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Split equality problem and multiplesets split equality problem for quasinonexpansive multivalued mappings
Journal of Inequalities and Applications volume 2014, Article number: 428 (2014)
Abstract
The multiplesets split equality problem (MSSEP) requires finding a point $x\in {\bigcap}_{i=1}^{N}{C}_{i}$, $y\in {\bigcap}_{j=1}^{M}{Q}_{j}$, such that $Ax=By$, where N and M are positive integers, $\{{C}_{1},{C}_{2},\dots ,{C}_{N}\}$ and $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{M}\}$ are closed convex subsets of Hilbert spaces ${H}_{1}$, ${H}_{2}$, respectively, and $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators. When $N=M=1$, the MSSEP is called the split equality problem (SEP). If let $B=I$, then the MSSEP and SEP reduce to the wellknown multiplesets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. Recently, some authors proposed many algorithms to solve the SEP and MSSEP. However, to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases. One of the purposes of this paper is to study the SEP and MSSEP for a family of quasinonexpansive multivalued mappings in the framework of infinitedimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP without the need to compute the projection on the closed convex sets.
1 Introduction and preliminaries
Throughout this paper, we assume that H is a real Hilbert space, C is a subset of H. Denote by $\mathit{CB}(H)$ the collection of all nonempty closed and bounded subsets of H and by $Fix(T)$ the set of the fixed points of a mapping T. The Hausdorff metric $\tilde{H}$ on $\mathit{CB}(H)$ is defined by
where $d(x,K):={inf}_{y\in K}d(x,y)$.
Definition 1.1 Let $R:H\to \mathit{CB}(H)$ be a multivalued mapping. An element $p\in H$ is said to be a fixed point of R, if $p\in Rp$. The set of fixed points of R will be denoted by $Fix(R)$. R is said to be

(i)
nonexpansive, if $\tilde{H}(Rx,Ry)\le \parallel xy\parallel $, $\mathrm{\forall}x,y\in H$;

(ii)
quasinonexpansive, if $Fix(R)\ne \mathrm{\varnothing}$ and $\tilde{H}(Rx,Ry)\le \parallel xy\parallel $, $\mathrm{\forall}x\in H$, $y\in Fix(R)$.
Let $\{{C}_{1},{C}_{2},\dots ,{C}_{N}\}$ and $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{M}\}$ be nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively, and let $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. Recall that the multiplesets split feasibility problem (MSSFP) is to find a point x satisfying the property:
if such a point exists. If $N=M=1$, then the MSSFP reduce to the wellknown split feasibility problem (SFP).
The SFP and MSSFP was first introduced by Censor and Elfving [1], and Censor et al. [2], respectively, which attracted many authors’ attention due to the applications in signal processing [1] and intensitymodulated radiation therapy [2]. Various algorithms have been invented to solve it (see [1–8], etc.).
Recently, Moudafi [9] proposed a new split equality problem (SEP): Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces, $C\subseteq {H}_{1}$, $Q\subseteq {H}_{2}$ be two nonempty closed convex sets, and let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators. Find $x\in C$, $y\in Q$ satisfying
When $B=I$, SEP reduces to the wellknown SFP.
Naturally, we propose the following multiplesets split equality problem (MSSEP) requiring to find a point $x\in {\bigcap}_{i=1}^{N}{C}_{i}$, $y\in {\bigcap}_{j=1}^{M}{Q}_{j}$, such that
where N and M are positive integers, $\{{C}_{1},{C}_{2},\dots ,{C}_{N}\}$ and $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{M}\}$ are closed convex subsets of Hilbert spaces ${H}_{1}$, ${H}_{2}$, respectively, and $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators.
In the paper [9], Moudafi give the alternating CQalgorithm and relaxed alternating CQalgorithm iterative algorithm for solving the split equality problem.
Let $S=C\times Q$ in $H={H}_{1}\times {H}_{2}$, define $G:H\to {H}_{3}$ by $G=[A,B]$, then ${G}^{\ast}G:H\to H$ has the matrix form
The original problem can now be reformulated as finding $w=(x,y)\in S$ with $Gw=0$, or, more generally, minimizing the function $\parallel Gw\parallel $ over $w\in S$. Therefore solving SEP (1.1) is equivalent to solving the following minimization problem:
In the paper [10], we use the wellknown Tychonov regularization to get some algorithms that converge strongly to the minimumnorm solution of the SEP.
Note that to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases.
The purpose of this paper is to introduce and study the following split equality problem for quasinonexpansive multivalued mappings in infinitely dimensional Hilbert spaces, i.e., to find $w=(x,y)\in C$ such that
where ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ are real Hilbert spaces, $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators, ${R}_{i}:{H}_{i}\to \mathit{CB}({H}_{i})$, $i=1,2$ are two quasinonexpansive multivalued mappings, $C=Fix({R}_{1})$, $Q=Fix({R}_{2})$. In the rest of this paper, we still use Γ to denote the set of solutions of SEP (1.3), and assume consistency of SEP so that Γ is closed, convex, and nonempty, i.e., $\mathrm{\Gamma}=\{(x,y)\in {H}_{1}\times {H}_{2},Ax=By,x\in C,y\in Q\}\ne \mathrm{\varnothing}$. The multiplesets split equality problem (MSSEP) for a family quasinonexpansive multivalued mappings in infinitely dimensional Hilbert spaces, i.e., to find $w=(x,y)\in C$ such that
where ${R}_{i}^{j}:{H}_{i}\to \mathit{CB}({H}_{i})$, $i=1,2$, $j=1,2,\dots ,N$ is a family of quasinonexpansive multivalued mappings, $C={\bigcap}_{j=1}^{N}Fix({R}_{1}^{j})$, $Q={\bigcap}_{j=1}^{N}Fix({R}_{2}^{j})$. In the rest of this paper, we use $\overline{\mathrm{\Gamma}}$ to denote the set of solutions of MSSEP (1.4), and assume consistency of MSSEP so that $\overline{\mathrm{\Gamma}}$ is closed, convex, and nonempty, i.e., $\overline{\mathrm{\Gamma}}=\{(x,y)\in {H}_{1}\times {H}_{2},Ax=By,x\in C,y\in Q\}\ne \mathrm{\varnothing}$.
In this paper, we study the SEP and MSSEP for a family of quasinonexpansive multivalued mappings in the framework of infinitedimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP not requiring to compute the projection on the closed convex sets.
We now collect some definitions and elementary facts which will be used in the proofs of our main results.
Definition 1.2 Let H be a Banach space.

(1)
A multivalued mapping $R:H\to \mathit{CB}(H)$ is said to be demiclosed at the origin if, for any sequence $\{{x}_{n}\}\subseteq H$ with ${x}_{n}$ converges weakly to x and $d({x}_{n},R{x}_{n})\to 0$, we have $x\in Rx$.

(2)
A multivalued mapping $R:H\to \mathit{CB}(H)$ is said to be semicompact if, for any bounded sequence $\{{x}_{n}\}\subseteq H$ with $d({x}_{n},R{x}_{n})\to 0$, there exists a subsequence $\{{x}_{{n}_{k}}\}$ such that $\{{x}_{{n}_{k}}\}$ converges strongly to a point $x\in H$.
Let X be a Banach space, C a closed convex subset of X, and $T:C\to C$ a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing}$. If $\{{x}_{n}\}$ is a sequence in C weakly converging to x and if $\{(IT){x}_{n}\}$ converges strongly to y, then $(IT)x=y$.
Lemma 1.4 [13]
Let H be a Hilbert space and $\{{w}_{n}\}$ a sequence in H such that there exists a nonempty set $S\subseteq H$ satisfying the following:

(i)
for every $w\in S$, ${lim}_{n\to \mathrm{\infty}}\parallel {w}_{n}w\parallel $ exists;

(ii)
any weakcluster point of the sequence $\{{w}_{n}\}$ belongs to S.
Then there exists $\tilde{w}\in s$ such that $\{{w}_{n}\}$ weakly converges to $\tilde{w}$.
Lemma 1.5 [10]
Let $T=I\gamma {G}^{\ast}G$, where $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$ with $\rho ({G}^{\ast}G)$ being the spectral radius of the selfadjoint operator ${G}^{\ast}G$ on H, $S=C\times Q$. Then we have the following:

(1)
$\parallel T\parallel \le 1$ (i.e., T is nonexpansive) and averaged;

(2)
$Fix(T)=\{(x,y)\in H,Ax=By\}$, $Fix({P}_{S}T)=Fix({P}_{S})\cap Fix(T)=\mathrm{\Gamma}$.
2 Iterative algorithm for SEP
In this section, we establish an iterative algorithm that converges strongly to a solution of SEP (1.3).
Algorithm 2.1 For an arbitrary initial point ${w}_{0}=({x}_{0},{y}_{0})$, the sequence $\{{w}_{n}=({x}_{n},{y}_{n})\}$ is generated by the iteration:
where ${\alpha}_{n}>0$ is a sequence in $(0,1)$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$ with $\rho ({G}^{\ast}G)$ being the spectral radius of the selfadjoint operator ${G}^{\ast}G$ on H, $R:{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2}$ by
and ${R}_{1}$, ${R}_{2}$ are quasinonexpansive multivalued mappings on ${H}_{1}$, ${H}_{2}$, respectively.
To prove its convergence we need the following lemma.
Lemma 2.2 Any sequence $\{{w}_{n}\}$ generated by Algorithm (2.1) is Féjermonotone with respect to Γ, namely for every $w\in \mathrm{\Gamma}$,
provided that ${\alpha}_{n}>0$ is a sequence in $(0,1)$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$.
Proof Let ${u}_{n}=(I\gamma {G}^{\ast}G){w}_{n}$ and taking $w\in \mathrm{\Gamma}$, by Lemma 1.5, $w\in Fix({P}_{S})\cap Fix(I\gamma {G}^{\ast}G)$, $Gw=0$ and we have
On the other hand, we have
Hence, we have
It follows that $\parallel {w}_{n+1}w\parallel \le \parallel {w}_{n}w\parallel $, $\mathrm{\forall}w\in \mathrm{\Gamma}$, $n\ge 1$. □
Theorem 2.3 If $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ and ${R}_{1}$, ${R}_{2}$ are demiclosed at the origin, then the sequence $\{{w}_{n}\}$ generated by Algorithm (2.1) converges weakly to a solution of SEP (1.3). In addition, if ${R}_{1}$, ${R}_{2}$ are semicompact, then $\{{w}_{n}\}$ converges strongly to a solution of SEP (1.3).
Proof For any solution of SEP w, according to Lemma 2.2, we see that the sequence $\parallel {w}_{n}w\parallel $ is monotonically decreasing and thus converges to some positive real. Since $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ and $0<\gamma <\lambda $, by (2.2), we can obtain
Since ${v}_{n}\in R{u}_{n}$, we can get $d({u}_{n},R{u}_{n})\le \parallel {u}_{n}{v}_{n}\parallel \to 0$.
From the Féjermonotonicity of $\{{w}_{n}\}$ it follows that the sequence is bounded. Denoting by $\tilde{w}$ a weakcluster point of $\{{w}_{n}\}$ let $v=0,1,2,\dots $ be the sequence of indices, such that ${w}_{{n}_{v}}$ converges weakly to $\tilde{w}$. Then, by Lemma 1.3, we obtain $G\tilde{w}=0$, and it follows that $\tilde{w}\in Fix(I\gamma {G}^{\ast}G)$.
Since ${R}_{1}$, ${R}_{2}$ are demiclosed at the origin, it is easy to check that R is demiclosed at the origin. Now, by setting ${u}_{n}=(I\gamma {G}^{\ast}G){w}_{n}$, it follows that ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$. Since $d({u}_{n},R{u}_{n})\to 0$, and R is demiclosed at the origin, we obtain $\tilde{w}\in FixR=C\times Q$, i.e., ${P}_{S}(\tilde{w})=\tilde{w}$. That is to say, $\tilde{w}\in Fix({P}_{S})$.
Hence $\tilde{w}\in Fix({P}_{S})\cap Fix(I\gamma {G}^{\ast}G)$. By Lemma 1.5, we find that $\tilde{w}$ is a solution of SEP (1.3).
The weak convergence of the whole sequence $\{{w}_{n}\}$ holds true since all conditions of the wellknown Opial lemma (Lemma 1.4) are fulfilled with $S=\mathrm{\Gamma}$.
Moreover, if ${R}_{1}$, ${R}_{2}$ are semicompact, it is easy to prove that R is semicompact, and since $d({u}_{n},R{u}_{n})\to 0$, we get the result that there exists a subsequence of $\{{u}_{{n}_{i}}\}\subseteq \{{u}_{n}\}$ such that ${u}_{{n}_{i}}$ converges strongly to ${w}^{\ast}$. Since ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$, we have ${w}^{\ast}=\tilde{w}$ and so ${u}_{{n}_{i}}$ converges strongly to $\tilde{w}\in \mathrm{\Gamma}$. From the Féjermonotonicity of $\{{w}_{n}\}$ and $\parallel {w}_{n+1}{u}_{n}\parallel =(1{\alpha}_{n})\parallel {u}_{n}{v}_{n}\parallel \to 0$, we can find that $\parallel {w}_{n}\tilde{w}\parallel \to 0$, i.e., $\{{w}_{n}\}$ converges strongly to a solution of the SEP (1.3). □
3 Iterative algorithm for MSSEP
In this section, we establish an iterative algorithm that converges strongly to a solution of the following MSSEP (1.4) for a family quasinonexpansive multivalued mappings in infinitely dimensional Hilbert spaces.
Let ${C}_{j}=Fix{R}_{1}^{j}$, ${Q}_{j}=Fix{R}_{2}^{j}$ and ${S}_{j}={C}_{j}\times {Q}_{j}$, $j=1,2,\dots ,N$, $S={\bigcap}_{j=1}^{N}{S}_{j}$. The original problem can now be reformulated as finding $w=(x,y)\in S$ with $Gw=0$, or, more generally, minimizing the function $\parallel Gw\parallel $ over $w\in S$.
Algorithm 3.1 For an arbitrary initial point ${w}_{0}=({x}_{0},{y}_{0})$, sequence $\{{w}_{n}=({x}_{n},{y}_{n})\}$ is generated by the iteration:
where $i(n)=n(modN)+1$, ${\alpha}_{n}>0$ is a sequence in $(0,1)$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$, ${R}_{i(n)}:{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2}$ by
and ${R}_{1}^{i(n)}$, ${R}_{2}^{i(n)}$ are a family of quasinonexpansive multivalued mappings on ${H}_{1}$, ${H}_{2}$, respectively.
The proof of the following lemma is similar to Lemma 1.5, and we omit its proof.
Lemma 3.2 Let $T=I\gamma {G}^{\ast}G$, where $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$. Then we have $Fix(T)=\{(x,y)\in H,Ax=By\}$, $Fix({P}_{\bigcap {S}_{j}}T)=Fix({P}_{\bigcap {S}_{j}})\cap Fix(T)=\overline{\mathrm{\Gamma}}$ and $\bigcap Fix({P}_{{S}_{j}}T)=\bigcap [Fix({P}_{{S}_{j}})\cap Fix(T)]=\overline{\mathrm{\Gamma}}$.
To prove its convergence we also need the following lemma.
Lemma 3.3 Any sequence $\{{w}_{n}\}$ generated by Algorithm (3.1) is the Féjermonotone with respect to $\overline{\mathrm{\Gamma}}$, namely for every $w\in \overline{\mathrm{\Gamma}}$,
provided that ${\alpha}_{n}>0$ is a sequence in $(0,1)$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$.
Proof Let ${u}_{n}=(I\gamma {G}^{\ast}G){w}_{n}$ and taking $w\in \overline{\mathrm{\Gamma}}$, by Lemma 3.2, $w\in Fix({P}_{{S}_{j}})\cap Fix(I\gamma {G}^{\ast}G)$, $\mathrm{\forall}N\ge i\ge 1$, $Gw=0$ and we have
On the other hand, in the same way as in the proof of Lemma 2.2, we have
Hence, we have
It follows that $\parallel {w}_{n+1}w\parallel \le \parallel {w}_{n}w\parallel $, $\mathrm{\forall}w\in \overline{\mathrm{\Gamma}}$, $n\ge 1$. □
Theorem 3.4 If $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$, then the sequence $\{{w}_{n}\}$ generated by Algorithm (3.1) converges weakly to a solution of MSSEP (1.4). In addition, if there exists $1\le j\le N$ such that ${R}_{1}^{j}$, ${R}_{2}^{j}$ are semicompact, then $\{{w}_{n}\}$ converges strongly to a solution of MSSEP (1.4).
Proof From (3.2), and the fact that $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$, we obtain
Therefore,
Since ${v}_{n}\in {R}_{i(n)}{u}_{n}$, we get $d({u}_{n},{R}_{i(n)}{u}_{n})\le \parallel {u}_{n}{v}_{n}\parallel \to 0$.
It follows from the Féjermonotonicity of $\{{w}_{n}\}$ that the sequence is bounded. Let $\tilde{w}$ be a weakcluster point of $\{{w}_{n}\}$. Take a subsequence $\{{w}_{{n}_{k}}\}$ of $\{{w}_{n}\}$ such that ${w}_{{n}_{k}}$ converges weakly to $\tilde{w}$. Then, by Lemma 1.3, we obtain $G\tilde{w}=0$, and it follows that $\tilde{w}\in Fix(I\gamma {G}^{\ast}G)$.
Now, by setting ${u}_{n}=(I\gamma {G}^{\ast}G){w}_{n}$, it follows that ${u}_{{n}_{k}}$ converges weakly to $\tilde{w}$.
Since
we have
On the other hand,
that is,
Thus, ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n+1}{u}_{n}\parallel =0$ and ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n+j}{u}_{n}\parallel =0$ for all $j=1,2,\dots ,N$.
It follows that, for any $j=1,2,\dots ,N$,
Hence, ${lim}_{n\to \mathrm{\infty}}d({u}_{n},{R}_{j}{u}_{n})=0$ for all $j=1,2,\dots ,N$. Since ${R}_{1}^{j}$, ${R}_{2}^{j}$ are demiclosed at the origin, it is easy to check that ${R}_{j}$ is demiclosed at the origin, and it follows that $\tilde{w}\in {\bigcap}_{j=1}^{N}Fix{R}_{j}=C\times Q$, i.e., ${P}_{S}(\tilde{w})=\tilde{w}$. That is to say $\tilde{w}\in Fix({P}_{S})$. Hence $\tilde{w}\in Fix({P}_{S})\cap Fix(I\gamma {G}^{\ast}G)$. By Lemma 3.2, we get that $\tilde{w}$ is a solution of the MSSEP (1.4).
The weak convergence of the whole sequence $\{{w}_{n}\}$ holds true since all conditions of the wellknown Opial lemma (Lemma 1.4) are fulfilled with $S=\overline{\mathrm{\Gamma}}$.
Moreover, if ${R}_{1}^{j}$, ${R}_{2}^{j}$ are semicompact, it is easy to prove that ${R}_{j}$ is semicompact, since $d({u}_{n},{R}_{j}{u}_{n})\to 0$, we find that there exists a subsequence of $\{{u}_{{n}_{i}}\}\subseteq \{{u}_{n}\}$ such that ${u}_{{n}_{i}}$ converges strongly to ${w}^{\ast}$. Since ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$, we have ${w}^{\ast}=\tilde{w}$ and so ${u}_{{n}_{i}}$ converges strongly to $\tilde{w}\in \mathrm{\Gamma}$. From the Féjermonotonicity of $\{{w}_{n}\}$ and $\parallel {w}_{n+1}{u}_{n}\parallel =(1{\alpha}_{n})\parallel {u}_{n}{v}_{n}\parallel \to 0$, we can see that $\parallel {w}_{n}\tilde{w}\parallel \to 0$, i.e., $\{{w}_{n}\}$ converges strongly to a solution of the MSSEP (1.4). □
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Acknowledgements
This research was supported by NSFC Grants No. 11071279; No. 11226125; No. 11301379.
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Wu, Y., Chen, R. & Shi, L.Y. Split equality problem and multiplesets split equality problem for quasinonexpansive multivalued mappings. J Inequal Appl 2014, 428 (2014) doi:10.1186/1029242X2014428
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Keywords
 split equality problem
 iterative algorithms
 converge strongly