Theorem 2.1
Let
\(H_{1}\)
and
\(H_{2}\)
be Hilbert spaces, and let
N
and
M
be set-valued maximal monotone mappings on
\(H_{1}\)
and
\(H_{2}\), respectively. Let
\(S_{i}\)
be nonexpansive mappings on
\(H_{1}\)
for all integers
\(i\geq 1\). Let
\(F:H_{1}\rightarrow H_{1}\)
be an
\(\mathcal{L}\)-Lipschitz continuous and
τ-strongly monotone mapping. Let
A
be a linear bounded operator from
\(H_{1}\)
to
\(H_{2}\), and let
\(A^{*}\)
be its adjoint operator. Assume that
\(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\cap \operatorname{SIP}(M,N) \neq \emptyset \). Let
\(\{ x_{n}\}\)
be a vector sequence in
\(H_{1}\)
generated by the iterative process
$$ \textstyle\begin{cases} x_{1}\in H_{1}, \\ y_{n}=\gamma _{n}\operatorname{Res}_{s_{n}}^{N}(x_{n}+\gamma A^{*}( \operatorname{Res}_{r_{n}}^{M}-I)Ax_{n})+(1-\gamma _{n})x_{n}, \\ x_{n+1}=\beta _{n}(I-\mu \alpha _{n}F)W_{n}y_{n}+(1-\beta _{n})x_{n},\quad n\geq 1, \end{cases} $$
where
γ
and
μ
are two positive real numbers, \(\{s_{n}\}\)
and
\(\{r_{n}\}\)
are two positive real number sequences, \(\{\alpha _{n} \}\), \(\{\beta _{n}\}\), and
\(\{\gamma _{n}\}\)
are real number sequences in
\((0,1)\). Suppose that
\(\gamma \in (0,\frac{1}{\|A\|^{2}})\), \(\mu \in (0, \frac{2 \tau }{\mathcal{L}^{2}})\), \(\liminf_{n\rightarrow \infty } s_{n}> 0\), \(\lim_{n\rightarrow \infty }|s_{n}-s_{n+1}|<\infty \), \(\liminf_{n\rightarrow \infty }r_{n}> 0\), \(\lim_{n\rightarrow \infty }|r_{n}-r_{n+1}|<\infty \), \(\sum_{n=1}^{ \infty }\alpha _{n}=\infty \), \(\{\beta _{n}\}\)
is number sequence in
\([\bar{\beta },\bar{\beta }']\), where
β̄
and
\(\bar{\beta }'\)
are two real numbers in
\((0,1)\), such that
\(\lim_{n\rightarrow \infty }|\beta _{n+1}- \beta _{n}|=0\), and
\(\{\gamma _{n}\}\)
is a sequence in
\([\bar{\gamma },1]\), where
\(\bar{\gamma }\in (0,1]\), such that
\(\lim_{n\rightarrow \infty }| \gamma _{n+1}- \gamma _{n}|=0\). Then the sequence
\(\{ x_{n}\}\)
converges strongly to
\(\widetilde{x}\in H_{1}\), which is a unique solution of the variational inequality
$$ \langle \widetilde{x} -y, F\widetilde{x} \rangle \leq 0,\quad \forall y\in \bigcap _{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N) . $$
Proof
The proof is split into four steps.
Step 1. We prove that \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\).
For any fixed \(p\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N)\), we conclude \(Ap=\operatorname{Res}_{r _{n}}^{M}Ap\), \(p=\operatorname{Res}_{s_{n}}^{N}p\), and \(p=S_{i}p\) for each \(i\geq 1\). Since Ap is a fixed point of \(\operatorname{Res} _{r_{n}}^{M}\) and \(\operatorname{Res}_{r_{n}}^{M}\) is a (firmly) nonexpansive mapping, we have
$$ \bigl\langle \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}, \operatorname{Res} _{r_{n}}^{M}Ax_{n}-Ap \bigr\rangle \leq \frac{ \Vert \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \Vert ^{2}}{2}. $$
(2.1)
Putting
$$ z_{n}=\operatorname{Res}_{s_{n}}^{N} \bigl(x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}\bigr), $$
(2.1) sends us to
$$ \begin{aligned}[b] & \Vert z_{n}-p \Vert ^{2} \\ &\quad \leq \bigl\Vert \gamma A^{*}\bigl(\operatorname{Res}_{r_{n}} ^{M}-I\bigr)Ax_{n}+(x_{n}-p) \bigr\Vert ^{2} \\ &\quad \leq \gamma ^{2} \Vert A \Vert ^{2} \bigl\Vert \bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2}+2\gamma \bigl\langle A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n},x_{n}-p \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\ &\quad =\gamma \bigl(\gamma \Vert A \Vert ^{2}-2\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}} ^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2} \\ &\qquad {}+2\gamma \bigl\langle \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax _{n}, \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ap \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\ &\quad \leq \gamma \bigl(\gamma \Vert A \Vert ^{2}-1\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}} ^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}, \end{aligned} $$
(2.2)
which leads to
$$ \begin{aligned}[b] \Vert y_{n}-p \Vert ^{2} &\leq \gamma _{n} \Vert z_{n}-p \Vert ^{2}+(1-\gamma _{n}) \Vert x_{n}-p \Vert ^{2} \\ &\leq \Vert x_{n}-p \Vert ^{2}-\gamma _{n}\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}).\end{aligned} $$
(2.3)
The restriction imposed on parameter γ tells us that \(\|y_{n}-p\|\leq \|x_{n}-p\|\). Since \(W_{n}\) is a nonexpansive mapping for each n, we find from Lemma 1.3 that
$$ \begin{aligned} \Vert x_{n+1}-p \Vert &\leq \beta _{n} \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)p-\mu \alpha _{n}Fp \bigr\Vert +(1- \beta _{n}) \Vert x_{n}-p \Vert \\ & \leq \beta _{n} \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)p \bigr\Vert + \mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \beta _{n}(1-\tau \alpha _{n}) \Vert W_{n}y_{n}-W_{n}p \Vert +\mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \beta _{n}(1-\tau \alpha _{n}) \Vert y_{n}-p \Vert +\mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \tau \alpha _{n}\beta _{n} \frac{ \Vert Fp \Vert \mu }{\tau }+(1- \tau \alpha _{n}\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \max \biggl\{ \frac{ \Vert Fp \Vert \mu }{ \tau }, \Vert x_{n}-p \Vert \biggr\} ,\end{aligned} $$
from which we conclude that \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\).
Step 2. We prove that \(\lim_{n\rightarrow \infty }\|x_{n+1}-x_{n}\|=0\). From resolvent equality (1.6) in Lemma 1.6 we see that
$$\begin{aligned} \Vert z_{n}-z_{n+1} \Vert \leq& \bigl\Vert \operatorname{Res}_{s_{n}}^{N} \rho _{n}- \operatorname{Res}_{s_{n+1}}^{N}\rho _{n} \bigr\Vert + \bigl\Vert \operatorname{Res}_{s _{n+1}}^{N} \rho _{n}-\operatorname{Res}_{s_{n+1}}^{N}\rho _{n+1} \bigr\Vert \\ \leq& \bigl\Vert \operatorname{Res}_{s_{n}}^{N}\rho _{n}-\operatorname{Res}_{s _{n+1}}^{N}\rho _{n} \bigr\Vert + \Vert \rho _{n}-\rho _{n+1} \Vert \\ = &\biggl\Vert \operatorname{Res}_{s_{n+1}}^{N} \biggl( \frac{s_{n+1}}{s_{n}}\rho _{n}+\biggl(1-\frac{s _{n+1}}{s_{n}}\biggr) \operatorname{Res}_{s_{n}}^{N}\rho _{n} \biggr)- \operatorname{Res}_{s_{n+1}}^{N}\rho _{n} \biggr\Vert + \Vert \rho _{n}-\rho _{n+1} \Vert \\ =& \biggl\Vert \biggl(\frac{s_{n+1}}{s_{n}}\rho _{n}+\biggl(1- \frac{s_{n+1}}{s_{n}}\biggr) \operatorname{Res}_{s_{n}}^{N} \rho _{n} \biggr)-\rho _{n} \biggr\Vert + \Vert \rho _{n}- \rho _{n+1} \Vert \\ \leq& \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert + \Vert \rho _{n+1}-\rho _{n} \Vert , \end{aligned}$$
(2.4)
where
$$ \rho _{n}=x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n}. $$
It is easy to see that
$$ \begin{aligned} & \bigl\Vert (x_{n+1}-x_{n})- \gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert \\ &\quad = \sqrt{ \Vert x _{n+1}-x_{n} \Vert ^{2}-2\gamma \bigl\langle x_{n+1}-x_{n}, A^{*} (Ax_{n+1}-Ax _{n})\bigr\rangle + \bigl\Vert \gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert ^{2}} \\ &\quad = \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) \Vert x_{n+1}-x_{n} \Vert ,\end{aligned} $$
which sends us to
$$ \begin{aligned}[b] &\Vert \rho _{n+1}-\rho _{n} \Vert \\ &\quad \leq \gamma \bigl\Vert A^{*} \bigl( \operatorname{Res} _{r_{n+1}}^{M}Ax_{n+1}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr) \bigr\Vert + \bigl\Vert (x _{n+1}-x_{n})-\gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert \\ &\quad \leq \Vert x_{n+1}-x _{n} \Vert +\gamma \Vert A \Vert \biggl\vert 1-\frac{r_{n+1}}{r_{n}} \biggr\vert \Big| \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert .\end{aligned} $$
(2.5)
Inequalities (2.4) and (2.5) yield
$$ \begin{aligned} \Vert z_{n}-z_{n+1} \Vert \leq{}& \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert + \Vert x_{n+1}-x_{n} \Vert \\ &{}+\gamma \Vert A \Vert \bigg|1-\frac{r_{n+1}}{r_{n}}\bigg| \Big| \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax _{n} \bigr\Vert , \end{aligned} $$
which further leads us to
$$\begin{aligned} \Vert y_{n}-y_{n+1} \Vert \leq& \gamma _{n} \Vert z_{n}-z_{n+1} \Vert +(1-\gamma _{n}) \Vert x _{n}-x_{n+1} \Vert + \vert \gamma _{n}-\gamma _{n+1} \vert \Vert z_{n+1}-x_{z+1} \Vert \\ \leq& \gamma _{n} \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s _{n}}^{N} \rho _{n} \bigr\Vert + \Vert x_{n+1}-x_{n} \Vert \\ &{} +\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r _{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ &{} + | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert . \end{aligned}$$
From Lemma 1.1 we arrive at
$$\begin{aligned}& \Vert W_{n+1}y_{n+1}-W_{n}y_{n} \Vert \\& \quad \leq \Vert W_{n+1}y_{n+1}-W_{n}y_{n+1} \Vert + \Vert W _{n}y_{n+1}-W_{n}y_{n} \Vert \\& \quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+ \Vert y_{n+1}-y_{n} \Vert \\& \quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+\gamma _{n} \biggl\vert 1-\frac{s _{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert \\& \qquad {} + \Vert x_{n+1}-x_{n} \Vert +\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1-\frac{r_{n+1}}{r _{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\& \qquad {}+ | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert , \end{aligned}$$
(2.6)
where Ψ is a bounded set containing \(\{y_{n}\}\). Inequality (2.6) ensures that
$$ \begin{aligned} & \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert \\ &\quad \leq \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1}-(I- \mu \alpha _{n+1}F)W _{n}y_{n} \bigr\Vert \\ &\qquad {} + \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)W _{n}y_{n} \bigr\Vert \\ &\quad \leq (1-\tau \alpha _{n+1}) \Vert W_{n+1}y_{n+1}- W_{n}y _{n} \Vert + \vert \alpha _{n+1}- \alpha _{n} \vert \Vert \mu FW_{n}y_{n} \Vert \\ &\quad \leq (1-\tau \alpha _{n+1}) \Vert W_{n+1}y_{n+1}- W_{n}y_{n} \Vert + \vert \alpha _{n+1}- \alpha _{n} \vert \Vert \mu FW_{n}y_{n} \Vert \\ &\quad \leq (1-\tau \alpha _{n+1}){ \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr] \\ &\qquad {}+ (1-\tau \alpha _{n+1}) \gamma _{n} \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s _{n}}^{N}\rho _{n} \bigr\Vert \\ & \qquad {}+ (1-\tau \alpha _{n+1}) \Vert x_{n+1}-x_{n} \Vert +(1-\tau \alpha _{n+1})\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r_{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ &\qquad {} +(1-\tau \alpha _{n+1}) | \gamma _{n}- \gamma _{n+1} | \Vert z_{n+1}-x _{z+1} \Vert + | \alpha _{n+1}-\alpha _{n}| \Vert \mu FW_{n}y_{n} \Vert . \end{aligned} $$
This further leads to
$$ \begin{aligned} & \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert - \Vert x_{n+1}-x_{n} \Vert \\ &\quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+ \gamma _{n} \biggl\vert 1-\frac{s _{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert \\ & \qquad {}+ \gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r_{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ & \qquad {}+ | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert +\bigl( \vert \alpha _{n+1} \vert + \vert \alpha _{n} \vert \bigr) \Vert \mu FW_{n}y _{n} \Vert .\end{aligned} $$
Using Lemma 1.2, the boundedness of operator A, and the restrictions on the parameter sequences \(\{\alpha _{n}\}\), \(\{\gamma _{n}\}\), \(\{s_{n}\}\), and \(\{r_{n}\}\), we obtain that
$$ \limsup_{n\rightarrow \infty } \bigl( \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y _{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert - \Vert x_{n+1}-x_{n} \Vert \bigr)\leq 0. $$
With the aid of Lemma 1.7, we conclude that
$$ \lim_{n\rightarrow \infty } \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-x_{n} \bigr\Vert =0. $$
(2.7)
Since \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty \), we also have
$$ \lim_{n\rightarrow \infty } \Vert W_{n}y_{n}-x_{n} \Vert =0. $$
(2.8)
From (2.7) we see that
$$ \lim_{n\rightarrow \infty } \Vert x_{n+1}-x_{n} \Vert =0. $$
(2.9)
Since \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\), we find that there is a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) that converges weakly to x̄.
Step 3. We prove that \(x\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S _{i})\cap \operatorname{SIP} (M,N)\).
Put
$$ \varphi _{n}=(I-\mu \alpha _{n}F)W_{n}y_{n}. $$
For any \(p\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\cap \operatorname{SIP} (M,N)\), we conclude from (2.3) that
$$\begin{aligned} \Vert \varphi _{n}-p \Vert ^{2} \leq &\bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)W_{n}p \bigr\Vert ^{2}-2 \mu \alpha _{n}\langle \varphi _{n}-p,Fp \rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert W_{n}y_{n}-W_{n}p \Vert ^{2}-2 \mu \alpha _{n}\langle \varphi _{n}-p,Fp\rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert y_{n}-p \Vert ^{2}-2\mu \alpha _{n} \langle \varphi _{n}-p,Fp \rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert x_{n}-p \Vert ^{2}-\gamma \bigl(1- \gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2} \bigl\Vert \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \bigr\Vert ^{2} \\ &{} +2\mu \alpha _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert . \end{aligned}$$
This shows us that
$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\beta _{n} \Vert \varphi _{n}-p \Vert ^{2}+(1-\beta _{n}) \Vert x_{n}-p \Vert ^{2} \\ \leq &\Vert x_{n}-p \Vert ^{2}-\beta _{n}\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2} \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax _{n} \bigr\Vert ^{2} \\ &{} +2\mu \alpha _{n}\beta _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert . \end{aligned}$$
It follows that
$$ \begin{aligned} &\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2}\beta _{n} \bigl\Vert Ax _{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) +2\mu \alpha _{n}\beta _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert .\end{aligned} $$
Limit (2.9) and the fact that \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty \) lead us to
$$ \lim_{n\rightarrow \infty } \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax _{n} \bigr\Vert =0. $$
(2.10)
Next, we have
$$\begin{aligned}& 2 \Vert z_{n}-p \Vert ^{2} \\& \quad \leq 2\bigl\langle \gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I\bigr)Ax_{n}+x_{n}-p, z_{n}-p\bigr\rangle \\& \quad = \gamma ^{2} \bigl\Vert A ^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2} +2\gamma \bigl\langle A ^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n},x_{n}-p \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\& \qquad {} - \bigl\Vert x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax _{n}-y_{n} \bigr\Vert ^{2}+ \Vert z_{n}-p \Vert ^{2} \\& \quad \leq \gamma ^{2} \Vert A \Vert ^{2} \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax_{n} \bigr\Vert ^{2} \\& \qquad {}+2\gamma \bigl(\bigl\langle \operatorname{Res}_{r_{n}}^{M} Ax_{n}-Ap, \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \bigr\rangle - \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax _{n} \bigr\Vert ^{2}\bigr) \\& \qquad {} + \Vert z_{n}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2} - \Vert z_{n}-x _{n} \Vert ^{2} -2\gamma \bigl\langle A^{*}\bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax _{n},x_{n}-z_{n} \bigr\rangle \\& \qquad {} - \bigl\Vert \gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}) \\& \quad \leq \Vert x_{n}-p \Vert ^{2}+ \Vert z_{n}-p \Vert ^{2}+2 \Vert A \Vert \gamma \Vert x_{n}-z_{n} \Vert \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax _{n}-Ax_{n} \bigr\Vert - \Vert x_{n}-z_{n} \Vert ^{2}, \end{aligned}$$
that is,
$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2} \leq \Vert x_{n}-p \Vert ^{2}+2 \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax_{n}-Ax_{n} \bigr\Vert - \Vert x_{n}-z_{n} \Vert ^{2}. \end{aligned} $$
This sends us to
$$\begin{aligned} \Vert \varphi _{n}-p \Vert ^{2} \leq& (1-\tau \alpha _{n})^{2} \Vert W_{n}y_{n}-W_{n}p \Vert ^{2}-2\mu \alpha _{n}\langle \varphi _{n}-p,Fp\rangle \\ \leq& (1- \tau \alpha _{n})^{2} \gamma _{n} \Vert z_{n}-p \Vert ^{2}+(1-\tau \alpha _{n})^{2}(1- \gamma _{n}) \Vert x_{n}-p \Vert ^{2} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert \\ \leq& (1-\tau \alpha _{n})^{2} \Vert x_{n}-p \Vert ^{2}+2(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax _{n}-Ax_{n} \bigr\Vert \\ &{} -(1-\tau \alpha _{n})^{2}\gamma _{n} \Vert x_{n}-z _{n} \Vert ^{2} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$
It follows that
$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\beta _{n} \Vert \varphi _{n}-p \Vert ^{2}+(1-\beta _{n}) \Vert x_{n}-p \Vert ^{2} \\ \leq& \Vert x_{n}-p \Vert ^{2}+2\beta _{n}(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax _{n}-Ax_{n} \bigr\Vert \\ &{} -\beta _{n}(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert x _{n}-z_{n} \Vert ^{2} +2\mu \alpha _{n}\beta _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$
Hence
$$\begin{aligned}& \beta _{n}(1-\tau \alpha _{n})^{2}\gamma _{n} \Vert x_{n}-z_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) +2 \Vert A \Vert \gamma \Vert z _{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax_{n}-Ax_{n} \bigr\Vert \\& \qquad {} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$
Using (2.9) and (2.10), we have that \(x_{n}-z_{n}\rightarrow 0\) as \(n\rightarrow \infty \), that is,
$$ \lim_{n\rightarrow \infty } \bigl\Vert x_{n}- \operatorname{Res}_{s_{n}}^{N}\bigl(x _{n}+ \gamma A^{*}\bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}\bigr) \bigr\Vert =0. $$
(2.11)
Since \(x_{n}-z_{n}\rightarrow 0\) as \(n\rightarrow \infty \), we have that \(\{z_{n}\}\) converges weakly to x̄. Further, \(\{z_{n_{i}}\}\) converges weakly to x̄ as \(i\rightarrow \infty \). The graphs of maximal monotone mappings are weakly-strongly closed. Observe that
$$ \frac{x_{n_{i}}-z_{n_{i}}}{s_{n_{i}}}+\gamma A^{*} \frac{ \operatorname{Res}_{r_{n_{i}}}^{M}Ax_{n_{i}}-Ax_{n_{i}}}{s_{n_{i}}} \in Nz_{n_{i}}. $$
So \(0\in N(\bar{x})\). Fixing a positive real number p, Lemma 1.6 yields that \(\|Ax_{n_{i}}-\operatorname{Res}_{p}^{M}Ax _{n_{i}}\|\rightarrow \) as \(i\rightarrow \infty \), which implies \(0\in M(A\bar{x})\).
We are now in a position to show that \(\bar{x}\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})=\operatorname{Fix}(W)\). We have
$$\begin{aligned} \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert \leq& \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert + \Vert W_{n_{i}}y_{n_{i}}-Wy_{n_{i}} \Vert \\ \leq& \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert +\sup _{x\in \varPsi } \Vert W_{n_{i}}x-Wx \Vert . \end{aligned}$$
Relations (2.8) and (2.11) yield that \(\lim_{i\to \infty }\|y_{n_{i}}-W_{n_{i}}y_{n_{i}}\|=0\). If \(\bar{x}\neq W\bar{x}\), then the Opial condition, Lemma 1.5, sends us to
$$\begin{aligned} \limsup_{i\to \infty } \Vert \bar{x}-y_{n_{i}} \Vert < & \limsup_{i\to \infty } \Vert W \bar{x}-y_{n_{i}} \Vert \\ \leq& \limsup_{i\to \infty }\bigl\{ \Vert Wy_{n_{i}}-y _{n_{i}} \Vert + \Vert W\bar{x}-Wy_{n_{i}} \Vert \bigr\} \\ \leq& \limsup_{i\to \infty } \Vert \bar{x}-y_{n_{i}} \Vert , \end{aligned}$$
a contradiction. Thus \(\bar{x}\in \operatorname{Fix}(W)\), that is, \(\bar{x}\in \bigcap^{\infty }_{i=1} \operatorname{Fix}(S_{i})\).
Step 4. We prove that the sequence \(\{x_{n}\}\) is strongly convergent.
Since F is strongly monotone and Lipschitz continuous, we get that the following variational inequality has a unique solution:
$$ \langle \widetilde{x} -y, F\widetilde{x}\rangle \leq 0,\quad \forall y\in \bigcap _{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N) . $$
Thus
$$ \limsup_{n\rightarrow \infty }\langle \widetilde{x}-\varphi _{n},F \widetilde{x}\rangle \leq 0. $$
(2.12)
Lemma 1.1 and Lemma 1.3 send us to
$$\begin{aligned}& \Vert x_{n+1}-\widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \Vert \varphi _{n}- \widetilde{x} \Vert ^{2} + (1-\beta _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \bigl( \bigl\Vert (I-\mu \alpha _{n} F)W_{n}y_{n}-(I-\mu \alpha _{n} F)\widetilde{x} \bigr\Vert ^{2} -2\mu \alpha _{n}\bigl\langle \varphi _{n}- \widetilde{x}, F( \widetilde{x})\bigr\rangle \bigr)+ (1-\beta _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \bigl( (1-\tau \alpha _{n})^{2} \Vert y_{n}- \bar{x} \Vert ^{2}-2 \mu \alpha _{n} \bigl\langle F (\widetilde{x}), \varphi _{n}- \widetilde{x} \bigr\rangle \bigr) + (1-\beta _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2} \\& \quad \leq (1-2\tau \beta _{n}\alpha _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2}+2\tau \beta _{n}\alpha _{n}\varPi _{n}, \end{aligned}$$
where \(\varPi =\frac{\mu }{\tau } \langle F \bar{x}, \widetilde{x}- \varphi _{n} \rangle +\frac{\tau \alpha _{n}}{2}\|x_{n}-\widetilde{x}\| ^{2}\). In light of Lemma 1.4, we find that \(\| x_{n}- \widetilde{x}\| \rightarrow 0\) as \(n\rightarrow \infty \). This completes the proof. □
From Theorem 2.1 we have the following subresult on split inclusion problem (1.3).
Corollary 2.1
Let
\(H_{1}\)
and
\(H_{2}\)
be Hilbert spaces, and let
N
and
M
be set-valued maximal monotone mappings on
\(H_{1}\)
and
\(H_{2}\), respectively. Let
\(F:H_{1}\rightarrow H_{1}\)
be an
\(\mathcal{L}\)-Lipschitz continuous and
τ-strongly monotone mapping. Let
A
be a linear bounded operator from
\(H_{1}\)
to
\(H_{2}\), and let
\(A^{*}\)
be its the adjoint operator. Assume that
\(\operatorname{SIP}(M,N) \neq \emptyset \). Let
\(\{ x_{n}\}\)
be the vector sequence in
\(H_{1}\)
generated by the iterative process
$$ x_{1}\in H_{1},\quad x_{n+1}=\beta _{n}(I-\mu \alpha _{n}F) \operatorname{Res}_{s_{n}}^{N} \bigl(x_{n}+\gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I\bigr)Ax_{n}\bigr)+(1-\beta _{n})x_{n},\quad n\geq 1, $$
where
γ
and
μ
are two positive real numbers, \(\{s_{n}\}\)
and
\(\{r_{n}\}\)
are two positive real number sequences, and
\(\{\alpha _{n}\}\)
and
\(\{\beta _{n}\}\)
are real number sequences in
\((0,1)\). Suppose that
\(\gamma \in (0,\frac{1}{\|A\|^{2}})\), \(\mu \in (0, \frac{2 \tau }{\mathcal{L}^{2}})\), \(\liminf_{n\rightarrow \infty } s_{n}> 0\), \(\lim_{n\rightarrow \infty }|s_{n}-s_{n+1}|<\infty \), \(\liminf_{n\rightarrow \infty }r_{n}> 0\), \(\lim_{n\rightarrow \infty }|r_{n}-r_{n+1}|<\infty \), \(\sum_{n=1}^{ \infty }\alpha _{n}=\infty \), \(\{\beta _{n}\}\)
is a number sequence in
\([\bar{\beta },\bar{\beta }']\), where
β̄
and
\(\bar{\beta }'\)
are two real numbers in
\((0,1)\), such that
\(\lim_{n\rightarrow \infty }|\beta _{n+1}- \beta _{n}|=0\). Then the sequence
\(\{ x_{n}\}\)
converges strongly to
\(\widetilde{x}\in H_{1}\), which is a unique solution of the variational inequality
\(\langle \widetilde{x} -y, F\widetilde{x} \rangle \leq 0\), \(\forall y\in \operatorname{SIP}(M,N)\).
Remark 2.1
In this paper, we investigated the descent iterative methods for split inclusion problem with a common fixed point constraint of an infinite family of nonexpansive mappings. It deserves mentioning that our method does not involve projections. A solution theorem of the problem was established in the framework of Hilbert spaces under some weak assumptions imposed on different mappings and control sequences.