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A new iterative algorithm for split solution problems of quasinonexpansive mappings
Journal of Inequalities and Applications volume 2015, Article number: 131 (2015)
Abstract
Some strong convergence algorithms are introduced to solve the split common fixed point problem for quasinonexpansive mappings. These results develop the related ones for fixed point iterative methods in the literature.
Introduction and preliminaries
Throughout this paper, let H be a real Hilbert space with zero vector θ, whose inner product and norm are denoted by \(\langle \cdot,\cdot\rangle\) and \(\Vert\cdot\Vert\), respectively. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let K be a nonempty closed convex subset of a Banach space E and T be a mapping from K into itself. In this paper, the set of fixed points of T is denoted by \(F(T)\). The symbols → and ⇀ denote strong and weak convergence, respectively.
Let \(T:K\rightarrow K\) be a mapping and K a subset of a Banach space E. T is called a nonexpansive mapping if, for all \(x,y\in K\), \(\Vert TxTy\Vert\leq\Vert xy\Vert\). T is called quasinonexpansive, if \(F(T)\neq\emptyset\) and for all \(x\in K\), \(p\in F(T)\), \(\Vert TxTp\Vert\leq\Vert xp\Vert\). For examples of quasinonexpansive mappings, see [1].
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. \(T_{1}:H_{1}\rightarrow H_{1}\), \(T_{2}:H_{2}\rightarrow H_{2}\) are two nonlinear operators with \(F(T_{1})\neq\emptyset\) and \(F(T_{2})\neq\emptyset\). \(A:H_{1}\rightarrow H_{2}\) is a bounded linear operator. The split fixed point problem for \(T_{1}\) and \(T_{2}\) is to
Let \(\Gamma=\{ x\in F(T_{1}): Ax\in F(T_{2})\}\) denote the solution set of the problem (1.1). The problem was proposed by Censor and Segal [2] in a finitedimensional space firstly. Next, Moudafi [3] studied the problem (1.1) in real Hilbert spaces; this generalized the problem (1.1) from a finitedimensional space to infinitedimensional Hilbert spaces. More precisely, the following result was obtained.
Theorem M
(see [3])
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. Given a bounded linear operator \(A:H_{1}\rightarrow H_{2}\), let \(U:H_{1}\rightarrow H_{1}\) and \(T:H_{2}\rightarrow H_{2}\) be two quasinonexpansive operators with \(F(U)\neq\emptyset\) and \(F(T)\neq\emptyset\). Assume that \(UI\) and \(TI\) are demiclosed at θ. Let \(\{x_{n}\}\) be generated by
where \(\beta\in(0,1)\), \(\{\alpha_{n}\}\subset(\delta,1\delta)\) for a small enough \(\delta>0\), \(\gamma\in(0,\frac{1}{\lambda\beta})\), and λ is the spectral radius of the operator \(A^{*}A\). Then \(\{x_{n}\}\) weakly converges to a split common fixed point \(x^{*}\in\{x^{*}\in F(U): Ax^{*}\in F(T)\}\).
It is well known that the split feasibility problem and the convex feasibility problem are useful to some areas of applied mathematics such as image recovery, convex optimization, and so on. According to [2], the split common fixed point problem (1.1) is a generalization of both these; also see [3]. This shows the split common fixed point problem (1.1) is important. Recently, some convergence theorems for the split common solution problems were given in [4–9]. We notice that Theorem M is a weak convergence theorem, and it is well known that a strong convergence theorem is always more convenient to use. Hence, the purpose of this paper is to give some algorithms for the problem (1.1), and establishes some strong convergence theorems. At the same time, we generalize the problem (1.1) to two countable families of quasinonexpansive mappings.
A mapping T is said to be demiclosed if, for any sequence \(\{x_{n}\}\) which weakly converges to y, and if the sequence \(\{Tx_{n}\}\) strongly converges to z, we have \(T(y)=z\); see [3].
Definition 1.1
Let K be a nonempty closed convex subset of a real Hilbert space and T a mapping from K into K. The mapping T is called zerodemiclosed if \(\{x_{n}\}\) in K satisfying \(\Vert x_{n}Tx_{n}\Vert\rightarrow0\) and \(x_{n}\rightharpoonup z\in K\) implies \(Tz=z\).
Proposition 1.1
Let K be a nonempty closed convex subset of a real Hilbert space with zero vector θ and T a mapping from K into K. Then the following statements hold.

(a)
T is zerodemiclosed if and only if \(IT\) is demiclosed at θ.

(b)
If T is a nonexpansive mapping and there is a bounded sequence \(\{x_{n}\}\subset H\) such that \(\Vert x_{n}Tx_{n}\Vert\rightarrow0\) as \(n\rightarrow0\), then T is zerodemiclosed.
Example 1.1
(see [4])
Let \(H=\mathbb{R}\) with the inner product defined by \(\langle x,y\rangle=xy\) for all \(x,y\in\mathbb{R}\) and the standard norm \(\cdot\). Let \(C:=[0,+\infty)\) and \(Tx=\frac{x^{2}+2}{1+x}\) for all \(x\in C\). Then T is a continuous zerodemiclosed quasinonexpansive mapping but not nonexpansive.
Example 1.2
(see [4])
Let \(H=\mathbb{R}\) with the inner product defined by \(\langle x,y\rangle=xy\) for all \(x,y\in\mathbb{R}\) and the standard norm \(\cdot\). Let \(C:=[0,+\infty)\). Let T be a mapping from C into C defined by
Then T is a discontinuous quasinonexpansive mapping but not zerodemiclosed.
The following results are important in this paper.
Let C be a closed convex subset of a real Hilbert space H. \(P_{C}\) denotes a metric projection of H onto C, it is well known that \(P_{C}(x)\) has the properties: for \(x\in H\), and \(z\in C\),
and
In a real Hilbert space H, it is also well known that
and
Strong convergence theorems
In this section, we construct some algorithms to solve the split common fixed point problem (1.1) for quasinonexpansive mappings.
Theorem 2.1
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. C is a nonempty closed convex subset of \(H_{1}\) and K a nonempty closed convex subset of \(H_{2}\). \(T_{1}: C\rightarrow H_{1}\) and \(T_{2}:H_{2}\rightarrow H_{2}\) are two quasinonexpansive mappings with \(F(T_{1})\neq\emptyset\) and \(F(T_{2})\neq\emptyset\). \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. Assume that \(T_{1} I\) and \(T_{2}I\) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A. \(\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\Vert A^{*}\Vert^{2}})\). Assume that \(\Gamma=\{p\in F(T_{1}): Ap\in F(T_{2})\}\neq\emptyset\), then \(x_{n} \rightarrow x^{*}\in \Gamma\) and \(Ax_{n} \rightarrow Ax^{*}\in F(T_{2})\).
Proof
It is easy to verify that \(C_{n}\) is closed for \(n\in\mathbb{N}\cup\{0\}\). We verify \(C_{n}\) is convex for \(n\in\mathbb{N}\cup\{0\}\). In fact, let \(v_{1},v_{2}\in C_{n+1}\), for each \(\lambda\in(0,1)\), we have
namely, \(\Vert y_{n}(\lambda v_{1}+(1\lambda)v_{2})\Vert\leq\Vert z_{n}(\lambda v_{1}+(1\lambda)v_{2})\Vert\). Similarly, we have \(\Vert z_{n}(\lambda v_{1}+(1\lambda)v_{2})\Vert\leq\Vert x_{n}(\lambda v_{1}+(1\lambda)v_{2})\Vert\); this shows \(\lambda v_{1}+(1\lambda)v_{2}\in C_{n+1}\) and \(C_{n+1}\) is a convex set for \(n\in\mathbb{N}\cup\{0\}\). Now we prove \(\Gamma\subset C_{n}\) for \(n\in\mathbb{N}\cup\{0\}\). Let \(p\in\Gamma\), then
Again from \(p\in\Gamma\), (2.1), and (2.3), it follows that
Hence, \(p\in C_{n}\) and \(\Gamma\subset C_{n}\) for \(n\in \mathbb{N}\cup\{0\}\).
Notice that \(\Gamma\subset C_{n+1}\subset C_{n}\) and \(x_{n+1}=P_{C_{n+1}}(x_{0})\subset C_{n}\), then
By (2.5), \(\{x_{n}\}\) is bounded. For \(n\in\mathbb{N}\), by (1.4), we have
which implies that \(0\leq\Vert x_{n}x_{n+1}\Vert^{2}\leq\Vert x_{n+1}x_{0}\Vert^{2}\Vert x_{0}x_{n}\Vert^{2}\). Thus \(\{\Vert x_{n}x_{0}\Vert\}\) is nondecreasing. Therefore, by the boundedness of \(\{ x_{n}\}\), \(\lim_{n\rightarrow\infty}\Vert x_{n}x_{0}\Vert\) exists. For \(m, n\in\mathbb{N}\) with \(m>n\), from \(x_{m}=P_{C_{m}}(x_{0})\subset C_{n}\) and (1.4), we have
By (2.5) and (2.6), \(\lim_{n\rightarrow\infty}\Vert x_{n}x_{m}\Vert=0\). So, \(\{x_{n}\}\) is a Cauchy sequence.
Let \(x_{n}\rightarrow x^{*}\). Since \(x_{n+1}=P_{C_{n+1}}(x_{0})\in C_{n+1}\subset C_{n}\), we have
Notice that \(\lambda(1 \lambda\Vert A^{*}\Vert^{2})>0\), from (2.3) and (2.7),
Again from (2.1) and (2.7), we have
Since \(x_{n}\rightarrow x^{*}\), from (2.7) we have \(z_{n}\rightarrow x^{*}\), which implies that \(z_{n}\rightharpoonup x^{*}\). By Proposition 1.1, we obtain \(x^{*}\in F(T_{1})\).
Next, we want to show \(Ax^{*}\in F(T_{2})\). Since A is a bounded linear operator, we know that \(\Vert Ax_{n}Ax^{*}\Vert\rightarrow0\) by \(x_{n}\rightarrow x^{*}\). Together with \(\Vert T_{2}Ax_{n}Ax_{n}\Vert\rightarrow0\) and \(T_{2}I\) being demiclosed at θ, we have \(Ax^{*}\in F(T_{2})\). Thus, \(x^{*}\in \Gamma\) and \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Gamma\). The proof is completed. □
Remark 2.1
If the quasinonexpansive mappings \(T_{1}\) and \(T_{2}\) are continuous, then the demiclosed property can be removed for the quasinonexpansive mappings \(T_{1}\) and \(T_{2}\) in Theorem 2.1.
Now, we consider the split fixed point problem for a finite family of quasinonexpansive mappings.
Lemma 2.1
(see [3])
Let \(T:H\rightarrow H\) be a quasinonexpansive mapping, and set \(T_{\alpha}:=(1\alpha)I+\alpha T\) for \(\alpha\in(0,1]\). Then \(\Vert T_{\alpha}xp\Vert\leq\Vert xp\Vert\alpha(1\alpha)\Vert T xx\Vert\), \(p\in F(T)\) and \(x\in H\). Moreover, \(F(T_{\alpha})=F(T)\).
Lemma 2.2
Let \(T_{1}, T_{2}:H\rightarrow H\) be two quasinonexpansive mappings and set \(S_{\xi_{1}}:=(1\xi_{1})I+\xi_{1}T_{1}\) and \(S_{\xi_{2}}:=(1\xi_{2})I+\xi_{2}T_{2}\) for \(\xi_{1}, \xi_{2}\in(0,1)\). Again let \(S=\tau S_{\xi_{1}}+(1\tau)S_{\xi_{2}}\) for \(\tau\in(0,1)\). Then S is a quasinonexpansive mapping, and \(F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})=\bigcap_{i=1}^{2}F(T_{i})\).
Proof
(1) It is easy to verify that \(\bigcap_{i=1}^{2}F(S_{\xi_{i}})=\bigcap_{i=1}^{2}F(T_{i})\). We only need to prove \(F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})\). Clearly, \(\bigcap_{i=1}^{2}F(S_{\xi_{i}})\subset F(S)\). On the other hand, for \(p\in F(S)\) and \(p_{1}\in\bigcap_{i=1}^{2}F(S_{\xi_{i}})\), we have
which yields \(\Vert T_{1}pp\Vert=\Vert T_{2} pp \Vert=0\), namely, \(p\in\bigcap_{i=1}^{2}F(T_{i})= \bigcap_{i=1}^{2}F(S_{\xi_{i}})\). So, \(F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})\).
(2) Let \(x\in H\) and \(p\in F(S)\). Then
So, S is a quasinonexpansive mapping. The proof is completed. □
Lemma 2.3
Let \(T_{1}, T_{2},\ldots, T_{k}:H\rightarrow H\) be k quasinonexpansive mappings and set \(S=\sum_{i=1}^{k}\tau_{i} S_{\xi_{i}}\), where \(\tau_{i}\in(0,1)\) satisfies \(\sum_{i=1}^{k}\tau_{i} =1\), \(S_{\xi_{i}}:=(1\xi_{i})I+\xi_{i}T_{i}\) for \(\xi_{i}\in(0,1)\), \(i=1,2,\ldots,k\). Then S is a quasinonexpansive mapping, and \(F(S)=\bigcap_{i=1}^{k}F(S_{\xi_{i}})=\bigcap_{i=1}^{k}F(T_{i})\).
Proof
Using mathematical induction, Lemma 2.3 is obtained by Lemma 2.2. □
Theorem 2.2
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. C is a nonempty closed convex subset of \(H_{1}\) and K a nonempty closed convex subset of \(H_{2}\). \(T_{1},\ldots, T_{k}: C\rightarrow H_{1}\) are k quasinonexpansive mappings with \(\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset\). \(G_{1},\ldots, G_{l}:H_{2}\rightarrow H_{2}\) are l quasinonexpansive mappings with \(\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset\). \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. Assume that \(T_{i}I\) (\(i=1,2,\ldots,k\)) and \(G_{j}I\) (\(j=1,2,\ldots, l\)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A, \(\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\A^{*}\^{2}})\). \(\tau_{i}\in(0,1)\) and \(\varepsilon_{j}\in(0,1)\) satisfy \(\sum_{i=1}^{k}\tau_{i} =1\) and \(\sum_{j=1}^{l}\varepsilon_{j} =1\), \(T_{\xi _{i}}:=(1\xi_{i})I+\xi_{i}T_{i}\) for \(\xi_{i}\in(0,1)\), \(i=1,2,\ldots,k\), \(G_{\theta_{j}}:=(1\theta_{j})I+\theta_{j}G_{j}\) for \(\theta_{j}\in(0,1)\), \(j=1,2,\ldots,l\). Assume that \(\Gamma=\{p\in\bigcap_{i=1}^{k}F(T_{i}): Ap\in\bigcap_{j=1}^{l}F(G_{j})\}\neq\emptyset\), then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
Proof
Let \(T=\sum_{i=1}^{k}\tau_{i} T_{\xi_{i}} \), \(S=\sum_{i=1}^{l}\varepsilon_{j} G_{\theta_{j}}\), by Lemma 2.3, \(F(T)=\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset\), and \(F(S)=\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset\). Moreover, T and S are quasinonexpansive mappings.
Next, we want to prove \(TI\) and \(SI\) are demiclosed at θ. By the hypothesis, \(T_{i}I\) (\(i=1,2,\ldots,k\)) and \(G_{j}I\) (\(j=1,2,\ldots, l\)) are demiclosed at θ. So, \(T_{\xi_{i}}I= \xi_{i}(T_{i}I)\) and \(G_{\theta_{j}}I=\theta_{j}(G_{j}I)\) are demiclosed at θ, and that \(TI=\sum_{i=1}^{k}\tau_{i} (T_{\xi_{i}}I)\) and \(SI=\sum_{i=1}^{l}\varepsilon_{j} (G_{\theta_{j}}I)\) are demiclosed at θ.
Thus, by Theorem 2.1, we obtain the desired result. The proof is completed. □
If \(C=H_{1}\) in Theorem 2.1 and Theorem 2.2, then we have the following corollaries.
Corollary 2.1
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. \(T_{1}: H_{1}\rightarrow H_{1}\) and \(T_{2}:H_{2}\rightarrow H_{2}\) are two quasinonexpansive mappings with \(F(T_{1})\neq\emptyset\) and \(F(T_{2})\neq\emptyset\). \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. Assume that \(T_{1} I\) and \(T_{2}I\) are demiclosed at θ. Let \(x_{0}\in H_{1}\), \(C_{0}=H_{1}\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A, \(\{\alpha_{n}\}\subset (0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\A^{*}\^{2}})\). Assume that \(\Gamma=\{p\in F(T_{1}): Ap\in F(T_{2})\}\neq\emptyset\), then the sequence \(\{x_{n}\}\) converges strongly to an element \(x^{*}\in\Gamma\).
Corollary 2.2
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. \(T_{1},\ldots, T_{k}: H_{1}\rightarrow H_{1}\) are k quasinonexpansive mappings with \(\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset\). \(G_{1},\ldots, G_{l}: H_{2}\rightarrow H_{2}\) are l quasinonexpansive mappings with \(\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset\). \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. Assume that \(T_{i}I\) (\(i=1,2,\ldots,k\)) and \(G_{j}I\) (\(j=1,2,\ldots, l\)) are demiclosed at θ. Let \(x_{0}\in H_{1}\), \(C_{0}=H_{1}\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A, \(\{\alpha_{n}\}\subset (0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\A^{*}\^{2}})\). Here \(\tau_{i}\in(0,1)\) and \(\varepsilon_{j}\in(0,1)\) satisfy \(\sum_{i=1}^{k}\tau_{i} =1\) and \(\sum_{j=1}^{l}\varepsilon_{j} =1\), \(T_{\xi _{i}}:=(1\xi_{i})I+\xi_{i}T_{i}\) for \(\xi_{i}\in(0,1)\), \(i=1,2,\ldots,k\), \(G_{\theta_{j}}:=(1\theta_{j})I+\theta_{j}G_{j}\) for \(\theta_{j}\in(0,1)\), \(j=1,2,\ldots,l\). Assume that \(\Gamma=\{p\in\bigcap_{i=1}^{k}F(T_{i}): Ap\in\bigcap_{j=1}^{l}F(G_{j})\}\neq\emptyset\), then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
If \(H_{1}=H_{2}:=H\) and A is an identity operator, then we have the following results by Theorems 2.1 and 2.2, respectively.
Corollary 2.3
Let H be a real Hilbert space. C is a nonempty closed convex subset of H. \(T_{1}: C\rightarrow H\) and \(T_{2}:H\rightarrow H\) are two quasinonexpansive mappings with \(\Gamma:=F(T_{1})\cap F(T_{2})\neq \emptyset\). Assume that \(T_{1} I\) and \(T_{2}I\) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). Then \(x_{n} \rightarrow x^{*}\in \Gamma\).
Corollary 2.4
Let H be a real Hilbert space. C is a nonempty closed convex subset of H. \(T_{1},\ldots, T_{k}: C\rightarrow H\) are k quasinonexpansive mappings with \(\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset\). \(G_{1},\ldots, G_{l}:H\rightarrow H\) are l quasinonexpansive mappings with \(\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots,k\)) and \(G_{j}I\) (\(j=1,2,\ldots, l\)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). \(\tau_{i}\in(0,1)\) and \(\varepsilon_{j}\in(0,1)\) satisfy \(\sum_{i=1}^{k}\tau_{i} =1\) and \(\sum_{j=1}^{l}\varepsilon_{j} =1\), \(T_{\xi _{i}}:=(1\xi_{i})I+\xi_{i}T_{i}\) for \(\xi_{i}\in(0,1)\), \(i=1,2,\ldots,k\), \(G_{\theta_{j}}:=(1\theta_{j})I+\theta_{j}G_{j}\) for \(\theta_{j}\in(0,1)\), \(j=1,2,\ldots,l\). Assume that \(\Gamma:=(\bigcap_{i=1}^{k}F(T_{i}))\cap( \bigcap_{j=1}^{l}F(G_{j}))\neq\emptyset\), then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
If \(C=H:=H_{1}=H_{2}\) and A is an identity operator, then we have the following results by Corollaries 2.3 and 2.4, respectively.
Corollary 2.5
Let H be a real Hilbert space. \(T_{1}, T_{2}: H\rightarrow H\) are two quasinonexpansive mappings with \(\Gamma:=F(T_{1})\cap F(T_{2})\neq \emptyset\). Assume that \(T_{1} I\) and \(T_{2}I\) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). Then \(x_{n} \rightarrow x^{*}\in \Gamma\).
Corollary 2.6
Let H be a real Hilbert space. \(T_{1},\ldots, T_{k}: H\rightarrow H\) are k quasinonexpansive mappings with \(\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset\). \(G_{1},\ldots, G_{l}:H\rightarrow H\) are l quasinonexpansive mappings with \(\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots,k\)) and \(G_{j}I\) (\(j=1,2,\ldots, l\)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). \(\tau_{i}\in(0,1)\) and \(\varepsilon_{j}\in(0,1)\) satisfy \(\sum_{i=1}^{k}\tau_{i} =1\) and \(\sum_{j=1}^{l}\varepsilon_{j} =1\), \(T_{\xi _{i}}:=(1\xi_{i})I+\xi_{i}T_{i}\) for \(\xi_{i}\in(0,1)\), \(i=1,2,\ldots,k\), \(G_{\theta_{j}}:=(1\theta_{j})I+\theta_{j}G_{j}\) for \(\theta_{j}\in(0,1)\), \(j=1,2,\ldots,l\). Assume that \(\Gamma:=(\bigcap_{i=1}^{k}F(T_{i}))\cap( \bigcap_{j=1}^{l}F(G_{j}))\neq\emptyset\), then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
Remark 2.2
The coefficient condition that \(\{\alpha_{n}\}\subset(\delta,1\delta)\) for a small enough \(\delta>0\) in Theorem M is replaced with \(\{\alpha_{n}\}\subset(0,\eta]\subset (0,1)\). This shows we can let \(\alpha_{n}=\frac{1}{n+1}\) in this paper, which is a natural choice.
Further generalization of the problem (1.1)
In Section 2, we gave a strong convergence algorithm for the problem (1.1). By the algorithm, we also considered the split solution problem for two finite families of quasinonexpansive mappings; see the algorithm (2.10). However, the algorithm (2.10) has an obvious drawback, in that the algorithm (2.10) will be invalid for two countable families of quasinonexpansive mappings. So, in this section, we introduce an algorithm for the split solution problem of two countable families of quasinonexpansive mappings. The following lemma can be found in [10].
Lemma
The unique solutions to the positive integer equation
are
where \([x]\) denotes the maximal integer that is not larger than x.
Theorem 3.1
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. C is a nonempty closed convex subset of \(H_{1}\). \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. \(\{T_{i}\}_{i=1}^{\infty}: C\rightarrow H_{1}\) and \(\{G_{i}\}_{i=1}^{\infty}: H_{2}\rightarrow H_{2}\) are two countable families of quasinonexpansive mappings with \(\Gamma=\{p\in\bigcap_{i=1}^{\infty}F(T_{i}): Ap\in\bigcap_{j=1}^{\infty}F(G_{j})\}\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots \)) and \(G_{j}I\) (\(j=1,2,\ldots \)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A, \(\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\A^{*}\^{2}})\). \(i_{n}\) satisfies (3.1), i.e. \(i_{n}=n\frac{(m1)m}{2}\) and \(m\geq i_{n}\) for \(n=1,2,\ldots \) . Then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
Proof
Just like the proof in Theorem 2.1, we can obtain the following facts (I)(IV):
(I) For \(p\in\Gamma\),
and
(II) We have \(\Gamma\subset C_{n}\) for \(n\in \mathbb{N}\cup\{0\}\). \(C_{n}\) is also closed and convex for \(n\in\mathbb{N}\cup\{0\}\).
(III) \(\{x_{n}\}\) is a Cauchy sequence and
(IV)
Now, for each \(i\in\mathbb{N}\), set \(K_{i}=\{k\geq1: k=i+\frac {(m1)m}{2}, m\geq i,m\in \mathbb{N}\}\). Since \(n=i_{n}+\frac {(m1)m}{2}\), \(m\geq i_{n}\), and \(m\in\mathbb{N} \) for \(n=1,2,\ldots \) , and the definition of \(K_{i}\), we have \(i_{k}\equiv i\) for \(k\in K_{i}\). Obviously, \(\{k\}\) is a subsequence of \(\{n\}\). Thus, for \(k\in K_{i}\) and \(i\in\mathbb{N}\), it follows from (3.8) that
Let \(x_{n}\rightarrow x^{*}\). From (3.7) we have \(z_{n}\rightarrow x^{*}\). By (3.9), we obtain \(x^{*}\in F(T_{i})\).
Next, we want to prove \(Ax^{*}\in F(G_{i})\). Since A is a bounded linear operator, \(\Vert Ax_{n}Ax^{*}\Vert\rightarrow0\) by \(x_{n}\rightarrow x^{*}\). Together with \(\Vert(G_{i}I)Ax_{k} \Vert\rightarrow0\), we have \(Ax_{n}\rightarrow Ax^{*}\in F(G_{i})\). Thus, \(x^{*}\in \Gamma\) and \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Gamma\). The proof is completed. □
If \(C=H_{1}\), then we have the following result by Theorem 3.1.
Corollary 3.1
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces. \(A: H_{1}\rightarrow H_{2}\) is a bounded linear operator. \(\{T_{i}\}_{i=1}^{\infty}: H_{1}\rightarrow H_{1}\) and \(\{G_{i}\}_{i=1}^{\infty}: H_{2}\rightarrow H_{2}\) are two countable families of quasinonexpansive mappings with \(\Gamma=\{p\in\bigcap_{i=1}^{\infty}F(T_{i}): Ap\in\bigcap_{j=1}^{\infty}F(G_{j})\}\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots \)) and \(G_{j}I\) (\(j=1,2,\ldots \)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=H_{1}\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator and \(A^{*}\) denotes the adjoint of A, \(\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)\), \(\lambda\in(0,\frac{1}{\A^{*}\^{2}})\). \(i_{n}\) satisfies (3.1), i.e. \(i_{n}=n\frac{(m1)m}{2}\) and \(m\geq i_{n}\) for \(n=1,2,\ldots \) . Then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
If \(H_{1}=H_{2}:=H\) and A is an identity operator, then we have the following results by Theorem 3.1 and Corollary 3.1, respectively.
Corollary 3.2
Let H be a real Hilbert space. C is a nonempty closed convex subset of H. \(\{T_{i}\}_{i=1}^{\infty}: C\rightarrow H\) and \(\{G_{i}\}_{i=1}^{\infty}: H\rightarrow H\) are two countable families of quasinonexpansive mappings with \(\Gamma:= (\bigcap_{i=1}^{\infty}F(T_{i}))\cap( \bigcap_{j=1}^{\infty}F(G_{j}))\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots \)) and \(G_{j}I\) (\(j=1,2,\ldots \)) are demiclosed at θ. Let \(x_{0}\in C\), \(C_{0}=C\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). \(i_{n}\) satisfies (3.1), i.e. \(i_{n}=n\frac{(m1)m}{2}\) and \(m\geq i_{n}\) for \(n=1,2,\ldots \) . Then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
Corollary 3.3
Let H be a real Hilbert space. \(\{T_{i}\}_{i=1}^{\infty}: H\rightarrow H\) and \(\{G_{i}\}_{i=1}^{\infty}: H\rightarrow H\) are two countable families of quasinonexpansive mappings with \(\Gamma=\{p\in(\bigcap_{i=1}^{\infty}F(T_{i}))\cap( \bigcap_{j=1}^{\infty}F(G_{j}))\}\neq\emptyset\). Assume that \(T_{i}I\) (\(i=1,2,\ldots \)) and \(G_{j}I\) (\(j=1,2,\ldots \)) are demiclosed at θ. Let \(x_{0}\in H\), \(C_{0}=H\), and \(\{x_{n}\}\) be a sequence generated in the following manner:
where P is a projection operator. \(\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)\), \(\lambda\in(0,1)\). \(i_{n}\) satisfies (3.1), i.e. \(i_{n}=n\frac{(m1)m}{2}\) and \(m\geq i_{n}\) for \(n=1,2,\ldots \) . Then the sequence \(\{x_{n}\}\) converges strongly to an element \(q\in\Gamma\).
Conclusion

(1)
We give strong convergence algorithms for the split common fixed point problem of quasinonexpansive mappings. Our results improve and generalize some wellknown results in [3, 11] and so on.

(2)
Although Theorem 3.1 gives a strong convergence algorithm for two countable families of quasinonexpansive mappings, the condition that each mapping must be demiclosed at θ is very strong. In addition, we guess the speed of convergence is not too fast for the algorithm (3.3). Therefore, the algorithm (3.3) should be improved further in the future.

(3)
The split common solution problem is a very interesting topic. It has received attention by many scholars. Many research articles have been published, for example, [12–21] and references therein.
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The Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206) is acknowledged.
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Li, R., He, Z. A new iterative algorithm for split solution problems of quasinonexpansive mappings. J Inequal Appl 2015, 131 (2015). https://doi.org/10.1186/s1366001506530
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MSC
 49J53
 65K10
 49M37
 90C25
Keywords
 split common fixed point
 iterative method
 strong convergence
 quasinonexpansive mapping