- Research
- Open access
- Published:
Weak and strong convergence of inertial algorithms for solving split common fixed point problems
Journal of Inequalities and Applications volume 2021, Article number: 26 (2021)
Abstract
In this paper, we propose two iterative schemes for approximating solutions of split common fixed point problems in multiple linear operators case. The first algorithm implements the Krasnosel’skiĭ–Mann iteration with an inertial effect for which the weak convergence is established under mild assumptions. With the tool of nearly contractive mappings, we introduce a viscosity-type iteration which ensures strong convergence. We apply our results to solve a multiple split monotone variational inclusion problem. A numerical example is given to demonstrate the efficiency of the proposed algorithms.
1 Introduction
Throughout this paper, let \(\mathcal{H},\mathcal{K}_{1},\mathcal{K}_{2},\ldots ,\mathcal{K}_{ r}\) be real Hilbert spaces. Given a not necessarily linear operator T from \(\mathcal{H}\) into \(\mathcal{H}\), we denote by \({\mathrm{Fix}}(T):=\{x\in \mathcal{H}\mid x=Tx\}\) the set of all fixed points of T. We are interested in the following problem [13].
General Split Common Fixed Point Problem (GSCFPP)
where \(U_{i}:\mathcal{H}\to \mathcal{H}\), \(T_{j}:\mathcal{K}_{j}\to \mathcal{K}_{j}\) are operators, and \(A_{j}:\mathcal{H}\to \mathcal{K}_{j}\) is a bounded linear operator for all \(i=1, 2,\ldots ,p\) and \(j=1, 2,\ldots ,r\).
If \(p=r=1\) then the GSCFPP is reduced to the split common fixed point problem.
Split Common Fixed Point Problem (SCFPP)
which was first introduced by Censor and Segal [12]. They considered the case of directed operators U and T. In [23], Moudafi solved the case of demicontractive mappings, and proposed the following algorithm:
under suitable conditions of parameters, weak convergence is guaranteed.
Several important inverse problems [5–12, 23], can be rewritten to the description of the GSCFPP. In order to solve the GSCFPP, Chen, Sahu, and Wong [13] proposed several weakly and strongly convergent schemes. In this paper, we study the convergence of the Krasnosel’skiĭ–Mann iteration and the nearly contractive viscosity-type iteration, both involving the inertial effect. The inertial terminology greatly improves the performance of the algorithm (see, e.g. [2, 16, 19, 24]). The inertial method was developed by Polyak [26] firstly. Álvarez and Attouch [1] employed Polyak’s idea, they constructed an algorithm, combined with the proximal point algorithm, named inertial proximal point algorithm as the following form:
where B is a maximal monotone operator (Sect. 4). Here, \(\vartheta _{n}(x_{n}-x_{n-1})\) is named the inertial term. It was proved that the inertial proximal point algorithm (1.1) converges weakly to a zero point of B if \(\{\lambda _{n}\}\) is non-decreasing and \(\{\vartheta _{n}\}\subset [0,1)\) satisfies
As in the common case that incorporating the inertial method in an algorithm greatly improves the performance numerically. We refer the related research [4, 14, 15, 20, 29–32] to readers.
In Sect. 2, we review some fundamental tools and results from the convex analysis. In Sect. 3, we construct algorithms for solving the GSCFPP and study their weak and strong convergence. In Sect. 4, we study the multiple split monotone variational inclusion problem. In Sect. 5, we provide a numerical example to demonstrate the performance of our algorithms.
2 Preliminaries
Let I be the identity operator on \(\mathcal{H}\). Given a sequence \(\{x_{n}\}_{n\in \mathbb{N}}\) in \(\mathcal{H}\) and \(x\in \mathcal{H}\). The notations “\(x_{n}\to x\)” and “\(x_{n}\rightharpoonup x\)” indicate the strong convergence to x and weak convergence of \(\{x_{n}\}\) to x, respectively. We denote by \(\omega _{w}(x_{n})\) the collection of all points x̄ such that there is a subsequence of \(\{x_{n}\}\) converges weakly to x̄.
Definition 2.1
A (not necessarily linear) operator \(T:\mathcal{H}\to \mathcal{H}\) is said to be
-
quasi-nonexpansive if \(\operatorname{Fix}(T)\neq \emptyset \) and \(\| Tx-z \|\leq \| x-z \|\), \(\forall x\in \mathcal{H}\), \(z \in \operatorname{Fix}(T)\);
-
firmly nonexpansive if \(\|Tx-Ty\|^{2} \leq \langle Tx-Ty,x-y \rangle \), \(\forall x,y\in \mathcal{H}\), or equivalently,
$$ \Vert Tx-Ty \Vert ^{2}\leq \Vert x-y \Vert ^{2}- \bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2},\quad \forall x,y\in \mathcal{H}; $$ -
α-strongly quasi-nonexpansive with \(\alpha >0\) if \(\operatorname{Fix}(T)\neq \emptyset \) and
$$ \langle x-Tx,z-x \rangle \leq \frac{-1-\alpha }{2} \Vert x-Tx \Vert ^{2}, \quad \forall x\in \mathcal{H}, z\in \operatorname{Fix}(T), $$or equivalently,
$$ \Vert Tx-z \Vert ^{2}\leq \Vert x-z \Vert ^{2}- \alpha \Vert x-Tx \Vert ^{2}, \quad \forall x\in \mathcal{H}, z\in \operatorname{Fix}(T); $$ -
β-demicontractive with \(\beta <1\) if \(\operatorname{Fix}(T)\neq \emptyset \) and
$$ \langle x-Tx,z-x \rangle \leq \frac{\beta -1}{2} \Vert x-Tx \Vert ^{2}, \quad \forall x\in \mathcal{H}, z\in \operatorname{Fix}(T), $$or equivalently,
$$ \Vert Tx-z \Vert ^{2}\leq \Vert x-z \Vert ^{2}+\beta \Vert x-Tx \Vert ^{2}, \quad \forall x\in \mathcal{H}, z\in \operatorname{Fix}(T). $$
Recall the metric projection \(P_{C}\) onto a nonempty, closed and convex subset C of \(\mathcal{H}\) is defined with that \(P_{C}(x)\) is the unique point in C such that \(\|x-P_{C}(x)\|=\inf_{w\in C}\|x-w\|\), or equivalently
It is well known that \(P_{C}\) is firmly nonexpansive, 1-strongly quasi-nonexpansive and also \((-1)\)-demicontractive.
Lemma 2.2
([21])
If \(T:\mathcal{H}\to \mathcal{H}\) is β-demicontractive, then the fixed point set Fix(T) of T is closed and convex.
Definition 2.3
Let \(T:\mathcal{H}\to \mathcal{H}\). We say that \(I-T\) is demiclosed at zero if for any sequence \(\{x_{n}\}\) in \(\mathcal{H}\) converging weakly to x and \(\{x_{n}-Tx_{n}\}\) converging strongly to 0, we have \((I-T)x=0\).
For example, when T is nonexpansive, \(I-T\) is demiclosed at zero.
Lemma 2.4
([25])
Let \(\mathcal{H}\) be a Hilbert space and \(\{x_{n}\}\) a sequence in \(\mathcal{H}\) such that there exists a nonempty set D of \(\mathcal{H}\) satisfying:
-
(a)
For every \(z\in D\), \(\lim_{n\to \infty }\|x_{n}-z\|\) exists.
-
(b)
\(\omega _{w}(x_{n})\subset D\).
Then there exists \(\bar{x}\in D\) such that \(x_{n}\rightharpoonup \bar{x}\)
Lemma 2.5
Let \(\{\varphi _{n}\}\), \(\{\delta _{n}\}\) be two nonnegative sequences, and \(\{\vartheta _{n}\}\) be a sequence in \([0,\vartheta ]\) where \(\vartheta \in [0,1)\). Assume
-
(a)
\(\varphi _{n+1}-\varphi _{n}\leq \vartheta _{n}(\varphi _{n}-\varphi _{n-1})+ \delta _{n}\), \(n\geq 1\),
-
(b)
\(\sum_{n=1}^{\infty }\delta _{n}<\infty \).
Then the sequence \(\{\varphi _{n}\}\) is convergent and \(\sum_{n=1}^{\infty }[\varphi _{n+1}-\varphi _{n}]_{+}< \infty \), where \([t]_{+}:=\max \{t,0\}\).
Lemma 2.6
Assume \(\{w_{n}\}\) is a sequence of nonnegative real numbers such that
where \(\{\nu _{n}\}\), \(\{u_{n}\}\) and \(\{\sigma _{n}\}\) are sequences of real numbers such that
-
(a)
\(\{\nu _{n}\}\subset (0,1)\) and \(\sum_{n=1}^{\infty }\nu _{n}=\infty \),
-
(b)
\(\lim_{n\to \infty }\sigma _{n}=0\),
-
(c)
\(\limsup_{k\to \infty }u_{n_{k}}\leq 0\) whenever \(\limsup_{k\to \infty }\eta _{n_{k}}= 0\) for any subsequence of \(\{n_{k}\}\) of \(\{n\}\).
Then \(\lim_{n\to \infty }w_{n}=0\).
3 Iterative algorithms for GSCFPP
We consider the following GSCFPP:
where \(U_{i}:\mathcal{H}\to \mathcal{H}\) an \(\alpha _{i}\)-strongly quasi-nonexpansive operator for \(i=1, 2,\ldots ,p\), and \(T_{j}:\mathcal{K}_{j}\to \mathcal{K}_{j}\) a \(\beta _{i}\)-demicontractive operator for \(j=1, 2,\ldots ,r\).
In this section, we develop two iterative algorithms for solving GSCFPP (3.1) when \(p=r\). For the case \(p\neq r\), Wang and Xu [33] set \(U_{p+1},U_{p+2},\ldots ,U_{r}\) to be the identity mapping when \(p< r\), and set \(T_{p+1},T_{p+2},\ldots ,T_{r}\) to be the identity mappings when \(p>r\). There is another option: if \(p< r\), we define \(U_{p+1}:=U_{1}\), \(U_{p+2}:=U_{2},\ldots \) , so on; while if \(p>r\), we repeat members of \(\{A_{j}\}_{j=1}^{r}\) and \(\{T_{j}\}_{j=1}^{r}\) in a similar way. In either way, (3.1) is in the case \(p=r\). Let Γ be the solution set of GSCFPP (3.1) and suppose \(\Gamma \neq \emptyset \) throughout this paper.
Lemma 3.1
([13])
The solution set of GSCFPP (3.1)
is closed and convex.
Definition 3.2
([13])
Let \(U_{i}:\mathcal{H}\to \mathcal{H}\) be an \(\alpha _{i}\)-strongly quasi-nonexpansive operator, \(T_{i}:\mathcal{K}_{i}\to \mathcal{K}_{i}\) be a \(\beta _{i}\)-demicontractive operator, \(A_{i}:\mathcal{H}\to \mathcal{K}_{i}\) be a bounded linear operator, \(A_{i}^{*}\) be the adjoint operator of \(A_{i}\), and \(I_{i}\) be the identity operator on \(\mathcal{K}_{i}\) for \(i=1, 2,\ldots ,p\). We define an operator \(S:\mathcal{H}\to \mathcal{H}\) as follows:
where \(\omega _{i}\in (0,1)\) with \(\sum_{i=1}^{p}\omega _{i}=1\).
Lemma 3.3
([13])
Let \(U_{i}:\mathcal{H}\to \mathcal{H}\) be an \(\alpha _{i}\)-strongly quasi-nonexpansive operator, \(T_{i}:\mathcal{K}_{i}\to \mathcal{K}_{i}\) be a \(\beta _{i}\)-demicontractive operator, \(A_{i}:\mathcal{H}\to \mathcal{K}_{i}\) be a bounded linear operator, and \(0<\gamma _{i}<\frac{1-\beta _{i}}{\|A_{i}\|^{2}}\) for \(i=1, 2,\ldots ,p\). Define a functional \(r:\mathcal{H}\to \mathbb{R}\) by
where \(V_{i}=I-\gamma _{i}A_{i}^{*}(I_{i}-T_{i})A_{i}\) for \(i=1, 2,\ldots ,p\). Then, for \(x\in \mathcal{H}\) and \(z\in \Gamma \), we have
Remark 3.4
In Lemma 3.3, we observe that \(r(x)\geq 0\) for all \(x\in \mathcal{H}\). Therefore, the operator S is quasi-nonexpansive.
Lemma 3.5
If \(z_{n}:=x_{n}+\vartheta _{n}(x_{n}-x_{n-1})\) where \(0\leq \vartheta _{n}<1\) for all \(n\in \mathbb{N}\), then, for \(z\in \mathcal{H}\),
Proof
Using the identity \(2\langle a,b \rangle =\|a\|^{2}+\|b\|^{2}-\|a-b\|^{2}\), we have
Hence, we obtain the desired result. □
3.1 Inertial Krasnosel’skiĭ–Mann algorithm
Algorithm 3.6
(iKMA: Inertial Krasnosel’skiĭ–Mann algorithm)
Let \(\vartheta _{n}\in [0,\vartheta ]\) with \(\vartheta \in [0,1)\), \(\{s_{n}\}\) is a sequence in \((0,1]\), and \(\omega _{i}\in (0,1)\) with \(\sum_{i=1}^{p}\omega _{i}=1\). Set
Theorem 3.7
Let \(U_{i}:\mathcal{H}\to \mathcal{H}\) be an \(\alpha _{i}\)-strongly quasi-nonexpansive operator, \(T_{i}:\mathcal{K}_{i}\to \mathcal{K}_{i}\) be a \(\beta _{i}\)-demicontractive operator, \(A_{i}:\mathcal{H}\to \mathcal{K}_{i}\) be a bounded linear operator, and \(0<\gamma _{i}<\frac{1-\beta _{i}}{\|A_{i}\|^{2}}\) for \(i=1, 2,\ldots ,p\). Assume that \(I-U_{i}\), \(I_{i}-T_{i}\) are demiclosed at 0 for all \(i=1, 2,\ldots ,p\), and \(\{s_{n}\}\) is a sequence in \([a,1]\) for some \(a>0\). Then the sequence \(\{x_{n}\}\) generated by Algorithm 3.6converges weakly to a point in Γ, provide that sequence \(\{\vartheta _{n}\}\) is chosen in \([0,\vartheta ]\) with \(\vartheta \in [0,1)\) such that
Proof
Let \(z\in \Gamma \). Combining Lemma 3.5, we have
Applying Lemma 2.5 to (3.6), we conclude that the sequence \(\{\|x_{n}-z\|\}\) is convergent. The condition (3.4) implies
It follows from the assumption on parameters and (3.5) that
Letting \(n\to \infty \) in (3.8), and observing the assumption \(s_{n}\geq a\) for all \(n\in \mathbb{N}\), we have \(r(z_{n})\to 0\). Since the coefficients in the finite sum (3.2) are all positive, we obtain
For any subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) which converges weakly to x̃. By (3.7), the subsequence \(\{z_{n_{k}}\}\) of \(\{z_{n}\}\) also converges weakly to x̃. Then \(A_{i}z_{n_{k}}\rightharpoonup A_{i}\tilde{x}\) for all \(i=1, 2,\ldots ,p\). Hence, using the demiclosedness of \(I-U_{i}\) and \(I_{i}-T_{i}\) for all \(i=1, 2,\ldots ,p\) in (3.9), we conclude that \(\omega _{w}(x_{n})\subset \Gamma \). It follows that \(\{x_{n}\}\) converges weakly to a point in Γ by Lemma 2.4. □
Remark 3.8
(a) Given \(\vartheta \in [0,1)\), and \(\{\varepsilon _{n}\}\) is any positive sequence such that \(\sum_{n=1}^{\infty }\varepsilon _{n}<+\infty \). The value of \(\vartheta _{n}\) can be chosen from \([0,\bar{\vartheta _{n}}]\) where
(b) In Theorem 3.7, the parameter \(s_{n}\) is a relax condition to the classical Krasnosel’skiĭ–Mann algorithm (see, e.g. [3, Theorem 5.15]). When \(s_{n}\equiv 1\), the Algorithm 3.6 turns to the Picard iteration.
3.1.1 Strong convergence: inertial NC-viscosity-type algorithm
Let f be a contraction on \(\mathcal{H}\). The viscosity approximation method proposed by Moudafi [22] generates a strongly convergent sequence:
which converges strongly to a fixed point \(x^{*}\) of T. In [34], Xu further proved that the above \(x^{*}\) also satisfies the following variational inequality:
provided that \(\{t_{n}\}\) fulfills certain conditions.
Recall that a sequence of mappings \(\{f_{n}\}\) from \(\mathcal{H}\) into \(\mathcal{H}\) is called a nearly contractive mappings with sequence \(\{(\kappa _{n}, a_{n})\}\) in \([0,1)\times [0,\infty )\) [18, 27, 28] if \(a_{n}\to 0\), and for any \(x,y\in \mathcal{H}\) and \(n\in \mathbb{N}\), we have
Here, we give two examples of nearly contractive mappings. Let \(\mathcal{H}=\mathbb{R}\), a sequence of mappings \(\{f_{n}\}\) defined by
Then it can be verified that \(|f_{n}(x)-f_{n}(y)|\leq \frac{1}{n+1}|x-y|+ \frac{3}{n+1}\), for all \(x,y\in \mathcal{H}\), \(n\in \mathbb{N}\). Also, if \(f_{n}\equiv f\) for all n, where f is a contraction on \(\mathcal{H}\), then \(f_{n}\) is a nearly contractive mappings with sequences \(\kappa _{n}\equiv \kappa \), \(a_{n}\equiv 0\).
Algorithm 3.9
(iNCVA: Inertial NC-Viscosity-type algorithm)
Let \(\{t_{n}\}\), \(\{s_{n}\}\) be two sequences in \((0,1)\), \(\vartheta _{n}\in [0,\vartheta ]\) with \(\vartheta \in [0,1)\), and \(\{f_{n}\}\) be a nearly contractive mappings with \(\{(\kappa _{n}, a_{n})\}\). Set
Lemma 3.10
Let \(U_{i}:\mathcal{H}\to \mathcal{H}\) be an \(\alpha _{i}\)-strongly quasi-nonexpansive operator, \(T_{i}:\mathcal{K}_{i}\to \mathcal{K}_{i}\) be a \(\beta _{i}\)-demicontractive operator, \(A_{i}:\mathcal{H}\to \mathcal{K}_{i}\) be a bounded linear operator, and \(0<\gamma _{i}<\frac{1-\beta _{i}}{\|A_{i}\|^{2}}\) for \(i=1, 2,\ldots ,p\). Given a contraction f with \(\kappa \in (0,1)\) and \(x^{*}=P_{\Gamma }f(x^{*})\). Assume that \(I-U_{i}\) and \(I_{i}-T_{i}\) are demiclosed at 0 for all i, and \(\{f_{n}\}\) is a nearly contractive mappings with \(\{(\kappa _{n}, a_{n})\}\) such that \(\kappa _{n}\to \kappa \). Assume that the following conditions are satisfied:
-
(C1)
\(t_{n}\in (0,1)\) such that \(\lim_{n\to \infty } t_{n}=0\) and \(\sum_{n=1}^{\infty } t_{n}=\infty \);
-
(C2)
\(0<\eta \leq s_{n}\leq \xi <1\) for all \(n\in \mathbb{N}\);
-
(C3)
\(\lim_{n\to \infty } f_{n}(x^{*})=f(x^{*})\);
-
(C4)
\(\lim_{n\to \infty }\frac{\vartheta _{n}}{t_{n}}\|x_{n}-x_{n-1} \|=0\).
Then the sequence \(\{x_{n}\}\) generated by Algorithm 3.9converges strongly to \(x^{*}=P_{\Gamma }f(x^{*})\).
Proof
Let \(y_{n}=t_{n}f_{n}(z_{n})+(1-t_{n})Sz_{n}\), we have
Therefore,
Since \(\lim_{n\to \infty } f_{n}(x^{*})=f(x^{*})\), \(\lim_{n\to \infty }\kappa _{n}=\kappa \) and \(\lim_{n\to \infty }a_{n}=0\), we conclude that the sequence \(\{ \frac{\|f_{n}(x^{*})-x^{*}\|+a_{n}}{1-\kappa _{n}} \} \) is bounded. On the other hand, the conditions \((C2)\) and \((C4)\) imply the sequence \(\{ \frac{\vartheta _{n}}{(1-\kappa _{n})s_{n}t_{n}}\|x_{n}-x_{n-1}\| \} \) is also bounded. Let an upper bound of
be M. Then we write (3.10) as
by induction, we have
Hence, \(\{\|x_{n}-x^{*}\|\}\) is bounded, so is \(\{\|z_{n}-x^{*}\|\}\) and \(\{\|y_{n}-x^{*}\|\}\). Also,
It follows that \(\{f_{n}(z_{n})\}\) is bounded. From the inequality
we have
This implies that
Then, together with the inequality
we have
Hence,
Using the inequality \(\|u+v\|^{2}\leq \|u\|^{2}+2\langle v,u+v\rangle \), we get
Combining (3.11) and (3.12), we have
Set \(w_{n}=\|x_{n}-x^{*}\|^{2}\), \(\tau _{n}=s_{n}(1-s_{n})\|z_{n}-y_{n}\|^{2}+s_{n}(1-t_{n})r(z_{n})\), and
Then we rewrite (3.13) as
where
Since \(\sum_{n=1}^{\infty }t_{n}=\infty \) and \((C2)\) holds, \(\lim_{n\to \infty }\kappa _{n}=\kappa \). It follows that
Because \(t_{n}\in (0,1)\) and \((C4)\), we see that \(\lim_{n\to \infty }\vartheta _{n}\|x_{n}-x_{n-1}\|=0\). Together with the boundedness of \(\{\|z_{n}-x^{*}\|\}\), \(\{\|y_{n}-x^{*}\|\}\) and \(\lim_{n\to \infty }t_{n}=0\), we have \(\lim_{n\to \infty }\sigma _{n}=0\).
In order to prove \(\lim_{n\to \infty }w_{n}=0\), with Lemma 2.6, it remains to show that \(\lim_{k\to \infty }\tau _{n_{k}}= 0\) implies \(\limsup_{k\to \infty }u_{n_{k}}\leq 0\) for any subsequence of \(\{n_{k}\}\) of \(\{n\}\). Let \(\{\tau _{n_{k}}\}\) be a subsequence of \(\{\tau _{n}\}\) such that \(\lim_{k\to \infty }\tau _{n_{k}}= 0\). Since \(\lim_{k\to \infty } t_{n_{k}}=0\), \(0<\eta \leq s_{n_{k}}\leq \xi <1\), we have
and \(\lim_{k\to \infty }r(z_{n_{k}})=0\), that is,
Taking a subsequence \(\{y_{n_{k_{j}}}\}\) of \(\{y_{n_{k}}\}\) such that
Since \(\{y_{n}\}\) is bounded, we further assume that \(\{y_{n_{k_{j}}}\}\) converges weakly to a point ỹ. It follows from (3.16) that \(\{z_{n_{k_{j}}}\}\) also converges weakly to the same point ỹ. Due to (3.17), we get \(\tilde{y}=U_{i}\tilde{y}\) and \(A_{i}\tilde{y}=T_{i}A_{i}\tilde{y}\) for all \(i=1, 2,\ldots ,p\) by the demiclosedness of \(\{I-U_{i}\}\) and \(\{I_{i}-T_{i}\}\). Then ỹ is a point in Γ. It follows, from equation (2.1), that
Together with \(\lim_{n\to \infty }a_{n}=0\), \(\lim_{n\to \infty }f_{n}(x^{*})=f(x^{*})\) and \(\lim_{n\to \infty }\frac{\vartheta _{n}}{t_{n}}\|x_{n}-x_{n-1} \|=0\), we conclude that
Hence, \(x_{n}\to x^{*}\) as \(n\to \infty \). □
Remark 3.11
The value of \(\vartheta _{n}\) can be chosen from \([0,\bar{\vartheta _{n}}]\), where
and \(\{\varepsilon _{n}\}\) is a positive sequence such that \(\lim_{n\to \infty }\varepsilon _{n}=0\).
Theorem 3.12
Let \(U_{i}:\mathcal{H}\to \mathcal{H}\) be an \(\alpha _{i}\)-strongly quasi-nonexpansive operator, \(T_{i}:\mathcal{K}_{i}\to \mathcal{K}_{i}\) be a \(\beta _{i}\)-demicontractive operator, \(A_{i}:\mathcal{H}\to \mathcal{K}_{i}\) be a bounded linear operator, and \(0<\gamma _{i}<\frac{1-\beta _{i}}{\|A_{i}\|^{2}}\) for \(i=1, 2,\ldots ,p\). Given a sequence \(\{v_{n}\}\) in \(\mathcal{H}\) strongly converges to some vector v. Assume that \(I-U_{i}\) and \(I_{i}-T_{i}\) are demiclosed at 0 for all i, and the following conditions are satisfied:
-
(C1)
\(t_{n}\in (0,1)\) such that \(\lim_{n\to \infty } t_{n}=0\) and \(\sum_{n=1}^{\infty } t_{n}=\infty \),
-
(C2)
\(0<\eta \leq s_{n}\leq \xi <1\) for all \(n\in \mathbb{N}\),
-
(C3’)
\(\lim_{n\to \infty }\frac{\vartheta _{n}}{t_{n}}\|x_{n}-x_{n-1} \|=0\).
Then the sequence \(\{x_{n}\}\) generated by the following:
converges strongly to \(P_{\Gamma }v\). In particularly, if \(v_{n}\to 0\), then the sequence \(\{x_{n}\}\) converges strongly to the solution \(x_{\min}\) which satisfies \(\|x_{\min }\|=\min \{\|x\|:x\in \Gamma \}\).
Proof
Define contraction mappings \(f(x):=v\), and \(f_{n}(x):=v_{n}\). Then \(\{f_{n}(x)\}\) satisfies the conditions in Lemma 3.10. Hence, the proof is done. □
Remark 3.13
When we set \(f_{n}\equiv f\) for some contraction mapping, the Algorithm 3.9 becomes the viscosity algorithm. The condition \((C3)\) in Lemma 3.10 is satisfied automatically.
4 Applications
Definition 4.1
A set-valued operator \(B:\mathcal{H}\to 2^{\mathcal{H}}\) is said to be
-
monotone if for all \((x,u),(y,v)\in \text{graph }B\), i.e., \(u\in Bx\) and \(v\in By\), we have
$$ \langle x-y, u-v \rangle \geq 0; $$ -
maximal monotone if it is monotone and its graph is not properly contained in another the graph of monotone operator;
-
η-strongly monotone if there exists \(\eta >0\) such that for all \((x,u),(y,v)\in \text{graph }B\), we have \(\langle x-y, u-v \rangle \geq \eta \|x-y\|^{2}\);
-
ν-inverse strongly monotone (or ν-cocoercive) if there exists \(\nu >0\) such that
$$ \langle x-y, u-v \rangle \geq \nu \Vert u-v \Vert ^{2}, $$for all \((x,u),(y,v)\in \text{graph }B\).
We denote \(\operatorname{zero}(B)=\{x\in \mathcal{H}\mid 0\in Bx\}\).
Definition 4.2
Let \(B:\mathcal{H}\to 2^{\mathcal{H}}\) be a set-valued operator and \(\lambda >0\). The resolvent of B is defined as follows:
Proposition 4.3
([3])
Let \(B_{1}:\mathcal{H}\to 2^{\mathcal{H}}\) be maximal monotone and \(B_{2}: \mathcal{H}\to \mathcal{H}\) is ν-inverse strongly monotone. Then, for \(\lambda \in (0,2\nu )\), the forward–backward operator \(J_{\lambda B_{1}}(I-\lambda B_{2})\) is \(\frac{2\nu }{4\nu -\lambda }\)-averaged, or equivalently,
for all \(x,y\in \mathcal{H}\).
Remark 4.4
As a consequence of Proposition 4.3, if the set \(\operatorname{zero}(B_{1}+B_{2})\) is non-empty, then \(\operatorname{zero}(B_{1}+B_{2})={\mathrm{Fix}}(J_{\lambda B_{1}}(I- \lambda B_{2}))\) and \(J_{\lambda B_{1}}(I-\lambda B_{2})\) is \(\frac{2\nu -\lambda }{2\nu }\)-strongly quasi-nonexpansive (\(-(\frac{2\nu _{2}-\lambda }{2\nu _{2}})\)-demicontractive) for \(\lambda \in (0,2\nu )\). Also, the forward–backward operator \(U=J_{\lambda B_{1}}(I-\lambda B_{2})\) is nonexpansive, thus \(I-U\) is demiclosed at zero.
We propose the multiple-set split monotone variational inclusion problem (MSSMVIP) as follows:
where \(F_{i}:\mathcal{H}\to 2^{\mathcal{H}}\), \(G_{j}:\mathcal{K}\to 2^{\mathcal{K}}\) are set-valued maximal monotone operators, \(f_{i} : \mathcal{H}\to \mathcal{H}\), \(g_{j} : \mathcal{K}\to \mathcal{K}\) are inverse strongly monotone operators for \(i=1,2,\ldots ,p\), \(j=1,2,\ldots ,r\), and \(A:\mathcal{H}\to \mathcal{K}\) is a bounded linear operator.
We remark that there are some cases, \(f_{i}(x)+F_{i}\) may not be a maximal monotone operator, even if it is so, the resolvent \(J_{\lambda (f_{i}(x)+F_{i})}\) might not be able to carry an analytic form easily.
Set \(U_{i}=J_{\lambda F_{i}}(I-\lambda f_{i})\) for all \(i=1,2,\ldots ,p\) and \(T_{j}=J_{\lambda G_{j}}(I-\lambda g_{j})\) for all \(j=1,2,\ldots ,r\), respectively. Hence, according Remark 4.4, the operator \(U_{i}\) is a strongly quasi-nonexpansive and the operator \(T_{j}\) is a demicontractive operator for all i, j. Also, \(I-U_{i}\) and \(I-T_{j}\) are demiclosed at zero. Then MSSMVIP (4.2) becomes to GSCFPP (3.1) where \(\mathcal{K}_{1}=\mathcal{K}_{2}=\cdots =\mathcal{K}_{r}= \mathcal{K}\) and \(A_{1}=A_{2}=\cdots =A_{r}=A\). Assume that \(p=r\) and the solution set of MSSMVIP (4.2) (denoted \(\Gamma _{\mathrm{M}}\)) is not empty. The associated operator is
where \(\omega _{i}\in (0,1)\) with \(\sum_{i=1}^{p}\omega _{i}=1\).
Theorem 4.5
Let \(\mathcal{H}\) and \(\mathcal{K}\) be two real Hilbert spaces and \(A :\mathcal{H}\to \mathcal{K}\) be a bounded linear operator. Let \(f_{i} : \mathcal{H}\to \mathcal{H}\) and \(g_{j} : \mathcal{K}\to \mathcal{K}\) be \(\nu _{i}\)-inverse strongly monotone for all \(i=1,2,\ldots ,p\) and \(\mu _{j}\)-inverse strongly monotone for all \(j=1,2,\ldots ,p\) on \(\mathcal{H}\) and \(\mathcal{K}\), respectively. Let \(\{F_{i}\}\), \(\{G_{j}\}\) be two families of maximal monotone operators. Set \(\mu :=\min \{\nu _{1},\nu _{2},\ldots ,\nu _{p},\mu _{1},\mu _{2}, \ldots ,\mu _{p}\}\), \(\lambda \in [0,2\mu ]\), and \(\gamma _{j}\in (0,\frac{2-\lambda /2\mu _{j}}{\|A\|^{2}})\) for \(j=1,2,\ldots ,p\).
-
(iKMA) If \(\{s_{n}\}\) is a sequence in \((a,1]\) for some \(a>0\), then the sequence \(\{x_{n}\}\) generated by
$$ \textstyle\begin{cases} x_{0},x_{1}\in \mathcal{H}, \\ z_{n}=x_{n}+\vartheta _{n}(x_{n}-x_{n-1}), \\ x_{n+1}=(1-s_{n})z_{n}+s_{n}Sz_{n},\quad n\in \mathbb{N}, \end{cases} $$converges weakly to a point in \(\Gamma _{\mathrm{M}}\) provided that the sequence \(\{\vartheta _{n}\}\subset [0,\vartheta ]\) with \(\vartheta \in [0,1)\) satisfies the following condition:
$$ \sum_{n=1}^{\infty }\vartheta _{n} \Vert x_{n}-x_{n-1} \Vert ^{2}< \infty . $$ -
(iNCVA) Given a contraction h with \(\kappa \in (0,1)\) and \(x^{*}=P_{\Gamma _{\mathrm{M}}}h(x^{*})\). Suppose that \(\{h_{n}\}\) is a sequence of nearly contractive mappings with \(\{(\kappa _{n}, a_{n})\}\) such that \(\kappa _{n}\to \kappa \), and the following conditions are satisfied:
-
(C1)
\(t_{n}\in (0,1)\) such that \(\lim_{n\to \infty } t_{n}=0\) and \(\sum_{n=1}^{\infty } t_{n}=\infty \),
-
(C2)
\(0<\eta \leq s_{n}\leq \xi <1\) for all \(n\in \mathbb{N}\),
-
(C3)
\(\lim_{n\to \infty } h_{n}(x^{*})=h(x^{*})\),
-
(C4)
\(\lim_{n\to \infty }\frac{\vartheta _{n}}{t_{n}}\|x_{n}-x_{n-1} \|=0\).
Then the sequence \(\{x_{n}\}\) generated by
$$ \textstyle\begin{cases} x_{0},x_{1}\in \mathcal{H}, \\ z_{n}=x_{n}+\vartheta _{n}(x_{n}-x_{n-1}), \\ x_{n+1}=(1-s_{n})z_{n}+s_{n}(t_{n}h_{n}(z_{n})+(1-t_{n})Sz_{n}), \quad \textit{for all } n\in \mathbb{N}, \end{cases} $$converges strongly to \(x^{*}\).
-
(C1)
5 Numerical experience
In this section, we utilize those results to demonstrate convergence of our algorithms for solving GSCFPP in finite dimensional Hilbert spaces. We present the behavior of these inertial-iterations in a synthetic experiment. The codes were written by MATLAB (R20013b) and run on a ASUS laptop with RAM 8 GB and Intel(R) Core(TM) i5-4200H CPU @ 2.80 GHz. We consider the following GSCFPP.
Example 5.1
Let \(p=r=3\), \(\mathcal{H}=\mathbb{R}^{2}\), \(\mathcal{K}_{1}=\mathcal{K}_{2}=\mathbb{R}^{2}\), and \(\mathcal{K}_{3}=\mathbb{R}^{3}\). The three closed balls in \(\mathcal{H}\) are as follows:
The three sets \(Q_{1}\), \(Q_{2}\), \(Q_{3}\) are setting as
The three matrices are
Checking the point \((2,2)^{T}\) is in the solution set
The iKMA is set as
The iNCVA is set as
where \(\vartheta _{n}\in [0,\vartheta ]\) with \(\vartheta \in [0,1)\), and the parameters \(\gamma _{j}=0.9\times 1/\|A_{j}\|^{2}\) for all \(j=1, 2,3\). The initial points are chosen by \(x_{0}=(0,3)^{T}\), \(x_{1}=(3,2)^{T}\) and \(v_{0}=(2,2)^{T}\). We present the behavior of the value of the function ϕ at each iteration, where ϕ is defined by
The Remark 3.8 illustrates whereby to choose the parameter sequence \(\{\vartheta _{n}\}\). For the Example 5.1, the parameter sequence \(\{\vartheta _{n}\}\) is chosen as follows:
where ϑ is a real number in \([0,1)\).
Figure 1 (a), (b) show that the performance of iKMA and iNCVA with \(\vartheta =0.8\) and \(\vartheta =0\) (i.e., without the inertial terms), respectively. To be properly presented, we use the semilog plotting. Figure 1 (c) demonstrates the inertial action of iKMA of each even terms.
6 Conclusion
We present weak convergence and strong convergence results with inertial method for the split common fixed point problem with multiple-sets, multiple-operators and the wide class of quasi-nonexpansive type operators. Numerical simulations show that the algorithms involving the inertial effect converge more quickly than without inertial terms.
Availability of data and materials
Not applicable.
References
Álvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Attouch, H., Cabot, A.: Convergence rates of inertial forward–backward algorithms. SIAM J. Optim. 28(1), 849–874 (2018)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Chen, H.Y., Sahu, D.R., Wong, N.C.: Iterative algorithms for solving multiple split common fixed problems in Hilbert spaces. J. Nonlinear Convex Anal. 19(2), 265–285 (2018)
Cholamjiak, P., Shehu, Y.: Inertial forward–backward splitting method in Banach spaces with application to compressed sensing. Appl. Math. 64, 409–435 (2019)
Cholamjiak, W., Cholamjiak, P., Suantai, S.: An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces. J. Fixed Point Theory Appl. 20(1), Paper No. 42, 17 pp. (2018)
Dang, Y., Sun, J., Xu, H.: Inertial accelerated algorithms for solving a split feasibility problem. J. Ind. Manag. Optim. 13(3), 1383–1394 (2017)
He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013, Article ID 942315 (2013)
Latif, A., Sahu, D.R., Ansari, Q.H.: Variable KM-like algorithms for fixed point problems and split feasibility problems. Fixed Point Theory Appl. 2014, 211, 20 (2014)
Lorenz, D., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)
Maingé, P.E.: Convergence theorem for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)
Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007)
Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)
Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 055007, 6 (2010)
Nesterov, Yu.E.: A method for solving the convex programming problem with convergence rate \(O(1/k^{2})\). Dokl. Akad. Nauk SSSR 269(3), 543–547 (1983)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)
Sahu, D.R., Wong, N.C., Yao, J.C.: A generalized hybrid steepest-descent method for variational inequalities in Banach spaces. Fixed Point Theory Appl. 2011, Article ID 754702 (2011)
Sahu, D.R., Wong, N.C., Yao, J.C.: A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Control Optim. 50, 2335–2354 (2012)
Shehu, Y., Cholamjiak, P.: Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo 56(1), Paper No. 4, 21 pp. (2019)
Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag. Optim. 14, 1595–1615 (2018)
Suantai, S., Pholasa, N., Cholamjiak, P.: Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 1081–1099 (2019)
Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. 19(4), 3029–3051 (2017)
Wang, F., Xu, H.K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011)
Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)
Acknowledgements
The author appreciates the worthy comments from reviewers.
Funding
This research is supported by the Taiwan MOST grants 108-2811-M-110-517.
Author information
Authors and Affiliations
Contributions
This entire work has been completed by the author. The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, HY. Weak and strong convergence of inertial algorithms for solving split common fixed point problems. J Inequal Appl 2021, 26 (2021). https://doi.org/10.1186/s13660-021-02556-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-021-02556-4