## 3 Results
\(0<s\leq s_{n}\leq k<\Big{(}\frac{q}{C_{q}\|A\|^{q}}\Big{)}^{\frac{1}{q-1}}\)_._  
_Then the sequence \(\{x_{n}\}\) and \(\{u_{n}\}\) converges strongly to a point \(\Pi_{F(T)\cap\mathcal{F}}u\)._  
Proof. : If \(\beta_{n}=\gamma_{n}=\delta_{n}=0\) for all \(n\geq 1\) in Theorem 1, then we obtain the desired conclusion. Next, we consider the mapping \(T:C\to C\) defined by \(T=T_{N}\circ T_{N-1}\circ\cdots\circ T_{1}\), where \(T_{i}\) for each \(i=1,2,\cdots,N\) is a right Bregman strongly quasi-nonexpansive mapping on \(E\). Using the results in [10], we have the following:  
**Corollary 3**. : _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 \(T=T_{N}\circ T_{N-1}\circ\cdots\circ T_{1}\), where \(T_{i}:C\to C\) for each \(i=1,2,\cdots N\) be a finite family of right Bregman strongly quasi-nonexpansive mappings such that \(F(T_{i})=\widehat{F}(T_{i})\
eq\emptyset\) and \(\big{(}\bigcap\limits_{i=1}^{N}F(T_{i})\big{)}\cap\mathcal{F}\
eq\emptyset\). Suppose that \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) are the sequences in \([0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}+\delta_{n}=1\), \(\alpha_{n}\leq b<1\), \((1-\alpha_{n})a<\gamma_{n}<\delta_{n}\), \(a\in(0,\frac{1}{2})\). For any fixed \(u\in C\), let the sequences \(\{x_{n}\}\) and \(\{u_{n}\}\) be iteratively generated by \(u_{0}\in E_{1}\) and_  
\[\left\{ \begin{array}{l}x_{n}=\Pi_{C}J_{E_{1}^{*}}^{q}\big{(}J_{E_{1}}^{p}(u_ {n})-s_{n}A^{*}J_{E_{2}}^{p}(I-P_{Q})Au_{n}\big{)},\\ u_{n+1}=\Pi_{C}J_{E_{1}^{*}}^{q}(\alpha_{n}J_{E_{1}}^{p}(u)+(1-\alpha_{n})( \beta_{n}J_{E_{1}}^{p}(u)+\gamma_{n}J_{E_{1}}^{p}(x_{n})+\delta_{n}J_{E_{1}}^ {p}(Tx_{n}))\big{)}\end{array} \right. \tag{45}\]  
_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. \(0<s\leq s_{n}\leq k<\Big{(}\frac{q}{C_{q}\|A\|^{q}}\Big{)}^{\frac{1}{q-1}}\)_._  
_Then the sequence \(\{x_{n}\}\) and \(\{u_{n}\}\) converges strongly to a point \(\Pi_{F(T)\cap\mathcal{F}}u\)._  
Proof. : If \(T=T_{N}\circ T_{N-1}\circ\cdots\circ T_{1}\) in Theorem 1, then we obtain the desired conclusion. **Corollary 4**. : _Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces and \(C\), \(Q\) be nonempty closed convex subsets of \(H_{1}\), \(H_{2}\), respectively. Let \(A:H_{1}\to H_{2}\) be a bounded linear operator and \(A^{*}:H_{2}\to H_{1}\) 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 \(F(T)=\widehat{F}(T)\
eq\emptyset\), \(I-T\) is demiclosed at zero and \(F(T)\cap\mathcal{F}\
eq\emptyset\). Suppose that \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) are the sequences in \([0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}+\delta_{n}=1\), \(\alpha_{n}\leq b<1\), \((1-\alpha_{n})a<\gamma_{n}<\delta_{n}\), \(a\in(0,\frac{1}{2})\). For any fixed \(u\in C\), let the sequences \(\{x_{n}\}\) and \(\{u_{n}\}\) be iteratively generated by \(u_{0}\in E_{1}\) and_  
\[\left\{ \begin{array}{l}x_{n}=P_{\mathrm{C}}\big{(}u_{n}-s_{n}A^{*}(I-P_{ \mathrm{Q}})Au_{n}\big{)},\\ u_{n+1}=P_{\mathrm{C}}\big{(}\alpha_{n}u+(1-\alpha_{n})(\beta_{n}u+\gamma_{n} x_{n}+\delta_{n}Tx_{n})\big{)}\end{array} \right. \tag{46}\]  
_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. \(0<s\leq s_{n}\leq k<\frac{2}{\|A\|^{2}}\)_._  
_Then the sequence \(\{x_{n}\}\) and \(\{u_{n}\}\) converges strongly to a point \(\Pi_{F(T)\cap\mathcal{F}}u\)._  
Proof.