## 3 Results
Again, we obtain  
\[\leq d_{p}(I_{E_{1}^{q}}^{q}[\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n} )(\beta_{n}I_{E_{1}}^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{ p}(Tx_{n}))],w)\]7 \[\leq d_{p}(I_{E_{1}^{q}}^{q}[\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n} )(\beta_{n}I_{E_{1}}^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{ p}(Tx_{n}))],w)\]8 \[\leq d_{p}(I_{E_{1}^{q}}^{q}[\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n} )(\beta_{n}I_{E_{1}}^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{ p}(Tx_{n}))],w)\]9 \[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]0  
Since \(\alpha_{n}+\delta_{n}\leq 1\) and \(\alpha_{n}\leq b<1\), we have  
\[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]1 \[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]2  
as \(n\to\infty\). By using \((c)\), we have  
\[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]3as \(n\to\infty\). Since \(T\) is right Bregman strongly quasi-nonexpansive, we obtain  
\[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]4  
which implies that  
\[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]5  
Since \(\{x_{n}\}\) is bounded and \(E\) is reflexive, there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) which converges weakly to \(\bar{x}\in C\). From (30), it follows that \(\bar{x}\in F(T)\) since \(F(T)=\widehat{F}(T)\). Next, we prove that \(A\bar{x}\in Q\), that is, \(\bar{x}\in\mathcal{F}\). Setting  
\[=V_{f}(\alpha_{n}I_{E_{1}}^{p}(u)+(1-\alpha_{n})(\beta_{n}I_{E_{1} }^{p}(u)+\gamma_{n}I_{E_{1}}^{p}(x_{n})+\delta_{n}I_{E_{1}}^{p}(Tx_{n})),w)\]6  
From (16), (22) and (24), it follows that  
\[ \begin{split} d_{p}(v_{n},x_{n})&=d_{p}(v_{n},\Pi_ {c}v_{n})\\ &\leq d_{p}(v_{n},w)-d_{p}(x_{n},w)\\ &\leq d_{\langle u_{n},w\rangle}-d_{p}(x_{n},w)\\ &\leq\alpha_{n}\mathcal{M}+d_{p}(x_{n-1},w)-d_{p}(x_{n},w)\to 0 \end{split}  \tag{31}\]  
as \(n\to\infty\), where \(\mathcal{M}>0\) and \(d_{p}(x_{n-1},w)+\langle I_{E_{1}}^{p}(u)-I_{E_{1}}^{p}(w),y_{n-1}-w\rangle \leq\mathcal{M}\). Hence we have  
\[\lim_{n\to\infty}\|x_{n}-v_{n}\|=0. \tag{32}\]  
From (22), it follows that, as \(n\to\infty\),  
\[ \begin{split}\Big{(}s_{n}-\frac{C_{q}(s_{n}\|A\|)^{q}}{q}\Big{)} \|z_{n}\|^{p}&\leq d(u_{n},w)-d_{p}(x_{n},w)\\ &\leq\alpha_{n}\mathcal{M}+d_{p}(x_{n-1},w)-d_{p}(x_{n},w)\to 0.\end{split}  \tag{33}\]  
Since  
\[s\Big{(}1-\frac{C_{q}k^{q-1}(\|A\|)^{q}}{q}\Big{)}\leq\Big{(}s_{n}-\frac{C_{q} (s_{n}\|A\|)^{q}}{q}\Big{)}, \tag{34}\]  
it follows that \(\|z_{n}\|^{p}\to 0\) as \(n\to\infty\), which implies that \(\|Au_{n}-P_{Q}(u_{n})\|\to 0\) as \(n\to\infty\). By the definition of \(v_{n}\), we have  
\[ \begin{split} 0&\leq\|J_{E_{1}}^{p}(v_{n})-J_{E_{1}}^{p} (u_{n})\\ &\leq s_{n}\|A^{*}\|\|J_{E_{2}}^{p}(Au_{n}-P_{Q}(Au_{n}))\|\\ &\leq\Big{(}\frac{q}{C_{q}\|A\|^{q}}\Big{)}^{\frac{1}{q-1}}\|A^{* }\|\|Au_{n}-P_{Q}(Au_{n})\|\to 0\end{split}  \tag{35}\]  
as \(n\to\infty\). Since \(J_{E_{1}}^{q}\) is norm to norm uniformly continuous on bounded subsets of \(E_{1}^{*}\), we obtain  
\[\lim_{n\to\infty}\|v_{n}-u_{n}\|=\lim_{n\to\infty}\|J_{E_{1}}^{q}(J_{E_{1}}^{ p}(v_{n}))-[PERSON]_{E_{1}}^{q}(J_{E_{1}}^{p}(u_{n}))\|\to 0 \tag{36}\]  
as \(n\to\infty\). From (3) and (36), we obtain  
\[\|x_{n}-u_{n}\|\leq\|x_{n}-y_{n}\|+\|y_{n}-u_{n}\|\to 0. \tag{37}\]as \(n\to\infty\). From (14), it follows that  
\[