Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces

Abstract

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.

Keywords

Split common fixed point / implicit rule / Bregman quasi-nonexpansive mapping / strong convergence / Banach space

1 Introduction

Iterative algorithms for variational inequality and fixed point problems are frontier problems in mathematics field. The theory of split feasibility problem and split common fixed point problem play important roles in many fields, such as data denoising, image reconstruction, signal processing, and inverse problem modeling. Algorithms for these problems have been discussed and studied by many authors, see [2,8,10,12,22].

Assume that H_1 and H_2 are Hilbert spaces, C and Q are nonempty closed convex subsets of H_1 and H_2, respectively. A:H_1\to H_2 is a linear operator, the adjoint of A denoted by A^{\ast }. Let S:H_1\to H_1 and T:H_2\to H_2 be two linear operators. We use F(S) and F(T) to represent the fixed point sets of S and T, respectively. The split common fixed point problem is to find a point x\in H_1 such that:

(1)\begin{array}{l}x\in F(S),Ax\in F(T).\end{array}

The solution set of the split common fixed point problem denoted by \mathrm{\Omega }, i.e., \mathrm{\Omega }=\{x:x \in F(S ),Ax \in F(T )\}. If S and T are identity operators, then the split common fixed point problem (1) is the split feasibility problem. So the split feasibility problem is a special case of the split common fixed point problem (1), which is to find the point x\in H_1 such that:

(2)\begin{array}{l}x\in C,Ax\in Q.\end{array}

Censor and Segal [3] introduced the following iterative algorithm to solve the split common fixed point problem. For any point x_0:

\begin{array}{l}{x}_{n+1}=S(I-r{A}^{\ast }(I-T)A{x}_n),\end{array}

where r\in (0 ,\frac{2}{\Vert A{\Vert }^2}). They showed that this sequence weakly converges to the solution of (1). Subsequently, this result has been extended to the study of the quasi-nonexpansive mappings and the semi-contraction operators as well as to a wider range of Banach spaces [13,20,27]. On the other hand, the Bregman distance function has become one of the important tools for studying fixed point problems and optimization problems in generalized Banach spaces. It can be applied to many aspects of the constructed iterative algorithms to solve split feasibility problems, variational inequality problems, optimization problems, fixed point problems, equilibrium problems, etc. Using Bregman distance and Bregman projection, many scholars have devoted themselves to solving the fixed point problem of Bregman nonlinear operators and analyzed the convergence of some related iterative algorithms, see [1, 6,25,26] and its references therein. In particular, Shehu [18] ensured strong convergence by using a Halpern iterative procedure, and introduced the following iterative algorithms for left Bregman strongly relative nonexpansive mapping:

\{\begin{array}{l}{x}_n={\mathrm{\Pi }}_C{J}_{E_1^{\ast }}^q(J_{E_1}^p{u}_n-t_n{A}^{\ast }{J}_{E_1}^p(I-P_Q)A{u}_n),\\ u_{n+1}={\mathrm{\Pi }}_C{J}_{E_1^{\ast }}^q({\alpha }_n{J}_{E_1}^{p}u+(1-{\alpha }_n)J_{E_1}^p{x}_n),\end{array}

where E_1 is p-uniformly convex and uniformly smooth real Banach space, and E_1^{\ast } is the dual space of E_1. {\mathrm{\Pi }}_C is said to be Bregman projection, and J is dual mapping. P_C and P_Q are metric projections on C and Q, respectively. They proved that the sequence u_n converges strongly to the solution of (2).

Recently, many authors have studied implicit rules of nonexpansive mapping, which are one of the important numerical methods for solving some ordinary differential equations, see [7,15]. For example, Xu et al. [24] proposed implicit midpoint rules for nonexpansive mappings in Hilbert spaces. Luo et al. [9] demonstrated strong convergence of viscosity implicit iterative algorithms in uniformly smooth Banach spaces. Pant et al. [16] established a modified implicit iterative algorithm in Hilbert space to solve the split feasibility problem and the fixed point problem.

Motivated and inspired by the above results, the purpose of this paper is to study the split common fixed point problem in Banach spaces. We give a modified implicit iterative algorithm of Bregman quasi-nonexpansive mappings. And we prove the strong convergence theorem of the iterative algorithm. Finally, we apply the results to the zero point problem and the equilibrium problem, which generalize and improve many other results [3,10,12,13,16,18,19,24].

2 Preliminaries

Let X be a real Banach space and X^{\ast } be the dual space of X. Let and \frac{1}{p}+\frac{1}{q}= 1. The dual mapping J_X^p: X\to 2^{X^{\ast }} is defined by

\begin{array}{l}{J}_X^p(x)=\{f\in X^{\ast }:⟨x,f⟩=\Vert x{\Vert }^q,\Vert f\Vert =\Vert x{\Vert }^{q-1}\},\mathrm{\forall }x\in X.\end{array}

Assume that X is p-uniformly convex and uniformly smooth. Then X^{\ast } is q-uniformly convex and uniformly smooth. The dual mapping J_X^p is one by one, single value and J_X^p= (J_{X^{\ast }}^q {)}^{-1}. Let g:X\to \mathbb{R} be a Gâteaux differentiable convex function. The Bregman distance of g is defined as

\begin{array}{l}{D}_g(x,y)=g(y)-g(x)-⟨g^{\prime }(x),y-x⟩,\mathrm{\forall }x,y\in X.\end{array}

The dual mapping J_X^p is actually the derivative of the function g_p(x) =\frac{1}{p}\Vert x{\Vert }^p, so the Bregman distance of g_p is

D_p(x,y)= \frac{1}{q}\Vert x{\Vert }^p-⟨J_E^{p}x ,y⟩ +\frac{1}{p}\Vert y{\Vert }^p.

From the definition of D_p(\cdot ,\cdot ), we obtain

\begin{array}{l}{D}_p(x,y)=D_p(x,z)+D_p(z,y)+⟨z-y,J_E^{p}x-J_E^{p}z⟩.\end{array}

Let E be a nonempty closed convex subset of X. The Bregman projection of g_p is defined by

\begin{array}{l}{\mathrm{\Pi }}_E={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{y\in E}{D}_p(x,y),x\in X.\end{array}

Then the Bregman projection can also be expressed by the following variational inequality:

⟨\overline{x}-{\mathrm{\Pi }}_{E}x,J_E^p x-J_E^p{\mathrm{\Pi }}_{E}x⟩\le 0, \mathrm{\forall }\overline{x}\in E.

The mapping T:E\to X is called Bregman quasi-nonexpansive if F(T)\ne \mathrm{\varnothing } and

\begin{array}{l}{D}_p(Tx,x^{\ast })\le D_p(x,x^{\ast }),\mathrm{\forall }x\in E,x^{\ast }\in F(T).\end{array}

A mapping T is said to be firmly nonexpansive if for any x,y\in E, we have

\begin{array}{l}⟨Tx-Ty,J_E^p(x-Tx)-J_E^p(y-Ty)⟩\ge 0.\end{array}

A point x^{\ast }\in E is called asymptotically fixed point of mapping T, if \{x_n\} is a sequence in E and weak convergence to x^{\ast }, such that \underset{n\to \mathrm{\infty }}{lim}\Vert T{x}_n-x_n\Vert =0. The set of asymptotic fixed points of T is defined by \hat{F}(T). We observe that the Bregman quasi-nonexpansive mappings are generalized operators, which is a generalization of the Bregman relative nonexpansive mappings, firmly nonexpansive as well as the Bregman nonexpansive mappings. If E=H and H is a real Hilbert space, then Bregman quasi-nonexpansive mappings reduce to quasi-nonexpansive mappings and firmly nonexpansive type mappings reduce to firmly nonexpansive mappings.

Next, we recall some lemmas which are needed in the proof of our main results.

Lemma 1 [28]　 Suppose that X is a smooth and strictly convex reflexive Banach space, and E is a nonempty closed convex subset of X. T:E\to X is a firmly nonexpansive type mapping. Then F(T) is a closed convex subset of X and \hat{F}(T)=F (T).

Lemma 2 [5]　 Suppose that E is a nonempty closed convex subset of X and f:X\to \mathbb{R} is a Legendre function. If T:E\to X is a Bregman quasi-nonexpansive mapping. Then F(T) is closed and convex.

Lemma 3 [23]　 Suppose that X is a q-uniformly smooth Banach space with a smoothing coefficient C_q. For any x,y\in X, the following inequality holds:

\begin{array}{l}\Vert x-y{\Vert }^q\le \Vert x{\Vert }^q-q⟨y,J_X^{q}x⟩+C_q\Vert y{\Vert }^q.\end{array}

Lemma 4 [4]　 The bifunction V_p:X\times X^{\ast } \to R is defined as

V_p(\overline{x},x)=\frac{1}{q}\Vert \overline{x}{\Vert }^q- ⟨\overline{x},x⟩+\frac{1}{p}\Vert x{\Vert }^p, \mathrm{\forall } x\in X,\overline{x}\in X^{\ast } .

Then V_p is negative and V_p(\overline{x},x)= D_p(J_{X^{\ast }}^q \overline{x},x). From the subdifferential inequality, we have

\begin{array}{l}{V}_p(\overline{x},x)+⟨\overline{y},J_{X^{\ast }}^q\overline{x}-x⟩\le V_p(\overline{x}+\overline{y},x),\mathrm{\forall }x\in X,\overline{x},\overline{y}\in X^{\ast }.\end{array}

Moreover, V_p is convex with respect to the first variable, so for all z\in X, we have

D_p\left(J_{X^{\ast }}^q\left(\sum _{i=1}^N{t}_i{J}_X^p(x_i)\right),z\right)\le \sum _{i=1}^N{t}_i{D}_p(x_i, z),

where \{x_i{\}}_{i=1}^N\subset X and \{t_i{\}}_{i=1}^N\subset (0,1 ) satisfies \sum _{i=1}^N{t}_i=1.

Lemma 5 [21]　 Let q\ge 1,r> 0 be two fixed real numbers. Then the Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing convex function g:{\mathbb{R}}^{\mathbb{+}}\to {\mathbb{R}}^{\mathbb{+}} such that for all x,y\in B_r, 0\le \lambda \le 1,

\begin{array}{l}\Vert \lambda x+(1-\lambda )y{\Vert }^q\le \lambda \Vert x{\Vert }^q+(1-\lambda )\Vert y{\Vert }^q-W_q(\lambda )g(\Vert x-y\Vert ),\end{array}

where W_q:= {\lambda }^q(1 -\lambda ) +\lambda (1-\lambda {)}^q, B_r:=\{x\in X ,\Vert x\Vert \le r\}.

Lemma 6 [17]　 Let X be a real smooth and uniformly convex Banach space, \{x_n\} and \{y_n\} be two sequences in X. Then \underset{n\to \mathrm{\infty }}{lim}{D}_p(x_n,y_n)=0 if and only if \underset{n\to \mathrm{\infty }}{lim}\Vert x_n-y_n\Vert =0.

Lemma 7 [11]　 Let \{a_n\} be a nonempty real sequence, and for all j\in \mathbb{N}, there be a_{n_j}\le a_{n_j+1}, where \{n_j\} is a subsequence of \{n\}. Then there exists a nondecreasing sequence \{m_i\}\subset \mathbb{N},i\in \mathbb{N} such that for m_i\to \mathrm{\infty }, we have

\begin{array}{l}{a}_{m_i}\le a_{m_i+1},a_i\le a_{m_i+1}.\end{array}

In fact, m_i= max\{k\le i :a_k\le a_{k+1}\}.

Lemma 8 [14]　 Let \{a_n\} be a nonempty real sequence satisfying:

\begin{array}{l}{a}_{n+1}\le (1-{\alpha }_n)a_n+{\alpha }_n{\delta }_n,n\ge 0,\end{array}

where \{{\alpha }_n\}\subset (0 ,1) and \{{\delta }_n\}\subset \mathbb{R} such that \sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }, \underset{n\to \mathrm{\infty }}{lim sup}{\delta }_n\le 0. Then \underset{n\to \mathrm{\infty }}{lim}{a}_n=0.

3 Main results

Theorem 1　 Let X_1 and X_2 be two p-uniformly convex and uniformly smooth real Banach spaces, A:X_1\to X_2 be a bounded linear operator with the adjoint operator A^{\ast }. Let S:X_1\to X_1 be a Bregman quasi-nonexpansive mapping and I-S be semiclosed at zero. T:X_2\to X_2 is a firmly nonexpansive mapping. Assume that \mathrm{\Omega }=\{x:x \in F(S ),Ax \in F(T )\}\ne \mathrm{\varnothing }. For any u,x_1\in X_1, the iteration sequence \{x_n\} is defined by:

(3)\begin{array}{l}\{\begin{array}{l}{z}_n=J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n),\\ w_n=J_{X_1^{\ast }}^q(J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n),\\ x_{n+1}=J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n)],\end{array}\end{array}

where \{{\lambda }_n\},\{ r_n\} are sequences of real numbers, \{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\},\{t_n\} are sequences in (0,1) that satisfy the following conditions:

(1) {\alpha }_n+{\beta }_n+{\delta }_n=1,\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0, \sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty };

(2) r_n\in \left(0,{\left({\displaystyle \frac{q}{C_q\Vert A{\Vert }^q}}\right)}^{\frac{1}{q-1}}\right);

(3) .

Then the sequence \{x_n\} generated by (3) converges strongly to x^{\ast }\in \mathrm{\Omega }, where x^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}u.

Proof　It follows from Lemma 1 and Lemma 2 that F(T) and F(S) are both closed and convex, so \mathrm{\Omega } is closed and convex. Then {\mathrm{\Pi }}_{\mathrm{\Omega }}u is well defined.

Suppose x^{\ast }\in \mathrm{\Omega }, which shows that x^{\ast }\in F(S), A x^{\ast }\in F(T). Let y_n=J_{X_1^{\ast }}^q(t_n J_{X_1}^p{w}_n+ (1- t_n)J_{X_1}^p S{w}_n). From (3) and Lemma 3, we have

(4)\begin{array}{rl}{D}_p(w_n,x^{\ast })& =D_p(J_{X_1^{\ast }}^q(J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n),x^{\ast })\\ & =\frac{1}{q}\Vert J_{X_1^{\ast }}^q(J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n){\Vert }^p+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -⟨J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n,x^{\ast }⟩\\ & =\frac{1}{q}\Vert J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n{\Vert }^q+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -⟨J_{X_1}^p{z}_n,x^{\ast }⟩+r_n⟨A^{\ast }{J}_{X_2}^p(I-T)A{z}_n,x^{\ast }⟩\\ & \le \frac{1}{q}\Vert J_{X_1}^p{z}_n{\Vert }^q-r_n⟨A^{\ast }{J}_{X_2}^p(I-T)A{z}_n,z_n⟩+\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\Vert J_{X_2}^p(I-T)A{z}_n{\Vert }^q\\ & +\frac{1}{p}\Vert x^{\ast }{\Vert }^p-⟨J_{X_1}^p{z}_n,x^{\ast }⟩+r_n⟨A^{\ast }{J}_{X_2}^p(I-T)A{z}_n,x^{\ast }⟩\\ & =D_p(z_n,x^{\ast })+\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\Vert (I-T)A{z}_n{\Vert }^p+r_n⟨A^{\ast }{J}_{X_2}^p(I-T)A{z}_n,x^{\ast }-z_n⟩.\end{array}

T is a firmly nonexpansive mapping, therefore

(5)\begin{array}{rl}⟨A^{\ast }{J}_{X_2}^p(I-T)A{z}_n,x^{\ast }-z_n⟩& =⟨J_{X_2}^p(I-T)A{z}_n,A{x}^{\ast }-A{z}_n⟩\\ & \le ⟨J_{X_2}^p(I-T)A{z}_n,A{x}^{\ast }-TA{z}_n⟩\\ & +⟨J_{X_2}^p(I-T)A{z}_n,TA{z}_n-A{z}_n⟩\\ & =\Vert (I-T)A{z}_n{\Vert }^p+⟨J_{X_2}^p(I-T)A{z}_n,A{x}^{\ast }-TA{z}_n⟩\\ & \le -\Vert (I-T)A{z}_n{\Vert }^p.\end{array}

From (4), (5) and condition (2), we obtain

(6)\begin{array}{rl}{D}_p(w_n,x^{\ast })& =D_p(z_n,x^{\ast })-\left(r_n-\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\right)\Vert (I-T)A{z}_n{\Vert }^p\\ & \le D_p(z_n,x^{\ast }).\end{array}

Furthermore, it follows from Lemma 4, we have

\begin{array}{rl}{D}_p(z_n,x^{\ast })& =D_p(J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n),x^{\ast })\\ & \le {\lambda }_n{D}_p(x_n,x^{\ast })+(1-{\lambda }_n)D_p(w_n,x^{\ast })\\ & \le {\lambda }_n{D}_p(x_n,x^{\ast })+(1-{\lambda }_n)D_p(z_n,x^{\ast }).\end{array}

Since , we have

(7)\begin{array}{l}{D}_p(z_n,x^{\ast })\le D_p(x_n,x^{\ast }).\end{array}

Combining (6) and (7), we have

\begin{array}{rl}{D}_p(y_n,x^{\ast })& =D_p(J_{X_1^{\ast }}^q(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n),x^{\ast })\\ & \le t_n{D}_p(w_n,x^{\ast })+(1-t_n)D_p(S{w}_n,x^{\ast })\\ & \le t_n{D}_p(w_n,x^{\ast })+(1-t_n)D_p(w_n,x^{\ast })\\ & \le D_p(x_n,x^{\ast }).\end{array}

Hence

\begin{array}{rl}{D}_p(x_{n+1},x^{\ast })& =D_p(J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n],x^{\ast })\\ & \le {\alpha }_n{D}_p(u,x^{\ast })+{\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(y_n,x^{\ast })\\ & \le (1-{\alpha }_n)D_p(x_n,x^{\ast })+{\alpha }_n{D}_p(u,x^{\ast })\\ & \le max\{D_p(x_1,x^{\ast }),D_p(u,x^{\ast })\}.\end{array}

This shows that D_p(x_n,x^{\ast }) is bounded, so \{x_n\} is bounded. Then \{w_n\}, \{z_n\}, \{y_n\} is also bounded.

It follows from Lemma 5 that

(8)\begin{array}{rl}{D}_p(y_n,x^{\ast })& =D_p(J_{X_1^{\ast }}^q(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n),x^{\ast })\\ & =\frac{1}{q}\Vert J_{X_1^{\ast }}^q(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n){\Vert }^p+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -⟨t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n,x^{\ast }⟩\\ & =\frac{1}{q}\Vert t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n{\Vert }^q+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -t_n⟨J_{X_1}^p{w}_n,x^{\ast }⟩-(1-t_n)⟨J_{X_1}^{p}S{w}_n,x^{\ast }⟩\\ & \le \frac{t_n}{q}\Vert J_{X_1}^p{w}_n{\Vert }^q+\frac{1-t_n}{q}\Vert J_{X_1}^{p}S{w}_n{\Vert }^q-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\\ & +\frac{1}{p}\Vert x^{\ast }{\Vert }^p-t_n⟨J_{X_1}^p{w}_n,x^{\ast }⟩-(1-t_n)⟨J_{X_1}^{p}S{w}_n,x^{\ast }⟩\\ & \le \frac{t_n}{q}\Vert w_n{\Vert }^p-t_n⟨J_{X_1}^p{w}_n,x^{\ast }⟩+\frac{t_n}{p}\Vert x^{\ast }{\Vert }^p-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\\ & +\frac{1-t_n}{q}\Vert S{w}_n{\Vert }^p-(1-t_n)⟨J_{X_1}^{p}S{w}_n,x^{\ast }⟩+\frac{1-t_n}{p}\Vert x^{\ast }{\Vert }^p\\ & =t_n{D}_p(w_n,x^{\ast })+(1-t_n)D_p(S{w}_n,x^{\ast })-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\\ & \le D_p(w_n,x^{\ast })-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert ).\end{array}

Similarly, according to (6), we have

\begin{array}{rl}{D}_p(w_n,x^{\ast })& \le D_p(z_n,x^{\ast })\\ & =D_p(J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n),x^{\ast })\\ & =\frac{1}{q}\Vert J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n){\Vert }^p+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -⟨{\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n,x^{\ast }⟩\\ & =\frac{1}{q}\Vert {\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n{\Vert }^q+\frac{1}{p}\Vert x^{\ast }{\Vert }^p\\ & -{\lambda }_n⟨J_{X_1}^p{x}_n,x^{\ast }⟩-(1-{\lambda }_n)⟨J_{X_1}^p{w}_n,x^{\ast }⟩\\ & \le \frac{{\lambda }_n}{q}\Vert J_{X_1}^p{x}_n{\Vert }^q+\frac{1-{\lambda }_n}{q}\Vert J_{X_1}^p{w}_n{\Vert }^q-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert )\\ & +\frac{1}{p}\Vert x^{\ast }{\Vert }^p-{\lambda }_n⟨J_{X_1}^p{x}_n,x^{\ast }⟩-(1-{\lambda }_n)⟨J_{X_1}^p{w}_n,x^{\ast }⟩\\ & \le \frac{{\lambda }_n}{q}\Vert x_n{\Vert }^p-{\lambda }_n⟨J_{X_1}^p{x}_n,x^{\ast }⟩+\frac{{\lambda }_n}{p}\Vert x^{\ast }{\Vert }^p-\frac{W_q({\lambda }_n)}{q}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert )\end{array}

\begin{array}{rl}& +\frac{1-{\lambda }_n}{q}\Vert w_n{\Vert }^p-(1-{\lambda }_n)⟨J_{X_1}^p{w}_n,x^{\ast }⟩+\frac{1-{\lambda }_n}{p}\Vert x^{\ast }{\Vert }^p\\ =& {\lambda }_n{D}_p(x_n,x^{\ast })+(1-{\lambda }_n)D_p(w_n,x^{\ast })-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert ),\end{array}

so

(9)D_p(w_n,x^{\ast })\le D_p(x_n,x^{\ast })-\frac{W_q(t_n)}{q{\lambda }_n}g( \Vert J_{X_1}^p x_n-J_{X_1}^p w_n\Vert ).

Using Lemma 4, we get

(10)\begin{array}{rl}{D}_p(x_{n+1},x^{\ast })& =D_p(J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n],x^{\ast })\\ & =V_p({\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n,x^{\ast })\\ & \le V_p({\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n-{\alpha }_n(J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast }),x^{\ast })\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & =V_p({\alpha }_n{J}_{X_1}^p{x}^{\ast }+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n,x^{\ast })\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & =D_p(J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^p{x}^{\ast }+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n{J}_{X_1}^p{y}_n],x^{\ast })\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\alpha }_n{D}_p(x^{\ast },x^{\ast })+{\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(y_n,x^{\ast })\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & ={\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(y_n,x^{\ast })+{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩.\end{array}

By (6), (7) and (8), we have

(11)\begin{array}{rl}{D}_p(x_{n+1},x^{\ast })& ={\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(y_n,x^{\ast })+{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n\left(D_p(w_n,x^{\ast })-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\right)\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n\left(D_p(x_n,x^{\ast })-\frac{W_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\right)\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le (1-{\alpha }_n)D_p(x_n,x^{\ast })-\frac{{\delta }_n{W}_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩,\end{array}

and

(12)\begin{array}{rl}{D}_p(x_{n+1},x^{\ast })& ={\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(y_n,x^{\ast })+{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(w_n,x^{\ast })+{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n\left(D_p(z_n,x^{\ast })-\left(r_n-\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\right)\Vert (I-T)A{z}_n{\Vert }^p\right)\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n\left(D_p(x_n,x^{\ast })-\left(r_n-\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\right)\Vert (I-T)A{z}_n{\Vert }^p\right)\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le (1-{\alpha }_n)D_p(x_n,x^{\ast })-{\delta }_n\left(r_n-\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\right)\Vert (I-T)A{z}_n{\Vert }^p\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩.\end{array}

Using (8), (9) and (10), we have

(13)\begin{array}{rl}{D}_p(x_{n+1},x^{\ast })& \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n{D}_p(w_n,x^{\ast })+{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le {\beta }_n{D}_p(x_n,x^{\ast })+{\delta }_n\left(D_p(x_n,x^{\ast })-\frac{W_q(t_n)}{q{\lambda }_n}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert )\right)\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩\\ & \le (1-{\alpha }_n)D_p(x_n,x^{\ast })-\frac{{\delta }_n{W}_q(t_n)}{q{\lambda }_n}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert )\\ & +{\alpha }_n⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩.\end{array}

Let's consider it in two cases.

Case 1: Suppose that for all n\ge \mathbb{N}, \{D_p(x_n, x^{\ast })\} is nonincreasing. This shows that \{D_p(x_n, x^{\ast })\} is convergent. Then

\begin{array}{l}\underset{n\to \mathrm{\infty }}{lim}{D}_p(x_{n+1},x^{\ast })-D_p(x_n,x^{\ast })=0.\end{array}

Therefore, from (11), (12), (13) and the condition (1), when n\to \mathrm{\infty }, there is

\begin{array}{rl}& \frac{{\delta }_n{W}_q(t_n)}{q}g(\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert )\to 0;\\ & {\delta }_n\left(r_n-\frac{C_q(r_n\Vert A\Vert {)}^q}{q}\right)\Vert (I-T)A{z}_n{\Vert }^p\to 0;\\ & \frac{{\delta }_n{W}_q(t_n)}{q{\lambda }_n}g(\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert )\to 0.\end{array}

From the constraints and the continuity of the function g, we get

\begin{array}{l}\underset{n\to \mathrm{\infty }}{lim}\Vert J_{X_1}^p{w}_n-J_{X_1}^{p}S{w}_n\Vert =0;\underset{n\to \mathrm{\infty }}{lim}\Vert (I-T)A{z}_n{\Vert }^p=0;\underset{n\to \mathrm{\infty }}{lim}\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert =0.\end{array}

Also by (3), when n\to \mathrm{\infty }, we have

\begin{array}{rl}\Vert J_{X_1}^p{z}_n-J_{X_1}^p{w}_n\Vert & =\Vert {\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n-J_{X_1}^p{w}_n\Vert \\ & ={\lambda }_n\Vert J_{X_1}^p{x}_n-J_{X_1}^p{w}_n\Vert \to 0,\end{array}

\begin{array}{rl}\Vert J_{X_1}^p{y}_n-J_{X_1}^p{w}_n\Vert & =\Vert t_n{J}_{X_1}^p{w}_n+(1-{\lambda }_n)J_{X_1}^{p}S{w}_n-J_{X_1}^p{w}_n\Vert \\ & =(1-t_n)\Vert J_{X_1}^{p}S{w}_n-J_{X_1}^p{w}_n\Vert \to 0.\end{array}

Further, we can get

\begin{array}{rl}\Vert J_{X_1}^p{x}_{n+1}-J_{X_1}^p{x}_n\Vert & ={\alpha }_n\Vert J_{X_1}^{p}u-J_{X_1}^p{x}_n\Vert +{\delta }_n\Vert J_{X_1}^p{y}_n-J_{X_1}^p{y}_n\Vert \\ & \le {\alpha }_n\Vert J_{X_1}^{p}u-J_{X_1}^p{x}_n\Vert +{\delta }_n\Vert J_{X_1}^p{y}_n-J_{X_1}^p{w}_n\Vert \\ & +{\delta }_n\Vert J_{X_1}^p{w}_n-J_{X_1}^p{x}_n\Vert \\ & \to 0.\end{array}

Since J_{X_1^{\ast }}^q is uniformly norm continuous on X_1^{\ast }, when n\to \mathrm{\infty },

(14)\begin{array}{l}\Vert w_n-S{w}_n\Vert \to 0;\Vert x_n-w_n\Vert \to 0;\Vert z_n-w_n\Vert \to 0;\Vert y_n-w_n\Vert \to 0;\Vert x_{n+1}-x_n\Vert \to 0.\end{array}

It shows that

(15)\begin{array}{l}\underset{n\to \mathrm{\infty }}{lim}\Vert z_n-x_n\Vert \le \underset{n\to \mathrm{\infty }}{lim}(\Vert z_n-w_n\Vert +\Vert w_n-x_n\Vert )=0;\\ \underset{n\to \mathrm{\infty }}{lim}\Vert y_n-x_n\Vert \le \underset{n\to \mathrm{\infty }}{lim}(\Vert y_n-w_n\Vert +\Vert w_n-x_n\Vert )=0.\end{array}

Since X_1 is a p-uniformly convex and uniformly smooth Banach space and \{x_n\} is bounded, there exits a subsequesce \{x_{n_j}\} of \{x_n\}, which weak convergence to \overline{x}. By (14) and (15), there exist \{z_n\} and \{w_n\} subsequesce \{z_{n_j}\}, \{w_{n_j}\} satisfying z_{n_j}\rightharpoonup \overline{x}, w_{n_j}\rightharpoonup \overline{x}, respectively. Because A is a bounded linear operator, so A{z}_{n_j}\rightharpoonup A\overline{x}. By \underset{n\to \mathrm{\infty }}{lim}\Vert (I-T)A{z}_n{\Vert }^p= 0, we know that \Vert (I-T)A{z}_n\Vert =0. Hence A\overline{x}\in \hat{F}(T)=F(T). And according to (14) that \Vert w_n-S w_n\Vert \to 0 and I-S semiclosed at zero, we have \overline{x}\in F(S), then \overline{x}\in \mathrm{\Omega }.

Next, we prove that \{x_n\} strong convergence to x^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}u. Formula (13) indicates that

(16)\begin{array}{l}{D}_p(x_{n+1},x^{\ast })\le (1-{\alpha }_n)D_p(x_n,x^{\ast })+{\alpha }_n(J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }).\end{array}

Because x_{n_j}\rightharpoonup \overline{x}, it is obtained from (14) that

(17)\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim sup}⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n+1}-x^{\ast }⟩& =\underset{j\to \mathrm{\infty }}{lim sup}⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{n_j+1}-x^{\ast }⟩\\ & =⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },\overline{x}-x^{\ast }⟩\le 0.\end{array}

By (16), (17) and Lemma 8, we get \underset{n\to \mathrm{\infty }}{lim}{D}_p(x_n,x^{\ast })=0. Thus by Lemma 6, x_n\to x^{\ast } ={\mathrm{\Pi }}_{\mathrm{\Omega }}u.

Case 2: Assume that D_p(x_n,x^{\ast }) does not eventually decrease monotonically. Then there exists a subsequence \{n_j\} such that

\begin{array}{l}{D}_p(x_{n_j},x^{\ast })\le D_p(x_{n_j+1},x^{\ast }),\mathrm{\forall }j\in \mathbb{N}.\end{array}

It follows from Lemma (7) that there exists a nondecreasing sequence \{m_i\}\subset \mathbb{N} such that for m_i\to \mathrm{\infty } and for all i\in \mathbb{N},

\begin{array}{l}{D}_p(x_{m_i},x^{\ast })\le D_p(x_{m_i+1},x^{\ast }),D_p(x_i,x^{\ast })\le D_p(x_{m_i+1},x^{\ast }).\end{array}

Following the similar proof in Case 1, we have

(18)\begin{array}{l}{D}_p(x_{m_i+1},x^{\ast })\le (1-{\alpha }_{m_i})D_p(x_{m_i},x^{\ast })+{\alpha }_{m_i}⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{m_i+1}-x^{\ast }⟩,\end{array}

and

(19)\begin{array}{l}\underset{n\to \mathrm{\infty }}{lim sup}⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{m_i+1}-x^{\ast }⟩\le 0.\end{array}

Since D_p(x_{m_i},x^{\ast })\le D_p(x_{m_i+1},x^{\ast } ), it follows from (18) that

\begin{array}{rl}0& \le D_p(x_{m_i+1},x^{\ast })-D_p(x_{m_i},x^{\ast })\\ & \le (1-{\alpha }_{m_i})D_p(x_{m_i},x^{\ast })+{\alpha }_{m_i}⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{m_i+1}-x^{\ast }⟩-D_p(x_{m_i},x^{\ast }),\end{array}

which implies that

\begin{array}{l}{D}_p(x_{m_i},x^{\ast })\le ⟨J_{X_1}^{p}u-J_{X_1}^p{x}^{\ast },x_{m_i+1}-x^{\ast }⟩.\end{array}

From (19), we get

\begin{array}{l}\underset{i\to \mathrm{\infty }}{lim}{D}_p(x_{m_i},x^{\ast })=0.\end{array}

By (18), we have \underset{i\to \mathrm{\infty }}{lim}{D}_p(x_{m_i+1},x^{\ast })=0. Because for all i\in \mathbb{N}, D_p(x_i,x^{\ast })\le D_p(x_{m_i+1},x^{\ast } ), we have \underset{i\to \mathrm{\infty }}{lim}{D}_p(x_i, x^{\ast })=0. Thus x_i strongly converges to x^{\ast }. Eventually, in both cases we get x_n\to x^{\ast }, and the proof is completed.

4 Application

4.1 Split fixed point problem and zero point problem

Suppose that B:X\to 2^{X^{\ast }} is a mapping, and G(B):= \{(x,y)\in X\times X^{\ast },y\in Bx\} is defined as the graph of B. B is said to be monotone, \mathrm{\forall }x,y\in \mathrm{d}\mathrm{o}\mathrm{m}(B), if x^{\ast }\in Bx, y^{\ast }\in By, we have ⟨x-y,x^{\ast }-y^{\ast }⟩\ge 0. We call B a maximal monotone operator if the graph of B is not a proper subset of the graph of any other monotone operator. The Bregman resolvent operator of B is defined by:

\begin{array}{l}Re{s}_B=(J_X^p+B{)}^{-1}\circ J_X^p.\end{array}

Since Re{s}_B is Bregman relatively nonexpansive and F(Re{s}_B)=B^{-1}(0), where B^{-1}(0)=\{x\in X ,0\in Bx \} is closed convex [1]. applying Theorem 1, we obtain the result for the zero point problem of maximal monotone operators.

Theorem 2　 Let X_1 and X_2 be two p-uniformly convex and uniformly smooth real Banach spaces, A:X_1\to X_2 be a bounded linear operator with the adjoint operator A^{\ast }. Let B:X_1\to 2^{X_1^{\ast }} be a maximal monotone mapping and T:X_2\to X_2 be a definite nonexpansive mapping. Assume that \mathrm{\Gamma }=\{x:x \in B^{-1}(0) ,Ax\in F (T)\}\ne \mathrm{\varnothing }. For any u,x_1\in X_1, the iteration sequence \{x_n\} is defined by:

(20)\begin{array}{l}\{\begin{array}{l}{z}_n=J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n),\\ w_n=J_{X_1^{\ast }}^q(J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-T)A{z}_n),\\ x_{n+1}=J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}Re{s}_B{w}_n)],\end{array}\end{array}

where \{{\lambda }_n\},\{ r_n\} are sequences of real numbers, \{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\},\{t_n\} are sequences in (0,1) satisfy the following conditions:

(1) {\alpha }_n+{\beta }_n+{\delta }_n=1,\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0, \sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty };

(2) r_n\in \left(0,{\left(\frac{q}{C_q\Vert A{\Vert }^q}\right)}^{\frac{1}{q-1}}\right);

(3)

Then the sequence \{x_n\} generated by (20) converges strongly to x^{\ast }\in \mathrm{\Omega }, where x^{\ast }={\mathrm{\Pi }}_{\mathrm{\Gamma }}u.

4.2 Split fixed point problem and equilibrium problem

Assuming that E is a nonempty closed convex subset of X. Let the bifunctional \mathrm{\Phi }:E\times E\to \mathbb{R} satisfy the following conditions :

(A1) for all x\in E, \mathrm{\Phi }(x,x) =0;

(A2) for all x,z\in E, \mathrm{\Phi } is monotonous, that is \mathrm{\Phi }(x,z )+\mathrm{\Phi }(z, x)\le 0;

(A3) for all x,y,z\in E, \underset{t\to 0}{lim sup}\mathrm{\Phi } (ty+ (1-t)x, z)\le \mathrm{\Phi }(x,z);

(A4) for all x\in E, \mathrm{\Phi }(x,\cdot ) is convex and lower semi-continuous.

The equilibrium problem with \mathrm{\Phi } is to find a point u\in E for all v\in E such that

\begin{array}{l}\mathrm{\Phi }(u,v)\ge 0.\end{array}

Lemma 9 [21]　 Assume that E is a nonempty closed convex subset of the uniformly convex and uniformly smooth real Banach space X. \mathrm{\Phi }:E\times E\to \mathbb{R} is a bifunctional satisfying condition (A1)−(A4). For r>0, x\in E, the mapping F_r is defined by:

F_r(x)=\left\{z\in E:\mathrm{\Phi }(z,y)+\frac{1}{r}⟨J_X^{p}z-J_X^{p}x,y-z⟩\ge 0,\mathrm{\forall }y\in E\right\}.

Then the following holds:

(1) F_r is single-valued;

(2) F_r is a Bregman firmly nonexpansive mapping;

(3) F(F_r)=\hat{F}(F_r)=EP (\mathrm{\Phi } );

(4) EP(\mathrm{\Phi }) is closed and convex;

(5) for all x\in X,q\in EP(\mathrm{\Phi }), D_g(q, F_r(x))+ D_g(F_r(x) ,x)\le D_g(q,x).

In Theorem 1, we let T=F_r. Then we obtain the result for the split fixed point problem and the equilibrium problem.

Theorem 3　 Let X_1 and X_2 be two p-uniformly convex and uniformly smooth real Banach spaces, and E be a nonempty closed convex subset of X_2. Let A:X_1\to X_2 be a bounded linear operator with the adjoint operator A^{\ast }. Let S:X_1\to X_1 be a Bregman quasi-nonexpansive mapping and I- S be semiclosed at zero. \mathrm{\Phi }: E\times E\to \mathbb{R} is a bifunctional satisfying condition (A1)−(A4). Assume that \mathrm{\Gamma } =\{x :x\in F(S) ,Ax\in E P(\mathrm{\Phi })\}\ne \mathrm{\varnothing }. For any u,x_1\in X_1, the iteration sequence \{x_n\} is defined by:

(21)\begin{array}{l}\{\begin{array}{l}{z}_n=J_{X_1^{\ast }}^q({\lambda }_n{J}_{X_1}^p{x}_n+(1-{\lambda }_n)J_{X_1}^p{w}_n),\\ w_n=J_{X_1^{\ast }}^q(J_{X_1}^p{z}_n-r_n{A}^{\ast }{J}_{X_2}^p(I-F_r)A{z}_n),\\ x_{n+1}=J_{X_1^{\ast }}^q[{\alpha }_n{J}_{X_1}^{p}u+{\beta }_n{J}_{X_1}^p{x}_n+{\delta }_n(t_n{J}_{X_1}^p{w}_n+(1-t_n)J_{X_1}^{p}S{w}_n)],\end{array}\end{array}

where \{{\lambda }_n\},\{ r_n\} are sequences of real numbers, \{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\},\{t_n\} are sequences in (0,1) satisfy the following conditions:

(1) {\alpha }_n+{\beta }_n+{\delta }_n=1,\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0, \sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty };

(2) r_n\in \left(0,{\left(\frac{q}{C_q\Vert A{\Vert }^q}\right)}^{\frac{1}{q-1}}\right);

(3)

Then the sequence \{x_n\} generated by (21) converges strongly to x^{\ast }\in \mathrm{\Omega }, where x^{\ast }={\mathrm{\Pi }}_{\mathrm{\Gamma }}u.

5 Conclusion

In this paper, we mainly study the problem of split common fixed points problem in p-uniformly convex and uniformly smooth real Banach spaces by using implicit rules, and we give the convergence analysis of the theorems. Under appropriate parameter conditions, we prove the iterative sequence (1) of Bregman quasi-nonexpansive mapping converges strongly to the solution of the split common fixed points problem. Finally, the results are applied to the zero point problem and the equilibrium problem. In Theorem 1, setting {\lambda }_n= \frac{1}{2} leads to the conclusion about the split common fixed point of the implicit midpoint rule, and setting T=P_Q leads to the strong convergence theorems about the split feasibility problem and the fixed point problem.

In this paper, we study in the framework of p-uniformly convex and uniformly smooth real Banach spaces, and extend some related results in Hilbert spaces, uniformly convex and 2-uniformly smooth Banach spaces. We note that Bregman relatively nonexpansive mappings satisfy \hat{F}(T)=F (T). In Theorem 1, the convergence result still holds if the mapping S is a Bregman relatively nonexpansive mapping or a firmly nonexpansive mapping. In Theorem 1, if X is a Hilbert space and {\lambda }_n= 1, then it is the main result of Padcharoen et al. [13].

The split common fixed point problem studied in this paper is more extensive than the split feasibility problem and the split feasibility fixed point problem. We know that the firmly nonexpansive mappings contain projection operators. If T=P_Q, Theorem (1) is still extend and improve the main results of Ma et al. [10], Pant et al. [16] and Shehuet al. [18]. The results and methods presented in this paper are also include some corresponding recent results [3,12,19,24].

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