Let
\(x, y \in C\) and
\(z \in \mathcal{F}\). First, we show that
\(( I-d_{1}D_{1} ) \) is a nonexpansive mapping. Since
\(D_{1}\) is an
α-inverse strongly monotone mapping, we obtain
$$\begin{aligned} \bigl\Vert (I-d_{1}D_{1})x-(I-d_{1}D_{1})y \bigr\Vert ^{2}&=\Vert x-y\Vert ^{2}-2d_{1} \langle x-y,D_{1}x-D_{1}y \rangle +d _{1}^{2} \Vert D_{1}x-D_{1}y\Vert ^{2} \\ & \leq \Vert x-y \Vert ^{2} + d_{1}(d_{1}-2 \alpha)\Vert D_{1}x-D _{1}y \Vert ^{2} \leq \Vert x-y \Vert ^{2}. \end{aligned}$$
Thus
\((I-d_{1}D_{1})\) is a nonexpansive mapping. By using the same method as above, we see that
\((I-d_{2}D_{2})\) is a nonexpansive mapping. Since
\(f_{1}\) is a
ρ-inverse strongly monotone mapping and
\(f_{2}\) is a firmly nonexpansive mapping. From Lemma
3.1(1), we have
\(( T^{F_{1}}_{r} ( I-rf_{1} ) ) \) and
\(( T^{F_{2}}_{s} ( I-sf_{2} ) ) \) are nonexpansive mappings. Since
\(z\in \bigcap^{N}_{i=1} F ( T_{i} ) \) and Lemma
3.3, we have
$$ \Biggl\Vert P_{C} \Biggl( I-\lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T _{i} ) \Biggr) \Biggr) y_{n}-z \Biggr\Vert ^{2} \leq \Vert y_{n}-z \Vert ^{2}. $$
(3.7)
Since
\(z \in VI(C,D_{1})\) and
\(z \in VI(C,D_{2})\) and using the property of
\((I-d_{1}D_{1})\) and
\((I-d_{2}D_{2})\), we get
$$\begin{aligned} \Vert y_{n}-z \Vert ^{2} &= \bigl\Vert P_{C} ( I-d_{1}D_{1} ) \bigl( au_{n}+ ( 1-a ) P_{C} ( I-d_{2}D_{2} ) u_{n} \bigr) -P _{C} ( I-d_{1}D_{1} ) z \bigr\Vert ^{2} \\ &\leq a\Vert u_{n}-z\Vert ^{2}+ ( 1-a ) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2} \end{aligned}$$
(3.8)
$$\begin{aligned} &\leq \Vert u_{n}-z \Vert ^{2}. \end{aligned}$$
(3.9)
Since
\(z \in \Omega \), we have
\(z=T^{F_{1}}_{r} ( I-rf_{1} ) z\) and
\(Az=T^{F_{2}}_{s} ( I-sf_{2} ) Az\). From Lemma
3.1(2) and
\(\gamma \in (0,1/L)\), we obtain
$$\begin{aligned} \Vert u_{n}-z \Vert ^{2} &= \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) \bigl( x_{n}+ \gamma A^{*} \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n} \bigr) -T^{F_{1}}_{r} ( I-rf_{1} ) z \bigr\Vert ^{2} \\ &\leq \Vert x_{n}-z\Vert ^{2}+\gamma ( L\gamma -1 ) \bigl\Vert \bigl( T ^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n} \bigr\Vert ^{2} \end{aligned}$$
(3.10)
$$\begin{aligned} &\leq \Vert x_{n}-z\Vert ^{2}. \end{aligned}$$
(3.11)
Using the definition of
\(x_{n}\), (
3.7), (
3.9) and (
3.11), we get
$$\begin{aligned} \Vert x_{n+1}-z\Vert = {}& \Biggl\Vert \alpha_{n}(u-z) + \beta_{n}(x_{n}-z) \\ &{}+ \gamma_{n} \Biggl( P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N} _{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n}-z \Biggr) \Biggr\Vert \\ \leq {}& \alpha_{n}\Vert u-z\Vert + \beta_{n}\Vert x_{n}-z\Vert + \gamma_{n}\Vert y_{n}-z \Vert \\ \leq {}& \alpha_{n}\Vert u-z\Vert + \beta_{n}\Vert x_{n}-z\Vert + \gamma_{n}\Vert u_{n}-z \Vert \\ \leq {}& \alpha_{n}\Vert u-z\Vert + (1-\alpha_{n}) \Vert x_{n}-z\Vert . \end{aligned}$$
Using induction, we can conclude that
$$\Vert x_{n}-z\Vert \leq \max \bigl\{ \Vert u-z\Vert , \Vert x_{1}-z\Vert \bigr\} $$
for all
\(n\geq 1\). This implies that the sequence
\(\{x_{n}\}\) is bounded and so are
\(\{y_{n}\}\) and
\(\{u_{n}\}\). From Lemma
3.1 (2) and
\(\gamma \in (0,1/L)\), we obtain
$$\begin{aligned} &\Vert u_{n}-u_{n-1} \Vert ^{2} \\ &\quad = \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) \bigl( x_{n}+\gamma A^{*} \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n} \bigr) \\ &\qquad {}- T^{F_{1}}_{r} ( I-rf_{1} ) \bigl( x_{n-1}+\gamma A^{*} \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n-1} \bigr) \bigr\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x_{n-1} \Vert ^{2}+ \gamma ( \gamma L-1) \bigl\Vert \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n}- \bigl( T ^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n-1} \bigr\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x_{n-1} \Vert ^{2}. \end{aligned}$$
(3.12)
Next, we show that
\(\lim_{n \rightarrow \infty } \Vert x_{n+1} - x_{n} \Vert =0\). According to Eq. (
3.12), we have
$$\begin{aligned} &\Vert x_{n+1}-x_{n}\Vert \\ &\quad = \Biggl\Vert \Biggl( \alpha_{n}u + \beta_{n}x_{n} + \gamma_{n}P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n} \Biggr) \\ &\qquad {}- \Biggl( \alpha_{n-1}u + \beta_{n-1}x_{n-1} + \gamma_{n-1}P _{C} \Biggl( I - \lambda_{n-1} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n-1} \Biggr) \Biggr\Vert \\ &\quad \leq \vert \alpha_{n}-\alpha_{n-1}\vert \Vert u \Vert +\beta _{n}\Vert x_{n}-x_{n-1}\Vert + \vert \beta_{n}-\beta_{n-1}\vert \Vert x_{n-1} \Vert + \gamma_{n}\Vert y_{n}-y_{n-1}\Vert \\ &\qquad {}+\lambda_{n} \Biggl\Vert \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y _{n}- \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y_{n-1} \Biggr\Vert \\ &\qquad {}+ \vert \lambda_{n}-\lambda_{n-1}\vert \Biggl\Vert \Biggl( \sum^{N}_{i=1}k _{i} ( I-T_{i} ) \Biggr) y_{n-1} \Biggr\Vert \\ &\qquad {}+ \vert \gamma_{n}-\gamma_{n-1}\vert \Biggl\Vert P_{C} \Biggl( I - \lambda_{n-1} \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n-1} \Biggr\Vert \\ &\quad \leq (1-\alpha_{n})\Vert x_{n}-x_{n-1} \Vert +\vert \alpha_{n}- \alpha_{n-1}\vert \Vert u \Vert + \vert \beta_{n}-\beta_{n-1}\vert \Vert x_{n-1} \Vert \\ &\qquad {}+\lambda_{n} \Biggl\Vert \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y _{n}- \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y_{n-1} \Biggr\Vert \\ &\qquad {}+\vert \lambda_{n}-\lambda_{n-1}\vert \Biggl\Vert \Biggl( \sum^{N}_{i=1}k _{i} ( I-T_{i} ) \Biggr) y_{n-1} \Biggr\Vert \\ &\qquad {}+ \vert \gamma_{n}-\gamma_{n-1}\vert \Biggl\Vert P_{C} \Biggl( I - \lambda_{n-1} \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n-1} \Biggr\Vert \\ &\quad \leq (1-\alpha_{n})\Vert x_{n}-x_{n-1} \Vert +\vert \alpha_{n}- \alpha_{n-1}\vert M+\vert \beta_{n}-\beta_{n-1}\vert M+\lambda_{n}M \\ &\qquad {}+\vert \lambda_{n}-\lambda_{n-1}\vert M+ \vert \gamma_{n}-\gamma _{n-1}\vert M, \end{aligned}$$
where
$$\begin{aligned} M :=&\max_{n \in \mathbb{N}} \Biggl\{ \Vert u\Vert , \Vert x_{n}\Vert , \Biggl\Vert \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y_{n+1}- \Biggl( \sum^{N}_{i=1}k _{i} ( I-T_{i} ) \Biggr) y_{n} \Biggr\Vert , \\ &{} \Biggl\Vert \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y _{n} \Biggr\Vert , \Biggl\Vert P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum ^{N}_{i=1}k _{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n} \Biggr\Vert \Biggr\} . \end{aligned}$$
From condition (i), (iii), (iv) and Lemma
2.6, we have
$$ \lim_{n \rightarrow \infty } \Vert x_{n+1}-x_{n} \Vert =0. $$
(3.13)
According to Eqs. (
3.7), (
3.9) and (
3.10), we have
$$\begin{aligned} \Vert x_{n+1}-z\Vert ^{2}\leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+\gamma_{n} \Biggl\Vert P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n}-z \Biggr\Vert ^{2} \\ &{}+\beta_{n}\Vert x_{n}-z\Vert ^{2}- \beta_{n}\gamma_{n} \Biggl\Vert x_{n}-P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n} \Biggr\Vert ^{2} \\ \leq{} & \alpha_{n}\Vert u-z\Vert ^{2}+ \beta_{n}\Vert x_{n}-z\Vert ^{2}+ \gamma_{n}\Vert y_{n}-z\Vert ^{2} \\ &{}-\beta_{n}\gamma_{n} \Biggl\Vert x_{n}-P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n} \Biggr\Vert ^{2} \end{aligned}$$
(3.14)
$$\begin{aligned} \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+ \beta_{n}\Vert x_{n}-z\Vert ^{2}+ \gamma_{n}\Vert u_{n}-z\Vert ^{2} \\ &{}-\beta_{n}\gamma_{n} \Biggl\Vert x_{n}-P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n} \Biggr\Vert ^{2} \\ \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2} +\gamma _{n}\gamma ( L\gamma -1 ) \bigl\Vert \bigl( T^{F_{2}}_{s} ( I-sf _{2} ) -I \bigr) Ax_{n} \bigr\Vert ^{2} \\ &{}-\beta_{n}\gamma_{n} \Biggl\Vert x_{n}-P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n} \Biggr\Vert ^{2}. \end{aligned}$$
(3.15)
This implies that
$$\begin{aligned} &\gamma_{n}\gamma ( 1-L\gamma ) \bigl\Vert \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n} \bigr\Vert ^{2} \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+\Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z\Vert +\Vert x_{n+1}-z\Vert \bigr). \end{aligned}$$
By using condition (i) and (
3.13), we have
$$ \lim_{n\rightarrow \infty } \bigl\Vert \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax _{n} \bigr\Vert =0. $$
(3.16)
By using the same method as (
3.16), we have
$$ \lim_{n\rightarrow \infty } \Biggl\Vert x_{n}-P_{C} \Biggl( I - \lambda_{n} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n} \Biggr\Vert =0. $$
(3.17)
Let
\(M_{n} = x_{n}+\gamma A^{*} ( T^{F_{2}}_{s} ( I-sf_{2} ) -I ) Ax_{n}\). Applying the inequality (
3.11), we have
$$ \Vert M_{n}-z \Vert \leq \Vert x_{n}-z\Vert . $$
(3.18)
Using the property of inverse strongly monotone operators and (
3.18), we have
$$\begin{aligned} \Vert u_{n}-z \Vert ^{2} = {}& \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) M _{n} -T^{F_{1}}_{r} ( I-rf_{1} ) z \bigr\Vert ^{2} \\ \leq {}& \bigl\Vert ( I-rf_{1} ) M_{n} - ( I-rf_{1} ) z \bigr\Vert ^{2} \\ ={} & \Vert M_{n}-z \Vert ^{2}-2r\langle M_{n}-z, f_{1}M_{n}-f_{1}z \rangle +r^{2}\Vert f_{1}M_{n}-f_{1}z \Vert ^{2} \\ \leq{} & \Vert x_{n}-z \Vert ^{2}+r ( r-2\rho ) \Vert f_{1}M_{n}-f_{1}z \Vert ^{2}. \end{aligned}$$
(3.19)
Substituting (
3.19) in (
3.15), we have
$$\begin{aligned} \Vert x_{n+1}-z\Vert ^{2} \leq {}& \alpha_{n} \Vert u-z\Vert ^{2}+\beta_{n}\Vert x_{n}-z \Vert ^{2} \\ &{}+\gamma_{n} \bigl( \Vert x_{n}-z \Vert ^{2}+r ( r-2\rho ) \Vert f_{1}M_{n}-f_{1}z \Vert ^{2} \bigr) \\ \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2}+\gamma _{n}r ( r-2\rho ) \Vert f_{1}M_{n}-f_{1}z \Vert ^{2}. \end{aligned}$$
That is,
$$\gamma_{n}r ( 2\rho -r ) \Vert f_{1}M_{n}-f_{1}z \Vert ^{2} \leq \alpha_{n}\Vert u-z\Vert ^{2}+\Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z\Vert +\Vert x_{n+1}-z\Vert \bigr). $$
According to condition (i) and (
3.13), we get
$$ \lim_{n \rightarrow \infty } \Vert f_{1}M_{n}-f_{1}z \Vert = 0. $$
(3.20)
By the property of firmly nonexpansive mappings, we have
$$\begin{aligned} \Vert u_{n}-z \Vert ^{2} = {}& \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) M _{n} -T^{F_{1}}_{r} ( I-rf_{1} ) z \bigr\Vert ^{2} \\ \leq {}& \bigl\langle u_{n}-z,(I-rf_{1})M_{n}-(I-rf_{1})z \bigr\rangle \\ = {}& \frac{1}{2} \bigl( \Vert u_{n}-z\Vert ^{2}+ \bigl\Vert (I-rf_{1})M_{n}-(I-rf_{1})z \bigr\Vert ^{2} \\ &{}- \bigl\Vert (u_{n}-z)- \bigl( (I-rf_{1})M_{n}-(I-rf_{1})z \bigr) \bigr\Vert ^{2} \bigr). \end{aligned}$$
(3.21)
That is,
$$\begin{aligned} \Vert u_{n}-z \Vert ^{2} \leq {}& \bigl\Vert (I-rf_{1})M_{n}-(I-rf_{1})z \bigr\Vert ^{2}- \bigl\Vert (u_{n}-M_{n})+r ( f_{1}M_{n}-f_{1}z ) \bigr\Vert ^{2} \\ \leq {}& \Vert M_{n}-z\Vert ^{2}- \bigl( \Vert u_{n}-M_{n}\Vert ^{2}+2r\langle u_{n}-M _{n}, f_{1}M_{n}-f_{1}z \rangle \\ &{}+ r^{2}\Vert f_{1}M_{n}-f_{1}z \Vert ^{2} \bigr) \\ \leq {}& \Vert M_{n}-z\Vert ^{2}-\Vert u_{n}-M_{n}\Vert ^{2}+2r\Vert u_{n}-M_{n}\Vert \Vert f_{1}M_{n}-f_{1}z \Vert \\ &{}- r^{2}\Vert f_{1}M_{n}-f_{1}z \Vert ^{2}. \end{aligned}$$
(3.22)
Substituting (
3.22) in (
3.15), we get
$$\begin{aligned} \Vert x_{n+1}-z\Vert ^{2} \leq {}& \alpha_{n} \Vert u-z\Vert ^{2}+\beta_{n}\Vert x_{n}-z \Vert ^{2}+\gamma_{n} \bigl( \Vert M_{n}-z \Vert ^{2}-\Vert u_{n}-M_{n}\Vert ^{2} \\ &{}+ 2r\Vert u_{n}-M_{n}\Vert \Vert f_{1}M_{n}-f_{1}z\Vert -r^{2}\Vert f_{1}M_{n}-f_{1}z\Vert ^{2} \bigr) \\ \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2}-\gamma _{n}\Vert u_{n}-M_{n}\Vert ^{2} \\ &{}+ 2r\gamma_{n}\Vert u_{n}-M_{n}\Vert \Vert f_{1}M_{n}-f_{1}z\Vert . \end{aligned}$$
It follows that
$$\begin{aligned} \gamma_{n}\Vert u_{n}-M_{n}\Vert ^{2} \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+ \Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z \Vert +\Vert x_{n+1}-z\Vert \bigr) \\ &{}+ 2r\gamma_{n}\Vert u_{n}-M_{n}\Vert \Vert f_{1}M_{n}-f_{1}z\Vert . \end{aligned}$$
From condition (i), (
3.13) and (
3.20), we ensure that
$$ \lim_{n \rightarrow \infty } \Vert u_{n}-M_{n} \Vert = 0. $$
(3.23)
From (
3.16) and (
3.23), we also have
$$\begin{aligned} \Vert u_{n}-x_{n}\Vert &\leq \Vert u_{n}-M_{n}\Vert +\Vert M_{n}-x_{n} \Vert \\ &= \Vert u_{n}-M_{n}\Vert + \bigl\Vert x_{n}+\gamma A^{*} \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n}-x_{n} \bigr\Vert \\ &\leq \Vert u_{n}-M_{n}\Vert +\gamma \Vert A\Vert \bigl\Vert \bigl( T^{F_{2}}_{s} ( I-sf_{2} ) -I \bigr) Ax_{n} \bigr\Vert . \end{aligned}$$
Then we have
$$ \lim_{n \rightarrow \infty } \Vert u_{n}-x_{n} \Vert = 0. $$
(3.24)
By using the same method as (
3.19), we have
$$ \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2} \leq \Vert x_{n}-z \Vert ^{2}+d _{2}(d_{2}-2\beta)\Vert D_{2}u_{n}-D_{2}z\Vert ^{2}. $$
(3.25)
Substituting (
3.8) and (
3.25) in (
3.14), we have
$$\begin{aligned} \Vert x_{n+1}-z\Vert ^{2} \leq{} & \alpha_{n} \Vert u-z\Vert ^{2}+\beta_{n}\Vert x_{n}-z \Vert ^{2}+\gamma_{n} \bigl( a\Vert u_{n}-z \Vert ^{2} \\ &{}+ ( 1-a ) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u _{n}-z \bigr\Vert ^{2} \bigr) \\ \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2} \\ &{}+ \gamma_{n} ( 1-a ) d_{2}(d_{2}-2\beta) \Vert D_{2}u_{n}-D_{2}z\Vert ^{2}. \end{aligned}$$
We can conclude that
$$\begin{aligned} &\gamma_{n} ( 1-a )d_{2}(2\beta -d_{2})\Vert D_{2}u_{n}-D_{2}z\Vert ^{2} \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+\Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z\Vert +\Vert x_{n+1}-z\Vert \bigr). \end{aligned}$$
According to condition (i) and (
3.13), we get
$$ \lim_{n \rightarrow \infty } \Vert D_{2}u_{n}-D_{2}z \Vert = 0. $$
(3.26)
Since
\(P_{C}\) is a firmly nonexpansive mapping and using the same method as (
3.21), we get
$$\begin{aligned} & \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2} \\ &\quad \leq \frac{1}{2} \bigl( \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2}+ \bigl\Vert ( I-d_{2}D_{2} ) u_{n}- ( I-d_{2}D_{2} ) z \bigr\Vert ^{2} \\ &\qquad {}- \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z- ( I-d_{2}D_{2} ) u_{n}+ ( I-d_{2}D_{2} ) z \bigr\Vert ^{2} \bigr). \end{aligned}$$
That is,
$$\begin{aligned} \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2} \leq {}& \Vert u_{n}-z \Vert ^{2}- \bigl\Vert \bigl(P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr)+d_{2} ( D_{2}u_{n}-D_{2}z ) \bigr\Vert ^{2} \\ \leq {}& \Vert x_{n}-z\Vert ^{2}- \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert ^{2} \\ &{}+ 2d_{2} \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert \Vert D_{2}u_{n}-D_{2}z \Vert \\ &{}- d_{2}^{2}\Vert D_{2}u_{n}-D_{2}z \Vert ^{2}. \end{aligned}$$
(3.27)
Substituting (
3.8) and (
3.27) in (
3.14), we have
$$\begin{aligned} &\Vert x_{n+1}-z\Vert ^{2} \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+ \beta_{n}\Vert x_{n}-z\Vert ^{2}+ \gamma_{n} \bigl( a\Vert u_{n}-z\Vert ^{2}+ ( 1-a ) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-z \bigr\Vert ^{2} \bigr) \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+ \beta_{n}\Vert x_{n}-z\Vert ^{2}+ \gamma_{n} \bigl( a\Vert x_{n}-z\Vert ^{2}+ ( 1-a ) \bigl( \Vert x_{n}-z\Vert ^{2} \\ &\qquad {}- \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert ^{2} + 2d_{2} \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert \Vert D_{2}u_{n}-D_{2}z \Vert \\ &\qquad {}- d_{2}^{2}\Vert D_{2}u_{n}-D_{2}z \Vert ^{2} \bigr) \bigr) \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2}-\gamma _{n}(1-a) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert ^{2} \\ &\qquad {}+ 2d_{2}\gamma_{n}(1-a) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert \Vert D_{2}u_{n}-D_{2}z\Vert . \end{aligned}$$
Therefore
$$\begin{aligned} &\gamma_{n}(1-a) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert ^{2} \\ & \quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+\Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z\Vert + \Vert x_{n+1}-z\Vert \bigr) \\ &\qquad {}+ 2d_{2}\gamma_{n}(1-a) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert \Vert D_{2}u_{n}-D_{2}z\Vert . \end{aligned}$$
From condition (i), (
3.13) and (
3.26), we get
$$ \lim_{n \rightarrow \infty } \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u _{n}-u_{n} \bigr\Vert = 0. $$
(3.28)
Let
\(k_{n} = au_{n}+ ( 1-a ) P_{C} ( I-d_{2}D_{2} ) u _{n}\). By using the same method as (
3.19), we have
$$ \Vert y_{n}-z\Vert ^{2} \leq \Vert x_{n}-z\Vert ^{2}+d_{1}(d_{1}-2 \alpha)\Vert D_{1}k_{n}-D_{1}z\Vert ^{2}. $$
(3.29)
Substituting (
3.29) in (
3.14), we have
$$\begin{aligned} &\Vert x_{n+1}-z\Vert ^{2} \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+ \beta_{n}\Vert x_{n}-z\Vert ^{2}+ \gamma_{n} \bigl( \Vert x_{n}-z\Vert ^{2}+d_{1}(d_{1}-2\alpha)\Vert D_{1}k_{n}-D_{1}z\Vert ^{2} \bigr) \\ &\quad \leq \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2}+d_{1}(d _{1}-2\alpha)\gamma_{n}\Vert D_{1}k_{n}-D_{1}z \Vert ^{2}. \end{aligned}$$
This implies that
$$d_{1}(2\alpha -d_{1})\gamma_{n}\Vert D_{1}k_{n}-D_{1}z\Vert ^{2} \leq \alpha _{n}\Vert u-z\Vert ^{2}+\Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z\Vert +\Vert x_{n+1}-z\Vert \bigr). $$
According to condition (i) and (
3.13), we have
$$ \lim_{n \rightarrow \infty } \Vert D_{1}k_{n}-D_{1}z \Vert = 0. $$
(3.30)
By using the same method as (
3.21), we have
$$\begin{aligned} \Vert y_{n}-z\Vert ^{2} \leq {}& \frac{1}{2} \bigl( \Vert y_{n}-z\Vert ^{2}+ \bigl\Vert ( I-d_{1}D_{1} ) k_{n}- ( I-d_{1}D_{1} ) z \bigr\Vert ^{2} \\ &{}- \bigl\Vert (y_{n}-k_{n})+d_{1}(D_{1}k_{n}-D_{1}z) \bigr\Vert ^{2} \bigr). \end{aligned}$$
That is,
$$\begin{aligned} \Vert y_{n}-z\Vert ^{2} \leq {}& \Vert k_{n}-z\Vert ^{2}- \bigl( \Vert y_{n}-k_{n} \Vert ^{2}+2d _{1}\langle y_{n}-k_{n},D_{1}k_{n}-D_{1}z \rangle \\ &{}+ d_{1}^{2}\Vert D_{1}k_{n}-D_{1}z \Vert ^{2} \bigr) \\ \leq {}& \Vert x_{n}-z\Vert ^{2}-\Vert y_{n}-k_{n}\Vert ^{2}+2d_{1}\Vert y_{n}-k_{n}\Vert \Vert D_{1}k_{n}-D_{1}z \Vert \\ &{}- d_{1}^{2}\Vert D_{1}k_{n}-D_{1}z \Vert ^{2}. \end{aligned}$$
(3.31)
Substituting (
3.31) in (
3.14), we have
$$\begin{aligned} \Vert x_{n+1}-z\Vert ^{2}\leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+\beta_{n}\Vert x_{n}-z\Vert ^{2}+\gamma_{n} \bigl( \Vert x_{n}-z\Vert ^{2}-\Vert y_{n}-k_{n} \Vert ^{2} \\ &{}+ 2d_{1}\Vert y_{n}-k_{n}\Vert \Vert D_{1}k_{n}-d_{1}z\Vert -d_{1}^{2} \Vert D_{1}k_{n}-D_{1}z \Vert ^{2} \bigr) \\ \leq{} & \alpha_{n}\Vert u-z\Vert ^{2}+(1- \alpha_{n})\Vert x_{n}-z\Vert ^{2}-\gamma _{n}\Vert y_{n}-k_{n}\Vert ^{2} \\ &{}+ 2\gamma_{n}d_{1}\Vert y_{n}-k_{n} \Vert \Vert D_{1}k_{n}-D_{1}z\Vert . \end{aligned}$$
(3.32)
This implies that
$$\begin{aligned} \gamma_{n}\Vert y_{n}-k_{n}\Vert ^{2} \leq {}& \alpha_{n}\Vert u-z\Vert ^{2}+ \Vert x_{n}-x_{n+1}\Vert \bigl( \Vert x_{n}-z \Vert +\Vert x_{n+1}-z\Vert \bigr) \\ &{}+ 2\gamma_{n}d_{1}\Vert y_{n}-k_{n} \Vert \Vert D_{1}k_{n}-D_{1}z\Vert . \end{aligned}$$
According to condition (i), (
3.13) and (
3.30), we get
$$ \lim_{n \rightarrow \infty } \Vert y_{n}-k_{n} \Vert = 0. $$
(3.33)
From (
3.28) and (
3.33)
$$\begin{aligned} \Vert y_{n}-u_{n}\Vert \leq {}& \Vert y_{n}-k_{n}\Vert +\Vert k_{n}-u_{n} \Vert \\ \leq {}& \Vert y_{n}-k_{n}\Vert +(1-a) \bigl\Vert P_{C} ( I-d_{2}D_{2} ) u_{n}-u_{n} \bigr\Vert , \end{aligned}$$
we conclude that
$$ \lim_{n \rightarrow \infty } \Vert y_{n}-u_{n} \Vert = 0. $$
(3.34)
By (
3.24) and (
3.34), we also conclude that
$$ \lim_{n \rightarrow \infty } \Vert y_{n}-x_{n} \Vert = 0. $$
(3.35)
To show this, choose a subsequence
\(\{ x_{n_{j}} \} \) of
\(\{ x_{n} \} \) such that
$$ \limsup_{n \rightarrow \infty } \langle u-z, x_{n} - z \rangle = \lim_{j \rightarrow \infty } \langle u-z, x_{n _{j}} - z \rangle. $$
(3.36)
Without loss of generality, we may assume that
\(x_{n_{j}} \rightharpoonup \omega \) as
\(j \rightarrow \infty \). From (
3.35), we obtain
\(y_{n_{j}} \rightharpoonup \omega \) as
\(j \rightarrow \infty \). From Lemma
2.3, we have
\(VI ( C,D_{1} ) = F ( P_{C}(I-d _{1}D_{1}) ) \). Assume that
\(\omega \notin VI ( C,D_{1} ) \), we have
\(\omega \neq P_{C}(I-d _{1}D_{1})\omega \). Using Opial’s condition, (
3.33), we obtain
$$\begin{aligned} \liminf_{j \rightarrow \infty } \Vert y_{n_{j}} - \omega \Vert < {}& \liminf_{j \rightarrow \infty } \bigl\Vert y_{n_{j}} - P_{C}(I-d_{1}D _{1})\omega \bigr\Vert \\ \leq{} & \liminf_{j \rightarrow \infty } \bigl( \bigl\Vert P_{C}(I-d_{1}D _{1})k_{n_{j}} - P_{C}(I-d_{1}D_{1})y_{n_{j}} \bigr\Vert \\ &{}+ \bigl\Vert P_{C}(I-d_{1}D_{1})y_{n_{j}} - P_{C}(I-d_{1}D_{1}) \omega \bigr\Vert \bigr) \\ \leq {}& \liminf_{j \rightarrow \infty } \bigl( \Vert k_{n_{j}} - y _{n_{j}}\Vert + \Vert y_{n_{j}} - \omega \Vert \bigr) \\ \leq{} & \liminf_{j \rightarrow \infty } \Vert y_{n_{j}} - \omega \Vert . \end{aligned}$$
This is a contradiction, so we have
$$ \omega \in VI ( C,D_{1} ). $$
(3.37)
From (
3.24), we have
\(u_{n_{j}} \rightharpoonup \omega \) as
\(j \rightarrow \infty \). By (
3.28) and using the same method as (
3.37), we obtain
$$ \omega \in VI ( C,D_{2} ). $$
(3.38)
Next, we show that
\(\omega \in \bigcap^{N}_{i=1} F ( T_{i} ) \). From Lemma
2.5, we have
$$\bigcap^{N}_{i=1} F ( T_{i} ) = F \Biggl( P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) \Biggr). $$
Assume that
\(\omega \notin \bigcap^{N}_{i=1} F ( T_{i} ) \), and that
\(\omega \neq P_{C} ( I - \lambda_{n_{j}} ( \sum^{N}_{i=1}k _{i} ( I-T_{i} ) ) ) \omega \). Using Opial’s condition, (
3.17) and (
3.35), we obtain
$$\begin{aligned} & \liminf_{j \rightarrow \infty } \Vert x_{n_{j}} - \omega \Vert \\ &\quad < \liminf_{j \rightarrow \infty } \Biggl\Vert x_{n_{j}} - P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) \omega \Biggr\Vert \\ &\quad \leq \liminf_{j \rightarrow \infty } \Biggl( \Biggl\Vert x_{n_{j}} - P _{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y _{n_{j}} \Biggr\Vert \\ &\qquad {}+ \Biggl\Vert P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) y_{n_{j}} - P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) x _{n_{j}} \Biggr\Vert \\ &\qquad {}+ \Biggl\Vert P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum^{N}_{i=1}k _{i} ( I-T_{i} ) \Biggr) \Biggr) x_{n_{j}} - P_{C} \Biggl( I - \lambda_{n_{j}} \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \Biggr) \omega \Biggr\Vert \Biggr) \\ &\quad \leq \liminf_{j \rightarrow \infty } \Biggl( \Vert y_{n_{j}}-x _{n_{j}} \Vert +\lambda_{n_{j}} \Biggl\Vert \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) y_{n_{j}} - \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) x_{n_{j}} \Biggr\Vert \\ &\qquad {}+ \Vert x_{n_{j}} - \omega \Vert +\lambda_{n_{j}} \Biggl\Vert \Biggl( \sum^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) x_{n_{j}}- \Biggl( \sum ^{N}_{i=1}k_{i} ( I-T_{i} ) \Biggr) \omega \Biggr\Vert \Biggr) \\ &\quad \leq \liminf_{j \rightarrow \infty } \Vert x_{n_{j}} - \omega \Vert . \end{aligned}$$
This is a contradiction, so we have
$$ \omega \in \bigcap^{N}_{i=1} F ( T_{i} ). $$
(3.39)
After that, we show that
\(\omega \in \Omega \). Assume
\(\omega \notin EP(F_{1},f_{1})\). Since
\(EP(F_{1},f_{1})=F(T^{F_{1}}_{r} ( I-rf _{1} ))\), we obtain
\(\omega \neq T^{F_{1}}_{r} ( I-rf_{1} ) \omega \). Using Opial’s condition and (
3.23), we get
$$\begin{aligned} \liminf_{j \rightarrow \infty } \Vert u_{n_{j}} - \omega \Vert < {}& \liminf_{j \rightarrow \infty } \bigl\Vert u_{n_{j}} - T^{F_{1}}_{r} ( I-rf_{1} ) \omega \bigr\Vert \\ \leq {}& \liminf_{j \rightarrow \infty } \bigl( \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) M_{n_{j}} - T^{F_{1}}_{r} ( I-rf_{1} ) u _{n_{j}} \bigr\Vert \\ &{}+ \bigl\Vert T^{F_{1}}_{r} ( I-rf_{1} ) u_{n_{j}} - T ^{F_{1}}_{r} ( I-rf_{1} ) \omega \bigr\Vert \bigr) \\ \leq {}& \liminf_{j \rightarrow \infty } \bigl( \Vert M_{n_{j}} - u _{n_{j}}\Vert +\Vert u_{n_{j}} - \omega \Vert \bigr) \\ \leq {}& \liminf_{j \rightarrow \infty } \Vert u_{n_{j}} - \omega \Vert . \end{aligned}$$
This is a contradiction, so we have
$$ \omega \in EP(F_{1},f_{1}). $$
(3.40)