# Multidirectional hybrid algorithm for the split common fixed point problem and application to the split common null point problem

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## Abstract

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence. The results of this paper improve and extend the recent results of Takahashi and Yao (Fixed Point Theory Appl 2015:87, 2015) and many others .

### Keywords

Split common fixed point problem Split common null point problem Fixed point Metric resolvent Multidirectional hybrid algorithm Duality mapping### Mathematics Subject Classification

47H05 47H09 47H10## Introduction and preliminaries

*A*and \(P_Q\) is the metric projection of \(H_2\) onto

*Q*. Furthermore, if \(D\cap A^{-1}Q\) is nonempty, then

*D*. Using such results regarding nonlinear operators and fixed points, many authors have studied the split feasibility problem in Hilbert spaces; see, for instance, Alsulami and Takahashi (2014), Byrne et al. (2012), Censor and Segal (2009), Moudafi (2010), Takahashi et al. (2015). Recently, Takahashi (2014) and Takahashi (2015) extended an equivalent relation as in (1) in Hilbert spaces to Banach spaces and then obtained strong convergence theorems for finding a solution of the split feasibility problem in Banach spaces. Very recently, using the hybrid method by Nakajo and Takahashi (2003) in mathematical programming, Alsulami et al. (2015) proved strong convergence theorems for finding a solution of the split feasibility problem in Banach spaces; see also Ohsawa and Takahashi (2003), Solodov and Svaiter (2000). Takahashi (2015) also obtained a result for finding a solution of the split feasibility problem in Banach space from the idea of the shrinking projection method by Takahashi et al. (2008). Takahashi and Yao (2015) presented the following hybrid iteration algorithm in a Hilbert space

*H*: for \(x_1 \in H\),

**Theorem TY**

*Let*

*H*

*be a Hilbert space and let*

*F*

*be a uniformly convex and smooth Banach space. Let*\(J_F\)

*be the duality mapping on*

*F*.

*Let*

*A*

*and*

*B*

*be maximal monotone operators of*

*H*

*into*\(2^H\)

*and*

*F*

*into*\(2^{F^*}\).

*such that*\(A^{-1}0 \ne \emptyset\)

*and*\(B^{-1}0\ne \emptyset\),

*respectively. Let*\(J_\lambda\)

*be the resolvent of*

*A*

*for*\(\lambda >0\)

*and let*\(Q_\mu\)

*be the metric resolvent of*

*B*

*for*\(\mu >0\).

*Let*\(T : H \rightarrow F\)

*be a bounded linear operator such that*\(T\ne 0\)

*and let*\(T^*\)

*be the adjoint operator of*

*T*.

*Suppose that*\(A^{-1}0\cap T^{-1}(B^{-1})0 \ne \emptyset\).

*Let*\(x_1 \in H\)

*and let*\(\{x_n\}\)

*be a sequence generated by (TY), where*\(\{\alpha _n\subset [0,1]\}\)

*and*\(\{\lambda _n\}, \{\mu _n\}\)

*satisfy the condition such that*

*for some*\(a,b,c \in R\).

*Then*\(\{x_n\}\)

*converges strongly to a point*\(z_0=P_{A^{-1}0\cap T^{-1}(B^{-1}0)}x_1\).

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence.

*E*be a real Banach space with norm \(\parallel \cdot \parallel\) and let \(E^*\) be the dual space of

*E*. We denote the value of \(y^* \in E^*\) at \(x\in E\) by \(\langle x,y^*\rangle\). A Banach space

*E*is uniformly convex if for any two sequences \(\{x_n\}\) and \(\{y_n\}\) in

*E*such that

*J*from

*E*into \(2^{E^*}\) is defined by

*E*is said to be Gateaux differentiable if for each \(x,y\in U\), the limit

*E*is called smooth. We know that

*E*is smooth if and only if

*J*is a single- valued mapping of

*E*into \(E^*\). We also know that

*E*is reflexive if and only if

*J*is surjective, and

*E*is strictly convex if and only if

*J*is one-to-one. Therefore, if

*E*is a smooth, strictly convex and reflexive Banach space, then

*J*is a single-valued bijection and in this case, the inverse mapping \(J^{-1}\) coincides with the duality mapping \(J_*\) on \(E^*\). For more details, see Takahashi (2009) and Takahashi (2000).

Let *C* be a nonempty, closed and convex subset of a strictly convex and reflexive Banach space *E*. Then we know that for any \(x\in E\), there exists a unique element \(z\in C\) such that \(\parallel x-z\parallel \le \parallel x-y \parallel\) for all \(y\in C\). Putting \(z=P_Cx\), we call \(P_C\) the metric projection of *E* onto *C*.

**Definition 1**

*E*be a metric space, let \(T: D(T)\rightarrow R(T)\) be a mapping with the domain

*D*(

*T*) and the range

*R*(

*T*). The mapping

*T*is said to be quasi-nonexpansive if

*F*(

*T*) is the nonempty fixed point set of

*T*.

**Definition 2**

*E*be a smooth Banach space, let \(S: D(T)\rightarrow R(T)\) be a mapping with the domain

*D*(

*T*) and the range

*R*(

*T*). The mapping

*S*is said to be second-type quasi-nonexpansive, if

*F*(

*S*) is the nonempty fixed point set of

*T*.

**Definition 3**

*E*,

*F*be two normed spaces and

*T*be a linear operator from

*E*into

*F*. The adjoint operator \(T^*: F^*\rightarrow E^*\) of

*T*is defined by

*E*and

*F*, respectively.

The adjoint spaces and adjoint operators are very important in the theory of functional analysis and applications. Not only is it an important theoretical subject but it is also a very useful tool in the functional analysis and topological theory.

**Definition 4**

Let *E* be a Banach space, let *C* be a nonempty, closed, and convex subset of *E*. Let \(\{T_n\}\) be sequence of mappings from *C* into itself with nonempty common fixed point set \(F=\cap _{n=1}^{\infty }F(T_n)\). The \(\{T_n\}\) is said to be uniformly closed if for any convergent sequence \(\{z_n\} \subset C\) such that \(\Vert T_nz_n-z_n\Vert \rightarrow 0\) as \(n\rightarrow \infty\), the limit of \(\{z_n\}\) belong to *F*.

## Main results

**Lemma 5**

*Let * *H* *be a Hilbert space, let* *C* *be a closed convex subset of* *H*, *and let* \(\{T_n\}\) *be a uniformly closed family of countable quasi-nonexpansive mappings from* *C* *into itself. Then the common fixed point set* *F* *is closed and convex.*

*Proof*

*F*is closed. Next we show that

*F*is also convex. For any \(x,y \in F\), let \(z=tx+(1-t)y\) for any \(t \in (0,1)\), we have

*n*. This implies \(z \in F\), therefore

*F*is convex. This completes the proof. \(\square\)

**Lemma 6**

*Let* *E* *be a smooth Banach space, let* *C* *be a closed convex subset of* *E*, *and let* \(\{S_n\}\) *be a uniformly closed family of countable second-type quasi-nonexpansive mappings from* *C* *into itself. Then the common fixed point set* *F* *is closed and convex.*

*Proof*

*F*is closed. Next we show that

*F*is also convex. For any \(x,y \in F\), let \(z=tx+(1-t)y\) for any \(t \in (0,1)\), we have

*F*is convex. This completes the proof. \(\square\)

**Lemma 7**

*Let*

*H*

*be a Hilbert space, let*

*C*

*be a nonempty closed convex subset of*

*H*

*and let*\(x \in E.\)

*Then*

Next, we present a new hybrid algorithm so-called the multidirectional hybrid algorithm for finding the common fixed point of a uniformly closed family of countable quasi-nonexpansive mappings and a uniformly closed family of countable second-type quasi-nonexpansive mappings.

**Theorem 8**

*Let*

*H*

*be a Hilbert space and let*

*E*

*be a uniformly convex and smooth Banach space. Let*

*J*

*be the duality mapping on*

*E*.

*Let*\(\{T_n\}: H\rightarrow H\)

*be a uniformly closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point set*\(\cap _{n=1}^{\infty }F(T_n)\)

*and*\(\{S_n\}: E\rightarrow E\)

*be a uniformly closed family of countable second-type quasi-nonexpansive mappings with the nonempty common fixed point sets*\(\cap _{n=1}^{\infty }F(S_n)\).

*Suppose that*\(F=(\cap _{n=1}^{\infty }F(T_n))\cap (T^{-1}(\cap _{n=1}^{\infty }F(S_n)))\ne \emptyset\).

*Let*\(T : H \rightarrow E\)

*be a bounded linear operator such that*\(T\ne 0\)

*and let*\(T^*\)

*be the adjoint operator of*

*T*.

*Let*\(x_{1,i}\in H, i=1,2,3, \ldots ,N\)

*and let*\(\{x_n\}\)

*and*\(\{z_n\}\)

*be two sequences generated by*

*where*\(\{r_n\}\)

*satisfy the condition such that*

*for some constants*

*a*,

*b*

*and*\(\lambda \in [0,1]\)

*is a constant. Then the following conclusions hold:*

(1) \(\{x_n\}\) and \(\{z_n\}\) *converge strongly to a point* \(w \in F\);

(2) *the limits* \(\lim _{n\rightarrow \infty }P_{C_n}x_{1,i}=P_{F}x_{1,i}, \ i=1,2,3, \ldots ,N\);

(3) \(w=\sum _{i=1}^{N}\lambda _i \lim _{n\rightarrow \infty } P_{C_n}x_{1,i}\).

*Proof*

*F*is nonempty, closed, and convex, there exist \(p_{1,i}=P_{F}x_{1,i}\) such that

*m*, that

*C*for all \(i=1,2,3, \ldots ,N\), therefore there exit two points \(p_i\in C\) such that

*n*. Also from Lemma 7, we have

By using Theorem 8 and setting \(N = 1\), we can get the following result.

**Theorem 9**

*Let*

*H*

*be a Hilbert space and let*

*E*

*be a uniformly convex and smooth Banach space. Let*

*J*

*be the duality mapping on*

*E*.

*Let*\(\{T_n\}: H\rightarrow H\)

*be a uniformly closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point set*\(\cap _{n=1}^{\infty }F(T_n)\)

*and*\(\{S_n\}: E\rightarrow E\)

*be a uniformly closed family of countable second-type quasi-nonexpansive mappings with the nonempty common fixed point sets*\(\cap _{n=1}^{\infty }F(S_n)\).

*Suppose that*\(F=(\cap _{n=1}^{\infty }F(T_n))\cap (T^{-1}(\cap _{n=1}^{\infty }F(S_n)))\ne \emptyset\).

*Let*\(T : H \rightarrow E\)

*be a bounded linear operator such that*\(T\ne 0\)

*and let*\(T^*\)

*be the adjoint operator of*

*T*.

*Let*\(x_1\in H\)

*and let*\(\{x_n\}\)

*be a sequence generated by*

*where*\(\{r_n\}\)

*satisfy the condition such that*

*for some constants*

*a*,

*b*.

*Then*\(\{x_n\}\)

*converges strongly to a point*\(z_0=P_{F}x_1\).

## Application for common null point problem

*E*be a Banach space, let

*A*be a multi-valued operator from

*E*to \(E^*\) with domain \(D(A)=\{z\in E: Az\ne \emptyset \}\) and range \(R(A)=\{z\in E: z\in D(A)\}\). An operator A is said to be monotone if

*A*is a maximal monotone operator, then \(A^{-1}0\) is closed and convex. The following result is also well-known.

**Theorem 10**

(Rockafellar 1970). *Let* *E* *be a reflexive, strictly convex and smooth Banach space and let* *A* *be a monotone operator from* *E* *to* \(E^{*}\). *Then* *A* *is maximal if and only if* \(R(J + rA)=E^*\). *for all* \(r>0\).

*E*be a reflexive, strictly convex and smooth Banach space, and let

*A*be a maximal monotone operator from

*E*to \(E^*\). Using Theorem 10 and strict convexity of

*E*, we obtain that for every \(r>0\) and \(x\in E\), there exists a unique \(x_r\) such that

*A*. We know that \(J_r\) is a nonexpansive mapping and \(A^{-1}0=F(J_r)\) for all \(r > 0\), see Takahashi (2000, 2009), Alber (1996).

**Lemma 11**

*Let*

*E*

*be a reflexive, strictly convex and smooth Banach space, and let*

*A*

*be a maximal monotone operator from*

*E*

*to*\(E^*\).

*Then*

*where*\(J_{r}\)

*is the resolvent of*

*A*.

From Lemma 11, we know that, \(J_r\) is a second-type quasi-nonexpansive mapping, where \(J_r\) is the resolvent of *A* with \(r>0.\)

**Definition 12**

Let *E* be a Banach space, let *C* be a nonempty, closed, and convex subset of *E*. Let \(\{T_n\}\) be sequence of mappings from *C* into itself with nonempty common fixed point set \(F=\cap _{n=1}^{\infty }F(T_n)\). The \(\{T_n\}\) is said to be uniformly weak closed if for any weak convergent sequence \(\{z_n\} \subset C\) such that \(\Vert T_nz_n-z_n\Vert \rightarrow 0\) as \(n\rightarrow \infty\), the weak limit of \(\{z_n\}\) belong to *F*.

A uniformly weak closed family of countable quasi-nonexpansive mappings must be a uniformly closed family of countable quasi-nonexpansive mappings.

**Theorem 13**

*Let* \(r_n\ge c >0\), *for some constant* *c*, *then* \(\{J^A_{r_n}\}_{n=0}^{\infty }\) *is a uniformly weak closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point sets* \(\bigcap _{n=0}^{\infty }F(J^A_{r_n})=A^{-1}0\).

*Proof*

*J*is uniformly norm-to-norm continuous on bounded sets, we obtain

*A*that

*A*, we obtain \(p\in A^{-1}0\). That is \(p\in \bigcap _{n=0}^{\infty }F(J^A_{r_n})\). This completes the proof. \(\square\)

**Theorem 14**

*Let*

*H*

*be a Hilbert space and let*

*F*

*be a uniformly convex and smooth Banach space. Let*\(J_F\)

*be the duality mapping on*

*F*.

*Let*

*A*

*and*

*B*

*be maximal monotone operators of*

*H*

*into*\(2^H\)

*and*

*F*

*into*\(2^F\)

*such that*\(A^{-1}0\ne \emptyset\)

*and*\(B^{-1}0\ne \emptyset\) ,

*respectively. Let*\(J_r\)

*be the resolvent of*

*A*

*for*\(r>0\)

*and let*\(Q_\mu\)

*be the metric resolvent of B for*\(\mu >0\).

*Let*\(T : H \rightarrow F\)

*be a bounded linear operator such that*\(T\ne 0\)

*and let*\(T^*\)

*be the adjoint operator of*

*T*.

*Suppose that*\(W=A^{-1}0\cap T^{-1}(B^{-1}0)\ne \emptyset\).

*Let*\(x_1\in H\)

*and let*\(\{x_n\}\)

*be a sequence generated by*

*where*\(\{r_n\}\)

*satisfy the condition such that*

*for some constants*

*a*,

*b*,

*c*.

*Then*\(\{x_n\}\)

*converges strongly to a point*\(z_0=P_{W}x_1\).

### Proof

**Theorem 15**

*Let*

*H*

*be a Hilbert space and let*

*F*

*be a uniformly convex and smooth Banach space. Let*\(J_F\)

*be the duality mapping on*

*F*.

*Let*

*A*

*and*

*B*

*be maximal monotone operators of*

*H*

*into*\(2^H\)

*and*

*F*

*into*\(2^F\)

*such that*\(A^{-1}0\ne \emptyset\)

*and*\(B^{-1}0\ne \emptyset\) ,

*respectively. Let*\(J_r\)

*be the resolvent of*

*A*

*for*\(r>0\)

*and let*\(Q_\mu\)

*be the metric resolvent of B for*\(\mu >0\).

*Let*\(T : H \rightarrow F\)

*be a bounded linear operator such that*\(T\ne 0\)

*and let*\(T^*\)

*be the adjoint operator of*

*T*.

*Suppose that*\(W=A^{-1}0\cap T^{-1}(B^{-1}0)\ne \emptyset\).

*Let*\(x_{1,i}\in H\)

*and let*\(\{x_n\}\)

*and*\(\{z_n\}\)

*be two sequences generated by*

*where*\(\{r_n\}\)

*satisfy the condition such that*

*for some constants*

*a*,

*b*

*and*\(\lambda \in [0,1]\)

*is a constant. Then the following conclusions hold:*

(1) \(\{x_n\}\) and \(\{z_n\}\) converge strongly to a point \(w \in W\);

(2) the limits \(\lim _{n\rightarrow \infty }P_{C_n}x_{1,i}=P_{W}x_{1,i}, \ i=1,2,3, \ldots ,N\);

(3) \(w=\sum _{i=1}^{N}\lambda _i \lim _{n\rightarrow \infty } P_{C_n}x_{1,i}\).

*Proof*

## Examples

It is easy to see that, a uniformly weak closed family \(\{T_n\}\) of countable quasi-nonexpansive mappings must be a uniformly closed family \(\{T_n\}\) of countable quasi-nonexpansive mappings. Next we will give an example which is a uniformly closed family of countable quasi-nonexpansive mappings, but not a uniformly weak closed family of countable quasi-nonexpansive mappings.

**Conclusion 16**

*Let*

*H*

*be a Hilbert space,*\(\{x_n\}_{n=1}^{\infty }\subset H\)

*be a sequence such that it converges weakly to a non-zero element*\(x_0\)

*and*\(\Vert x_i-x_j\Vert \ge 1\)

*for any*\(i\ne j\).

*Define a sequence of mappings*\(T_n: H\rightarrow H\)

*as follows*

*where*\(L_n\le 1\)

*and*\(\lim _{n\rightarrow \infty }L_n=1\).

*Then*\(\{T_n\}\)

*is a uniformly closed family of countable quasi-nonexpansive mappings with the common fixed point set*\(F=\{0\}\),

*but not a uniformly weak closed family of countable quasi-nonexpansive mappings.*

### Proof

*N*such that \(z_n\ne x_m\), for any \(n, m >N\). Then \(T_nz_n=-z_n\) for \(n>N\), it follows from \(\Vert z_n-T_nz_n\Vert \rightarrow 0\) that \(2z_n\rightarrow 0\) and hence \(z_0 \in F\). From the definition of \(\{T_n\}\), we have

## Conclusion

In the multidirectional iteration algorithm, the \(C_n\) is a closed convex set, and \(F\subset C_n\) for any \(n\ge 1\). If we use one initial \(x_{1,1}\), the projection point \(x_n=P_{C_n}x_{1,1}\) belongs to the boundary of the \(C_n\). If we use *N* initials \(x_{1,1}, x_{1,2}, x_{1,3}, \ldots , x_{1,N}\), the element \(x_{n}=\sum _{i=1}^{N}\lambda _i P_{C_n}x_{1,i}\) belongs to the interior of the \(C_n\). In general, the distance \(d(\sum _{i=1}^{N}\lambda _i P_{C_n}x_{1,i}, F)\) is less than the distance \(d(P_{C_n}x_{1,1}, F)\), so the multidirectional iteration algorithm can accelerate the convergence speed of iterative sequence \(\{x_n\}\). We give a simple experimental example in the following.

### Example

## Notes

### Authors' contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

### Competing interests

The authors declare that they have no competing interests.

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