## 3 Results  
Now, we give our main results in this paper. **Theorem 1**. : _Let \(E_{1}\), \(E_{2}\) be 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 \(F(T)=\widetilde{F}(T)\
eq\emptyset\) 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_  
\[\left\{ \begin{array}{l}x_{n}=\Pi_{C}J_{E_{1}^{*}}^{q}\big{(}I_{E_{1}}^{p}(u_ {n})-s_{n}A^{*}J_{E_{2}}^{p}(I-P_{Q})Au_{n}\big{)},\\ u_{n+1}=\Pi_{C}J_{E_{1}^{*}}^{q}(\alpha_{n}J_{E_{1}}^{p}(u)+(1-\alpha_{n})( \beta_{n}J_{E_{1}}^{p}(u)+\gamma_{n}J_{E_{1}}^{p}(x_{n})+\delta_{n}J_{E_{1}}^{ p}(Tx_{n}))\big{)}\end{array} \right. \tag{20}\]  
_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 \(\{u_{n}\}\) converge strongly to a point \(\Pi_{F(T)\cap\mathcal{F}}u\)._  
Proof. : Firstly, we prove that the sequences \(\{x_{n}\}\) and \(\{u_{n}\}\) are bounded. Setting \(z_{n}:=Au_{n}-P_{Q}(Au_{n})\) for each \(n\geq 1\). From (14), it follows that, for any \(w\in F(T)\cap\mathcal{F}\),  
\[ \begin{split}\langle J_{E_{2}}^{p}(z_{n}),Au_{n}-Aw\rangle& =\|Au_{n}-P_{Q}(Au_{n})\|^{p}+\langle J_{E_{2}}^{p}(z_{n}),P_{Q}(Au_ {n})-Aw\rangle\\ &\geq\|Au_{n}-P_{Q}(Au_{n})\|^{p}\\ &=\|z_{n}\|^{p}.\end{split}  \tag{21}\]  
So, from (21) and (12), it follows that  
\[ \begin{split}& d_{p}(x_{n},w)\\ &\leq d_{p}(J_{E_{1}^{*}}^{q}|J_{E_{1}}^{p}(u_{n})-s_{n}A^{*}J_{E_ {2}}^{p}(I-P_{Q})Au_{n}],w)\\ &=\frac{1}{q}\|J_{E_{1}}^{p}(u_{n})-s_{n}A^{*}J_{E_{2}}^{p}(z_{n} )\|^{q}-\langle J_{E_{1}}^{p}(u_{n}),w\rangle+s_{n}\langle J_{E_{2}}^{p}(z_{n} ),Aw\rangle+\frac{1}{p}\|w\|^{p}\\ &\leq\|J_{E_{1}}^{p}(u_{n})\|-s_{n}\langle Au_{n}J_{E_{2}}^{p}(z_ {n})\rangle+\frac{C_{q}(s_{n}\|A\|)^{q}}{q}\|J_{E_{2}}^{p}(z_{n})\|^{p}\\ &\quad-\langle J_{E_{1}}^{p}(u_{n}),w\rangle+\frac{1}{p}\|w\|^{p }+s_{n}\langle [PERSON]_{E_{2}}^{p}(z_{n})\rangle\\ &=\frac{1}{q}\|u_{n}\|^{p}-\langle [PERSON]_{E_{1}}^{p}(u_{n}),w\rangle +\frac{1}{q}\|w\|^{p}+s_{n}\langle [PERSON]_{E_{2}}^{p}(z_{n}),[PERSON]_{n}\rangle\\ &\quad+\frac{C_{q}(s_{n}\|A\|)^{q}}{q}\|z_{n}\|^{p}\\ &=d_{p}(u_{n},w)+s_{n}\langle [PERSON]_{E_{2}}^{p}(z_{n}),Aw-Au_{n} \rangle+\frac{C_{q}(s_{n}\|A\|)^{q}}{q}\|z_{n}\|^{p}\\ &\leq d_{p}(u_{n},w)-(s_{n}-\frac{C_{q}(s_{n}\|A\|)^{q}}{q})\|z_ {n}\|^{p}.\end{split}