An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces

Abstract

In this paper, we propose an iterative technique with residual vectors for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of a split inclusion problem (SIP) with a way of selecting the stepsizes without prior knowledge of the operator norm in the framework of p-uniformly convex and uniformly smooth Banach spaces. Then strong convergence of the proposed algorithm to a common element of the above two sets is proved. As applications, we apply our result to find the set of common fixed points of a family of mappings which is also a solution of the SIP. We also give a numerical example and demonstrate the efficiency of the proposed algorithm. The results presented in this paper improve and generalize many recent important results in the literature.

1 Introduction

Let $H_1$ and $H_2$ be two Hilbert spaces. Let $B_1:H_1\multimap H_1$ and $B_2:H_2\multimap H_2$ be two set-valued maximal monotone operators and $A:H_1\rightarrow H_2$ be a bounded linear operator. Moudafi (2011) introduced the following so-called split inclusion problem (SIP):

$$\begin{aligned} \text {Find}~~x^*\in H_1~~\text {such that}~~0\in B_1(x^*)~~\text {and}~~0\in B_2(Ax^*). \end{aligned}$$

(1.1)

The set of solutions of problem (1.1) is denoted by $\Gamma$, i.e., $\Gamma :=\{x^*\in H_1:x^*\in B_{1}^{-1}(0)~~\text {and}~~Ax^*\in B_{2}^{-1}(0)\}$. In fact, we know that the split inclusion problem is a generalization of the inclusion problem and the split feasibility problem. Next, we provide some special cases of SIP (1.1).

• Let $f:H_1\rightarrow {\mathbb {R}}\cup \{+\infty \}$ and $g:H_2\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be proper, lower semicontinuous and convex functions. If we take $B_1=\partial f$ and $B_2=\partial g$, where $\partial f$ and $\partial g$ are the subdifferential of f and g, then the SIP (1.1) becomes the following so-called proximal split feasibility problem:

  $$\begin{aligned} \text {Find}~~x^*\in \text {argmin}~f~~\text {such that}~~Ax^*\in \text {argmin}~g, \end{aligned}$$

  (1.2)

  where argmin $f=\{x\in H_1: f(x)\le f(y),~\forall y\in H_1\}$ and argmin $g=\{x\in H_2: g(x)\le g(y),~\forall y\in H_2\}$. In particular, if we take $f(x)=\frac{1}{2}\Vert Mx-b\Vert ^2$ and $g(x) =\frac{1}{2}\Vert Nx-c\Vert ^2$, where M and N are matrices, and $b,c\in H_1$, then the SIP (1.2) becomes the least square problem. This problem has been intensively studied, especially, in Hilbert spaces; see for instance (Moudafi and Thakur 2014).

• Let C and Q be nonempty, closed, and convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. If $B_1 = N_C$, $B_2 = N_Q$, where $N_C$ and $N_Q$ are the normal cones of C and Q, respectively, then the SIP (1.2) becomes the following so-called split feasibility problem:

  $$\begin{aligned} \text {Find}~~x^*\in C~~\text {such that}~~Ax^*\in Q. \end{aligned}$$

  (1.3)

This problem was first introduced, in a finite dimensional Hilbert space, by Censor and Elfving (1994) for modeling inverse problems in radiation therapy treatment planning which arise from phase retrieval and in medical image reconstruction, especially intensity modulated therapy (Censor et al. 2006).

To solve the SIP (1.1), Byrne et al. (2011) gave the following convergence theorem in infinite dimensional Hilbert spaces:

Theorem 1.1

Let $H_1$ and $H_2$ be real Hilbert spaces, $A : H_1\rightarrow H_2$ be a bounded linear operator with its adjoint operator $A^*$. Let $B_1 : H_1\multimap H_1$ and $B_2 : H_2\multimap H_2$ be set-valued maximal monotone mappings, $\lambda >0$ and $\gamma \in \big (0,\frac{2}{\Vert A\Vert ^2}\big )$. Suppose that $\Gamma \ne \emptyset$. For given $x_1\in H_1$, let $\{x_n\}$ be the sequence defined by

$$\begin{aligned} x_{n+1}=J_{\lambda }^{B_1}(x_n-\gamma A^*(I-J_{\lambda }^{B_2})Ax_n),~~\forall n\ge 1. \end{aligned}$$

(1.4)

Then $\{x_n\}$ converges weakly to an element $x^*\in \Gamma$.

In order to obtain strong convergence, Kazmi and Rizvi (2014) proposed an algorithm for solving SIP (1.1) with fixed points of a nonexpansive mapping T. They obtained the following result:

Theorem 1.2

Let $H_1$ and $H_2$ be real Hilbert spaces. Let $A : H_1\rightarrow H_2$ be a bounded linear operator and $f : H_1 \rightarrow H_1$ be a contraction mapping with a constant $\alpha \in (0, 1)$. Let $B_1 : H_1\multimap H_1$ and $B_2 : H_2\multimap H_2$ be set-valued maximal monotone mappings, $\lambda >0$. Let $T : H_1 \rightarrow H_1$ be a nonexpansive mapping such that $F(T)\cap \Gamma \ne \emptyset$. For a given $x_1\in H_1$ arbitrarily, let the iterative sequences $\{u_n\}$ and $\{x_n\}$ be generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} u_n=J_{\lambda }^{B_1}(x_n-\gamma A^*(I-J_{\lambda }^{B_2})Ax_n),\\ x_{n+1}=\alpha _n f(x_n)+(1-\alpha _n)Tu_n,~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(1.5)

where $\gamma \in \big (0,\frac{2}{\Vert A\Vert ^2}\big )$ and $\{\alpha _n\}$ is a sequence in (0, 1) such that $\lim _{n\rightarrow \infty }\alpha _n=0$, $\sum _{n=1}^{\infty }\alpha _n=\infty$ and $\sum _{n=1}^{\infty }|\alpha _{n+1}-\alpha _n|<\infty$. Then the sequences $\{u_n\}$ and $\{x_n\}$ both converge strongly to $x^*\in F(T)\cap \Gamma$, where $x^*=P_{F(T)\cap \Gamma }f(x^*)$.

On the other hand, Takahashi and Takahashi (2016) first introduced the SIP outside Hilbert spaces. Let $E_1$ and $E_2$ be two Banach spaces. Let $B_1:E_1\multimap E_1$ and $B_2:E_2\multimap E_2$ be two set-valued maximal monotone operators and $A:E_1\rightarrow E_2$ be a bounded linear operator. They proposed the SIP in Banach spaces as follows:

$$\begin{aligned} \text {Find}~~x^*\in E_1~~\text {such that}~~0\in B_1(x^*)~~\text {and}~~0\in B_2(Ax^*). \end{aligned}$$

(1.6)

In recent years, many authors have constructed several iterative methods for solving SIP (see, e.g., Sitthithakerngkiet et al. 2018; Takahashi and Takahashi 2016; Takahashi 2015, 2017; Takahashi and Yao 2015; Suantai et al. 2018; Jailoka and Suantai 2017; Ogbuisi and Mewomo 2017; Alofi et al. 2016).

Very recently, Alofi et al. (2016) introduced an algorithm based on Halpern’s iteration for solving SIP (1.1) in a uniformly convex and smooth Banach space. They proved the following strong convergence theorem:

Theorem 1.3

Let H be a Hilbert space and let E be a uniformly convex and smooth Banach space. Let $J_E$ be the duality mapping on E. Let $B_1:H\multimap H$ and $B_2:E\multimap E^*$ be maximal monotone operators, respectively. Let $J_{\lambda }^{B_1}$ be the resolvent of $B_1$ for $\lambda >0$ and let $J_{\mu }^{B_2}$ be the metric resolvent of B for $\mu > 0$. Let $A:H\rightarrow E$ be a bounded linear operator with its adjoint $A^*$ such that $A\ne 0$. Suppose that $\Gamma \ne \emptyset$. Let $\{u_n\}$ be a sequence in H such that $u_n\rightarrow u$. Let $x_1\in H$ and let $\{x_n\}\subset H$ be a sequence generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} y_n=\alpha _nu_n+(1-\alpha _n)J_{\lambda _n}^{B_1}(x_n-\lambda _n A^*(I-J_{\mu _n}^{B_2})Ax_n),\\ x_{n+1}=\beta _nx_n+(1-\beta _n)y_n,~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(1.7)

where $\{\lambda _n\}$, $\{\mu _n\}\subset (0,\infty )$, $\{\alpha _n\}\subset (0,1)$ and $\{\beta _n\}\subset (0,1)$ satisfy the following conditions:

$$\begin{aligned}&\lim _{n\rightarrow \infty }\alpha _n=0 \quad \text {and}\quad \sum _{n=1}^{\infty }\alpha _n=\infty \\&\quad 0<a\le \lambda _n\Vert A\Vert ^2\le b<2,~~~0<k\le \mu _n,~~0<c\le \beta _n\le d<1, \end{aligned}$$

for some $a,b,c,d\in {\mathbb {R}}$. Then $\{x_n\}$ converges strongly to $x^*\in \Gamma$, where $x^*=P_{\Gamma }u$.

However, it is observed that several iterative methods suggested require the computation of the norm of the bounded linear operator $\Vert A\Vert$, which may not be calculated easily in general. In this work, motivated by the previous works, we introduce an iterative technique with residual vectors for solving the fixed point problem of a relatively nonexpansive mapping and SIP with a way of selecting the step sizes without prior knowledge of the operator norm in the framework of p-uniformly convex and uniformly smooth Banach spaces. We prove its strong convergence of proposed algorithm to a common element of the set fixed points of a relatively nonexpansive mapping and the solutions of the SIP. As applications, we apply our result to finding the set of common fixed points of a family of mappings which is also a solution of the SIP. We also give some numerical examples and demonstrate the efficiency of the proposed algorithm. The results obtained in this paper improve and generalize many known results in the literature.

2 Preliminaries

Let E and $E^*$ be real Banach spaces and the dual space of E, respectively. Let $E_1$ and $E_2$ be real Banach spaces and let $A : E_1 \rightarrow E_2$ be a bounded linear operator with its adjoint operator $A^* : E_{2}^{*}\rightarrow E_{1}^{*}$ which is defined by

$$\begin{aligned} \langle A^*{\bar{y}},x\rangle :=\langle {\bar{y}},Ax\rangle ,~~\forall x\in E_1,~~{\bar{y}}\in E_{2}^{*}. \end{aligned}$$

The modulus of convexity of E is the function $\delta _E:(0,2]\rightarrow [0,1]$ defined by

$$\begin{aligned} \begin{array}{lll} \delta _E(\epsilon )=\inf \bigg \{1-\frac{\Vert x+y\Vert }{2}:\Vert x\Vert =\Vert y\Vert =1, \Vert x-y\Vert \ge \epsilon \bigg \}. \end{array} \end{aligned}$$

The modulus of smoothness of E is the function $\rho _E:{\mathbb {R}}^{+}:=[0,\infty )\rightarrow {\mathbb {R}}^{+}$ defined by

$$\begin{aligned} \begin{array}{lll} \rho _E(\tau )=\bigg \{\frac{\Vert x+\tau y\Vert +\Vert x- \tau y\Vert }{2}-1:\Vert x\Vert =\Vert y\Vert =1\bigg \}. \end{array} \end{aligned}$$

Definition 2.1

A Banach space E is said to be

• 1. uniformly convex if $\delta _E(\epsilon )>0$ for all $\epsilon \in (0,2]$;

• 2. p-uniformly convex (or to have a modulus of convexity of power type p) if there is a $c_p > 0$ such that $\delta _E(\epsilon )\ge c_p\epsilon ^p$ for all $\epsilon \in (0,2]$;

• 3. uniformly smooth if $\lim _{\tau \rightarrow 0}\frac{\rho _{E}(\tau )}{\tau }=0$;

• 4. q-uniformly smooth if there exists a $c_q > 0$ such that $\rho _E(\tau )\le c_q\tau ^q$ for all $\tau >0$.

From the Definition 2.1, we observe that every p-uniformly convex space is uniformly convex and if E is q-uniformly smooth, then E is also uniformly smooth. It is known that (Agarwal et al. 2009)

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} E~\text {is}~p\text {-uniformly convex if and only if}~E^*~\text {is}~q\text {-uniformly smooth},\\ E~\text {is}~q\text {-uniformly smooth if and only if}~E^*~\text {is}~p\text {-uniformly convex}, \end{array} \end{array}\right. } \end{aligned}$$

(2.1)

where $p\ge 2$ and $1<q\le 2$ are conjugate exponents, i.e., p, q satisfy $\frac{1}{p} + \frac{1}{q} = 1$ (see Xu and Roach 1991). For the sequence spaces $\ell _p$, Lebesgue spaces $L_p$ and Sobolev spaces $W_{p}^{m}$, we also know that (Agarwal et al. 2009; Hanner 1956; Xu and Roach 1991)

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} \ell _p,~L_p~\text {and}~W_{p}^{m}~\text {are}~2\text {-uniformly convex and }p\text {-uniformly smooth}~\text {with}~1<p\le 2,\\ \ell _p,~L_p~\text {and}~W_{p}^{m}~\text {are}~q\text {-uniformly convex and }2\text {-uniformly smooth}~\text {with}~2\le q<\infty . \end{array} \end{array}\right. } \end{aligned}$$

Definition 2.2

A continuous strictly increasing function $\varphi :{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ is said to be a gauge if $\varphi (0)=0$ and $\lim _{t\rightarrow \infty }\varphi (t)=\infty$.

Definition 2.3

The mapping $J_{\varphi }^{E}:E\multimap E^*$ associated with a gauge function $\varphi$ defined by

$$\begin{aligned} J_{\varphi }^{E} (x)=\{f\in E^*:\langle x,f\rangle =\Vert x\Vert \varphi (\Vert x\Vert ),~\Vert f\Vert =\varphi (\Vert x\Vert ),~~\forall x\in E\}, \end{aligned}$$

is called the duality mapping with gauge $\varphi$, where $\langle \cdot ,\cdot \rangle$ denotes the duality pairing between E and $E^*$.

If $\varphi (t)=t$, then $J_{\varphi }^{E}= J_2^{E}=J$ is the normalized duality mapping. In particular, $\varphi (t)=t^{p-1}$, where $p>1$, the duality mapping $J_{\varphi }^{E}=J_{p}^{E}$ is called the generalized duality mapping which is defined by

$$\begin{aligned} \begin{array}{lcl} J_{p}^{E}(x)=\{f\in E^*:\langle x,f\rangle =\Vert x\Vert ^p,\Vert f\Vert =\Vert x\Vert ^{p-1}\}. \end{array} \end{aligned}$$

It is well known that if E is uniformly smooth, the generalized duality mapping $J_{p}^{E}$ is norm to norm uniformly continuous on bounded subsets of E (see Reich 1981). Furthermore, $J_{p}^{E}$ is one-to-one, single-valued and satisfies $J_{p}^{E}=(J_{q}^{E^*})^{-1}$, where $J_{q}^{E^*}$ is the generalized duality mapping of $E^*$ (see Reich 1992; Cioranescu 1990 for more details).

For a gauge $\varphi$, the function $\Phi :{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ defined by

$$\begin{aligned} \Phi (t)=\int _{0}^{t}\varphi (s)\mathrm{d}s \end{aligned}$$

is a continuous convex strictly increasing differentiable function on ${\mathbb {R}}^{+}$ with $\Phi '(t)=\varphi (t)$ and $\lim _{t\rightarrow \infty }\frac{\Phi (t)}{t}=\infty$. Therefore, $\Phi$ has a continuous inverse function $\Phi ^{-1}$.

We next recall the Bregman distance, which was introduced and studied in Bregman (1967).

Definition 2.4

Let E be a real smooth Banach space. The Bregman distance $D_\varphi (x,y)$ between x and y in E is defined by

$$\begin{aligned} D_\varphi (x,y)=\Phi (\Vert y\Vert )-\Phi (\Vert x\Vert )-\langle J_\varphi (x), y-x\rangle . \end{aligned}$$

We note that the Bregman distance $D_\varphi$ does not satisfy the well-known properties of a metric because $D_\varphi$ is not symmetric and does not satisfy the triangle inequality. Moreover, the Bregman distance has the following important properties:

$$\begin{aligned}&D_\varphi (x,y)= D_\varphi (x,z)+ D_\varphi (z,y)+\langle J_{\varphi }^{E}x-J_{\varphi }^{E}z,z-y\rangle , \end{aligned}$$

(2.2)

$$\begin{aligned}&D_\varphi (x,y)+ D_\varphi (y,x)=\langle J_{\varphi }^{E}x-J_{\varphi }^{E}y,x-y\rangle , \end{aligned}$$

(2.3)

for all $x,y,z\in E$.

In the case $\varphi (t)=t^{p-1}$, where $p > 1$, the distance $D_\varphi =D_p$ is called the p-Lyapunov function which was studied in Bonesky et al. (2008) and it is given by

$$\begin{aligned} D_p(x,y)=\frac{1}{q}\Vert x\Vert ^p-\langle J_{p}^{E}x,y\rangle +\frac{1}{p}\Vert y\Vert ^p, \end{aligned}$$

where p, q are conjugate exponents. For the p-uniformly convex space, the Bregman distance has the following relation (see Schöpfer et al. 2008):

$$\begin{aligned} \tau \Vert x-y\Vert ^p\le D_p(x,y)\le \langle J_{p}^{E}x-J_{p}^{E}y,x-y\rangle , \end{aligned}$$

(2.4)

where $\tau >0$ is some fixed number. If $p=2$, we get

$$\begin{aligned} D_2(x,y):=\phi (x,y)=\Vert x\Vert ^2-2\langle Jx,y\rangle +\Vert y\Vert ^2, \end{aligned}$$

where $\phi$ is called the Lyapunov function which was introduced by Alber (1993, 1996).

The following Lemma can be obtained from Theorem 2.8.17 of Agarwal et al. (2009) (see also Lemma 5 of Kuo and Sahu 2013).

Lemma 2.5

Let $p>1$, $r>0$ and E be a Banach space. Then the following statements are equivalent:

• (i) E is uniformly convex;

• (ii) There exists a strictly increasing convex function $g_{r}^{*}:{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ with $g_{r}^{*}(0)=0$ such that

  $$\begin{aligned} \big \Vert \sum _{k=1}^{N}\alpha _kx_k\big \Vert ^p\le \sum _{k=1}^{N}\alpha _k\Vert x_k\Vert ^p-\alpha _i\alpha _jg_{r}^{*}(\Vert x_i-x_j\Vert ), \end{aligned}$$

  for all $i,j\in \{1,2,\ldots ,N\}$, $x_k\in B_r:=\{x\in E:\Vert x\Vert \le r\}$, $\alpha _k\in (0,1)$ with $\sum _{k=1}^{N}\alpha _k=1$, where $k\in \{1,2,\ldots ,N\}$.

Lemma 2.6

(Xu 1991) Let $1<q\le 2$ and E be a Banach space. Then the following statements are equivalent:

• (i) E is q-uniformly smooth;

• (ii) there is a constant $\kappa _q>0$ which is called the q-uniform smoothness coefficient of E such that for all $x,y\in E$

  $$\begin{aligned} \begin{array}{lcl} \Vert x-y\Vert ^q\le \Vert x\Vert ^q-q\langle y,J_q^{E}(x)\rangle +\kappa _q\Vert y\Vert ^q. \end{array} \end{aligned}$$

  (2.5)

In what follows, we shall use the following notations: $x_n\rightarrow x$ means that $\{x_n\}$ converges strongly to x and $x_n\rightharpoonup x$ means that $\{x_n\}$ converges weakly to x. Let C be a closed and convex subset of E and let T be a mapping from C into itself. We denote F(T) by the set of all fixed points of T, i.e., F(T) = $\{x\in C:x=Tx\}$. A point $z\in C$ is called an asymptotic fixed point (Reich 1996) of T, if there exists a sequence $\{x_n\}$ in C which converges weakly to z and $\lim _{n\rightarrow \infty }\Vert x_n-Tx_n\Vert =0$. We denote ${\widehat{F}}(T)$ by the set of asymptotic fixed points of T. A mapping $T : C\rightarrow C$ is called Bregman relatively nonexpansive (Butnariu et al. 2001, 2003; Censor and Reich 1996; Matsushita and Takahashi 2005), if the following conditions are satisfied:

• (R1) $F(T)={\widehat{F}}(T)\ne \emptyset$;

• (R2) $D_p(Tx,z)\le D_p(x,z),~~\forall z\in F(T),~\forall x\in C$.

Let E be a p-uniformly convex Banach space which is also uniformly smooth. Following Censor and Lent (1981) and Alber (1993), we make use of the function $V_p:E^*\times E\rightarrow {\mathbb {R}}^{+}$ which is defined by

$$\begin{aligned} V_p(x^*,x)=\frac{1}{q}\Vert x^*\Vert ^q-\langle x^*,x\rangle +\frac{1}{p}\Vert x\Vert ^p \end{aligned}$$

(2.6)

for all $x\in E$ and $x^*\in E^*$, where p, q are conjugate exponents. Then $V_p$ is nonnegative and convex in the first variable. It is observed that

$$\begin{aligned} V_p(x^*,x)=D_p(J_{q}^{E^*}(x^*),x) \end{aligned}$$

(2.7)

for all $x\in E$ and $x^*\in E^*$. In addition,

$$\begin{aligned} V_p(x^*,x)\le V_p(x^*-y^*,x)+\langle J_{q}^{E^*}(x^*)-x,y^*\rangle \end{aligned}$$

(2.8)

for all $x\in E$ and $x^*\in E^*$.

Lemma 2.7

(Bonesky et al. 2008) Let $p > 1$ and E be a real p-uniformly convex and uniformly smooth Banach space. For $x\in E$ and a sequence $\{x_n\}$ in E. Then, $\lim _{n\rightarrow \infty }D_p(x_n,x)=0\Longleftrightarrow \lim _{n\rightarrow \infty }\Vert x_n-x\Vert =0$.

Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Then we know that for any $x\in E$, there exists a unique element $z\in C$ such that

$$\begin{aligned} D_p(x,z)=\min _{y\in C}D_p(x,y). \end{aligned}$$

(2.9)

The mapping $\Pi _C:E\rightarrow C$ defined by $z=\Pi _{C}x$ is called the generalized projection of E onto C.

Lemma 2.8

(Kuo and Sahu 2013) Let C be a nonempty, closed and convex subset of a p-uniformly convex and uniformly smooth Banach space E and let $x \in E$. Then the following assertions hold:

• (i) $z=\Pi _{C}x$ if and only if $\langle J_{p}^{E}(x)-J_{p}^{E}(z),y-z\rangle \le 0$, $\forall y\in C$.

• (ii) $D_p(\Pi _Cx,y)+D_p(x,\Pi _Cx)\le D_p(x,y)$, $\forall y\in C$.

Let $B:E\multimap E^*$ be a mapping. The effective domain of B is denoted by D(B), i.e., $D(B)=\{x\in E : Bx\ne \emptyset \}$. A multi-valued mapping B is said to be monotone if

$$\begin{aligned} \langle u-v,x-y\rangle \ge 0,~~\forall x,y\in D(B),~u\in Bx~\text {and}~v\in By. \end{aligned}$$

(2.10)

A monotone operator B on E is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on E.

Let E be a p-uniformly convex and uniformly smooth Banach space and let $B:E\multimap E^*$ be a maximal monotone operator. Then, for $x\in E$ and $\lambda > 0$, we define a mapping $Q_{\lambda }^{B}:E\rightarrow D(B)$ by

$$\begin{aligned} Q_{\lambda }^{B}(x):=(I+\lambda (J_{p}^{E})^{-1}B)^{-1}(x)~~\text {for all}~~x\in E. \end{aligned}$$

(2.11)

This mapping is called the metric resolvent of B for $\lambda >0$. The set of null points of B is defined by $B^{-1}(0)=\{z\in E: 0\in Bz\}$. We know that $B^{-1}(0)$ is closed and convex (see Takahashi 2000). We see that

$$\begin{aligned} 0\in J_{p}^{E}(Q_{\lambda }^{B}(x)-x)+\lambda BQ_{\lambda }^{B}(x). \end{aligned}$$

(2.12)

Further, $F(Q_{\lambda }^{B})=B^{-1}(0)$ for $\lambda >0$ (see Zeidler 1984). From Kuo and Sahu (2013), we also know that

$$\begin{aligned} \langle Q_{\lambda }^{B}(x)-Q_{\lambda }^{B}(y),J_{p}^{E}(x-Q_{\lambda }^{B}(x))-J_{p}^{E}(y-Q_{\lambda }^{B}(y))\rangle \ge 0, \end{aligned}$$

(2.13)

for all $x, y \in E$ and if $B^{-1}(0)\ne \emptyset$, then

$$\begin{aligned} \langle J_{p}^{E}(x-Q_{\lambda }^{B}(x)),Q_{\lambda }^{B}(x)-z\rangle \ge 0, \end{aligned}$$

(2.14)

for all $x\in E$ and $z\in B^{-1}(0)$.

In addition, we can define a single-valued mapping $R_{\lambda }^{B}:E\rightarrow D(B)$ so-called the resolvent of B by (Kohsaka and Takahashi 2005)

$$\begin{aligned} R_{\lambda }^{B}(x):=(J_{p}^{E}+\lambda B)^{-1}J_{p}^{E}(x) \quad \text {for all}~~x\in E. \end{aligned}$$

It is known that $R_{\lambda }^{B}$ is a relatively nonexpansive mapping and $F(R_{\lambda }^{B})=B^{-1}(0)$ for $\lambda >0$ (see Kuo and Sahu 2013).

Lemma 2.9

(Kohsaka and Takahashi 2005) Let $B:E\multimap E^*$ be a maximal monotone operator with $B^{-1}(0)\ne \emptyset$ and let $R_{\lambda }^{B}$ be a resolvent operator of B for $\lambda >0$. Then

$$\begin{aligned} D_p (R_{\lambda }^{B}(x),z)+ D_p (R_{\lambda }^{B}(x),x)\le D_p (x,z), \end{aligned}$$

for all $x\in E$ and $z\in B^{-1}(0)$.

The following Theorem is proved by Kohsaka and Takahashi (see Kohsaka and Takahashi 2005, Lemma 7.2).

Lemma 2.10

(Kohsaka and Takahashi 2005) Let $B:E\multimap E^*$ be a monotone operator. Then B is maximal if and only if for each $\lambda >0$,

$$\begin{aligned} R(J_{p}^{E}+\lambda B)=E^*, \end{aligned}$$

where $R(J_{p}^{E}+\lambda B)$ is the range of $J_{p}^{E}+\lambda B$.

The following lemma was proved by Suantai et al. (2018).

Lemma 2.11

Let $E_1$ and $E_2$ be uniformly convex and smooth Banach spaces. Let $A:E_1\rightarrow E_2$ be a bounded linear operator with the adjoint operator $A^*$. Let $R_{\lambda }^{B_1}$ be the resolvent operator of a maximal monotone operator $B_1$ for $\lambda _1>0$ and $Q_{\lambda _{2}}^{B_2}$ be a metric resolvent of a maximal monotone operator $B_2$ for $\lambda _2>0$. Suppose that $\Gamma \ne \emptyset$. Let $r>0$ and $x^*\in E_1$. Then $x^*$ is a solution of problem (1.6) if and only if

$$\begin{aligned} x^*=R_{\lambda _1}^{B_1}(J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(x^*)-rA^*J_{p}^{E_2}(I-Q_{\lambda _{2}}^{B_2})Ax^*)). \end{aligned}$$

Lemma 2.12

Let E be a real p-uniformly convex and uniformly smooth Banach spaces. Suppose that $x\in E$ and $\{x_n\}$ is a sequence in E. Then the following statements are equivalent:

• (a) $\{D_p(x_n,x)\}$ is bounded;

• (b) $\{x_n\}$ is bounded.

Proof

For the implication $(a)\Longrightarrow (b)$ was proved in Reich and Sabach (2010). For the converse implication $(b)\Longrightarrow (a)$, we assume that $x\in E$ and $\{x_n\}$ are bounded. From (2.4), we observe that

$$\begin{aligned} D_p(x_n,x)\le & {} \langle J_{p}^{E}x_n-J_{p}^{E}x,x_n-x\rangle \\\le & {} \Vert J_{p}^{E}x_n-J_{p}^{E}x\Vert \Vert x_n-x\Vert \\\le & {} M, \end{aligned}$$

for all $n\in {\mathbb {N}}$, where $M=\sup _{n\ge 1}\{\Vert x_n\Vert ,\Vert x_n\Vert ^{p-1},\Vert x\Vert ,\Vert x\Vert ^{p-1}\}$. This implies that $\{D_p(x_n,x)\}$ is bounded. $\square$

Lemma 2.13

(Reich 1979) Assume that $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$\begin{aligned} a_{n+1}\le (1-\gamma _n)a_n+\gamma _n\delta _n,~~\forall n\ge 1, \end{aligned}$$

where $\{\gamma _n\}$ is a sequence in (0, 1) and $\{\delta _n\}$ is a sequence in ${\mathbb {R}}$ such that $\lim _{n\rightarrow \infty }\gamma _n=0$, $\sum _{n=1}^{\infty }\gamma _n=\infty$ and $\limsup _{n\rightarrow \infty }\delta _n\le 0$. Then $\lim _{n\rightarrow \infty }a_n=0$.

Lemma 2.14

(Maingé 2008) Let $\{\Gamma _n\}$ be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence $\{\Gamma _{n_i}\}$ of $\{\Gamma _n\}$ which satisfies $\Gamma _{n_i}<\Gamma _{n_{i}+1}$ for all $i\in {\mathbb {N}}$. Define the sequence $\{\tau (n)\}_{n\ge n_0}$ of integers as follows:

$$\begin{aligned} \tau (n)=\max \{k\le n:\Gamma _k<\Gamma _{k+1}\}, \end{aligned}$$

where $n_0\in {\mathbb {N}}$ such that $\{k\le n_0:\Gamma _k<\Gamma _{k+1}\}\ne \emptyset$. Then, the following hold:

• (i) $\tau ({n_0})\le \tau ({n_0+1})\le \ldots$ and $\tau (n)\rightarrow \infty$;

• (ii) $\Gamma _{\tau _n}\le \Gamma _{\tau (n)+1}$ and $\Gamma _n\le \Gamma _{\tau (n)+1}$, $\forall n\ge n_0$.

Lemma 2.15

Let E be a real p-uniformly convex and uniformly smooth Banach space. Let $z,x_k\in E$ $(k=1,2,\ldots ,N)$ and $\alpha _k\in (0,1)$ with $\sum _{k=1}^{N}\alpha _k=1$. Then, we have

$$\begin{aligned} D_p \bigg (J_{q}^{E^*}\bigg (\sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k)\bigg ),z\bigg )\le \sum _{k=1}^{N}\alpha _k D_p (x_k,z)-\alpha _i\alpha _jg_{r}^{*}\big (\Vert J_{p}^{E}(x_i)-J_{p}^{E}(x_j)\Vert \big ), \end{aligned}$$

for all $i,j\in \{1,2,\ldots ,N\}$.

Proof

Since p-uniformly convex, hence it is uniformly convex. From Lemma 2.5, we have

$$\begin{aligned}&D_p \bigg (J_{q}^{E^*}\bigg (\sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k)\bigg ),z\bigg )\\&\quad =V_p\bigg (\sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k),z\bigg )\\&\quad =\frac{1}{q}\big \Vert \sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k)\big \Vert ^q-\left\langle \sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k),z\right\rangle +\frac{1}{p}\Vert z\Vert ^p\\&\quad \le \frac{1}{q}\sum _{k=1}^{N}\alpha _k\Vert J_{p}^{E}(x_k)\Vert ^q-\alpha _i\alpha _jg_{r}^{*}(\Vert J_{p}^{E}(x_i)-J_{p}^{E}(x_j)\Vert )-\left\langle \sum _{k=1}^{N}\alpha _kJ_{p}^{E}(x_k),z\right\rangle +\frac{1}{p}\Vert z\Vert ^p\\&\quad =\frac{1}{q}\sum _{k=1}^{N}\alpha _k\Vert J_{p}^{E}(x_k)\Vert ^q-\sum _{k=1}^{N}\alpha _k\langle J_{p}^{E}(x_k),z\rangle +\frac{1}{p}\Vert z\Vert ^p-\alpha _i\alpha _jg_{r}^{*}(\Vert J_{p}^{E}(x_i)-J_{p}^{E}(x_j)\Vert )\\&\quad = \sum _{k=1}^{N}\alpha _k D_p (x_k,z)-\alpha _i\alpha _jg_{r}^{*}(\Vert J_{p}^{E}(x_i)-J_{p}^{E}(x_j)\Vert ), \end{aligned}$$

for all $i,j\in \{1,2,\ldots ,N\}$. This completes the proof. $\square$

3 Algorithm and strong convergence theorem

In this section, we introduce an iterative algorithm for finding a common element of the set of solutions of split inclusion problem (1.6) and the set of fixed points of a Bregman relatively nonexpansive mapping. More specifically, we assume the following assumptions:

• $E_1$ and $E_2$ are p-uniformly convex and uniformly smooth Banach spaces;

• $B_1:E_1\multimap E_{1}^{*}$ and $B_2:E_2\multimap E_{2}^{*}$ are maximal monotone operators such that $B_{1}^{-1}(0)\ne \emptyset$ and $B_{2}^{-1}(0)\ne \emptyset$, respectively;

• $R_{\lambda _{1}}^{B_1}$ is the resolvent operator of a maximal monotone $B_1$ for $\lambda _1>0$ and $Q_{\lambda _{2}}^{B_2}$ is the metric resolvent operator of a maximal monotone $B_2$ for $\lambda _{2}>0$;

• $A:E_1\rightarrow E_2$ is a bounded linear operator with its adjoint operator $A^*:E_{2}^{*}\rightarrow E_{1}^{*}$;

• $T:E_1\rightarrow E_1$ is a Bregman relatively nonexpansive mapping such that $F(T)={\widehat{F}}(T)\ne \emptyset$;

• The set of solution of SIP is consistent, i.e., $\Gamma \ne \emptyset$;

• $\Omega :=F(T)\cap \Gamma \ne \emptyset$;

• $\epsilon _n$ denotes the residual vector in $E_1$ such that $\lim _{n\rightarrow \infty }\epsilon _n=u\in E_1$.

Algorithm 3.1

Choose an initial guess $u_1\in E_1$; let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} x_n=R_{\lambda _1}^{B_1}(J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _{2}}^{B_2})Au_n))\\ u_{n+1}=J_{q}^{E_{1}^{*}}(\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n)),~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(3.1)

where $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are sequences in (0, 1) such that $\alpha _n+\beta _n+\gamma _n=1$. Suppose that stepsize $\lambda _n$ is a bounded sequence chosen in such a way that

$$\begin{aligned} 0<\epsilon \le \lambda _n\le \bigg (\frac{q\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p}{\kappa _q\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q}-\epsilon \bigg )^{\frac{1}{q-1}},~~ n\in N, \end{aligned}$$

(3.2)

for some $\epsilon >0$, where the index set $N:=\{n\in {\mathbb {N}}:(I-Q_{\lambda _2}^{B_2})Au_n\ne 0\}$ and $\lambda _n=\lambda$ ($\lambda$ being any nonnegative value), otherwise. Note that the choice in (3.2) of the stepsize $\lambda _n$ is independent of the norms $\Vert A\Vert$.

Lemma 3.2

Let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by Algorithm 3.1. Then, $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ are bounded.

Proof. By the choice of $\lambda _n$, we observe that

$$\begin{aligned}&\lambda _{n}^{q-1}\le \frac{q\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p}{\kappa _q\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q}-\epsilon \nonumber \\ ~~\Longleftrightarrow & {} ~~\kappa _q\lambda _{n}^{q-1}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\le \Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p-\epsilon \kappa _q\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\nonumber \\\Longleftrightarrow & {} \frac{\epsilon \kappa _q}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\le \Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p-\frac{\kappa _q\lambda _{n}^{q-1}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q.\nonumber \\ \end{aligned}$$

(3.3)

Let $z\in \Omega$. From (2.14), we observe that

$$\begin{aligned}&\langle J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,Au_n-Az\rangle \nonumber \\&\quad =\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p+\langle J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,Q_{\lambda _2}^{B_2}(Au_n)-Az\rangle \nonumber \\&\quad \ge \Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p. \end{aligned}$$

(3.4)

Set $v_n:=J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n)$ for all $n\ge 1$. By (3.4) and Lemma 2.6, we have

$$\begin{aligned}&D_p (x_n,z)\le D_p(v_n,z)\nonumber \\&\quad = D_p \big (J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n),z\big )\nonumber \\&\quad = \frac{1}{q}\Vert J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n)\Vert ^p-\langle J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,z\rangle +\frac{1}{p}\Vert z\Vert ^p\nonumber \\&\quad = \frac{1}{q}\Vert J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n)\Vert ^q-\langle J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,z\rangle +\frac{1}{p}\Vert z\Vert ^p\nonumber \\&\quad \le \frac{1}{q}\Vert J_{p}^{E_1}(u_n)\Vert ^q-\lambda _n\langle Au_n,J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\rangle +\frac{\kappa _q\lambda _{n}^{q}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q-\langle J_{p}^{E_1}(u_n),z\rangle \nonumber \\&\qquad +\lambda _n\langle J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,Az\rangle +\frac{1}{p}\Vert z\Vert ^p\nonumber \\&\quad =\frac{1}{q}\Vert u_n\Vert ^p-\langle J_{p}^{E_1}(u_n),z\rangle \nonumber \\&\qquad +\frac{1}{p}\Vert z\Vert ^p+\lambda _n\langle J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,Az-Au_n\rangle +\frac{\kappa _q\lambda _{n}^{q}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\nonumber \\&\quad = D_p (u_n,z)+\lambda _n\langle J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n,Az-Au_n\rangle +\frac{\kappa _q\lambda _{n}^{q}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\nonumber \\&\quad \le D_p (u_n,z)-\lambda _n\bigg (\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p-\frac{\kappa _q\lambda _{n}^{q-1}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\bigg ), \end{aligned}$$

(3.5)

which implies by (3.3) that

$$\begin{aligned} D_p (x_n,z)\le D_p (u_n,z). \end{aligned}$$

Since $\lim _{n\rightarrow \infty }\epsilon _n=u\in E_1$, which implies that $\{\epsilon _n\}$ is bounded, then from Lemma 2.12, we have $\{ D_p (\epsilon _n,z)\}$ is bounded. So there exists a constant $K > 0$ such that $D_p (\epsilon _n,z)\le K$ for all $n\ge 1$. From Lemma 2.15, we have

$$\begin{aligned} D_p (x_{n+1},z)\le & {} D_p (u_{n+1},z)\nonumber \\= & {} D_p (J_{q}^{E_{1}^{*}}(\alpha _nJ_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n)),z)\nonumber \\\le & {} \alpha _n D_p (\epsilon _n,z)+\beta _n D_p (x_n,z)+\gamma _n D_p (Tx_n,z)\nonumber \\&-\beta _n\gamma _ng_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\\le & {} \alpha _n D_p (\epsilon _n,z)+(1-\alpha _n) D_p (x_n,z)-\beta _n\gamma _ng_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\\le & {} \alpha _n K+(1-\alpha _n) D_p (x_n,z)\nonumber \\\le & {} \max \{K, D_p (x_n,z)\}\nonumber \\&\vdots&\nonumber \\\le & {} \max \{K, D_p (x_1,z)\}. \end{aligned}$$

(3.6)

By induction, we have $\{ D_p (x_n,z)\}$ is bounded. Hence, $\{x_n\}$ is bounded and so are $\{u_n\}$ and $\{Au_n\}$.

Theorem 3.3

Let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by Algorithm 3.1. Suppose that the following conditions hold:

• (C1) $\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

• (C2) $0<k \le \beta _n\gamma _n\le 1$ for some $k\in (0,1)$.

Then $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ converge strongly to $x^*=\Pi _{\Omega }u$, where $\Pi _{\Omega }$ is the generalized projection from $E_1$ onto $\Omega$.

Proof

Let $x^*=\Pi _{F(T)\cap \Gamma }u$. From (2.7) and (3.6), we have

$$\begin{aligned}&D_p(x_{n+1},x^*)\nonumber \\&\quad \le D_p(u_{n+1},x^*)\nonumber \\&\quad = V_p\big (\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n),x^*\big )\nonumber \\&\quad \le V_p(\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n)-\alpha _n(J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),x^*))\nonumber \\&\qquad +\,\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle \nonumber \\&\quad = V_p(\alpha _nJ_{p}^{E_1}(x^*)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n),x^*)+\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle \nonumber \\&\quad = D_p(J_{q}^{E_{1}^{*}}(\alpha _nJ_{p}^{E_1}(x^*)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n)),x^*)\nonumber \\&\qquad +\,\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle \nonumber \\&\quad \le \alpha _n D_p(x^*,x^*)+\beta _nD_p(x_n,x^*)+\gamma _nD_p(Tx_n,x^*)-\beta _n\gamma _ng_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\&\qquad +\,\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle \nonumber \\&\quad \le (1-\alpha _n)D_p(x_n,x^*)-\beta _n\gamma _ng_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\&\qquad +\,\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle . \end{aligned}$$

(3.7)

We now divide the proof into two cases:

Case 1 Suppose that there exists $n_0\in {\mathbb {N}}$ such that $\{ D_p (x_n,x^*)\}_{n=n_0}^{\infty }$ is non-increasing. So we have $\{ D_p (x_n,x^*)\}_{n=1}^{\infty }$ converges and it is bounded. From (3.7), we have

$$\begin{aligned} 0\le & {} kg_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\\le & {} \beta _n\gamma _ng_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )\nonumber \\\le & {} D_p (x_n,x^*)- D_p (x_{n+1},x^*)+\alpha _n \langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1} (x^*),u_{n+1}-x^*\rangle . \end{aligned}$$

(3.8)

This implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }g_{r}^{*}(\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert )=0. \end{aligned}$$

By the property of $g_{r}^{*}$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert =0. \end{aligned}$$

(3.9)

Since $J_{q}^{E_{1}^{*}}$ is uniformly norm-to-norm continuous on bounded subsets of $E_{1}^{*}$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_n-Tx_n\Vert =0. \end{aligned}$$

(3.10)

By Lemma 2.7, we also have

$$\begin{aligned} \lim _{n\rightarrow \infty }D_p(x_n,Tx_n)=0. \end{aligned}$$

(3.11)

By the boundedness of $\{x_n\}$ and the reflexivity of $E_1$, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup {\hat{x}}\in E_1$. From (3.10), we obtain ${\hat{x}}\in {\widehat{F}}(T)= F(T)$. From (3.3), (3.5) and (3.6), we see that

$$\begin{aligned} \frac{\epsilon ^2\kappa _q}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2}) Au_n\Vert ^q\le & {} \lambda _n\bigg (\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p- \frac{\kappa _q\lambda _{n}^{q-1}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{ \lambda _2}^{B_2})Au_n\Vert ^q\bigg )\\\le & {} D_p(u_n,{\hat{x}})-D_p(x_n,{\hat{x}})\\\le & {} \alpha _{n-1} D_p (\epsilon _{n-1},{\hat{x}})+ D_p (x_{n-1},{\hat{x}})- D_p (x_n,{\hat{x}})\rightarrow 0~~\text {as}~~n\rightarrow \infty , \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert =0. \end{aligned}$$

(3.12)

From (3.5) and (3.6), we have

$$\begin{aligned} \epsilon \Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p\le & {} \lambda _n\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p\nonumber \\\le & {} D_p(u_n,{\hat{x}})-D_p(x_n,{\hat{x}})+\frac{\kappa _q \lambda _{n}^{q}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\nonumber \\\le & {} \alpha _{n-1} D_p (\epsilon _{n-1},{\hat{x}})+ D_p (x_{n-1},{\hat{x}})- D_p (x_n,{\hat{x}})\nonumber \\&+\frac{\kappa _q\lambda _{n}^{q}}{q}\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q\rightarrow 0 \quad \text {as}~~n\rightarrow \infty . \end{aligned}$$

Hence

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert Au_n-Q_{\lambda _2}^{B_2}Au_n\Vert =0. \end{aligned}$$

(3.13)

Then, we have

$$\begin{aligned} \Vert J_{p}^{E_1}(v_n)-J_{p}^{E_1}(u_n)\Vert\le & {} \lambda _n\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert \nonumber \\\le & {} \lambda _n\Vert A^*\Vert \Vert J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert \nonumber \\\le & {} \lambda _n\Vert A^*\Vert \Vert Au_n-Q_{\lambda _2}^{B_2}Au_n\Vert ^{p-1}\rightarrow 0 \quad \text {as}~~n\rightarrow \infty , \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert J_{p}^{E_1}(v_n)-J_{p}^{E_1}(u_n)\Vert =0. \end{aligned}$$

(3.14)

Since $J_{q}^{E_{1}^{*}}$ is norm-to-norm uniformly continuous on bounded subsets of $E_{1}^{*}$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert v_n-u_n\Vert =0. \end{aligned}$$

(3.15)

By Lemma 2.9 and (3.6), we have

$$\begin{aligned} D_p(x_n,v_n)= & {} D_p(R_{\lambda _1}^{B_1}v_n,v_n)\nonumber \\\le & {} D_p(v_n,{\hat{x}})-D_p(x_n,{\hat{x}})\nonumber \\\le & {} D_p(u_n,{\hat{x}})-D_p(x_n,{\hat{x}})\nonumber \\\le & {} \alpha _{n-1}D_p(\epsilon _{n-1},{\hat{x}})+D_p(x_{n-1},{ \hat{x}})-D_p(x_n,{\hat{x}})\rightarrow 0 \quad \text {as}~~n\rightarrow \infty . \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert R_{\lambda _1}^{B_1}v_n-v_n\Vert =\lim _{n\rightarrow \infty }\Vert x_n-v_n\Vert =0. \end{aligned}$$

(3.16)

Since $x_{n_i}\rightharpoonup {\hat{x}}\in E_1$, we also have $v_{n_i}\rightharpoonup {\hat{x}}\in E_1$. From (3.16), we get ${\hat{x}}\in F(R_{\lambda _1}^{B_1})\in B_{1}^{-1}(0)$.

From (3.15) and (3.16), we obtain

$$\begin{aligned} \Vert x_n-u_n\Vert \le \Vert x_n-v_n\Vert +\Vert v_n-u_n\Vert \rightarrow 0 \quad \text {as}~~n\rightarrow \infty . \end{aligned}$$

(3.17)

Since $x_{n_i}\rightharpoonup {\hat{x}}\in E_1$ and from (3.17), we also get $u_{n_i}\rightharpoonup {\hat{x}}\in E_1$. From (2.14), we have

$$\begin{aligned} \Vert (I-Q_{\lambda _2}^{B_2})A{\hat{x}}\Vert ^p= & {} \langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}\rangle \nonumber \\= & {} \langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),A{\hat{x}}-Au_{n_i}\rangle \nonumber \\&+\,\langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),Au_{n_i}-Q_{\lambda _2}^{B_2}Au_{n_i}\rangle \nonumber \\&+\,\langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),Q_{\lambda _2}^{B_2}Au_{n_i}-Q_{\lambda _2}^{B_2}A{\hat{x}}\rangle \nonumber \\\le & {} \langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),A{\hat{x}}-Au_{n_i}\rangle \nonumber \\&+\,\langle J_{p}^{E_2}(A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}),Au_{n_i}-Q_{\lambda _2}^{B_2}Au_{n_i}\rangle . \end{aligned}$$

(3.18)

Since A is continuous, we have $Au_{n_i}\rightharpoonup A{\hat{x}}$ as $i\rightarrow \infty$. Then, we have

$$\begin{aligned} \Vert A{\hat{x}}-Q_{\lambda _2}^{B_2}A{\hat{x}}\Vert =0, \end{aligned}$$

that is, $A{\hat{x}}=Q_{\lambda _2}^{B_2}A{\hat{x}}$. This shows that $A{\hat{x}}\in F(Q_{\lambda _2}^{B_2})=B_{2}^{-1}(0)$. So ${\hat{x}}\in \Gamma$. Therefore, we conclude that ${\hat{x}}\in \Omega :=F(T)\cap \Gamma$.

Now, we see that

$$\begin{aligned} D_p(u_{n+1},x_n)\le & {} D_p(J_{q}^{E_{1}^{*}}(\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(Tx_n)),x_n)\\\le & {} \alpha _nD_p(\epsilon _n,x_n)+\beta _nD_p(x_n,x_n)+\gamma _nD_p(Tx_n,x_n)\rightarrow 0~~\text {as}~~n\rightarrow \infty , \end{aligned}$$

and hence

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_{n+1}-x_n\Vert =0. \end{aligned}$$

(3.19)

So, we have

$$\begin{aligned} \Vert u_{n+1}-u_n\Vert \le \Vert u_{n+1}-x_n\Vert +\Vert x_n-u_n\Vert \rightarrow 0~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(3.20)

We now choose a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),x_n-x^*\rangle =\lim _{i\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),x_{n_i}-x^*\rangle , \end{aligned}$$

where $x^*=\Pi _{\Omega }u$. From (3.17) and Lemma 2.8, we get

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),u_n-x^*\rangle= & {} \limsup _{n\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),x_n-x^*\rangle \nonumber \\= & {} \lim _{i\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),x_{n_i}-x^*\rangle \nonumber \\= & {} \langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),{\hat{x}}-x^*\rangle \le 0. \end{aligned}$$

From (3.20), we also have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),u_{n+1}-x^*\rangle \le 0. \end{aligned}$$

(3.21)

By (3.7), we note that

$$\begin{aligned} D_p(x_{n+1},x^*)\le & {} (1-\alpha _n)D_p(x_n,x^*)+\alpha _n\langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1}(x^*),u_{n+1}-x^*\rangle \\= & {} (1-\alpha _n)D_p(x_n,x^*)+\alpha _n\langle J_{p}^{E_1}(\epsilon _n)-J_{p}^{E_1}(u),u_{n+1}-x^*\rangle \\&+\alpha _n\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),u_{n+1}-x^*\rangle . \end{aligned}$$

Since $\epsilon _n\rightarrow u$ implies $J_{p}^{E_1}(\epsilon _n)\rightarrow J_{p}^{E_1}(u)$. Considering this together with (3.21), we conclude by Lemma 2.13 that $D_p(x_n,x^*)\rightarrow 0$ as $n\rightarrow \infty$. Therefore,S $x_n\rightarrow x^*\in \Omega$.

Case 2 Suppose that there exists a subsequence $\{\Gamma _{n_i}\}$ of $\{\Gamma _n\}$ such that $\Gamma _{n_i}< \Gamma _{n_i+1}$ for all $i\in {\mathbb {N}}$. Let us define a mapping $\tau :{\mathbb {N}}\rightarrow {\mathbb {N}}$ by

$$\begin{aligned} \tau (n)=\max \{k\le n:\Gamma _k<\Gamma _{k+1}\}. \end{aligned}$$

Then, by Lemma 2.14, we obtain

$$\begin{aligned} \Gamma _{\tau (n)}\le \Gamma _{\tau (n)+1}~~\text {and}~~\Gamma _n\le \Gamma _{\tau (n)+1}. \end{aligned}$$

Put $\Gamma _n=D_p(x_n,x^*)$ for all $n\in {\mathbb {N}}$. Then, we have from (3.6) that

$$\begin{aligned} 0\le & {} \lim _{n\rightarrow \infty }(D_p(x_{{\tau (n)}+1},x^*)-D_p(x_{\tau (n)},x^*))\\\le & {} \lim _{n\rightarrow \infty }(D_p(\epsilon _{\tau (n)},x^*)+(1-\alpha _{\tau (n)})D_p(x_{\tau (n)},x^*)-D_p(x_{\tau (n)},x^*))\\= & {} \lim _{n\rightarrow \infty }\alpha _{\tau (n)}\big (D_p(\epsilon _{\tau (n)},x^*)-D_p(x_{\tau (n)},x^*))=0, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }(D_p(x_{{\tau (n)}+1},x^*)-D_p(x_{\tau (n)},x^*))=0. \end{aligned}$$

(3.22)

Following the proof line in Case 1, we can show that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\Vert x_{\tau (n)}-Tx_{\tau (n)}\Vert =0,\\&\lim _{n\rightarrow \infty }\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2}Au_{\tau (n)}\Vert =0,\\&\lim _{n\rightarrow \infty }\Vert Au_{\tau (n)}-Q_{\lambda _2}^{B_2}Au_{\tau (n)}\Vert =0,\\&\lim _{n\rightarrow \infty }\Vert x_{\tau (n)}-v_{\tau (n)}\Vert =\lim _{n\rightarrow \infty }\Vert x_{\tau (n)}-u_{\tau (n)}\Vert =0 \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_{\tau (n)+1}-u_{\tau (n)}\Vert =0. \end{aligned}$$

Furthermore, we can show that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),u_{\tau (n)+1}-x^*\rangle \le 0. \end{aligned}$$

From (3.7), we have

$$\begin{aligned} D_p(x_{{\tau (n)}+1},x^*)\le & {} (1-\alpha _{\tau (n)})D_p(x_{\tau (n)},x^*)+\alpha _{\tau (n)}\langle J_{p}^{E_1}(\epsilon _{\tau (n)})-J_{p}^{E_1}(x^*),u_{{\tau (n)}+1}-x^*\rangle , \end{aligned}$$

which implies that

$$\begin{aligned} \alpha _{\tau (n)}D_p(x_{\tau (n)},x^*)\le & {} D_p(x_{\tau (n)},x^*)-D_p(x_{{\tau (n)}+1},x^*)\nonumber \\&+\alpha _{\tau (n)}\langle J_{p}^{E_1}(\epsilon _{\tau (n)})-J_{p}^{E_1}(x^*),u_{{\tau (n)}+1}-x^*\rangle . \end{aligned}$$

Since $\Gamma _{\tau (n)}\le \Gamma _{\tau (n)+1}$ and $\alpha _{\tau (n)}>0$, we get

$$\begin{aligned} D_p(x_{{\tau (n)}},x^*)\le & {} \langle J_{p}^{E_1}(\epsilon _{\tau (n)})-J_{p}^{E_1}(x^*),u_{{\tau (n)}+1}-x^*\rangle \\= & {} \langle J_{p}^{E_1}(\epsilon _{\tau (n)})-J_{p}^{E_1}(u),u_{{\tau (n)}+1}-x^*\rangle +\langle J_{p}^{E_1}(u)-J_{p}^{E_1}(x^*),u_{{\tau (n)}+1}-x^*\rangle . \end{aligned}$$

Since $\epsilon _{\tau (n)}\rightarrow u$ implies $J_{p}^{E_1}(\epsilon _{\tau (n)})\rightarrow J_{p}^{E_1}(u)$. Hence, $\lim _{n\rightarrow \infty }D_p(x_{{\tau (n)}},x^*)=0$. From (3.22), we obtain

$$\begin{aligned} D_p(x_n,x^*)\le D_p(x_{{\tau (n)+1}},x^*)=&D_p(x_{{\tau (n)}},x^*)+(D_p(x_{{\tau (n)+1}},x^*)\\&-D_p(x_{{\tau (n)}},x^*))\rightarrow 0~~\text {as}~~n\rightarrow \infty , \end{aligned}$$

which implies that $D_p(x_n,x^*)\rightarrow 0$. That is $x_n\rightarrow x^*$ as $n\rightarrow \infty$. This completes the proof. $\square$

We consequently obtain the following result in Hilbert spaces:

Corollary 3.4

Let $H_1$ and $H_2$ be Hilbert spaces. Let $B_1:H_1\multimap H_{1}$ and $B_2:H_2\multimap H_{2}$ be maximal monotone operators such that $B_{1}^{-1}(0)\ne \emptyset$ and $B_{2}^{-1}(0)\ne \emptyset$, respectively. Let $R_{\lambda _{1}}^{B_1}$ be the resolvent operator of a maximal monotone $B_1$ for $\lambda _1>0$ and let $Q_{\lambda _{2}}^{B_2}$ be the metric resolvent operator of a maximal monotone $B_2$ for $\lambda _{2}>0$. Let $A:H_1\rightarrow H_2$ be a bounded linear operator with its adjoint operator $A^*:H_{2}\rightarrow H_{1}$. Let $T:H_1\rightarrow H_1$ be a relatively nonexpansive mapping such that $F(T)=\widehat{F}(T)\ne \emptyset$. Assume that $\Omega :=F(T)\cap \Gamma \ne \emptyset$. Choose an initial guess $u_1\in H_1$; let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} x_n=R_{\lambda _1}^{B_1}(u_n-\lambda _n A^*(I-Q_{\lambda _{2}}^{B_2})Au_n)\\ u_{n+1}=\alpha _n\epsilon _n+\beta _nx_n+\gamma _nTx_n,~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(3.23)

where $\{\epsilon _n\}\subset H_1$ is a residual vector such that $\epsilon _n\rightarrow u$, and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are sequences in (0, 1) such that $\alpha _n+\beta _n+\gamma _n=1$. Suppose that stepsize $\lambda _n$ is a bounded sequence chosen in such a way that

$$\begin{aligned} 0<\epsilon \le \lambda _n\le \frac{2\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^2}{\Vert A^*(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^2}-\epsilon ,~~ n\in N, \end{aligned}$$

(3.24)

for some $\epsilon >0$, where the index set $N:=\{n\in {\mathbb {N}}:(I-Q_{\lambda _2}^{B_2})Au_n\ne 0\}$ and $\lambda _n=\lambda$ ($\lambda$ being any nonnegative value), otherwise. Suppose that the following conditions hold:

• (C1) $\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

• (C2) $0<k \le \beta _n\gamma _n\le 1$ for some $k\in (0,1)$.

Then $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ converge strongly to $x^*=\Pi _{\Omega }u$.

4 Convergence theorems for a family of mappings

In this section, we apply our result to the common fixed point problems of a family of mappings.

4.1 A countable family of relatively nonexpansive mappings

Definition 4.1

(Aoyama et al. 2007) Let C be a subset of a real p-uniformly convex and uniformly smooth Banach space E. Let $\{T_n\}_{n=1}^{\infty }$ be a sequence of mappings of C in to E such that $\bigcap _{n=1}^{\infty }F(T_n)\ne \emptyset$. Then $\{T_n\}_{n=1}^{\infty }$ is said to satisfy the AKTT-condition if, for any bounded subset B of C,

$$\begin{aligned} \sum _{n=1}^{\infty }\sup _{z\in B}\{\Vert J_{p}^{E}(T_{n+1}z)-J_{p}^{E}(T_nz)\Vert \}<\infty . \end{aligned}$$

As in Suantai et al. (2012), we can prove the following Proposition:

Proposition 4.2

Let C be a nonempty, closed and convex subset of a real p-uniformly convex and uniformly smooth Banach space E. Let $\{T_n\}_{n=1}^{\infty }$ be a sequence of mappings of C such that $\bigcap _{n=1}^{\infty }F(T_n)\ne \emptyset$ and $\{T_n\}_{n=1}^{\infty }$ satisfies the AKTT-condition. Suppose that for any bounded subset B of C. Then there exists the mapping $T : B \rightarrow E$ such that

$$\begin{aligned} Tx=\lim _{n\rightarrow \infty }T_nx,~~\forall x\in B \end{aligned}$$

(4.1)

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{z\in B}\Vert J_{p}^{E}(Tz)-J_{p}^{E}(T_nz)\Vert =0. \end{aligned}$$

In the sequel, we say that $(\{T_n\}, T )$ satisfies the AKTT-condition if $\{T_n\}_{n=1}^{\infty }$ satisfies the AKTT-condition and T is defined by (4.1) with $\bigcap _{n=1}^{\infty }F(T_n)=F(T)$.

Theorem 4.3

Let $E_1$ and $E_2$ be p-uniformly convex and uniformly smooth Banach spaces. Let $B_1:E_1\multimap E_{1}^{*}$ and $B_2:E_2\multimap E_{2}^{*}$ be maximal monotone operators such that $B_{1}^{-1}(0)\ne \emptyset$ and $B_{2}^{-1}(0)\ne \emptyset$, respectively. Let $R_{\lambda _{1}}^{B_1}$ be the resolvent operator of a maximal monotone $B_1$ for $\lambda _1>0$ and let $Q_{\lambda _{2}}^{B_2}$ be the metric resolvent operator of a maximal monotone $B_2$ for $\lambda _{2}>0$. Let $A:E_1\rightarrow E_2$ be a bounded linear operator with its adjoint operator $A^*:E_{2}^{*}\rightarrow E_{1}^{*}$. Let $\{T_n\}_{n=1}^{\infty }$ be a countable family of Bregman relatively nonexpansive mappings on $E_1$ such that $F(T_n)={\widehat{F}}(T_n)$ for all $n\ge 1$. Assume that $\Omega :=\bigcap _{n=1}^{\infty }F(T_n)\cap \Gamma \ne \emptyset$. Choose an initial guess $u_1\in E_1$, and let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} x_n=R_{\lambda _1}^{B_1}(J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _{2}}^{B_2})Au_n))\\ u_{n+1}=J_{q}^{E_{1}^{*}}(\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+\gamma _nJ_{p}^{E_1}(T_nx_n)),~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(4.2)

where $\{\epsilon _n\}\subset E_1$ is a residual vector such that $\epsilon _n\rightarrow u$, and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are sequences in (0, 1) such that $\alpha _n+\beta _n+\gamma _n=1$. Suppose that stepsize $\lambda _n$ is a bounded sequence chosen in such a way that

$$\begin{aligned} 0<\epsilon \le \lambda _n\le \bigg (\frac{q\Vert (I-Q_{\lambda _2}^{B_2})Au_n\Vert ^p}{\kappa _q\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q}-\epsilon \bigg )^{\frac{1}{q-1}},~~ n\in N, \end{aligned}$$

(4.3)

for some $\epsilon >0$, where the index set $N:=\{n\in {\mathbb {N}}:(I-Q_{\lambda _2}^{B_2})Au_n\ne 0\}$ and $\lambda _n=\lambda$ ($\lambda$ being any nonnegative value), otherwise. Suppose that the following conditions hold:

• (C1) $\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

• (C2) $0<k \le \beta _n\gamma _n\le 1$ for some $k\in (0,1)$.

Suppose in addition that $(\{T_n\}_{n=1}^{\infty }, T )$ satisfies the AKTT-condition and $F(T) = {\widehat{F}}(T)$. Then $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ converge strongly to $x^*=\Pi _{\Omega }u$, where $\Pi _{\Omega }$ is the generalized projection from $E_1$ onto $\Omega$.

Proof

To this end, it suffices to show that $\lim _{n\rightarrow \infty }\Vert x_n-Tx_n\Vert =0$. By following the method of proof in Theorem 3.3, we can show that $\{x_n\}$ is bounded and $\lim _{n\rightarrow \infty }\Vert x_n-T_nx_n\Vert =0$. Since $J_{p}^{E_1}$ is uniformly continuous on bounded subsets of $E_1$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_nx_n)\Vert =0. \end{aligned}$$

By Proposition 4.2, we see that

$$\begin{aligned} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(Tx_n)\Vert\le & {} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_nx_n)\Vert +\Vert J_{p}^{E_1}(T_nx_n)-J_{p}^{E_1}(Tx_n)\Vert \\\le & {} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_nx_n)\Vert \\&+\sup _{x\in \{x_n\}}\Vert J_{p}^{E_1}(T_nx)-J_{p}^{E_1}(Tx)\Vert \rightarrow 0~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

Since $J_{q}^{E_{1}^{*}}$ is norm-to-norm uniformly continuous on bounded subsets of $E_{1}^{*}$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_n-Tx_n\Vert =0. \end{aligned}$$

This completes the proof. $\square$

4.2 A semigroup of relatively nonexpansive mappings

Definition 4.4

A one-parameter family ${\mathcal {S}} = \{T_t\}_{t\ge 0}$ from E into E is said to be a nonexpansive semigroup if it satisfies the following conditions:

• (S1) $T_0x=x ~~\text {for all} ~~x\in E$;

• (S2) $T_{s+t}=T_sT_t$ for all $s,t\ge 0$;

• (S3) for each $x\in C$ the mapping $t\mapsto T_tx$ is continuous;

• (S4) for each $t \ge 0$, $T_t$ is nonexpansive, i.e.,

  $$\begin{aligned} \Vert T_tx-T_ty\Vert \le \Vert x-y\Vert ,~\forall x,y\in E. \end{aligned}$$

Remark 4.5

We denote by $F({\mathcal {S}})$ the set of all common fixed points of ${\mathcal {S}}$, i.e., $F({\mathcal {S}})=\bigcap _{t\ge 0}F(T_t)$.

We now give some examples of semigroup operator. The following classical examples were one of the main sources for the development of semigroup theory (see Engel and Nagel 2000):

Example 4.6

Let E be a real Banach space and let ${\mathcal {L}}(E)$ be the space of all bounded linear operators on E. For $A\in {\mathcal {L}}(E)$ and define a bounded linear operator $T_t$ by

$$\begin{aligned} T_t:=e^{tA}=\sum _{n=0}^{\infty }\frac{t^nA^n}{n!}, \end{aligned}$$

for $t\ge 0$. Then, the operator $T_t$ is a semigroup on E.

Example 4.7

Let $E:=L^{p}({\mathbb {R}}^n)$, $1\le p<\infty$. Consider the initial value problem for the heat equation:

(4.4)

where $\Delta =\sum _{i=1}^{n}\frac{\partial ^2}{\partial x_{i}^{2}}$ is the Laplacian operator on E. We can solve the heat equation using Fourier transform and the solution (4.4) can be written as follows:

$$\begin{aligned} u(x,t)=\frac{1}{\sqrt{(4\pi t)^n}}\int _{{\mathbb {R}}^n}e^{\frac{-\Vert s-\xi \Vert ^2}{4t}}f(\xi )d\xi , \end{aligned}$$

where $t>0$, $s\in {\mathbb {R}}^n$ and $f\in E$. Then, we can write the solution u(x, t) in the form of convolution integral as follows:

$$\begin{aligned} u(x,t)=(K_t*f)(x), \end{aligned}$$

where $K_t$ is heat kernel given by $K_t(x)=\frac{1}{\sqrt{(4\pi t)^n}}e^{\frac{-\Vert x\Vert ^2}{4t}}$. Then the solution of (4.4) can be written as follows:

$$\begin{aligned} T_tf(x):=u(x,t)=(K_t*f)(x). \end{aligned}$$

We can check that the operator $T_tf(x)$ is a semigroup on E.

Definition 4.8

A one-parameter family ${\mathcal {S}} = \{T_t\}_{t\ge 0}:E\rightarrow E$ is said to be a family of uniformly Lipschitzian mappings if there exists a bounded measurable function $L_t:(0,\infty )\rightarrow [0,\infty )$ such that

$$\begin{aligned} \Vert T_tx-T_ty\Vert \le L_t\Vert x-y\Vert ,~\forall x,y\in E. \end{aligned}$$

We now first give the following definition:

Definition 4.9

A one-parameter family ${\mathcal {S}} = \{T_t\}_{t\ge 0}:E\rightarrow E$ is said to be a Bregman relatively nonexpansive semigroup if it satisfies (S1), (S2), (S3) and the following conditions:

• (a) $F({\mathcal {S}})={\widehat{F}}({\mathcal {S}})\ne \emptyset$;

• (b) $D_p (T_tx,z)\le D_p (x,z),~~\forall x\in E,~z\in F({\mathcal {S}})$ and $t\ge 0$.

Using idea in Aleyner and Censor (2005), Aleyner and Reich (2005) and Benavides et al. (2002), we define the following concept:

Definition 4.10

A continuous operator semigroup ${\mathcal {S}}=\{T_t\}_{t\ge 0}:E\rightarrow E$ is said to be uniformly asymptotically regular (in short, u.a.r.) if for all $s\ge 0$ and any bounded subset B of E such that

$$\begin{aligned} \lim _{t\rightarrow \infty }\sup _{x\in B}\Vert J_{p}^{E}(T_tx)- J_{p}^{E}\big (T_sT_sx\big )\Vert =0. \end{aligned}$$

Theorem 4.11

Let $E_1$ and $E_2$ be p-uniformly convex and uniformly smooth Banach spaces. Let $B_1:E_1\multimap E_{1}^{*}$ and $B_2:E_2\multimap E_{2}^{*}$ be maximal monotone operators such that $B_{1}^{-1}(0)\ne \emptyset$ and $B_{2}^{-1}(0)\ne \emptyset$, respectively. Let $R_{\lambda _{1}}^{B_1}$ be the resolvent operator of a maximal monotone $B_1$ for $\lambda _1>0$ and let $Q_{\lambda _{2}}^{B_2}$ be the metric resolvent operator of a maximal monotone $B_2$ for $\lambda _{2}>0$. Let $A:E_1\rightarrow E_2$ be a bounded linear operator with its adjoint operator $A^*:E_{2}^{*}\rightarrow E_{1}^{*}$. Let ${\mathcal {S}}=\{T_t\}_{t\ge 0}$ be a u.a.r. Bregman relatively nonexpansive semigroup of uniformly Lipschitzian mappings on $E_1$ into $E_1$ with a bounded measurable function $L_t:(0,\infty )\rightarrow [0,\infty )$ such that $F({\mathcal {S}}):=\bigcap _{h\ge 0}F(T_h)\ne \emptyset$. Assume that $\Omega :=F({\mathcal {S}})\cap \Gamma \ne \emptyset$. Choose an initial guess $u_1\in E_1$; let $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ be sequences generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} x_n=R_{\lambda _1}^{B_1}(J_{q}^{E_{1}^{*}}(J_{p}^{E_1}(u_n)-\lambda _n A^*J_{p}^{E_2}(I-Q_{\lambda _{2}}^{B_2})Au_n))\\ u_{n+1}=J_{q}^{E_{1}^{*}}(\alpha _n J_{p}^{E_1}(\epsilon _n)+\beta _nJ_{p}^{E_1}(x_n)+ \gamma _nJ_{p}^{E_1}(T_{t_n}x_n)),~~\forall n\ge 1, \end{array} \end{array}\right. } \end{aligned}$$

(4.5)

where $\{\epsilon _n\}\subset E_1$ is a residual vector such that $\epsilon _n\rightarrow u$, and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are sequences in (0, 1) such that $\alpha _n+\beta _n+\gamma _n=1$. Suppose that stepsize $\lambda _n$ is a bounded sequence chosen in such a way that

$$\begin{aligned} 0<\epsilon \le \lambda _n\le \bigg (\frac{q\Vert (I-Q_{\lambda _2}^{B_2}) Au_n\Vert ^p}{\kappa _q\Vert A^*J_{p}^{E_2}(I-Q_{\lambda _2}^{B_2})Au_n\Vert ^q}-\epsilon \bigg )^{\frac{1}{q-1}},~~ n\in N, \end{aligned}$$

(4.6)

for some $\epsilon >0$, where the index set $N:=\{n\in {\mathbb {N}}:(I-Q_{\lambda _2}^{B_2})Au_n\ne 0\}$ and $\lambda _n=\lambda$ ($\lambda$ being any nonnegative value), otherwise. Suppose that the following conditions hold:

• (C1) $\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

• (C2) $0<k \le \beta _n\gamma _n\le 1$ for some $k\in (0,1)$;

• (C3) $\{t_n\}\subset (0,\infty )$ with $\lim _{n\rightarrow \infty }t_n=\infty$.

Then $\{x_n\}_{n=1}^{\infty }$ and $\{u_n\}_{n=1}^{\infty }$ converge strongly to $x^*=\Pi _{\Omega }u$, where $\Pi _{\Omega }$ is the generalized projection from $E_1$ onto $\Omega$.

Proof

We only have to show that $\lim _{n\rightarrow \infty }\Vert x_n-T_tx_n\Vert =0$ for all $t\ge 0$. By following the method of proof in Theorem 3.3, we can show that $\{x_n\}$ is bounded and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_n-T_{t_n}x_n\Vert =0. \end{aligned}$$

(4.7)

Since $\{T_t\}_{t\ge 0}$ is a uniformly of Lipschitzian mappings with a bounded measurable function $L_t$. Then, we have

$$\begin{aligned} \Vert T_tT_{t_n}x_n-T_tx_n\Vert\le & {} L_t\Vert T_{t_n}x_n-x_n\Vert \\\le & {} \sup _{t\ge 0}\{L_t\}\Vert T_{t_n}x_n-x_n\Vert \rightarrow 0 \quad \text {as}~~n\rightarrow \infty . \end{aligned}$$

Since $J_{p}^{E_1}$ is uniformly norm-to-norm continuous on bounded subsets of $E_1$, then we also have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert J_{p}^{E_1}(T_tT_{t_n}x_n)-J_{p}^{E_1}(T_tx_n)\Vert =0. \end{aligned}$$

(4.8)

For each $t\ge 0$, we note that

$$\begin{aligned} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_tx_n\big )\Vert\le & {} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_{t_n}x_n)\Vert +\Vert J_{p}^{E_1}(T_{t_n}x_n)-J_{p}^{E_1}(T_tT_{t_n}x_n)\Vert \\&+\Vert J_{p}^{E_1}(T_tT_{t_n}x_n)-J_{p}^{E_1}(T_tx_n)\Vert \\\le & {} \Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_{t_n}x_n)\Vert +\Vert J_{p}^{E_1}(T_tT_{t_n}x_n)-J_{p}^{E_1}(T_tx_n)\Vert \\&+\sup _{x\in \{x_n\}}\Vert J_{p}^{E_1}(T_{t_n}x)-J_{p}^{E_1}(T_tT_{t_n})x\Vert . \end{aligned}$$

Since $\{T_t\}_{t\ge 0}$ is a u.a.r. Bregman relatively nonexpansive semigroup with $\lim _{n\rightarrow \infty }t_n=\infty$, then from (4.7) and (4.8), we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert J_{p}^{E_1}(x_n)-J_{p}^{E_1}(T_tx_n)\Vert =0 \end{aligned}$$

for all $t\ge 0$. Since $J_{q}^{E_{1}^{*}}$ is uniformly norm-to-norm continuous on bounded subsets of $E_{1}^{*}$, we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_n-T_tx_n\Vert =0. \end{aligned}$$

This completes the proof. $\square$

5 Numerical experiments

Table 1 Numerical results of Algorithm 5.2 with different choices of N and M

Fig. 1

The convergence behavior of $E_n$ for $N = 50$ and $M=50$

In this section, we give some numerical examples to support our main theorem.

Example 5.1

For each ${\mathbf {x}}=(x_1,x_2,\ldots ,x_N)\in {\mathbb {R}}^{N}$. Let $f:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\cup \{+\infty \}$ and $g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be defined by

$$\begin{aligned} f({\mathbf {x}}):=\Vert {\mathbf {x}}\Vert _2~~\text {and}~~g({\mathbf {x}})=-\sum _{i=1}^{N}\log x_i. \end{aligned}$$

Then, we have

$$\begin{aligned} \text {prox}_{\lambda f}({\mathbf {x}}) {\left\{ \begin{array}{ll} \begin{array}{ll} \bigg (1-\frac{\lambda }{\Vert {\mathbf {x}}\Vert _2}\bigg )x;~~~\Vert {\mathbf {x}}\Vert _2\ge \lambda \\ 0;~~~~\Vert {\mathbf {x}}\Vert _2<\lambda \end{array} \end{array}\right. } \end{aligned}$$

(5.1)

and

$$\begin{aligned} \text {prox}_{\lambda g}({\mathbf {x}})_i=\frac{x_i+\sqrt{x_{i}^{2}+4\lambda }}{2} \end{aligned}$$

for $i=1,2,3,\ldots ,N$. Let a mapping $T:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$ be defined by

$$\begin{aligned} T{\mathbf {x}}=(2-x_1,2-x_2,2-x_3,\ldots ,2-x_N). \end{aligned}$$

We aim to solve the following SIP and the fixed point problem: find $x^*\in \Gamma \cap F(T)$, i.e., find $x^*\in$ argmin f such that $Ax^*\in \arg$min g and $x^*$ is a fixed point of T, where A is a real $N\times M$ matrix. So our iterative scheme (3.1) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} {\mathbf {x}}_n=\text {prox}_{\lambda _1}^{f}\big [{\mathbf {u}}_n-\lambda _n A^t(A{\mathbf {u}}_n-\text {prox}_{\lambda _2}^{g}(A{\mathbf {u}}_n))\big ]\\ {\mathbf {u}}_{n+1}=\alpha _n\epsilon _n+\beta _n{\mathbf {x}}_n+\gamma _nT{\mathbf {x}}_n,~~\forall n\ge 1. \end{array} \end{array}\right. } \end{aligned}$$

(5.2)

Let $\lambda _1=\lambda _2=1$, $\alpha _n=\frac{1}{20n+1}$, $\beta _n=0.5$, $\gamma _n=\frac{10n-0.5}{20n+1}$ and $\lambda _n=\frac{\Vert A{\mathbf {u}}_n-\text {prox}_{\lambda _2}^{g}(A{\mathbf {u}}_n)\Vert ^2}{\Vert A^t(A{\mathbf {u}}_n-\text {prox}_{\lambda _2}^{g}(A{\mathbf {u}}_n))\Vert ^2}$. The stopping criterion is defined by $E_{n}=\Vert u_{n+1}-u_{n}\Vert <10^{-6}$. The matrix A is generated from a normal distribution with mean zero and one variance. For an initial guess ${\mathbf {x}}_1\in {\mathbb {R}}^{N}$ and residual vector $\epsilon _n\in {\mathbb {R}}^{N}$ randomly, we obtain the following numerical results, given in Table 1 and Figs. 1, 2, 3, 4 and 5:

Fig. 2

The convergence behavior of $E_n$ for $N = 100$ and $M=50$

Fig. 3

The convergence behavior of $E_n$ for $N = 200$ and $M=200$

Fig. 4

The convergence behavior of $E_n$ for $N = 150$ and $M=300$

Fig. 5

The convergence behavior of $E_n$ for $N = 500$ and $M=1000$