## 3 Results
\begin{split}\alpha_{m_{k}}d_{p}(x_{m_{k}},\widehat{x})& \leq d_{p}(x_{m_{k}},\widehat{x})-d_{p}(x_{m_{k}+1},\widehat{x})+ \alpha_{m_{k}}\langle J_{E_{1}}^{p}(u)-J_{E_{1}}^{p}(\widehat{x}),y_{m_{k}}- \widehat{x}\rangle\\ &\leq\alpha_{m_{k}}\langle J_{E_{1}}^{p}(u)-J_{E_{1}}^{p}( \widehat{x}),y_{m_{k}}-\widehat{x}\rangle\end{split}  \tag{41}\]  
Since \(\alpha_{m_{k}}>0\), we obtain  
\[d_{p}(x_{m_{k}},\widehat{x})\leq\langle J_{E_{1}}^{p}(u)-J_{E_{1}}^{p}( \widehat{x}),y_{m_{k}}-\widehat{x}\rangle. \tag{42}\]  
Then, from (39), it follows that \(d_{p}(x_{m_{k}},\widehat{x})\to 0\) as \(k\to\infty\). This together with (40), we obtain \(d_{p}(x_{m_{k}+1},\widehat{x})\to 0\) as \(k\to\infty\). Since \(d_{p}(x_{k},\widehat{x})\leq(x_{m_{k}+1},\widehat{x})\) for all \(k\in\mathbb{N}\), we have \(x_{k}\to\widehat{x}\) as \(k\to\infty\), which implies that \(x_{n}\to\widehat{x}\) as \(k\to\infty\). Therefore, from the above two cases, we conclude that \(\{x_{n}\}\) converges strongly to \(\widehat{x}=\Pi_{P(T)\cap\mathcal{F}}u\). This completes the proof. Corollary 1 ([19]): _Let \(E_{1}\), \(E_{2}\) be two \(p\)-uniformly convex real Banach spaces which are also uniformly smooth and \(C\), \(Q\) be nonempty closed convex subsets of \(E_{1}\), \(E_{2}\), respectively. Let \(A:E_{1}\to E_{2}\) be a bounded linear operator and \(A^{*}:E_{2}^{*}\to E_{1}^{*}\) be the adjoint of \(A\). Suppose that the problem (SFP) has a nonempty solution set \(\mathcal{F}\). Let \(T:C\to C\) be a right Bregman strongly quasi-nonexpansive mapping such that \(F(T)=\widehat{F}(T)\
eq\emptyset\) and \(F(T)\cap\mathcal{F}\
eq\emptyset\). Suppose that \(\{\beta_{n}\},\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) are the sequences in \((0,1)\) such that \(\beta_{n}+\gamma_{n}+\delta_{n}=1\), \(\beta_{n}\leq b<1\), \((1-\beta_{n})a<\gamma_{n}<\delta_{n}\), \(a\in(0,\frac{1}{2})\). For any fixed \(u\in C\), let the sequences \(\{x_{n}\}\) and \(\{u_{n}\}\) be iteratively generated by \(u_{0}\in E_{1}\) and_  
\[\left\{ \begin{array}{l}x_{n}=\Pi_{C}J_{E_{1}}^{q}(J_{E_{1}}^{p}(u_{n})-s_{n}A ^{*}J_{E_{2}}^{p}(I-P_{Q})Au_{n}),\\ u_{n+1}=\Pi_{C}J_{E_{1}}^{q}(\beta_{n}J_{E_{1}}^{p}(u)+\gamma_{n}J_{E_{1}}^{p}( x_{n})+\delta_{n}J_{E_{1}}^{p}(Tx_{n}))\end{array} \right. \tag{43}\]  
_for each \(n\geq 1\). Suppose the following condition is satisfied:_  
_(a)_: \(\lim\limits_{n\to\infty}\beta_{n}=0\)_;_  
_(b)_: \(\sum\limits_{n=1}^{\infty}\beta_{n}=\infty\)_;_  
_(c)_: \(0<s\leq s_{n}\leq k<\Big{(}\frac{q}{C_{q}\|A\|^{q}}\Big{)}^{\frac{1}{q-1}}\)_._  
_Then the sequence_ \(\{x_{n}\}\) _and_ \(\{u_{n}\}\) _converges strongly to a point_ \(\Pi_{F(T)\cap\mathcal{F}}u\)_._  
Proof: If \(\alpha_{n}=0\) for all \(n\geq 1\) in Theorem 1, then we obtain the desired conclusion. **Corollary 2**. : _Let \(E_{1}\), \(E_{2}\) be two \(p\)-uniformly convex real Banach spaces which are also uniformly smooth and \(C\), \(Q\) be nonempty closed convex subsets of \(E_{1}\), \(E_{2}\), respectively. Let \(A:E_{1}\to E_{2}\) be a bounded linear operator and \(A^{*}:E_{2}^{*}\to E_{1}^{*}\) be the adjoint of \(A\). Suppose that the problem (SFP) has a nonempty solution set \(\mathcal{F}\). Let \(T:C\to C\) be a right Bregman strongly quasi-nonexpansive mapping such that \(F(T)=\widehat{F}(T)\
eq\emptyset\) and \(F(T)\cap\mathcal{F}\
eq\emptyset\). Suppose that \(\{\alpha_{n}\}\) is a sequences in \((0,1)\) such that \(\alpha_{n}\leq b<1\), \(a\in(0,\frac{1}{2})\). For any fixed \(u\in C\), let the sequences \(\{x_{n}\}\) and \(\{u_{n}\}\) be iteratively generated by \(u_{0}\in E_{1}\) and_  
\[\left\{ \begin{array}{l}x_{n}=\Pi_{C}J_{E_{1}^{*}}^{q}\big{(}I_{E_{1}}^{p}(u_ {n})-s_{n}A^{*}J_{E_{2}}^{p}(I-P_{Q})Au_{n}\big{)},\\ u_{n+1}=\Pi_{C}J_{E_{1}^{*}}^{q}(\alpha_{n}J_{E_{1}}^{p}(u)+(1-\alpha_{n})J_{E_ {1}}^{p}(Tx_{n})\big{)}\end{array} \right. \tag{44}\]  
_for each \(n\geq 1\). Suppose the following conditions are satisfied:_  
1. \(\lim\limits_{n\to\infty}\alpha_{n}=0\)_;_
2. \(\sum\limits_{n=1}^{\infty}\alpha_{n}=\infty\)_;_
3.