## 3 Main Results
Indeed, noticing \(q_{n}=P_{\mathbb{C}}(I-\alpha_{n}\mu A_{2})T^{n}y_{n}\) for all \(n\geq 0\), we obtain from (7) that for each \(p\in\Omega\),  
\[\langle(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-P_{\mathbb{C}}(I-\alpha_{n}\mu A_{2})T ^{n}y_{n},p-q_{n}\rangle\leq 0,\]  
which hence leads to  
\[ \begin{array}{ll}\|q_{n}-p\|^{2}&=\langle P_{\mathbb{C}}(I-\alpha_{n}\mu A_{2 })T^{n}y_{n}-(I-\alpha_{n}\mu A_{2})T^{n}y_{n},q_{n}-p\rangle\\ &\quad+\langle(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-p,q_{n}-p\rangle\\ &\leq\langle(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-p,q_{n}-p\rangle\\ &=\langle(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-(I-\alpha_{n}\mu A_{2})p,q_{n}-p \rangle-\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle\\ &\leq\frac{1}{2}(1-\alpha_{n}\tau)^{2}\|T^{n}y_{n}-p\|^{2}+\frac{1}{2}\|q_{n}- p\|^{2}-\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle.\end{array} \]  
It follows from (1) that  
\[ \begin{array}{ll}\|q_{n}-p\|^{2}&\leq(1-\alpha_{n}\tau)(1+\theta_{n})^{2}\| y_{n}-p\|^{2}-2\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle\\ &\leq(1-\alpha_{n}\tau)\|y_{n}-p\|^{2}+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2} -2\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle.\end{array}  \tag{24}\]  
From (15) and (24), we get  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}\\ \leq\beta_{n}\|f(y_{n})-f(p)\|^{2}+(1-\beta_{n})\|q_{n}-p\|^{2}+2\beta_{n} \langle f(p)-p,x_{n+1}-p\rangle\\ \leq\alpha_{n}\delta\|y_{n}-p\|^{2}+(1-\alpha_{n}\tau)\|y_{n}-p\|^{2}+\theta_{ n}(2+\theta_{n})\|y_{n}-p\|^{2}\\ \quad-2\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle+2\beta_{n}\langle f(p)-p,x_{ n+1}-p\rangle\\ =[1-\alpha_{n}(\tau-\delta)]\|y_{n}-p\|^{2}+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2} -2\alpha_{n}\langle\mu A_{2}p,q_{n}-p\rangle\\ \quad+2\beta_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ \leq[1-\alpha_{n}(\tau-\delta)][\alpha_{n}\|x_{n}-p\|+(1-\sigma_{n})(\|z_{n}- p\|+\delta_{n}\|A_{1}p\|)]^{2}\\ \quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}-2\alpha_{n}\langle\mu A_{2}p,q_{ n}-p\rangle+2\beta_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ \leq[1-\alpha_{n}(\tau-\delta)][\alpha_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})(\|z_{ n}-p\|+\delta_{n}\|A_{1}p\|)^{2}]\\ \quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}-2\alpha_{n}\langle\mu A_{2}p,q_{ n}-p\rangle+2\beta_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ \leq[1-\alpha_{n}(\tau-\delta)][\alpha_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})\|z_{ n}-p\|^{2}+\delta_{n}\|A_{1}p\|(2\|z_{n}-p\|+\delta_{n}\|A_{1}p\|)]\\ \quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}-2\alpha_{n}\langle\mu A_{2}p,q_{ n}-p\rangle+2\beta_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ \leq[1-\alpha_{n}(\tau-\delta)][\alpha_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})\|z_{ n}-p\|^{2}]+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|+\alpha_{n}\|A_{1}p\|)\\ \quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|\mu A_{2}p\|\|q_{ n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|).\end{array}  \tag{25}\]  
We now note that \(q=P_{\mathbb{C}}(p-\mu_{2}B_{2}p)\), \(v_{n}=P_{\mathbb{C}}(u_{n}-\mu_{2}B_{2}u_{n})\) and \(z_{n}=P_{\mathbb{C}}(v_{n}-\mu_{1}B_{1}v_{n})\). Then \(z_{n}=Gu_{n}\). By Lemma 5, we have  
\[ \begin{array}{ll}\|v_{n}-q\|^{2}&=\|P_{\mathbb{C}}(u_{n}-\mu_{2}B_{2}u_{n})-P _{\mathbb{C}}(p-\mu_{2}B_{2}p)\|^{2}\\ &\leq\|u_{n}-p-\mu_{2}(B_{2}u_{n}-B_{2}p)\|^{2}\\ &\leq\|u_{n}-p\|^{2}-\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}\end{array}  \tag{26}\]  
and  
\[ \begin{array}{ll}\|z_{n}-p\|^{2}&=\|P_{\mathbb{C}}(v_{n}-\mu_{1}B_{1}v_{n})-P _{\mathbb{C}}(q-\mu_{1}B_{1}q)\|^{2}\\ &\leq\|v_{n}-q-\mu_{1}(B_{1}v_{n}-B_{1}q)\|^{2}\\ &\leq\|v_{n}-q\|^{2}-\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}.\end{array}  \tag{27}\]  
Substituting (26) for (27), we obtain from (13) that  
\[ \begin{array}{ll}\|z_{n}-p\|^{2}&\leq\|u_{n}-p\|^{2}-\mu_{2}(2\beta-\mu_{2})\| B_{2}u_{n}-B_{2}p\|^{2}-\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}\\ &\leq\|p-x_{n}\|^{2}-\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}-\mu_{1}(2 \alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}.\end{array}