Abstract
In this paper, we introduce a new algorithm which combines the Mann iteration and the inertial method for solving split common fixed point problems. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. As a consequence, we obtain weak convergence theorems for split variational inequality problems for inverse strongly monotone operators, and split common null point problems for maximal monotone operators. Finally, for supporting the convergence of the proposed algorithms we also consider several preliminary numerical experiments on a test problem.
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1 Introduction
Throughout this paper, H denotes a real Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \) and the induced norm \(\Vert \cdot \Vert \), I the identity operator on H, \(\mathbb {N}\) the set of all natural numbers and \(\mathbb {R}\) the set of all real numbers. For a self-operator T on H, \(\mathrm{Fix}(T)\) denotes the set of all fixed points of T.
Let \(H_1\) and \(H_2\) be real Hilbert spaces and let \(A:H_1 \rightarrow H_2\) be a bounded linear operator. Let \(\{U_j\}_{j=1}^t:H_1 \rightarrow H_1\) and \(\{T_i\}_{i=1}^t:H_2 \rightarrow H_2\) be two finite families of operators, where \(t, r \in \mathbb {N}\). The split common fixed point problem (SCFPP) is formulated as finding a point \(x^*\in H_1\) such that
Let \(\varGamma \) be the solution set of the SCFPP (1.1), that is
In particular, if \(t = r = 1\), the SCFPP (1.1) reduces to finding a point \(x^*\in H_1\) such that
The above problem is usually called the two-set SCFPP.
In recent years, the SCFPP (1.1) and the two-set SCFPP (1.3) have been studied and extended by many authors, see for instance [10, 11, 13, 14, 16, 18, 20, 22, 24, 29, 32, 33, 37, 38, 42, 43, 45,46,47,48]. It is known that the SCFPP includes the multiple-set split feasibility problem and split feasibility problem as special case. In fact, let \(\{C_j\}_{j=1}^t\) and \(\{Q_i\}_{i=1}^r\) be two finite families of nonempty closed convex subsets in \(H_1\) and \(H_2\), respectively. Let \(U_j=P_{C_j}\) and \(T_i=P_{Q_i}\); then the SCFPP (1.1) becomes the multiple-set split feasibility problem (MSSFP) as follows:
When \(t=r=1\) then the MSSFP (1.4) is reduced to the split feasibility problem (SFP) which is described as finding a point \(x^*\in H_1\) satisfying the following property:
The SFP (1.5) was first introduced by Censor and Elfving [15] for modelling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP (1.5) can also be used to model the intensity-modulated radiation therapy. For more details, the readers are referred to Xu [47] and the references therein.
In 2002, Byrne [10] introduced the so-called CQ algorithm for solving the SFP (1.5). For and \(x_0\in H_1\) and define \(\{x_n\}\) as
where \(0<\gamma < \dfrac{2}{\rho (A^*A)}\) and where \(P_C\) denotes the projector onto C and \(\rho (A^*A)\) is the spectral radius of the operator \(A^*A.\) It is known that the CQ algorithm converges weakly to a solution of the SFP (1.5) if such a solution exists.
The two-set SCFPP (1.3) was first introduced by Censor and Segal [18] in finite-dimensional Hilbert spaces. When U and T are directed operators, Censor and Segal [18] proposed and proved the convergence of the following algorithm:
where \( \gamma \in (0,\dfrac{2}{\lambda })\) with \(\lambda =\rho (A^*A)\). By making use of the product space technique, Censor and Segal [18] introduced a parallel iterative algorithm for the general SCFPP (1.1), which the sequence \(\{x_n\}\) defined by the following algorithm:
where \(\{\alpha _j\}_{j=1}^t\), \(\{\beta _j\}_{j=1}^r\) are nonnegative constants, \(0<\gamma <\dfrac{2}{L}\) with \(L:=\sum _{j=1}^t\alpha _j+\lambda \sum _{i=1}^r\beta _i\). They proved that the sequence \(\{x_n\}\) converges strongly to a solution of SCFPP (1.1).
Now, let us mention an inertial type algorithm. We know that the problem of finding a zero of a maximal monotone operator A on a real Hilbert H is formulated as
One of the fundamental approaches to solving it is the proximal method, which generates the next iteration \(x_{n+1}\) by solving the subproblem
where \(x_n\) is the current iteration and \(\lambda _n\) is a regularization parameter, see [9, 40]. In 2001, Attouch and Alvarez [1] applied an inertial technique to the algorithm (1.10) to construct an inertial proximal method for solving (1.9). It works as follows: given \(x_{n-1},~x_n\in H\) and two parameters \(\theta _n\in [0,1) ,\lambda _n>0\), find \(x_{n+1}\in H\) such that
which can be written equivalently to the following:
where \(J_{\lambda _n}^A\) is the resolvent of A with parameter \(\lambda _n\) and the inertia is induced by the term \(\theta _n(x_n-x_{n-1})\) and it can be regarded as procedure of seeding up the convergence properties (see, e.g., [1, 36]).
It is well known that the proximal iteration (1.10) may be interpreted as an implicit one-step discretization method for the evolution differential inclusion
where a.e. stands for almost everywhere. While the inspiration for (1.12) comes from the implicit discretization of the differential system of the second-order in time, namely
where \(\rho >0\) is a damping or a friction parameter. It gives rise to various numerical methods related to the inertial terminology, all those methods had nice convergence properties (see [1,2,3,4,5,6,7,8, 19, 21, 25,26,27,28, 30, 34, 35, 44]) by incorporating second-order information.
Motivated by the inertial method, in this paper, we apply the inertial technique to the Mann algorithm (see [31, 39]) to propose an inertial algorithm to solve the split common fixed point problem (1.1). Our algorithm is of the form
Under approximate conditions, we show that the sequence \(\{x_n\}\) generated by (1.15) converges weakly to some solution of SCFPP (1.1).
This paper is organized as follows: in Sect. 2, we recall some definitions and preliminary results for further use. Section 3 deals with analyzing the convergence of the proposed algorithm. In Sect. 4, we use our result to solve split variational inequality problems and split common null point problems. Finally, in Sect. 5 we perform several numerical examples to support the convergence of our algorithm.
2 Preliminaries
Let C be a nonempty closed convex subset of H. The weak convergence of \(\{x_n\}_{n=1}^{\infty }\) to x is denoted by \(x_{n}\rightharpoonup x\) as \(n\rightarrow \infty \), while the strong convergence of \(\{x_n\}_{n=1}^{\infty }\) to x is written as \(x_n\rightarrow x\) as \(n\rightarrow \infty .\)
For every point \(x\in H\), there exists the unique nearest point in C, denoted by \(P_Cx\) such that \(\Vert x-P_Cx\Vert \le \Vert x-y\Vert \ \forall y\in C\). \(P_C\) is called the metric projection of H onto C. It is known that \(P_C\) is nonexpansive.
Lemma 2.1
[23] Let C be a nonempty closed convex subset of H. Given \(x\in H\) and \(z\in C\). Then \(z=P_Cx\Longleftrightarrow \langle x-z,z-y\rangle \ge 0 \ \quad \forall y\in C.\)
Lemma 2.2
[23] Let C be a closed and convex subset of H, \(x\in H\). Then
-
(i)
\(\Vert P_Cx-P_Cy\Vert ^2\le \langle P_C x-P_C y,x-y\rangle \quad \forall y\in C\);
-
(ii)
\(\Vert P_C x-y\Vert ^2\le \Vert x-y\Vert ^2-\Vert x-P_Cx\Vert ^2 \quad \forall y\in C\).
For properties of the metric projection, the interested reader could be referred to Section 3 in [23].
Definition 2.3
Assume that \(T:H\rightarrow H\) is a nonlinear operator with \(\mathrm{Fix}(T)\ne \emptyset \). Then \(I-T\) is said to be demiclosed at zero if for any \(\{x_n\}\) in H, the following implication holds:
Definition 2.4
Let \(T:H\rightarrow H\) be an operator with \(\mathrm{Fix}(T)\ne \emptyset .\) Then
-
\(T:H\rightarrow H\) is called directed if
$$\begin{aligned} \langle z-Tx,x-Tx\rangle \le 0 \ \ \quad \forall z\in \mathrm{Fix}(T), x\in H, \end{aligned}$$(2.1)or equivalently
$$\begin{aligned} \Vert Tx-z\Vert ^2\le \Vert x-z\Vert ^2-\Vert x-Tx\Vert ^2 \ \ \quad \forall z\in \mathrm{Fix}(T), x\in H; \end{aligned}$$(2.2) -
\(T:H\rightarrow H\) is called quasi-nonexpansive if
$$\begin{aligned} \Vert Tx-z\Vert \le \Vert x-z\Vert \ \ \quad \forall z\in \mathrm{Fix}(T), x\in H; \end{aligned}$$(2.3) -
\(T:H\rightarrow H\) is called \(\beta \)-demicontractive with \(0\le \beta <1\) if
$$\begin{aligned} \Vert Tx-z\Vert ^2\le \Vert x-z\Vert ^2+\beta \Vert (I-T)x\Vert ^2\ \ \quad \forall z\in \mathrm{Fix}(T), x\in H, \end{aligned}$$(2.4)or equivalently
$$\begin{aligned} \langle Tx-x,x-z\rangle \le \dfrac{\beta -1}{2}\Vert x-Tx\Vert ^2 \ \ \ \forall z\in \mathrm{Fix}(T), x\in H. \end{aligned}$$(2.5)or equivalently
$$\begin{aligned} \langle Tx-z,x-z\rangle \le \Vert x-z\Vert ^2+ \dfrac{\beta -1}{2}\Vert x-Tx\Vert ^2 \ \ \quad \forall z\in \mathrm{Fix}(T), x\in H. \end{aligned}$$(2.6)
Lemma 2.5
[1] Let \(\{\varphi _n\} \), \(\{\delta _n\}\) and \(\{\alpha _n\}\) be sequences in \([0,+\infty )\) such that
and there exists a real number \(\alpha \) with \(0\le \alpha _n\le \alpha <1\) for all \(n\in \mathbb {N}\). Then the following hold:
-
(i)
\(\sum _{n=1}^{+\infty }[\varphi _n-\varphi _{n-1}]_{+}<+\infty \), where \([t]_{+}:=\max \{t,0\}\);
-
(ii)
there exists \(\varphi ^*\in [0,+\infty )\) such that \(\lim _{n\rightarrow +\infty }\varphi _n=\varphi ^*.\)
Lemma 2.6
(Opial 1967) Let C be a nonempty set of H and \(\{x_n\}\) be a sequence in H such that the following two conditions hold:
-
(i)
for every \(x\in C\), \(\lim _{n\rightarrow \infty }\Vert x_n-x\Vert \) exists;
-
(ii)
every sequential weak cluster point of \(\{x_n\}\) is in C. Then \(\{x_n\}\) converges weakly to a point in C.
Lemma 2.7
[43] Let \(U:H \rightarrow H\) be \(\beta \)-demicontractive with \(\mathrm{Fix}(U)\ne \emptyset \) and set \(U_\lambda =(1-\lambda )I+\lambda U\), \(\lambda \in (0,1-\beta )\): then
-
(a)
\(\mathrm{Fix}(U)=\mathrm{Fix}(U_\lambda );\)
-
(b)
\(\Vert U_\lambda x-z\Vert ^2\le \Vert x-z\Vert ^2-\lambda (1-\beta -\lambda )\Vert (I-U)x\Vert ^2 \ \quad \forall x\in H, z\in \mathrm{Fix}(U);\)
-
(c)
\(\mathrm{Fix}(U)\) is a closed convex subset of H.
Corollary 2.8
[25] Given an integer \(N\ge 1\). Assume that for each \(i=1,\ldots ,N\), \(\{T_i\}:H\rightarrow H\) is a \(k_i\)-demicontractive operator such that \(\cap _{i=1}^N \mathrm{Fix}(T_i)\ne \emptyset \). Assume that \(\{w_i\}_{i=1}^N\) is a finite sequence of positive numbers such that \(\sum _{i=1}^N w_i=1.\) Setting \(U=\sum _{i=1}^N w_i T_i\), then the following results hold:
-
(i)
\(\mathrm{Fix}(U)=\cap _{i=1}^N \mathrm{Fix}(T_i)\).
-
(ii)
U is \(\lambda \)-demicontractive operator, where \(\lambda =\max \{k_i | i=1,\ldots ,N\}.\)
-
(iii)
\(\langle x-Ux,x-z\rangle \ge \dfrac{1-\lambda }{2}\sum _{i=1}^N w_i \Vert x-T_i x\Vert ^2\) for all \(x\in H\) and \(z\in \cap _{i=1}^N \mathrm{Fix}(T_i).\)
3 Main results
Lemma 3.1
For \(t\in \mathbb {N}\), let \(\{T_i\}_{i=1}^t:H_2\rightarrow H_2\) be a finite family of \(\mu \)-demicontractive operators such that \(\cap _{i=1}^t \mathrm{Fix}(T_i)\ne \emptyset \). Let \(A: H_1\rightarrow H_2\) be a linear bounded operator with \(L=\Vert A^*A\Vert \). For a positive real number \(\gamma ,\) define the operator \(V: H_1\rightarrow H_1\) by
where \(\{\eta _i\}_{i=1}^t\subset (0,1)\) and \(\sum _{i=1}^t \eta _i=1\). Then we have the following:
-
(a)
for all \(x\in H_1\) and \( z\in A^{-1}(\cap _{i=1}^t \mathrm{Fix}(T_i))\),
$$\begin{aligned} \Vert Vx-z\Vert ^2\le \Vert x-z\Vert ^2-\dfrac{1}{\gamma L}(1-\mu -\gamma L)\Vert (I-V)x\Vert ^2. \end{aligned}$$(3.2) -
(b)
for all \(x\in H_1\) and \( z\in A^{-1}(\cap _{i=1}^t \mathrm{Fix}(T_i))\),
$$\begin{aligned} \Vert Vx-z\Vert ^2\le \Vert x-z\Vert ^2-\sum _{i=1}^t \eta _i\gamma (1-\mu -\gamma L)\Vert (I-T_i)Ax\Vert ^2. \end{aligned}$$(3.3) -
(c)
\(x\in \mathrm{Fix}(V)\) if and only if \(Ax\in \cap _{i=1}^t \mathrm{Fix}(T_i)\) provided that \(\gamma \in (0,\dfrac{1-\mu }{L})\).
Proof
-
(a)
Given \(x\in H_1\) and \(z\in A^{-1}(\cap _{i=1}^t \mathrm{Fix}(T))\). For each \(i=1,\ldots ,t\), we have
$$\begin{aligned} \langle A^*(I-T_i)Ax,x-z\rangle&=\langle (I-T_i)Ax,Ax-Az\rangle \nonumber \\&\ge \dfrac{1-\mu }{2}\Vert (I-T_i)Ax\Vert ^2. \end{aligned}$$(3.4)On the other hand,
$$\begin{aligned} \Vert A^*(I-T_i)Ax\Vert ^2&=\langle A^*(I-T_i)Ax,A^*(I-T_i)Ax\rangle \nonumber \\&=\langle (I-T_i)Ax,AA^*(I-T_i)Ax\rangle \nonumber \\&\le L \Vert (I-T_i)Ax\Vert ^2. \end{aligned}$$(3.5)Since (3.4) and (3.5), for each \(i=1,\ldots ,t\) we obtain
$$\begin{aligned} \langle A^*(I-T_i)Ax,x-z\rangle \ge \dfrac{1-\mu }{2L}\Vert A^*(I-T_i)Ax\Vert ^2. \end{aligned}$$(3.6)By the convexity of \(\Vert .\Vert ^2\) we have
$$\begin{aligned} \Vert Vx-z\Vert ^2&=\left\| x-\sum _{i=1}^t\eta _i\gamma A^*(I-T_i)Ax-z\right\| ^2\\&=\left\| \sum _{i=1}^t\eta _i(x-z-\eta _i\gamma A^*(I-T_i)Ax)\right\| ^2\\&\le \sum _{i=1}^t\eta _i\Vert (x-z-\eta _i\gamma A^*(I-T_i)Ax)\Vert ^2\\&=\Vert x-z\Vert ^2\\&\quad -\sum _{i=1}^t\eta _i[2\gamma \langle x-z,A^*(I-T_i)Ax\rangle -\gamma ^2\Vert A^*(I-T_i)Ax\Vert ^2]\\&\le \Vert x-z\Vert ^2\\&\quad -\sum _{i=1}^t\eta _i\left[ 2\gamma \dfrac{1-\mu }{2L}\Vert A^*(I-T_i)Ax\Vert ^2-\gamma ^2\Vert A^*(I-T_i)Ax\Vert ^2\right] \\&=\Vert x-z\Vert ^2\\&\quad -\sum _{i=1}^t\eta _i\left[ \dfrac{\gamma }{L}(1-\mu -\gamma L)\Vert A^*(I-T_i)Ax\Vert ^2\right] \\&\le \Vert x-z\Vert ^2-\dfrac{1}{\gamma L}(1-\mu -\gamma L)\left\| \sum _{i=1}^t\eta _i\gamma A^*(I-T_i)Ax\right\| ^2\\&= \Vert x-z\Vert ^2-\dfrac{1}{\gamma L}(1-\mu -\gamma L)\Vert (I-V)x\Vert ^2. \end{aligned}$$ -
(b)
Given \(x\in H_1\) and \( z\in A^{-1}(\cap _{i=1}^t \mathrm{Fix}(T_i))\), by the convexity of \(\Vert .\Vert ^2\) we have
$$\begin{aligned} \Vert Vx-z\Vert ^2&= \sum _{i=1}^t\eta _i\Vert x+\gamma A^*(T_i-I)Ax-z\Vert ^2\\&\le \Vert x-z\Vert ^2+\sum _{i=1}^t\eta _i[2\gamma \langle x-z,A^*(T_i-I)Ax\rangle \\&\quad +\gamma ^2\Vert A^*(T_i-I)Ax\Vert ^2]\\&= \Vert x-z\Vert ^2+\sum _{i=1}^t\eta _i[2\gamma \langle Ax-Az,(T_i-I)Ax\rangle \\&\quad +\gamma ^2\langle AA^*(T_i-I)Ax,(T_i-I)Ax\rangle ]\\&=\Vert x-z\Vert ^2+\sum _{i=1}^t\eta _i[\gamma (-1+\mu )\Vert (T_i-I)Ax\Vert ^2\\&\quad +\gamma ^2 \Vert AA^*\Vert \Vert (T_i-I)Ax\Vert ^2]\\&= \Vert x-z\Vert ^2-\sum _{i=1}^t\eta _i \gamma (1-\mu -\gamma L)(T_i-I)Ax\Vert ^2. \end{aligned}$$ -
(c)
It is obvious that if \(Ax\in \cap _{i=1}^t \mathrm{Fix}(T_i)\) then \(x\in \mathrm{Fix}(V)\). We show the converse: let \(x\in \mathrm{Fix}(V)\) and \(z\in A^{-1}(\mathrm{Fix}(T))\); we have
$$\begin{aligned} \Vert x-z\Vert ^2=\Vert Vx-z\Vert ^2\le \Vert x-z\Vert ^2-\sum _{i=1}^t\eta _i \gamma (1-\mu -\gamma L)\Vert (T_i-I)Ax\Vert ^2. \end{aligned}$$Since \(\gamma \in (0,\dfrac{1-\mu }{L})\), we obtain \((T_i-I)Ax=0\) for all \(i=1,2,\ldots ,t\), that is, \(Ax\in \cap _{i=1}^t\mathrm{Fix}(T).\)
Lemma 3.2
For \(t, r\in \mathbb {N}\), let \(\{T_i\}_{i=1}^t:H_2\rightarrow H_2\) be a finite family of \(\mu \)-demicontractive operators such that \(\cap _{i=1}^t \mathrm{Fix}(T_i)\ne \emptyset \) and \(\{U_j\}_{j=1}^r:H_1\rightarrow H_1\) be a finite family of quasi-nonexpansive operators such that \(\cap _{j=1}^r \mathrm{Fix}(U_j)\ne \emptyset \). Assume that \(\{I-U_j\}_{j=1}^r\) and \(\{I-T_i\}_{i=1}^t\) are demiclosed at zero. Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Suppose \(\varGamma \ne \emptyset \). Let \(S:H_1\rightarrow H_1\) be define by
where \(\gamma \in (0,\dfrac{1-\mu }{L})\) and \(\{w_j\}_{j=1}^r \subset (0,1), \{\eta _i\}_{i=1}^t\subset (0,1)\) with \(\sum _{j=1}^rw_j =1\) and \(\sum _{i=1}^t \eta _i=1\). Then the following statements hold:
-
(a)
The operator S is quasi-nonexpansive;
-
(b)
\(\mathrm{Fix}(S)=\varGamma \);
-
(c)
\(I-S\) is demiclosed at zero.
Proof
Let \(V:=I+\sum _{i=1}^t\eta _i\gamma A^*(T_i-I)A\). We can rewrite the operator S as follows:
We will show that
-
(i)
\(\{U_jV\}_{j=1}^r\) is a finite family of 0-demicontractive operators.
-
(ii)
\(\cap _{j=1}^r \mathrm{Fix}(U_jV)=\varGamma \).
-
(iii)
for each \(j=1,\ldots ,r\) then \(I-U_jV\) is demiclosed at zero.
By Lemma 3.1 then V is quasi-nonexpansive. Therefore, for each \(j=1,\ldots ,r\) the operator \(U_jV\) is quasi-nonexpansive.
Next, we show that for each \(j=1,\ldots ,r\) then
Indeed, it suffices to show that for each \(j=1,\ldots ,r\), \(\mathrm{Fix}(U_jV)\subset \mathrm{Fix}(U)\cap \mathrm{Fix}(V).\) Let \(p\in \mathrm{Fix}(U_jV)\); it is enough to show that \(p\in \mathrm{Fix}(V)\). We take \(z\in \mathrm{Fix}(U_j)\cap \mathrm{Fix}(V)\); we have
This implies that \(\sum _{i=1}^t\eta _i \gamma (1-\mu -\gamma L) \Vert (T_i-I)Ap\Vert ^2=0\), that is \(T_i(Ap)=Ap\) for all \(i=1,\ldots ,t\); this implies that \(Ap\in \cap _{i=1}^t\mathrm{Fix}(T_i)\), by Lemma 3.1 then \(p\in \mathrm{Fix} (V)\). Therefore, \(\mathrm{Fix}(U_j)\cap \mathrm{Fix}(V)=\mathrm{Fix}(U_jV)\) for all \(j=1,\ldots ,r.\) We now show that
By Lemma 3.1 we have
Finally, we show that for each \(j=1,\ldots ,r\), \(I-U_jV\) is demiclosed at zero. Let \(\{x_n\}\subset H_1\) be a sequence such that \(x_n\rightharpoonup z\in H_1\) and \(U_jVx_n-x_n\rightarrow 0\). We have
This implies that
By Lemma 3.1 we have
This implies that
that is \(Vx_n-x_n\rightarrow 0.\) On the other hand,
Since \(x_n\rightharpoonup z\), we have \(Vx_n\rightharpoonup z\) and by the demiclosedness of \(U_j\) we have \(z\in \mathrm{Fix}(U_j)\).
By Lemma 3.1 we have
which implies that
Therefore, \((I-T_i)Ax_n\rightarrow 0\) for all \(i=1,\ldots ,t\). Since \(Ax_n\rightharpoonup Az\) and the demiclosedness of \(T_i\) we get \(Az\in \mathrm{Fix}(T_i)\) for all \(i=1,\ldots ,t\), that is \(Az\in \cap _{i=1}^t \mathrm{Fix}(T_i)\). By Lemma 3.1 we get \(z \in \mathrm{Fix}(V)\). Therefore, \(z\in \mathrm{Fix}(U_j)\cap \mathrm{Fix}(V)=\mathrm{Fix}(U_jV).\)
Now, we will prove lemma:
By Claim (i) and Lemma 2.8 we obtain \(Sx=\sum _{j=1}^rw_j U_jVx\) is quasi-nonexpansive and \(\mathrm{Fix}(S)=\cap _{j=1}^t \mathrm{Fix}(U_jV)=\varGamma .\)
Finally, we show that \(I-S\) is demiclosed at zero. Indeed, let \(\{x_n\}\subset H_1\) be a sequence such that \(x_n\rightharpoonup z\in H_1\) and \(\Vert x_n-Sx_n\Vert \rightarrow 0\). Let \(p\in \mathrm{Fix}(S)\); by Lemma 2.8 we have
This implies that, for each \(j=1,\ldots ,t\) we have \(\Vert x_n-U_jVx_n\Vert \rightarrow 0\) as \(n\rightarrow \infty \). By the demiclosedness of \(I-U_jV\) we have \(z\in \mathrm{Fix}(U_jV)\). Therefore, \(z\in \cap _{j=1}^t \mathrm{Fix}(U_jV)=\mathrm{Fix}(S).\)
Theorem 3.3
For \(t, r\in \mathbb {N}\), let \(\{T_i\}_{i=1}^t:H_2\rightarrow H_2\) be a finite family of \(\mu \)-demicontractive operators such that \(\cap _{i=1}^t \mathrm{Fix}(T_i)\ne \emptyset \) and \(\{U_j\}_{j=1}^r:H_1\rightarrow H_1\) be a finite family of quasi-nonexpansive operators such that \(\cap _{j=1}^r \mathrm{Fix}(U_j)\ne \emptyset \). Assume that \(\{I-U_j\}_{j=1}^r\) and \(\{I-T_i\}_{i=1}^t\) are demiclosed at zero. Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Suppose \(\varGamma \ne \emptyset \). Let \(\{x_n\}\subset H_1\) be a sequence generated by
where \(\gamma \in (0,\dfrac{1-\mu }{L})\) and \(\{w_j\}_{j=1}^r \subset (0,1), \{\eta _i\}_{i=1}^t\subset (0,1)\) with \(\sum _{j=1}^rw_j =1\), \(\sum _{i=1}^t \eta _i=1\). Assume the sequences \(\{\alpha _n\}, \{\beta _n\} \) satisfy the following conditions:
-
(a)
\(0<a\le \beta _n\le b<\dfrac{1}{2}\), for some \(a,b>0.\)
-
(b)
\(\{\alpha _n\}\) is non-decreasing and \(0\le \alpha _n< \alpha <\dfrac{\epsilon }{2\epsilon +1}\), where \(\epsilon :=\dfrac{1-b}{b}\).
Then \(\{x_n\}\) converges weakly to an element of \(\varGamma \).
Proof
Let \(S:=\sum _{j=1}^rw_j U_j(I+\sum _{i=1}^t\eta _i\gamma A^*(T_i-I)A)\); then the sequence \(\{x_n\}\) can be rewritten as follows:
By Lemma 3.2, we have that S is quasi-nonexpansive. Let \(p\in \varGamma \); since the definition of the sequence \(\{x_n\}\) and Lemma 2.7 we have
By (3.12), we have
By the definition of the sequence \(\{y_n\}\), we have
From (3.15) and (3.16) we have
On the other hand, we have
From (3.17) and (3.19) we obtain
Note that since \(\epsilon >1\), we have
Put \(\Gamma _n:= \Vert x_n-p\Vert ^2-\alpha _n\Vert x_{n-1}-p\Vert ^2+\alpha _n (\epsilon +1)\Vert x_n-x_{n-1}\Vert ^2\). We have
where \(M:=\epsilon -(2\epsilon +1)\alpha >0.\) Therefore, we get
This implies that the sequence \(\{\Gamma _n\}\) is nonincreasing. On the other hand, we have
This implies that
We also have
From (3.23) and (3.24) we obtain
Since (3.22) we get
This implies
By (3.25) we obtain
We have
This implies that
From (3.25) and (3.28) we obtain
From (3.25), (3.18) and Lemma 2.5 we have
Since (3.30) and (3.16) we obtain
This implies that the sequences \(\{x_n\} \) and \(\{y_n\}\) are bounded. Since (3.13) we have
This implies that
Now, we show that the sequence \(\{x_n\}\) converges weakly to \(q\in \varGamma .\) Indeed, since \(\{x_n\}\) is bounded we assume that there exists a subsequence \(\{x_{n_j}\}\) of \(\{x_n\}\) such that \(x_{n_j}\rightharpoonup q\in H\). Since (3.29) we have \(y_{n_j}\rightharpoonup q\), by (3.33) and the demiclosedness of \(I-S\) we obtain \(q\in \mathrm{Fix}(S)=\varGamma .\)
Therefore, we proved the following:
-
(i)
For every \(p\in \varGamma \), then \(\lim _{n\rightarrow \infty }\Vert x_n-p\Vert ^2\) exists;
-
(ii)
Every sequential weak cluster point of the sequence \(\{x_n\}\) is in \(\varGamma \).
By Lemma 2.6 then sequence \(\{x_n\}\) converges weakly to \(q\in \varGamma .\)
Corollary 3.4
Let \(U:H_1\rightarrow H_1\) be a quasi-nonexpansive operator and \(T:H_2\rightarrow H_2\) be a \(\mu \)-demicontractive operator that both \(I-U\) and \(I-T\) are demiclosed at zero. Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Suppose \(\varGamma \ne \emptyset \). Let \(\{x_n\}\subset H_1\) be a sequence generated by
where \(\gamma \in (0,\dfrac{1-\mu }{L})\) and the sequences \(\{\alpha _n\}\), \(\{\beta _n\}\) satisfying the conditions:
-
(a)
\(0<a\le \beta _n\le b<\dfrac{1}{2}\), for some \(a,b>0.\)
-
(b)
\(\{\alpha _n\}\) is non-decreasing and \(0\le \alpha _n< \alpha <\dfrac{\epsilon }{2\epsilon +1}\), where \(\epsilon :=\dfrac{1-b}{b}\). Then \(\{x_n\}\) converges weakly to an element of \(\varGamma \).
Now applying Corollary 3.4 with \(\alpha _n=0\), we obtain the following result:
Corollary 3.5
Let \(U:H_1\rightarrow H_1\) be a quasi-nonexpansive operator and \(T:H_2\rightarrow H_2\) be a \(\mu \)-demicontractive operator that both \(I-U\) and \(I-T\) are demiclosed at zero. Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Suppose \(\varGamma \ne \emptyset \). Let \(\{x_n\}\subset H_1\) be a sequence generated by
where \(\gamma \in (0,\dfrac{1-\mu }{L})\) and the sequence \(\{\beta _n\}\) satisfying the condition \(0<a\le \beta _n\le b<1\), for some \(a,b>0.\) Then \(\{x_n\}\) converges weakly to an element of \(\varGamma \).
Next, we give an application of Theorem 3.3 to solve the MSSFP (1.4).
Corollary 3.6
Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Let \(\{C_j\}_{j=1}^r\) and \(\{Q_i\}_{i=1}^t\) be finite family of nonempty, closed and convex subsets of \(H_1\) and \(H_2\), respectively. Define a sequence \(\{x_n\}\subset H_1\) as the following:
where \(\{\gamma \}\in (0,\dfrac{1}{L})\) with \(L=\Vert A^*A\Vert \) and the parameters \(\{\alpha _n\}_{n=1}^\infty , \{\beta _n\}_{n=1}^\infty ,\) \( \{w_j\}_{j=1}^r, \{\eta _i\}_{i=1}^r\) satisfying the same conditions as in Theorem 3.3. Then the sequence \(\{x_n\}\) converges weakly to a solution of the MSSFP (1.4) whenever its solution set is nonempty.
4 Applications
4.1 The split variational inequality problem
Given operators \(f: H_1\rightarrow H_1\), \(g: H_2\rightarrow H_2\) and a bounded linear operator \(A: H_1\rightarrow H_2\) and nonempty closed convex subsets \(C\subset H_1\) and \(Q\subset H_2\), the split variational inequality problem (SVIP) is the problem of finding a point \(x^*\in \mathrm{VIP}(C,f) \) such that \(Ax^*\in \mathrm{VIP}(Q,g)\) that is,
This is equivalent to the problem as follows:
where \(\lambda >0\). We denote the set of solutions by \(\mathrm{SVIP}(A,C,Q,f,g)\). Therefore, SVIP can be viewed as SCFP.
In [17], Censor et al. considered the multiple-set split variational inequality problem (MSSVIP), which is formulated as follows: Let \(H_1\) and \(H_2\) be two real Hilbert spaces. Given a bounded linear operator \(A:H_1\rightarrow H_2\), operators \(\{f_j\}_{j=1}^r: H_1\rightarrow H_1\) and \(\{g_i\}_{i=1}^t:H_2\rightarrow H_2.\) Let \(\{C_j\}_{j=1}^r\) be a finite family of nonempty closed and covex subsets of \(H_1\) and \(\{Q_i\}_{i=1}^t\) be a finite family of nonempty closed and covex subsets of \(H_2\). The multiple set split variational inequality problem is formulated as follows:
that is
This is equivalent to the problem as follows:
where \(\lambda >0\). Let \(\Omega \) be the solution set of the MSSVIP
We recall that, an operator \(T: H\rightarrow H\) is called inverse strongly monotone if there exists a positive real number \(\alpha \) such that
It is known that, if \(\lambda \in (0,2\alpha ]\), then \(I-\lambda T\) is a nonexpansive operator.
Theorem 4.1
Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \). Let \(\{C_j\}_{j=1}^r\) and \(\{Q_i\}_{i=1}^t\) be finite family of nonempty, closed and covex subsets of \(H_1\) and \(H_2\), respectively. For each \(j=1,\ldots ,r\), \(f_j\) is \(\alpha _j-\) ism operator in \(H_1\) and for each \(i=1,\ldots ,t\), \(g_i\) is \(\beta _i\)-ism operator in \(H_2\). Set \(\alpha :=\min \{\alpha _1,\ldots ,\alpha _r, \beta _1,\ldots ,\beta _t\}\). Assume that \(\Omega \ne \emptyset \). Let \(\{x_n\}\subset H_1\) be a sequence generated by
where \(\gamma \in (0,\dfrac{1}{L})\), \(\lambda \in (0,2\alpha ]\) and \(\{w_j\}_{j=1}^r \subset (0,1), \{\eta _i\}_{i=1}^t\subset (0,1)\) with \(\sum _{j=1}^rw_j =1\), \(\sum _{i=1}^t \eta _i=1\). Assume the sequences \(\{\alpha _n\}, \{\beta _n\} \) satisfy the following conditions:
-
(a)
\(0<a\le \beta _n\le b<\dfrac{1}{2}\), for some \(a,b>0.\)
-
(b)
\(\{\alpha _n\}\) is non-decreasing and \(0\le \alpha _n< \alpha <\dfrac{\epsilon }{2\epsilon +1}\), where \(\epsilon :=\dfrac{1-b}{b}\).
Then \(\{x_n\}\) converges weakly to an element of \(\Omega \).
Proof
For each \(j=1,\ldots ,r\), set \(U_j=P_{C_j}(I-\lambda f_j)\) and for each \(i=1,\ldots ,t\), set \(T_i=P_{Q_i}(I-\lambda g_i)\). We can rewrite the sequence \(\{x_n\}\) as
Since \(\lambda \in (0,2\alpha ]\), we obtain \(\{U_j\}_{j=1}^r\) and \(\{T_i\}_{i=1}^t\) are finite families of nonexpansive operators. From Theorem 3.3 we get the proof.
4.2 The split common null point problem
Recently, Byrne et al. [12] introduced the split common null point problem (SCNPP) for set-valued maximal monotone operators in Hilbert spaces. Given set-valued operators \(F_j : H_1 \rightarrow H_1,\) \(1 \le j \le r\), and \(G_i : H_2 \rightarrow H_2,\) \(1 \le i\le t\), respectively, and the bounded linear operators \(A : H_1\rightarrow H_2\), the SCNPP is formulated as follows:
Recall that \(B: H\rightarrow 2^H\) is said to be monotone if
where \(B(D):=\{x\in H, Bx\ne \emptyset \}.\)
A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator.
For a maximal monotone operator \(B: H\rightarrow 2^H\) and \(\lambda >0\), we can define a single-valued operator:
It is known that \(J^B_\lambda \) is firmly nonexpansive and \(0\in B(x)\) iff \(x\in \mathrm{Fix}(J^B_\lambda )\), see [41].
Therefore, the problem (4.10) is equivalent to the problem as follows:
where \(\lambda >0\), that is, the SCNPP reduces to the SCFPP. Let \(\Omega \) be the solution set of the SCNPP.
Applying Theorem 3.3, we obtain the following result.
Theorem 4.2
Let \(A: H_1\rightarrow H_2\) be a bounded linear operator with \(L=\Vert A^*A\Vert \) and \(\lambda >0\). For each \(j=1,\ldots ,r\), \(F_j:H_1\rightarrow H_1\) is maximal monotone operator and for each \(i=1,\ldots ,t\), \(G_i:H_2\rightarrow H_2\) is maximal monotone operator. Assume that \(\Omega \ne \emptyset \). Let \(\{x_n\}\subset H_1\) be a sequence generated by
where \(\gamma \in (0,\dfrac{1}{L})\) and \(\{w_j\}_{j=1}^r \subset (0,1), \{\eta _i\}_{i=1}^t\subset (0,1)\) with \(\sum _{j=1}^rw_j =1\), \(\sum _{i=1}^t \eta _i=1\). Assume the sequences \(\{\alpha _n\}, \{\beta _n\} \) satisfy the following conditions:
(a) \(0<a\le \beta _n\le b<\dfrac{1}{2}\), for some \(a,b>0.\)
(b) \(\{\alpha _n\}\) is non-decreasing and \(0\le \alpha _n< \alpha <\dfrac{\epsilon }{2\epsilon +1}\), where \(\epsilon :=\dfrac{1-b}{b}\).
Then \(\{x_n\}\) converges weakly to an element of \(\Omega \).
5 Numerical examples
In this section, we consider a simple example in support of the convergence of Algorithm (3.11) in Theorem 3.3. The example considered here is as follows:
where \(C_i\subset \mathfrak {R}^N\), \(Q_j\subset \mathfrak {R}^M\) (\(i=1,\ldots , r\) and \(j=1,\ldots , t\)) are closed convex sets, and \(A\in \mathfrak {R}^{M\times N}\) is a matrix. We set \(U_i=P_{C_i}\) and \(T_j=P_{Q_j}\) for all i, j; then the considered problem is equivalent to SCFFP (1.1) and Algorithm (3.11) can be applied. We have used the function
to study the convergence of Algorithm (3.11). Note that \(\mu =0\) and \(L=||A^*A||\); we choose \(\lambda =0.5/L\), \(w_i=1/r\), \(\eta _j=1/t\). Set \(a=0.1\), \(b=0.45\), \(\epsilon =(1-b)/b\) and \(\alpha =0.9 \epsilon /(2\epsilon +1)\). Consider the parameter \(\beta _n\) as \(\beta _n=a+(b-a)/n\). Five sequences of \(\left\{ \alpha _n\right\} \) are used in all the experiments as
The starting point is \(x_0=(1,1,\ldots ,1)^T\).
For experiments, we consider, for each i, \(C_i\) is a ball containing the original point and centered at \(a_i\in \mathfrak {R}^N\) and the radius \(r_i>0\); for each j, \(Q_j\) is a box also containing the original point, and defined by
All the data and matrix A are generated randomly. The projection on \(C_i\) is inherently explicit, and the projection on \(Q_j\) is computed by the function quadprog in Matlab 7.0. All programs are computed on a PC Desktop Intel(R) Core(TM) i5-3210M CPU @ 2.50 GHz, RAM 2.00 GB.
Figures 1, 2, 3, 4 and 5 describe the behavior of \(f(y_n)\) generating by Algorithm (3.11) for given parameter sequences in the first 1000 iterations. In these figures, the y-axis represents the value of \(f(y_n)\) while the x-axis represents the number of iterations (# iterations). Moreover, we do not show the value of \(f(x_n)\), but for \(f(y_n)\) because, see Algorithm (3.11), we do not want to compute additionally the projections \(P_{C_i}(x_n)\) and \(P_{Q_j}(Ax_n)\).
From these figures, we see that the larger the dimension of spaces or the numbers of subproblems, the slower the convergence of the algorithm. Moreover, the better the convergence of \(f(y_n)\), the more the sequence \(\alpha _n\) increases fastly.
6 Conclusions
The paper has proposed an algorithm, called Inertial-Mann method, for solving the split common fixed point problem in Hilbert spaces. Under some suitable conditions imposed on parameters, we have proved the weak convergence of the algorithm. The two applications of the proposed algorithm to SVIPs and SCNPPs have been presented in the paper. Several numerical experiments on a simple example have been also performed to support for the obtained result.
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Acknowledgements
The authors are grateful to the anonymous referee for valuable suggestions which helped to improve the manuscript. The first author was partially supported by Vietnam Institute for Advanced Study in Mathematics (VIASM). The second author was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project 101.01-2017.315.
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Thong, D.V., Van Hieu, D. An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. 19, 3029–3051 (2017). https://doi.org/10.1007/s11784-017-0464-7
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DOI: https://doi.org/10.1007/s11784-017-0464-7