Let
\(p\in{ \varOmega }\). In terms of the firm nonexpansivity of
\(P_{C}\) and the
\(\zeta_{j}\)-inverse-strong monotonicity of
\(F_{j}\) for
\(j=1,2\), we obtain from
\(\nu_{j}\in(0,2\zeta_{j})\),
\(j=1,2\), and (
3.4)
$$\begin{aligned} \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} =& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})p \bigr\Vert ^{2} \\ \leq& \bigl\langle (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p,\tilde{y}_{n,N}-\tilde{p} \bigr\rangle \\ =&\frac{1}{2} \bigl[ \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p \bigr\Vert ^{2}+\Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} \\ &{} - \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p-(\tilde{y}_{n,N}-\tilde{p}) \bigr\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} \\ &{}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-\nu_{2}(F_{2}y_{n,N}-F_{2}p)-(p- \tilde{p}) \bigr\Vert ^{2} \bigr] \\ =&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{2} \bigl\langle (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}),F_{2}y_{n,N}-F_{2}p \bigr\rangle - \nu^{2}_{2}\Vert F_{2}y_{n,N}-F_{2}p \Vert ^{2} \bigr] \end{aligned}$$
and
$$\begin{aligned} \Vert z_{n}-p\Vert ^{2} =& \bigl\Vert P_{C}(I-\nu_{1}F_{1})\tilde{y}_{n,N}-P_{C}(I- \nu_{1}F_{1})\tilde{p} \bigr\Vert ^{2} \\ \leq& \bigl\langle (I-\nu_{1}F_{1})\tilde{y}_{n,N}-(I- \nu_{1}F_{1})\tilde{p},z_{n}-p \bigr\rangle \\ =&\frac{1}{2} \bigl[ \bigl\Vert (I-\nu_{1}F_{1}) \tilde{y}_{n,N}-(I-\nu_{1}F_{1})\tilde{p} \bigr\Vert ^{2}+\Vert z_{n}-p\Vert ^{2} \\ &{} - \bigl\Vert (I-\nu_{1}F_{1})\tilde{y}_{n,N}-(I- \nu_{1}F_{1})\tilde{p}-(z_{n}-p) \bigr\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}- \bigl\Vert ( \tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{1} \bigl\langle F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p},(\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\rangle - \nu^{2}_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert z_{n}-p\Vert ^{2}- \bigl\Vert ( \tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{1} \bigl\langle F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p},(\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\rangle \bigr]. \end{aligned}$$
Thus, we have
$$\begin{aligned} \|\tilde{y}_{n,N}-\tilde{p}\|^{2} \leq&\|y_{n,N}-p \|^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{2} \bigl\langle (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}),F_{2}y_{n,N}-F_{2}p \bigr\rangle \\ &{}- \nu^{2}_{2}\|F_{2}y_{n,N}-F_{2}p \|^{2} \end{aligned}$$
(3.28)
and
$$\begin{aligned} \|z_{n}-p\|^{2} \leq&\|y_{n,N}-p\|^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{}+2\nu_{1}\|F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\| \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert . \end{aligned}$$
(3.29)
Consequently, from (
3.4), (
3.24) and (
3.28), it follows that
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} - \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert \tilde{y}_{n,N}- \tilde{p}\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert y_{n,N}-p\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-(1-\beta_{n}) \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert , \end{aligned}$$
which yields
$$\begin{aligned}& (1-d) \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq(1-\beta_{n}) \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-\Vert x_{n+1}-p \Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+ \alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert . \end{aligned}$$
Since
\(\lim_{n\to\infty}\alpha_{n}=0\),
\(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and
\(\{x_{n}\}\),
\(\{y_{n}\}\),
\(\{y_{n,N}\}\), and
\(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (
3.22) that
$$ \lim_{n\to\infty} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert =0. $$
(3.30)
Furthermore, from (
3.4), (
3.24), and (
3.29), it follows that
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert y_{n,N}-p\Vert ^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-(1-\beta_{n}) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert , \end{aligned}$$
which leads to
$$\begin{aligned}& (1-d) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq(1-\beta_{n}) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-\Vert x_{n+1}-p \Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+ \alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert . \end{aligned}$$
Since
\(\lim_{n\to\infty}\alpha_{n}=0\),
\(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and
\(\{x_{n}\}\),
\(\{z_{n}\}\),
\(\{y_{n}\}\),
\(\{y_{n,N}\}\), and
\(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (
3.22) that
$$ \lim_{n\to\infty} \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert =0. $$
(3.31)
Note that
$$\Vert y_{n,N}-z_{n}\Vert \leq \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert + \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert . $$
Hence from (
3.30) and (
3.31) we get
$$ \lim_{n\to\infty}\|y_{n,N}-z_{n}\|=\lim _{n\to\infty}\|y_{n,N}-Gy_{n,N}\| =0. $$
(3.32)
□