In this section, we propose a formal statement of our present algorithm. Review the multiple-sets split equality problem (MSSEP), without loss of generality, suppose \(t>r\) in (1.3) and define \(Q_{r+1}=Q_{r+2}=\cdots =Q_{t}=H_{2}\). Hence, MSSEP (1.3) is equivalent to the following problem:
$$\begin{aligned} \mbox{find}\quad x\in{\bigcap_{i=1}^{t}}C_{i}\quad \mbox{and}\quad y\in{\bigcap_{j=1}^{t}}Q_{j} \quad\mbox{such that } Ax=By. \end{aligned}$$
(3.1)
Moreover, set \(S_{i}=C_{i}\times{Q_{i}}\in{H}=H_{1}\times{H_{2}}\ (i=1,2,\ldots ,t)\), \(S={\bigcap_{i=1}^{t}}S_{i}\), \(G=[A,-B]:H\rightarrow{H_{3}}\), the adjoint operator of G is denoted by \(G^{*}\), then the original problem (3.1) reduces to
$$\begin{aligned} \mbox{finding}\quad w=(x,y)\in{S}\quad \mbox{such that } Gw=0. \end{aligned}$$
(3.2)
Theorem 3.1
Let
\(\Omega\neq\emptyset\)
be the solution set of MSSEP (3.2). For an arbitrary initial point
\(w_{0}\in{S}\), the iterative sequence
\(\{w_{n}\}\)
can be given as follows:
$$\begin{aligned} \textstyle\begin{cases} v_{n}=P_{S}\{(1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}\},\\ w_{n+1}=P_{S}\{w_{n}-\mu_{n}G^{*}Gv_{n}+\lambda_{n}(v_{n}-w_{n})\}, \end{cases}\displaystyle \end{aligned}$$
(3.3)
where
\(\{\alpha_{n}\}_{n=0}^{\infty}\)
is a sequence in
\([0,1]\)
such that
\(\lim_{n\rightarrow\infty}\alpha_{n}=0, and \sum_{n=1}^{\infty}\alpha _{n}=\infty\), and
\(\{\gamma_{n}\}_{n=0}^{\infty}\), \(\{\lambda_{n}\}_{n=0}^{\infty}\), \(\{\mu_{n}\}_{n=0}^{\infty}\)
are sequences in
H
satisfying the following conditions:
$$\begin{aligned} \textstyle\begin{cases} {\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}, \qquad\lim_{n\rightarrow\infty }(\gamma_{n+1}-\gamma_{n})=0;\\ \lambda_{n}\in{(0,1)},\qquad \lim_{n\rightarrow\infty}(\lambda_{n+1}-\lambda _{n})=0;\\ {\mu_{n}}\leq\frac{2}{\rho(G^{*}G)}\lambda_{n}, \qquad\lim_{n\rightarrow\infty}(\mu _{n+1}-\mu_{n})=0;\\ \sum_{n=1}^{\infty}(\frac{\gamma_{n}}{\lambda_{n}})< \infty. \end{cases}\displaystyle \end{aligned}$$
(3.4)
Then
\(\{w_{n}\}\)
converges strongly to a solution of MSSEP (3.2).
Proof
In view of the property of the projection, we infer \(\hat{w}=P_{S}(\hat {w}-tG^{*}G\hat{w})\) for any \(t>0\). Further, from the condition in (3.4), we get that \({\mu_{n}}\leq\frac{2}{\rho(G^{*}G)}\lambda_{n}\), it follows that \(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\) is nonexpansive. Hence,
$$\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl\{ w_{n}-\mu_{n}G^{*}Gv_{n}+ \lambda_{n}(v_{n}-w_{n})\bigr\} -P_{S} \bigl\{ \hat {w}-tG^{*}G\hat{w}\bigr\} \bigr\Vert \\ &\quad=\biggl\Vert P_{S}\biggl\{ (1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}\biggr\} -P_{S}\biggl\{ (1-\lambda_{n})\hat{w} +\lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)\hat{w}\biggr\} \biggr\Vert \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\biggl\Vert \biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G \biggr)v_{n}-\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)\hat{w}\biggr\Vert \\ &\quad \leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\Vert v_{n}-\hat{w}\Vert . \end{aligned}$$
(3.5)
Since \(\alpha_{n}\rightarrow{0}\) as \(n\rightarrow\infty\) and from the condition in (3.4), \({\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}\), it follows that \(\alpha_{n}\leq{1-\frac{\gamma_{n}\rho(G^{*}G)}{2}}\) as \(n\rightarrow\infty\), that is, \(\frac{\gamma_{n}}{1-\alpha_{n}}\in{(0,\frac {2}{\rho{(G^{*}G)}})}\). We deduce that
$$\begin{aligned} & \Vert v_{n}-\hat{w}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl\{ (1-\alpha_{n})w_{n}- \gamma_{n}G^{*}Gw_{n}\bigr\} -P_{S}\bigl(\hat{w}-tG^{*}G \hat {w}\bigr)\bigr\Vert \\ &\quad\leq(1-\alpha_{n}) \biggl(w_{n}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}Gw_{n} \biggr)-\biggl\{ \alpha _{n}\hat{w}+(1-\alpha_{n}) \biggl( \hat{w}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr\} \\ &\quad\leq\biggl\Vert -\alpha_{n}\hat{w}+(1-\alpha_{n}) \biggl[w_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}G^{*}Gw_{n}-\hat{w}+ \frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr]\biggr\Vert , \end{aligned}$$
(3.6)
which is equivalent to
$$\begin{aligned} \Vert v_{n}-\hat{w}\Vert \leq\alpha_{n}\Vert {-\hat{w}} \Vert +(1-\alpha_{n})\Vert w_{n}-\hat{w}\Vert . \end{aligned}$$
(3.7)
Substituting (3.7) in (3.5), we obtain
$$\begin{aligned} \Vert w_{n}-\hat{w}\Vert &\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\bigl(\alpha_{n}\Vert {-\hat{w}}\Vert +(1- \alpha_{n})\Vert w_{n}-\hat{w}\Vert \bigr) \\ &\leq(1-\lambda_{n}\alpha_{n})\Vert w_{n}- \hat{w}\Vert +\lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert \\ &\leq\max\bigl\{ \Vert w_{n}-\hat{w}\Vert ,\Vert {-\hat{w}}\Vert \bigr\} . \end{aligned}$$
By induction,
$$\Vert w_{n}-\hat{w}\Vert \leq{\max\bigl\{ \Vert w_{0}- \hat{w}\Vert ,\Vert {-\hat{w}}\Vert \bigr\} }. $$
Consequently, \(\{w_{n}\}\) is bounded, and so is \(\{v_{n}\}\).
Let \(T=2P_{S}-I\). From Proposition 2.2, one can know that the projection operator \(P_{S}\) is monotone and nonexpansive, and \(2P_{S}-I\) is nonexpansive.
Therefore,
$$\begin{aligned} w_{n+1} =&\frac{I+T}{2}\biggl[(1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(1-\frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr] \\ =&\frac{I-\lambda_{n}}{2}w_{n}+\frac{\lambda_{n}}{2}\biggl(I- \frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}+\frac{T}{2}\biggl[(1- \lambda_{n})w_{n}+\lambda_{n}\biggl(I- \frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr], \end{aligned}$$
that is,
$$\begin{aligned} w_{n+1}=\frac{1-\lambda_{n}}{2}w_{n}+\frac{1+\lambda_{n}}{2}b_{n}, \end{aligned}$$
(3.8)
where \(b_{n}=\frac{\lambda_{n}(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G)v_{n}+T[(1-\lambda_{n})w_{n}+\lambda_{n}(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G)v_{n}]}{1+\lambda_{n}}\).
Indeed,
$$\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq \frac{\lambda_{n+1}}{1+\lambda_{n+1}}\biggl\Vert \biggl(I-\frac{\mu_{n+1}}{\lambda _{n+1}}G^{*}G \biggr)v_{n+1}-\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert +\biggl\vert \frac{\lambda _{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \\ &\qquad{}\times \biggl\Vert \biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert +\frac{T}{1+\lambda _{n+1}}\biggl\{ (1-\lambda_{n+1})w_{n+1}+ \lambda_{n+1}\biggl(I-\frac{\mu _{n+1}}{\lambda_{n+1}}G^{*}G\biggr)v_{n+1} \\ & \qquad{}-\biggl[(1-\lambda_{n})w_{n}+\lambda_{n} \biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr]\biggr\} +\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \\ & \qquad{}\times \biggl\Vert T\biggl[(1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}\biggr] \biggr\Vert . \end{aligned}$$
(3.9)
For convenience, let \(c_{n}=(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G)v_{n}\). By Lemma 2.5 in Shi et al. [1], it follows that \((I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G)\) is nonexpansive and averaged. Hence,
$$\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq\frac{\lambda_{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert +\biggl\vert \frac {\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \Vert c_{n} \Vert \\ & \qquad{}+\frac{T}{1+\lambda_{n+1}}\bigl\{ (1-\lambda_{n+1})w_{n+1}+\lambda _{n+1}c_{n+1}-\bigl[(1-\lambda_{n})w_{n}+ \lambda_{n}c_{n}\bigr]\bigr\} \\ & \qquad{}+\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1-\lambda _{n})w_{n}+\lambda_{n}c_{n} \bigr]\bigr\Vert \\ &\quad\leq\frac{\lambda_{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert +\biggl\vert \frac {\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \Vert c_{n} \Vert \\ &\qquad{}+\frac{1-\lambda_{n+1}}{1+\lambda_{n+1}}\Vert w_{n+1}-w_{n}\Vert + \frac{\lambda _{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert + \frac{\lambda_{n}-\lambda _{n+1}}{1+\lambda_{n+1}}\Vert w_{n}\Vert \\ & \qquad{}+\frac{\lambda_{n+1}-\lambda_{n}}{1+\lambda_{n+1}}\Vert c_{n}\Vert +\biggl\vert \frac {1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1- \lambda_{n})w_{n}+\lambda _{n}c_{n}\bigr] \bigr\Vert . \end{aligned}$$
(3.10)
Moreover,
$$\begin{aligned} & \Vert c_{n+1}-c_{n}\Vert \\ &\quad=\biggl\Vert \biggl(I-\frac{\mu_{n+1}}{\lambda_{n+1}}G^{*}G\biggr)v_{n+1}- \biggl(I-\frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert \\ &\quad\leq \Vert v_{n+1}-v_{n}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl[(1-\alpha_{n+1})w_{n+1}- \gamma_{n}G^{*}Gw_{n+1}\bigr]-P_{S}\bigl[(1-\alpha _{n})w_{n}-\gamma_{n}G^{*}Gw_{n}\bigr]\bigr\Vert \\ &\quad\leq\bigl\Vert \bigl(I-\gamma_{n+1}G^{*}G\bigr)w_{n+1}- \bigl(I-\gamma_{n+1}G^{*}G\bigr)w_{n}+(\gamma _{n}- \gamma_{n+1})G^{*}Gw_{n}\bigr\Vert \\ & \qquad{}+\alpha_{n+1}\Vert {-w_{n+1}}\Vert +\alpha_{n} \Vert w_{n}\Vert \\ &\quad\leq \Vert w_{n+1}-w_{n}\Vert +\vert \gamma_{n}-\gamma_{n+1}\vert \bigl\Vert G^{*}Gw_{n} \bigr\Vert +\alpha_{n+1}\Vert { -w_{n+1}}\Vert + \alpha_{n}\Vert w_{n}\Vert . \end{aligned}$$
(3.11)
Substituting (3.11) in (3.10), we infer that
$$\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq\biggl\vert \frac{\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda _{n}}\biggr\vert \Vert c_{n}\Vert +\frac{\lambda_{n}-\lambda_{n+1}}{1+\lambda_{n+1}}\Vert w_{n}\Vert + \frac {\lambda_{n+1}-\lambda_{n}}{1+\lambda_{n+1}}\Vert c_{n}\Vert \\ & \qquad{}+\Vert w_{n+1}-w_{n}\Vert +\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1- \lambda_{n})w_{n}+\lambda_{n}c_{n}\bigr] \bigr\Vert \\ & \qquad{}+\vert \gamma_{n}-\gamma_{n+1}\vert \Vert w_{n}\Vert +\alpha_{n+1}\Vert {-w_{n+1}}\Vert + \alpha_{n}\Vert w_{n}\Vert . \end{aligned}$$
(3.12)
By virtue of \(\lim_{n\rightarrow\infty}(\lambda_{n+1}-\lambda_{n})=0\), it follows that \(\lim_{n\rightarrow\infty} \vert \frac{\lambda_{n+1}}{1+\lambda_{n+1}}-\frac {\lambda_{n}}{1+\lambda_{n}}\vert =0\). Moreover, \(\{w_{n}\}\) and \(\{v_{n}\}\) are bounded, and so is \(\{c_{n}\}\). Therefore, (3.12) reduces to
$$\begin{aligned} \lim\sup_{n\rightarrow\infty}\bigl(\Vert b_{n+1}-b_{n} \Vert -\Vert w_{n+1}-w_{n}\Vert \bigr)\leq {0}. \end{aligned}$$
(3.13)
Applying (3.13) and Lemma 2.4, we get
$$\begin{aligned} \lim_{n\rightarrow\infty} \Vert b_{n}-w_{n}\Vert =0. \end{aligned}$$
(3.14)
Combining (3.14) with (3.8), we obtain
$$\lim_{n\rightarrow\infty} \Vert x_{n+1}-x_{n}\Vert =0. $$
Using the convexity of the norm and (3.5), we deduce that
$$\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\biggl\Vert -\alpha_{n}\hat {w}\\ &\qquad{}+(1-\alpha_{n})\biggl[w_{n}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}Gw_{n}- \biggl(\hat{w}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr]\biggr\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}\\ &\qquad{}+(1-\alpha_{n})\lambda_{n}\biggl[\Vert w_{n}-\hat{w}\Vert ^{2} +\frac{\gamma_{n}}{1-\alpha_{n}}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}-\frac {2}{\rho(G^{*}G)}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert ^{2}\biggr] \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}+ \lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}+ \lambda_{n}\gamma _{n}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}- \frac{2}{\rho(G^{*}G)}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat {w}\bigr\Vert ^{2}, \end{aligned}$$
which implies that
$$\begin{aligned} & \lambda_{n}\gamma_{n}\biggl(\frac{2}{\rho(G^{*}G)}- \frac{\gamma_{n}}{1-\alpha_{n}}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert ^{2} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}-\Vert w_{n+1}-\hat{w}\Vert ^{2}+\lambda_{n} \alpha_{n}\Vert {-\hat {w}}\Vert ^{2} \\ &\quad \leq \Vert w_{n+1}-w_{n}\Vert \bigl(\Vert w_{n}-\hat{w}\Vert +\Vert w_{n+1}-\hat{w}\Vert \bigr)+ \lambda _{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}. \end{aligned}$$
Since \(\lim\inf_{n\rightarrow\infty}\lambda_{n}\gamma_{n}(\frac{2}{\rho (G^{*}G)}-\frac{\gamma_{n}}{1-\alpha_{n}})>0\), \(\lim_{n\rightarrow\infty }\alpha_{n}=0\) and \(\lim_{n\rightarrow\infty} \Vert w_{n+1}-w_{n}\Vert =0\), we infer that
$$\begin{aligned} \lim_{n\rightarrow\infty}\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert =0. \end{aligned}$$
(3.15)
Applying Proposition 2.2 and the property of the projection \(P_{S}\), one can easily show that
$$\begin{aligned} & \Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad=\bigl\Vert P_{S}\bigl[(1-\alpha_{n})w_{n}- \gamma_{n}G^{*}Gw_{n}\bigr]-P_{S}\bigl[\hat{w}-\gamma _{n}G^{*}G\hat{w}\bigr]\bigr\Vert ^{2} \\ &\quad\leq\bigl\langle {(1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr),v_{n}-\hat{w}}\bigr\rangle \\ &\quad=\frac{1}{2}\bigl\{ \bigl\Vert w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr)-\alpha_{n}w_{n} \bigr\Vert ^{2}+\Vert v_{n}-\hat{w}\Vert ^{2} \\ & \qquad{}-\bigl\Vert (1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr)-v_{n}+\hat{w}\bigr\Vert ^{2}\bigr\} \\ &\quad\leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+2\alpha_{n}\Vert {-w_{n}}\Vert \bigl\Vert w_{n}-\gamma _{n}G^{*}Gw_{n}-\bigl(\hat{w}- \gamma_{n}G^{*}G\hat{w}\bigr)-\alpha_{n}w_{n}\bigr\Vert \\ & \qquad{}+\Vert v_{n}-\hat{w}\Vert ^{2}-\bigl\Vert w_{n}-v_{n}-\gamma_{n}G^{*}G(w_{n}- \hat{w})-\alpha_{n}w_{n}\bigr\Vert ^{2}\bigr\} \\ &\quad \leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+\alpha_{n}M+\Vert v_{n}-\hat{w}\Vert ^{2}-\Vert w_{n}-v_{n}\Vert ^{2} \\ &\qquad{}+2 \gamma_{n}\bigl\langle {w_{n}-v_{n},G^{*}G(w_{n}- \hat{w})}\bigr\rangle \\ & \qquad{}+2\alpha_{n}\langle{w_{n},w_{n}-v_{n}} \rangle-\bigl\Vert \gamma_{n}G^{*}G(w_{n}-\hat {w})+ \alpha_{n}w_{n}\bigr\Vert ^{2}\bigr\} \\ &\quad\leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+\alpha_{n}M+\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\qquad{}-\Vert w_{n}-v_{n}\Vert ^{2}+2 \gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & \qquad{}+2\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert \bigr\} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}+ \alpha_{n}M-\Vert w_{n}-v_{n}\Vert ^{2}+4\gamma_{n}\Vert w_{n}-v_{n} \Vert \bigl\Vert G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & \qquad{}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert , \end{aligned}$$
(3.16)
where \(M>0\) satisfies
$$M\geq{\sup_{k}\bigl\{ 2\Vert {-w_{n}}\Vert \bigl\Vert w_{n}-\gamma_{n}G^{*}Gw_{n}-\bigl(\hat{w}- \gamma_{n}G^{*}G\hat {w}\bigr)-\alpha_{n}w_{n}\bigr\Vert \bigr\} }. $$
From (3.5) and (3.16), we get
$$\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}- \lambda_{n}\Vert w_{n}-v_{n}\Vert ^{2}+\alpha_{n}M+4\gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert \gamma_{n}G^{*}G(w_{n}- \hat{w})\bigr\Vert \\ & \qquad{}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert , \end{aligned}$$
which means that
$$\begin{aligned} \lambda_{n}\Vert w_{n}-v_{n}\Vert ^{2}\leq{}&\Vert w_{n+1}-w_{n}\Vert \bigl(\Vert w_{n}-\hat{w}\Vert +\Vert w_{n+1}-\hat{w}\Vert \bigr)+ \alpha_{n}M \\ & {}+4\gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert \gamma_{n}G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & {}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert . \end{aligned}$$
Since \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\lim_{n\rightarrow\infty }\Vert w_{n+1}-w_{n}\Vert =0\) and \(\lim_{n\rightarrow\infty} \Vert G^{*}Gw_{n}-G^{*}G\hat {w}\Vert =0\), we infer that
$$\lim_{n\rightarrow\infty} \Vert w_{n}-v_{n}\Vert =0. $$
Finally, we show that \(w_{n}\rightarrow{\hat{w}}\). Using the property of the projection \(P_{S}\), we derive
$$\begin{aligned} & \Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad=\biggl\Vert P_{S}\biggl[(1-\alpha_{n}) \biggl(w_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}G^{*}Gw_{n}\biggr) \biggr]\\ &\qquad{}-P_{S}\biggl[\alpha_{n}\hat{w}+(1- \alpha_{n}) \biggl(\hat{w}-\frac{\gamma _{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr] \biggr\Vert ^{2} \\ &\quad\leq\biggl\langle (1-\alpha_{n}) \biggl(I-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G \biggr) (w_{n}-\hat {w})-\alpha_{n}\hat{w},v_{n}- \hat{w}\biggr\rangle \\ &\quad\leq(1-\alpha_{n})\Vert w_{n}-\hat{w}\Vert \Vert v_{n}-\hat{w}\Vert +\alpha_{n}\langle\hat {w}, \hat{w}-v_{n}\rangle \\ &\quad\leq\frac{1-\alpha_{n}}{2}\bigl(\Vert w_{n}-\hat{w}\Vert ^{2}+\Vert v_{n}-\hat{w}\Vert ^{2}\bigr)+\alpha _{n}\langle\hat{w},\hat{w}-v_{n}\rangle, \end{aligned}$$
which equals
$$\begin{aligned} \Vert v_{n}-\hat{w}\Vert ^{2}\leq\frac{1-\alpha_{n}}{1+\alpha_{n}} \Vert w_{n}-\hat{w}\Vert ^{2}+\frac{2\alpha_{n}}{1-\alpha_{n}}\langle \hat{w},\hat{w}-v_{n}\rangle. \end{aligned}$$
(3.17)
It follows from (3.5) and (3.17) that
$$\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\biggl\{ \frac{1-\alpha _{n}}{1+\alpha_{n}}\Vert w_{n}-\hat{w}\Vert ^{2}+\frac{2\alpha_{n}}{1-\alpha_{n}}\langle\hat {w}, \hat{w}-v_{n}\rangle\biggr\} \\ &\quad\leq\biggl(1-\frac{2\alpha_{n}\lambda_{n}}{1+\alpha_{n}}\biggr)\Vert w_{n}-\hat{w}\Vert ^{2}+\frac {2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle\hat{w},\hat{w}-v_{n}\rangle. \end{aligned}$$
(3.18)
Since \(\frac{\gamma_{n}}{1-\alpha_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}\), we observe that \(\alpha_{n}\in{(0,1-\frac{\gamma_{n}\rho(G^{*}G)}{2})}\), then
$$\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\in{\biggl(0,\frac{2\lambda_{n}(2-\gamma _{n}\rho(G^{*}G))}{\gamma_{n}\rho(G^{*}G)}\biggr)}, $$
that is to say,
$$\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle\hat{w},\hat{w}-v_{n}\rangle \leq \frac{2\lambda_{n}(2-\gamma_{n}\rho(G^{*}G))}{\gamma_{n}\rho(G^{*}G)}\langle \hat{w},\hat{w}-v_{n}\rangle. $$
By virtue of \(\sum_{n=1}^{\infty}(\frac{\lambda_{n}}{\gamma_{n}})<\infty\), \({\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}\) and \(\langle\hat{w},\hat {w}-v_{n}\rangle\) is bounded, we obtain \(\sum_{n=1}^{\infty}(\frac{2\lambda _{n}(2-\gamma_{n}\rho(G^{*}G))}{\gamma_{n}\rho_{n}(G^{*}G)})\langle\hat{w},\hat {w}-v_{n}\rangle<\infty\), which implies that
$$\sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle \hat {w},\hat{w}-v_{n}\rangle\leq\infty. $$
Moreover,
$$\begin{aligned} \sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle \hat {w},\hat{w}-v_{n}\rangle=\sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda _{n}}{1+\alpha_{n}}\frac{1+\alpha_{n}}{1-\alpha_{n}}\langle\hat{w},\hat {w}-v_{n} \rangle, \end{aligned}$$
(3.19)
it follows that all the conditions of Lemma 2.5 are satisfied. Combining (3.18), (3.19) and Lemma 2.5, we can show that \(w_{n}\rightarrow \hat{w}\). This completes the proof. □