapping \(T : C \rightarrow C\) by
Next we prove that T is a Ï„strict pseudocontractive mapping and \(F(T)\neq\emptyset\), therefore it is a Ï„quasistrict pseudocontractive mapping.
In fact, for any \(x, y \in C\), we have the following.
Case 1. If \(\prod_{i=1}^{\infty}x_{i}< 0\) and \(\prod_{i=1}^{\infty}y_{i}< 0\), then we have \(T x=x\), \(T y=y\), and so inequality (2.1) holds.
Case 2. If \(\prod_{i=1}^{\infty}x_{i}< 0\) and \(\prod_{i=1}^{\infty }y_{i}\geq 0\), then we have that \(T x = x\), \(Ty= y\). This implies that
Therefore inequality (2.1) holds.
Case 3. If \(\prod_{i=1}^{\infty}x_{i}\geq0\) and \(\prod_{i=1}^{\infty }y_{i}\geq 0\), then we have \(T x=(1)x\), \(Ty=(1)y\). Hence we have
Thus inequality (2.1) still holds. Therefore the mapping defined by (2.4) is a Ï„strict pseudocontractive mapping. Since \(\theta= (0, 0, 0,\ldots, 0, 0, \ldots) \in C\) and \(T(\theta) = \theta\), the set \(F(T)\) of fixed points of T in C is nonempty. Therefore T is also a Ï„quasistrict pseudocontractive mapping.
Definition 2.4

(1)
Let \(T: C\to C\) 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 \(\x_{n}  Tx_{n}\ \to0\), \(x=Tx\).

(2)
Let E be a Banach space. E is said to have the Opial property if for any sequence \(\{x_{n}\}\) in E with \(x_{n} \rightharpoonup x^{*}\), for any \(y\in E\) with \(y \neq x^{*}\), we have
$$\liminf_{n \to\infty}\bigl\Vert x_{n}  x^{*}\bigr\Vert < \liminf_{n \to\infty}\x_{n}  y\. $$
Definition 2.5
A mapping \(T:C\to C\) is said to be semicompact if for any bounded sequence \(\{x_{n}\}\subset C\) such that \(\x_{n}Tx_{n}\\to0\) (\(n\to\infty\)), there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{j}}\}\) converges strongly to \(x^{*}\in C\).
Lemma 2.6
Let \(\{a_{n}\}\), \(\{b_{n}\}\) and \(\{\delta_{n}\}\) be sequences of nonnegative real numbers satisfying
If \(\sum_{n=1}^{\infty}\delta_{n}<\infty\) and \(\sum_{n=1}^{\infty}b_{n}<\infty\), then the limit \(\lim_{n\to\infty}a_{n}\) exists.
3 Main results
Throughout this section, we assume that:

(1)
\(E_{1}\) is a real uniformly convex and 2uniformly 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)
\(A:E_{1} \to E_{2}\) is a bounded linear operator and \(A^{*}\) is the adjoint of A.

(4)
\(S: E_{1}\to E_{1}\) is an \(\{l_{n}\}\)asymptotical nonexpansive mapping with \(\{l_{n}\} \subset(1, \infty)\) and \(l_{n} \to1\). \(T: E_{2}\to E_{2}\) is a Ï„quasistrict pseudocontractive mapping with \(F(S)\neq\emptyset \) and \(F(T)\neq\emptyset\), and T is demiclosed at zero.
Remark
It follows from condition (1) that \(E_{1}\) is a real smooth, strictly convex and reflexive Banach space. Therefore, as is well known, the normalized duality mapping \(J_{1}: E_{1} \to2^{E_{1}^{*}}\) is singlevalued, onetoone and onto. And \(J_{1}^{1}: E_{1}^{*} \to2^{E_{1}}\) is also singlevalued, onetoone and onto.
We are now in a position to introduce and study the following split common fixed point problem (SCFP) for an asymptotical nonexpansive mapping S and a Ï„quasipseudocontractive mapping T in the setting of two Banach spaces, i.e., to find
where \(F(S)\) and \(F(T)\) are the sets of fixed points of S and T, respectively. In the sequel we use Î“ to denote the set of solutions of (SCFP) (3.1) for mappings S and T, that is,
Theorem 3.1
Let \(E_{1}\), \(E_{2}\), A, S, T and \(\{l_{n}\}\) be the same as above. For each \(x_{1}\in E_{1}\), let \(\{x_{n}\}\) be the sequence generated by
where \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) with \(\liminf_{n\to\infty}\alpha_{n}(1\alpha_{n})>0\), Î³ is a positive constant satisfying \(0<\gamma<\min\{\frac{12k^{2}}{\A\^{2}},\frac{1\tau}{\A\^{2}}\}\), \(\{l_{n}\}\) is a sequence in \([1,+\infty)\) with \(L=\sup_{n\geq1}\{l_{n}\}\) and \(\sum_{n=1}^{\infty}(l_{n}1)<\infty\).

(I)
If \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\emptyset\) (the set of solutions of (SCFP) (3.1)), then the sequence \(\{x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma\).

(II)
In addition, if \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\emptyset\) and S is semicompact, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in\Gamma\).
Proof
Now we prove conclusion (I).
We divide the proof into four steps.
Step 1. We first show that the limit \(\lim_{n\to\infty}\x_{n}p\\) exists for each \(p\in\Gamma\).
For any given \(p\in\Gamma\), we have \(p\in F(S)\) and \(Ap\in F(T)\). It follows from (3.3) and Lemma 2.2 that
It follows from (3.3), (3.4) and Lemma 2.1 that
From the condition \(\sum_{n=1}^{\infty}(l_{n}1)<\infty\) and Lemma 2.6 we know that \(\lim_{n\to\infty}\x_{n}p\\) exists. This implies that the sequence \(\{x_{n}\}\) is bounded.
Step 2. We prove that \(\lim_{n\to\infty}\x_{n+1}x_{n}\=0\) and \(\lim_{n\to\infty}\z_{n+1}z_{n}\=0\).
From (3.5) we have
Letting \(n\to\infty\) in the above inequality, we obtain
and
By virtue of Lemma 2.1 and the property of g, we know that the sequences \(\{z_{n}\}\) and \(\{S^{n} z_{n}\}\) are bounded and
It follows from (3.3) that
From (3.7) and (3.8) we have that
Similarly,
In view of (3.7), (3.10) and (3.11), we get
In addition, since
from (3.7) we obtain
Step 3. We prove that \(\lim_{n\to\infty}\z_{n}Sz_{n}\=0\).
By the assumption that S is an asymptotical nonexpansive mapping and \(L=\sup_{n\geq1}\{l_{n}\}\), it follows from (3.8) and (3.12) that
i.e.,
Step 4. We prove that \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\).
Since \(E_{1}\) is uniformly convex, it is reflexive. By the boundedness of \(\{x_{n}\}\), there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup x^{*}\). By virtue of (3.14), this implies that the subsequence \(\{z_{n_{i}}\}\) of \(\{z_{n}\}\) converges weakly to \(x^{*}\), too. From (3.15) we obtain
S is an asymptotical nonexpansive mapping, it is demiclosed at zero. Hence \(x^{*}\in F(S)\).
On the other hand, since A is a bounded linear operator, we know that \(\{Ax_{n_{i}}\}\) converges weakly to \(Ax^{*}\). From (3.7) we have
Since T is demiclosed at zero, we have \(Ax^{*}\in F(T)\). This together with \(x^{*}\in F(S)\) shows that \(x^{*}\in\Gamma\).
Now we prove that \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\).
In fact, if there exists another subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{j}}\}\) converges weakly to \(y^{*}\in\Gamma\), by virtue of Step 1 and the assumption that \(E_{1}\) has the Opial property, we have
This is a contradiction. Therefore \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\). The proof of conclusion (I) is completed.
Next, we prove conclusion (II).
Since \(\lim_{n\to\infty}\z_{n}Sz_{n}\=0\) and S is semicompact, there exists a subsequence \(\{z_{n_{k}}\}\) of \(\{z_{n}\}\) such that \(\{z_{n_{k}}\}\) converges strongly to \(\mu^{*}\in E_{1}\). By using (3.14), we know that the subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) converges strongly to \(\mu^{*}\), too. Due to \(\{x_{n}\}\) converging weakly to \(x^{*}\), we have \(\mu^{*}=x^{*}\). Since \(\lim_{n\to\infty}\x_{n}x^{*}\\) exists and \(\lim_{n_{k}\to\infty}\x_{n_{k}}x^{*}\=0\), we know that \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Gamma\). This completes the proof of conclusion (II).â€ƒâ–¡
4 Application to hierarchical variational inequality problem in Banach spaces
In this section we shall utilize the results presented in Section 3 to study the 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\ \le\x  y\\), \(\forall y \in K\). Putting \(z = P_{K} x\), we call \(P_{K}\) the metric projection of E onto K.
Lemma 4.1
[21]
Let E be a smooth, strictly convex and reflexive Banach space, and K be a nonempty, closed and convex subset of E. Let \(x \in E\) and \(z \in K\). Then the following conditions are equivalent:

(i)
\(z = P_{K} x\);

(ii)
\(\langle z  y, J(x  z)\rangle\ge0\), \(\forall y \in K\),
where J is the normalized duality mapping on E.
Definition 4.2
Let E be a smooth, strictly convex and reflexive Banach space, and K be a nonempty, closed and convex subset of E. Let \(S: K \to K\) be a nonlinear mapping with \(F(S)\) being a nonempty closed and convex subset of K and \(V: K \to K\) be a nonlinear mapping. The socalled hierarchical variational inequality problem for a mapping S with respect to a mapping V in Banach spaces is to find \(x^{*} \in F(S)\) such that
By Lemma 4.1, the hierarchical variational inequality problem in Banach space (4.1) is equivalent to the following fixed point equation:
Letting \(C = F(S)\) and \(Q = F(P_{F(S)}V)\) (the fixed point set of \(P_{F(S)} V\)) and \(A = I\) (the identity mapping on E), then the hierarchical variational inequality problem (4.1) for a mapping S with respect to a mapping V in Banach space is equivalent to the following split common fixed point problem in Banach space:
Therefore the set of solutions \(\Gamma_{1}\) of hierarchical variational inequality problem (4.1) is just the set of solutions of split common fixed point problem (4.3).
Hence from Theorem 3.1 we have the following.
Theorem 4.3
Let E be a real uniformly convex and 2uniformly smooth Banach space having the Opial property and the best smoothness constant k satisfying \(0< k<\frac{1}{\sqrt{2}}\). Let \(S: E\to E\) be an \(\{l_{n}\}\)asymptotical nonexpansive mapping with \(F(S)\neq\emptyset\). Let \(V: E \to E\) be a mapping such that the mapping \(T: = P_{F(S)}V\) is a Ï„quasistrict pseudocontractive mapping with \(F(T) \neq \emptyset\) and T is demiclosed at zero. For each \(x_{1}\in E\), let \(\{x_{n}\}\) be the sequence generated by
where \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) with \(\liminf_{n\to\infty}\alpha_{n}(1\alpha_{n})>0\), Î³ is a positive constant satisfying \(0<\gamma<\min\{(12k^{2}), (1\tau)\}\), \(\{l_{n}\}\) is a sequence in \([1,+\infty)\) with \(L=\sup_{n\geq1}\{l_{n}\}\) and \(\sum_{n=1}^{\infty}(l_{n}1)<\infty\).

(I)
If \(\Gamma_{1}\) (the set of solutions of hierarchical variational inequality problem (4.1)) is nonempty, then the sequence \(\{ x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma_{1}\).

(II)
In addition, if \(\Gamma_{1}\) is nonempty and S is semicompact, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in\Gamma_{1}\).
Proof
Taking \(E_{1} = E_{2} = E\), \(A = I\), \(T = P_{F}(S)\circ V \) in Theorem 3.1, and noting that in this case \(J_{1}= J_{2} = J\) (where J is the normalized duality mapping on E), therefore the conclusion of Theorem 4.3 can be obtained from Theorem 3.1 immediately.â€ƒâ–¡
5 A numerical example
Throughout this section we assume that \(E_{1} = E_{2}\) is a real Hilbert space \(l^{2}\), B is the unit ball in \(l^{2}\), and \(D =\{x=(x_{1},x_{2},\ldots ,x_{n},\ldots) \in l^{2}: x_{i}\in R^{1},i=1,2,\ldots\}\) is an orthogonal subspace of \(l^{2}\) (i.e., \(\forall x, y \in C\), we have \(\langle x,y\rangle=0\)). Define \(C : = B \cap D\). It is easy to know that C is a nonempty closed and convex subset of \(l^{2}\). Let \(S:C\rightarrow C\) be a mapping defined by
where \(\{a_{i}\}\) is a sequence in \((0,1)\) such that \(\prod_{i=2}^{\infty}\alpha_{i}=\frac{1}{2}\). Let \(T: C \to C\) be a mapping defined by
for each \(x =(x_{1}, x_{2},\ldots, x_{n},\ldots)\in C\). In (2.3) and (2.4), we have proved that \(S: C \to C\) is \(\{l_{n}\}\)asymptotically nonexpansive with \(l_{1} =2\), \(l_{n} = 2\prod_{i=2}^{n}a_{i}\), \(n\geq2\) and \(F(S) = \{0\}\), while \(T: C\to C\) is a Ï„quasistrict pseudocontractive mapping with \(F(T) = \{0\}\). Hence from Theorem 3.1 we can obtain the following.
Theorem 5.1
Let \(E_{1} = E_{2} = l^{2}\), \(C= B \cap D\) be the same as above. Let \(S: C \to C\) be the \(\{l_{n}\}\)asymptotically nonexpansive mapping and \(T: C \to C \) be the Ï„quasistrict pseudocontractive mapping defined by (5.1) and (5.2), respectively. Let \(A: C \to C\) be a bounded linear operator and \(A^{*}\) be the adjoint of A. For each \(x_{1}\in C\), let \(\{x_{n}\}\) be the sequence generated by
where \(P_{C}\) is the projection from \(l^{2}\) onto C, \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) with \(\liminf_{n\to\infty}\alpha_{n}(1\alpha_{n})>0\), and Î³ is a positive constant satisfying \(0<\gamma<\frac{1\tau}{\A\^{2}}\). If \(\sum_{n=1}^{\infty}(l_{n}1)<\infty\), then the conclusions of Theorem 3.1 hold.
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Acknowledgements
This work was supported by the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (No. 2015JY0165) and the Scientific Research Project of Yibin University (No. 2013YY06) and the National Natural Science Foundation of China (No. 11361070).
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Tang, J., Chang, Ss., Wang, L. et al. On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces. J Inequal Appl 2015, 305 (2015). https://doi.org/10.1186/s136600150832z
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DOI: https://doi.org/10.1186/s136600150832z