## 1 Introduction  
Let \(E_{1}\), \(E_{2}\) be Banach spaces and \(C\), \(Q\) be nonempty closed convex subsets of \(E_{1}\) and \(E_{2}\), respectively. Let \(A\colon E_{1}\to E_{2}\) be a bounded linear operator. The split feasibility problem (shortly, (SFP)) is as follows:  
\[\text{Find }\;x\in C\;\;\text{such that }\;Ax\in Q. \tag{1}\]  
We denote the solution set of the problem (SFP) by \(\mathcal{F}:=\{x\in C:Ax\in Q\}=C\cap A^{-1}(Q)\). It is worth mentioning that (SFP) in finite-dimensional spaces was first introduced by [PERSON] and [PERSON] [1] for modelling inverse problems which arise from phase retrievals and medical image reconstruction. Note that, in finite dimensional Hilbert spaces, the strong convergence of a sequence is equivalent to the weak convergence and the boundedness of a sequence implies that there exists a strongly convergent subsequence. However, in infinite dimensional Hilbert spaces, the strong convergence of a sequence is not equivalent to the weak convergence and the boundedness of a sequence implies that there exists a weakly convergent subsequence. So, for some algorithms, we can prove only strong convergence theorems in finite dimensional Hilbert spaces, but we can prove weak and strong convergence theorems in infinite dimensional Hilbert spaces. In [2], [PERSON] presented a new method \(\{x_{n}\}\), which is called the CQ-algorithm for solving the problem (SFP) that does not involve matrix inverses, defined as follows:  
For any \(x_{0}\in C\) and \(n\geq 1\),  
\[x_{n+1}=P_{C}(x_{n}+\gamma A^{T}(P_{Q}-I)x_{n}), \tag{2}\]  
where \(P_{C}\) and \(P_{Q}\) is the orthogonal projections onto \(C\) and \(Q\), respectively, \(\gamma\in(0,\frac{2}{L})\), \(L\) is the largest eigenvalue of the matrix \(A^{T}A\) and \(I\) is the identity matrix. After that many authors [3; 4; 5; 6; 7] study extend some iterative algorithms from Hilbert spaces to Banach spaces by using [PERSON]'s technic as follows:  
In solving the problem (SFP) in \(p\)-uniformly convex real Banach spaces which are also uniformly smooth, [PERSON] et al. [8] proposed the following algorithm \(\{x_{n}\}\) defined as follows:  
For any \(x_{1}\in E_{1}\) and \(n\geq 1\),  
\[x_{n+1}=\Pi_{C}J_{E_{1}}^{*}(J_{E_{1}}(x_{n})-s_{n}A^{*}J_{E_{2}}(Ax_{n}-P_{Q} (Ax_{n}))), \tag{3}\]  
where \(\Pi_{C}\) denotes the Bregman projection and \(J\) the duality mapping. Clearly, the algorithm (3) covers [PERSON]'s CQ algorithm (2), which is a gradient-projection method (GPM) in convex minimization as a special case. The duality mapping of \(E_{1}\) is sequentially weak-to-weak-continuous (see [8]) in Banach spaces such as the classical \(L_{p}\)\((2<p<\infty)\) spaces. In [9], [PERSON] modified the algorithm (3) and proved strong convergence theorems for the following multiple-sets split feasibility problem (MSSFP):  
\[\text{Find }\;x\in\bigcap_{i=1}^{r}C_{i}\;\text{ such that }\;Ax\in\bigcap_{j=1+r}^{r+s}Q_{j}, \tag{4}\]  
where \(r,s\) are two given integers, \(C_{i}\), \(i=1,2,3,\cdots,r\), is a closed convex subset in \(E_{1}\) and \(Q_{j}\), \(j=r+1,\cdots,r+s\), is a closed convex subset in \(E_{2}\). He defined the following: for each \(n\in\mathbb{N}\),  
\[T_{n}(x)= \begin{cases}\Pi_{C_{i}(n)}(x),&1\leq i(n)\leq r,\\ J_{E_{1}}^{*}[J_{E_{1}}(x)-s_{n}A^{*}J_{E_{2}}(Ax-P_{Q_{|n}}(Ax))],&r+1\leq i( n)\leq r+s,\end{cases}  \tag{5}\]  
where \(i:\mathbb{N}\rightarrow\mathcal{I}\) is the cyclic control mapping \(i(n)=n\operatorname{mod}(r+s)+1\) and \(t_{n}\) satisfies  
\[0<s\leq s_{n}\leq\frac{q}{C_{q}\|A\|^{q}}^{\frac{1}{q-1}} \tag{6}\]  
with a constant \(C_{q}\) and proposed the following algorithm \(\{x_{n}\}\) defined as follows: For any \(x_{1}=\bar{x}\) and \(n\geq 1\),  
\[ \begin{cases}y_{n}=T_{n}x_{n},\\ D_{n}=\{w\in E_{1}:d_{p}(y_{n},w)\leq d_{p}(x_{n},w)\},\\ E_{n}=\{w\in E_{1}:\langle x_{n}-w,I_{p}(\bar{x})-J_{p}(x_{n})\geq 0\},\\ x_{n+1}=\Pi_{D_{n}\cap E_{n}}(\bar{x}).\end{cases}