1 Introduction
Let
\(H_{1}\),
\(H_{2}\), and
\(H_{3}\) be real Hilbert spaces. The multiple-set split equality common fixed-point problem (MSECFP) is to find
\(x^{*}, y^{*}\) with the property
$$ x^{*} \in \bigcap^{p}_{i=1}F(U_{i}), \qquad y^{*}\in \bigcap_{j=1}^{r}F(T_{j}) \quad \text{such that } Ax^{*} = By^{*}, $$
(1.1)
where
\(p,r\geq1\) are integers,
\(\{U_{i}\}_{i=1}^{p}:H_{1}\rightarrow H_{1} \) and
\(\{T_{j}\}^{r}_{j=1}:H_{2}\rightarrow H_{2} \) are nonlinear operators,
\(A: H_{1}\rightarrow H_{2}\) and
\(B: H_{2}\rightarrow H_{3}\) are two bounded linear operators. If
\(U_{i}\)
\((1\leq i\leq p)\) and
\(T_{j}\)
\((1\leq j\leq r)\) are projection operators, then the MSECFP is reduced to the multiple-set split equality problem (MSEP):
$$ \mbox{finding}\quad x^{*} \in \bigcap^{p}_{i=1}C_{i} \quad \text{and}\quad y^{*}\in \bigcap_{j=1}^{r}Q_{j} \quad \text{such that } Ax^{*} = By^{*}, $$
(1.2)
where
\(\{C_{i}\}_{i=1}^{p}\) and
\(\{Q_{j}\}_{j=1}^{r}\) are nonempty closed convex subsets of real Hilbert spaces
\(H_{1}\) and
\(H_{2}\), respectively. When
\(p=r=1\), the MSECFP and MSEP become the split equality common fixed-point problem (SECFP) and split equality problem (SEP), respectively, which were first put forward by Moudafi [
1]. These allow asymmetric and partial relations between the variables
x and
y. They are applied in many situations, for instance, in game theory and in intensity-modulated radiation therapy (see [
2] and [
3]).
If
\(H_{2}=H_{3}\) and
\(B=I\), then MSECFP (
1.1) reduces to the multiple-set split common fixed-point problem (MSCFP):
$$ \mbox{finding}\quad x^{*}\in \bigcap^{p}_{i=1} F(U_{i})\quad \text{such that } Ax^{*}\in \bigcap ^{r}_{j=1} F(T_{j}) $$
(1.3)
and MSEP (
1.2) reduces to the multiple-set split feasibility problem (MSFP):
$$ \mbox{finding}\quad x^{*} \in \bigcap^{p}_{i=1}C_{i} \quad \text{such that } Ax^{*}\in \bigcap_{j=1}^{r}Q_{j}. $$
(1.4)
They play significant roles in dealing with problems in image restoration, signal processing, and intensity-modulated radiation therapy [
3‐
6]. With
\(p=r=1\), MSCFP (
1.3) is known as the split common fixed-point problem (SCFP) and MSFP (
1.4) is known as the split feasibility problem (SFP). Many iterative algorithms have been developed to solve the MSCFP and the MSFP. See, for example, [
7‐
14] and the references therein.
Note that the SFP can be formulated as a fixed-point equation
$$ P_{C} \bigl(I-\gamma A^{\ast}(I-P_{Q})A \bigr)x^{\ast}= x^{\ast}, $$
(1.5)
where
\(P_{C}\) and
\(P_{Q}\) are the (orthogonal) projections onto
C and
Q, respectively,
\(\gamma>0\) is any positive constant, and
\(A^{\ast}\) denotes the adjoint of
A. This implies that we can use fixed-point algorithms (see [
15‐
21]) to solve SFP. Byrne [
22] proposed the so-called CQ algorithm which generates a sequence
\(\{x_{k}\}\):
$$ x_{n+1}=P_{C} \bigl(x_{n}-\gamma A^{\ast}(I-P_{Q})Ax_{n} \bigr), $$
(1.6)
where
\(\gamma \in (0,2/ \lambda)\) with
λ being the spectral radius of the operator
\(A^{\ast}A\). The CQ algorithm is efficient when
\(P_{C}\) and
\(P_{Q}\) are easily calculated. However, if
C and
Q are complex sets, for example, the fixed-point sets, the efficiency of the CQ algorithm will be affected because the projections onto such convex sets are generally hard to be accurately calculated. To solve the SCFP of nonexpansive operators, Censor and Segal [
23] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:
$$ x_{n+1}=U \bigl(x_{n}-\gamma A^{\ast}(I-T)Ax_{n} \bigr),\quad n\in N, $$
(1.7)
where
\(\gamma\in(0, \frac{2}{\lambda})\) with
λ being the largest eigenvalue of the matrix
\(A^{\ast}A\).
For solving the constrained MSFP, Censor et al. [
6] introduced the following proximity function:
$$ g(x):=\frac{1}{2}\sum_{i=1}^{p} \alpha_{i} \Vert x-P_{C_{i}}x \Vert ^{2}+ \frac{1}{2}\sum_{j=1}^{r} \beta_{j} \bigl\Vert Ax-P_{Q_{j}}(Ax) \bigr\Vert ^{2}, $$
(1.8)
where
\(\alpha_{i}>0\)
\((1\leq i\leq p)\),
\(\beta_{j}>0\)
\((1\leq j\leq r)\), and
\(\sum_{i=1}^{p}\alpha_{i}+\sum_{j=1}^{r} \beta_{j}=1\). Then
$$\nabla g(x)=\sum_{i=1}^{p} \alpha_{i}(x-P_{C_{i}}x)+\sum_{j=1}^{r} \beta_{j} A^{\ast}\bigl(Ax-P_{Q_{j}}(Ax) \bigr), $$
and they proposed the following projection method:
$$ x_{n+1}=P_{\Omega}\bigl(x_{n}-\gamma \nabla g(x_{n}) \bigr), $$
(1.9)
where Ω is the constrained set,
\(0<\gamma_{L}\leq \gamma\leq \gamma_{U}<\frac{2}{L}\), and
L is the Lipschitz constant of ∇
g.
For solving MSCFP (
1.3) of directed operators, Censor and Segal [
23] introduced a parallel iterative algorithm as follows:
$$ x_{n+1}=x_{n}-\gamma \Biggl[\sum _{i=1}^{p}\alpha_{i} \bigl(x_{n}-U_{i}(x_{n}) \bigr)+\sum_{j=1}^{r}\beta_{j}A^{\ast}\bigl(Ax_{n}-T_{j}(Ax_{n}) \bigr) \Biggr], $$
(1.10)
where
\(\{\alpha_{i}\}_{i=1}^{p}\),
\(\{\beta_{j}\}_{j=1}^{r}\) are nonnegative constants,
\(0<\gamma<2/L\) with
\(L = \sum_{i=1}^{p}\alpha_{i}+\lambda\sum_{j=1}^{r}\beta_{j}\) and
λ being the largest eigenvalue of
\(A^{\ast}A\). They obtained the convergence of iterative algorithm (
1.6).
Wang and Xu [
24] proposed the following cyclic iterative algorithm for MSCFP (
1.3) of directed operators:
$$ x_{n+1}=U_{[n]_{1}}(x_{n}+\gamma A^{\ast}(T_{[n]_{2}}-I)Ax_{n}, $$
(1.11)
where
\(0<\gamma< 2/\rho(A^{\ast}A)\),
\([n]_{1} := n (\operatorname {mod}p)\), and
\([n]_{2} := n (\operatorname {mod}r)\). They proved the weak convergence of the sequence
\(\{x_{n}\}\) generated by (
1.7).
For solving MSCFP (
1.3), Tang and Liu [
25] introduced inner parallel and outer cyclic iterative algorithm:
$$ x_{n+1}=U_{[n]_{1}} \Biggl(x_{n}+ \gamma_{n}\sum_{j=1}^{r} \eta_{j}A^{*}(T_{j}-I)Ax_{n} \Biggr) $$
(1.12)
and outer parallel and inner cyclic iterative algorithm:
$$ x_{n+1}=\sum_{i=1}^{p} \omega_{i}U_{i} \bigl(x_{n}+\gamma_{n}A^{*}(T_{[n]_{2}}-I)Ax_{n} \bigr) $$
(1.13)
for directed operators
\(\{U_{i}\}_{i=1}^{p}\) and
\(\{T_{j}\}_{j=1}^{r}\), where
\([n]_{1}=n(\operatorname {mod}p)\),
\([n]_{2}=n(\operatorname {mod}r)\),
\(0< a\leq \gamma_{n}\leq b< 2/\rho(A^{\ast}A)\),
\(\{\eta_{j}\}_{j=1}^{r}\),
\(\{\omega_{i}\}_{i=1}^{p}\subset(0,1)\) with
\(\sum_{j=1}^{r}\eta_{j}=1\) and
\(\sum_{i=1}^{p}\omega_{i}=1\). They obtained the weak convergence of the above two mixed iterative sequences to solve MSCFP (
1.3) of directed operators.
The SEP proposed by Moudafi [
1] is to
$$ \mbox{find}\quad x^{*} \in C,\qquad y^{*}\in Q\quad \text{such that } Ax^{*} = By^{*}, $$
(1.14)
which can be written as the following minimization problem:
$$ \min_{x\in C,y\in Q}\frac{1}{2} \Vert Ax-By \Vert ^{2}. $$
(1.15)
Assume that the solution set of the SEP is nonempty. By the optimality conditions, Moudafi [
1] obtained the following fixed-point formulation:
\((x^{\ast},y^{\ast})\) solves the SEP if and only if
$$ \textstyle\begin{cases} x^{\ast}=P_{C}(x^{\ast}-\gamma A^{\ast}(Ax^{\ast}-By^{\ast})),\\ y^{\ast}=P_{Q}(y^{\ast}+\beta B^{\ast}(Ax^{\ast}-By^{\ast})), \end{cases} $$
(1.16)
where
γ,
\(\beta>0\). Therefore, for solving the SECP of firmly quasi-nonexpansive operators, Moudafi [
1] introduced the following alternating algorithm:
$$ \textstyle\begin{cases} x_{n+1}=U(x_{n}-\gamma_{n}A^{\ast}(Ax_{n}-By_{n})),\\ y_{n+1}=T(y_{n}+\gamma_{n}B^{\ast}(Ax_{n+1}-By_{n})), \end{cases} $$
(1.17)
where a nondecreasing sequence
\(\gamma_{n}\in (\varepsilon,\min(\frac{1}{\lambda_{A}},\frac{1}{\lambda_{B}})-\varepsilon)\),
\(\lambda_{A}\),
\(\lambda_{B}\) stand for the spectral radius of
\(A^{\ast}A\) and
\(B^{\ast}B\), respectively. In [
26], Moudafi and Al-Shemas introduced the following simultaneous iterative method:
$$ \textstyle\begin{cases} x_{n+1}=U(x_{n}-\gamma_{n}A^{\ast}(Ax_{n}-By_{n})),\\ y_{n+1}=T(y_{n}+\gamma_{n}B^{\ast}(Ax_{n}-By_{n})), \end{cases} $$
(1.18)
where
\(\gamma_{n}\in(\varepsilon,\frac{2}{\lambda_{A}+\lambda_{B}}-\varepsilon)\),
\(\lambda_{A}\),
\(\lambda_{B}\) stand for the spectral radius of
\(A^{\ast}A\) and
\(B^{\ast}B\), respectively. Recently, many iterative algorithms have been developed to solve the SEP, SECFP, and MSEP. See, for example, [
27‐
34] and the references therein. Note that in algorithms (
1.17) and (
1.18), the determination of the step size
\(\{\gamma_{n}\}\) depends on the operator (matrix) norms
\(\Vert A \Vert \) and
\(\Vert B \Vert \) (or the largest eigenvalues of
\(A^{\ast}A\) and
\(B^{\ast}B\)). To overcome this shortage, we introduce parallel and cyclic iterative algorithms with self-adaptive step size to solve MSECFP (
1.1) governed by firmly quasi-nonexpansive operators. We also propose two mixed iterative algorithms which combine the process of cyclic and parallel iterative methods and do not need the norms of bounded linear operators. As applications, we obtain several iterative algorithms to solve MSEP (
1.2).
3 Parallel and cyclic iterative algorithms
In this section, we introduce parallel and cyclic iterative algorithms and prove the weak convergence for solving MSECFP (
1.1) of firmly quasi-nonexpansive operators. In our algorithms, the selection of the step size does not need any prior information of the operator norms
\(\Vert A \Vert \) and
\(\Vert B \Vert \).
In what follows, we adopt the following assumptions:
(A1)
The problem is consistent, namely its solution set Γ is nonempty;
(A2)
Both \(U_{i}\) and \(T_{j}\) are firmly quasi-nonexpansive operators, and both \(I-U_{i}\) and \(I-T_{j}\) are demiclosed at origin (\(1\leq i\leq p\), \(1\leq j\leq r\)).
(A3)
The sequences \(\{\alpha_{n}^{i}\}_{i=1}^{p}\), \(\{\beta_{n}^{j}\}_{j=1}^{r}\subset[0,1]\) such that \(\sum_{i=1}^{p}\alpha_{n}^{i}=1\) and \(\sum_{j=1}^{r}\beta_{n}^{j}=1\) for every \(n\geq0\), \(j(n)=n(\operatorname {mod}r)+1\), \(i(n)=n(\operatorname {mod}p)+1\).
Next, we propose the cyclic iterative algorithm for solving MSECFP (
1.1) of firmly quasi-nonexpansive operators.
Now, we give applications of Theorem
3.1 and Theorem
3.2 to solve MSEP (
1.2). Assume that the solution set
S of MSEP (
1.2) is nonempty. Since the orthogonal projection operator is firmly nonexpansive, by Lemma
2.4 we have the following results for solving MSEP (
1.2).
4 Mixed cyclic and parallel iterative algorithms
Now, for solving MSECFP (
1.1) of firmly quasi-nonexpansive operators, we introduce two mixed iterative algorithms which combine the process of cyclic and simultaneous iterative methods. In our algorithms, the selection of the step size does not need any prior information of the operator norms
\(\Vert A \Vert \) and
\(\Vert B \Vert \), and the weak convergence is proved. We go on making use of assumptions (A1)–(A3).
Next, we propose another mixed cyclic and parallel iterative algorithm for solving MSECFP (
1.1) of firmly quasi-nonexpansive operators.
Similar to the proof of Theorem
4.1, we can get the following result.
Finally, we obtain two mixed iterative algorithms to solve MSEP (
1.2). Assume that the solution set
S of MSEP (
1.2) is nonempty.
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