Hybrid Projection Methods for Bregman Totally Quasi-D-Asymptotically Nonexpansive Mappings

Abstract

In this paper, a new iterative scheme by hybrid projection method is proposed for a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings. Conditions ensuring strong convergence are imposed to common elements of set of common fixed points of the mappings and set of common solutions to a system of generalized mixed equilibrium problems in a reflexive Banach space. These results extend many important recent ones in the literature.

1 Introduction

Let C be a closed convex subsets of Banach space E. Let f be a bifunction from $C\times C$ to $\mathbb {R}$, $\varphi :C\rightarrow \mathbb {R}$ be a real-valued function, and $A:C\rightarrow E^*$ be a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find $z\in C$ such that

$$\begin{aligned} f(z,y)+\langle Az,y-z\rangle +\varphi (y)-\varphi (z)\ge 0,\forall y\in C. \end{aligned}$$

(1.1)

The set of solution of (1.1) is denoted by ${\textit{GMEP}}(f,\varphi )$, i.e.,

$$\begin{aligned} {\textit{GMEP}}(f,\varphi )=\{z\in C\big |f(z,y)+\langle Az,y-z\rangle +\varphi (y)-\varphi (z)\ge 0,\forall y\in C\}. \end{aligned}$$

Special cases: (I) If $A=0$, then the problem (1.1) is equivalent to find $z\in C$ such that

$$\begin{aligned} f(z,y)+\varphi (y)-\varphi (z)\ge 0,\forall y\in C, \end{aligned}$$

(1.2)

which is called the mixed equilibrium problem. The set of solutions of (1.2) is denoted by ${ MEP}(f,\varphi )$.

(II) If $f=0$, then the problem (1.1) is equivalent to find $z\in C$ such that

$$\begin{aligned} \langle Az,y-z\rangle +\varphi (y)-\varphi (z)\ge 0,\forall y\in C, \end{aligned}$$

(1.3)

which is called the mixed variational inequality of Browder type. The set of solutions of (1.3) is denoted by $VI(C,A,\varphi )$. In particular, VI(C, A, 0) is denoted by VI(C, A).

(III) If $\varphi =0$, then the problem (1.1) is equivalent to find $z\in C$ such that

$$\begin{aligned} f(z,y)+\langle Az,y-z\rangle \ge 0,\forall y\in C, \end{aligned}$$

(1.4)

which is called the generalized equilibrium problem. The set of solutions of (1.4) is denoted by ${ GEP}(f)$.

(IV) If $A=0$, $\varphi =0$, then the problem (1.1) is equivalent to find $z\in C$ such that

$$\begin{aligned} f(z,y)\ge 0,\forall y\in C, \end{aligned}$$

(1.5)

which is called the equilibrium problem. The set of solutions of (1.5) is denoted by ${ EP}(f)$.

The generalized mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases(see, for example, [1]). Some methods have been proposed to solve the generalized mixed equilibrium problem (see, for example, [1–11]). Numerous problems in physics, optimization, and economics help find a solution of problem (1.5).

It is well known that, in an infinite-dimensional Hilbert space, only weak convergence theorems for the segmenting Mann iteration were established even for nonexpansive mappings. Attempts to modify the segmenting Mann iteration for nonexpansive mappings and asymptotically nonexpansive mappings by hybrid projection algorithms have recently been made so that strong convergence theorems are obtained; see, for example, ([6–26], and references therein).

In 2010, Petrot et al. [6] introduced the following hybrid iterative scheme for approximation of a common fixed point of two relatively quasi-nonexpansive mappings, which is also a solution to generalized mixed equilibrium problem in a uniformly smooth and uniformly convex real Banach space:

$$\begin{aligned} \left\{ \begin{aligned} x&_0\in C\;\;{\text {chosen}}\;{\text {arbitrarily}},\\ y&_n=J^{-1}(\delta _n Jx_n+(1-\delta _n)Jz_n),\\ z&_n=J^{-1}(\alpha _n Jx_n+\beta _n JTx_n+\gamma _n JSx_n),\\ f&(u_n,y)+\varphi (y)-\varphi (u_n)+\langle Au_n,y-u_n\rangle +\frac{1}{r_n}\langle y-u_n,Ju_n-Jx\rangle \ge 0,\forall y\in C,\\ C&_n=\{z\in C:\phi (z,u_n)\le \phi (z,x_n)\},\\ Q&_n=\{z\in C:\langle x_n-z,Jx_0-Jx_n\rangle \ge 0\},\\ x&_{n+1}=\Pi _{C_n\bigcap Q_n}x_0. \end{aligned} \right. \end{aligned}$$

They proved strong convergence theorem to a common element of set of common fixed points of S and T and set of solutions to the generalized mixed equilibrium problem.

In [7], Martinez-Yanes and Xu introduced the following iterative scheme for a single nonexpansive mapping T in a Hilbert space H:

$$\begin{aligned} \left\{ \begin{aligned} x&_0\in C\;\;{\text {chosen}}\;{\text {arbitrarily}},\\ y&_n=\alpha _n x_0+(1-\alpha _n)Tx_n,\\ C&_n=\{z\in C:||z-y_n||^2\le ||z-x_n||^2+\alpha _n(||x_0||^2+2\langle x_n-x_0,z\rangle )\},\\ Q&_n=\{z\in C:\langle x_n-z,x_0-x_n\rangle \ge 0\},\\ x&_{n+1}=P_{C_n\bigcap Q_n}x_0, \end{aligned} \right. \end{aligned}$$

where $P_C$ denotes the metric projection of H onto a closed and convex subset C of H. They proved that if $\{\alpha _n\}\subset (0,1)$ and $\lim _{n\rightarrow \infty }\alpha _n=0$, then the sequence $\{x_n\}$ converges strongly to $P_{F(T)}x_0$.

Recently, Qin et al. [8] extended the results of Martinez-Yanes and Xu [7] from Hilbert spaces to Banach spaces and proved the following result: Let C be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E and let $T_i:C\rightarrow C$ be two quasi-$\phi$-nonexpansive mappings for $i=1,2$. Define a sequence $\{x_n\}$ in C by the following algorithm:

$$\begin{aligned} \left\{ \begin{aligned} x&_0\in C\;\;{\text {chosen}}\;{\text {arbitrarily}},\\ C&_1=C,\;x_1=\Pi _{C_1}x_0,\\ y&_n=J^{-1}(\beta _n^{(0)}Jx_n+\beta _n^{(1)}JT_1x_n+\beta _n^ {(2)}JT_2x_n),\\ f&(u_n,y)+\frac{1}{r_n}\langle y-u_n,Ju_n-Jy_n\rangle \ge 0,\forall y\in C,\\ C&_{n+1}=\{z\in C_n:\phi (z,u_n)\le \phi (z,x_n)\},\\ x&_{n+1}=\Pi _{C_{n+1}}x_0. \end{aligned} \right. \end{aligned}$$

(1.6)

If $F:=F(T_1)\bigcap F(T_2)\bigcap { EP}(f)$ is nonempty, they proved that the sequence $\{x_n\}$ converges strongly to $\Pi _{F}x_0$.

In 2013, Zhu et al. [12] introduced the following hybrid projection algorithm:

$$\begin{aligned} \left\{ \begin{aligned} x&_0\in C\;\;{\text {chosen}}\;{\text {arbitrarily}},\\ C&_1=C,\;x_1=\textit{Proj}_{C_1}^gx_0,\\ y&_n=\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(Tx_n)],\\ f&(u_n,y)+\langle y-u_n,\triangledown g(u_n)-\triangledown g(y_n)\rangle \ge 0,\forall y\in C,\\ C&_{n+1}=\{z\in C_n:D(z,u_n)\le D(z,x_n)\},\\ x&_{n+1}=\textit{Proj}_{C_{n+1}}^gx_0 \end{aligned} \right. \end{aligned}$$

(1.7)

for every $n\ge 0$ and T is Bregman strongly nonexpansive mapping. They proved under the appropriate conditions on the parameters that the sequence $\{x_n\}$ generated by (1.9) converges strongly to a common element of the set of solutions of an equilibrium problem $\textit{EP}(f)$ and the set of fixed points of Bregman strongly nonexpansive mappings T in a reflexive Banach space, where $\textit{Proj}_{F}^g(x_0)$ is the Bregman projection of E onto F.

In [13], Pang et al. prove weak convergence theorems for the sequences produced by Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces.

In 1967, Bregman [27] discovered an elegant and effective technique for using the so-called Bregman distance function $D(\cdot ,\cdot )$ (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [12–27, 30] and the references therein).

In 2010, Reich and Sabach [21] introduced the concept of Bregman strongly nonexpansive mapping and studied the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces. In 2012, Suantai et al. [15] also considered the strong convergence for Bregman strongly nonexpansive mappings in reflexive Banach spaces.

Motivated by the above-mentioned results and the on-going research, the purpose of this paper is to introduce a new hybrid projection algorithm based on the shrinking projection method and prove strong convergence theorem for approximation of a common element of the set of common fixed point of a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings (which contains Bregman strongly nonexpansive mapping, Bregman relatively nonexpansive mapping, and Bregman quasi-D-nonexpansive mapping as its special case), and the set of solutions to system of generalized mixed equilibrium problems under a limit condition only in the framework of reflexive Banach spaces. As applications, we apply our results to zero-point problem of maximal monotone mappings in reflexive Banach spaces. The results presented in the paper improve and extend the corresponding results in [6–9, 12, 22, 33] and others.

2 Preliminaries

Throughout this paper, let E be a Banach space and let $E^*$ be the topological dual of E. For all $x\in E$ and $x^*\in E^*$, we denote the value of $x^*$ at x by $\langle x,x^*\rangle$ and the set of fixed points of T by F(T). Let $g:E\rightarrow \mathbb {R}\bigcup \{+\infty \}$ be a proper convex and lower semicontinuous function. Denote the domain of g by dom$\,g$, i.e., dom$\,g$=$\{x\in E:g(x)<+\infty \}$. The Fenchel conjugate of g is the function $g^*:E^*\rightarrow (-\infty ,+\infty ]$ defined by $g^*(\zeta )=\sup _{x\in E}\{\langle \zeta ,x\rangle -g(x)\}$. The normalized duality mapping $J:E\rightarrow 2^{E^*}$ is defined by $J(x)=\{x^*\in E^*:\langle x,x^*\rangle =||x||^2=||x^*||^2\}$. If E is reflexive, we know from [28] that $(\triangledown g)^{-1}=\triangledown g^*$, $\triangledown g=(\triangledown g^*)^{-1}$, ran$\triangledown g$=dom$\triangledown g^*=$int(dom$\,g^*$), and ran$\triangledown g^*=$dom$\triangledown g=$int(dom$\,g$).

Definition 2.1

[23, 27] Let $g:E\rightarrow \mathbb {R}$ be a Gŝteaux differentiable and convex function. The function $D(\cdot ,\cdot ):dom\,g\times int(dom\,g)\rightarrow [0,+\infty )$ defined by $D(y,x)=g(y)-g(x)-\langle y-x,\triangledown g(x)\rangle$ is called the Bregman distance with respect to g.

It follows from the strict convexity of g that $D(x,y)\ge 0$ for all x, y in E. However, $D(\cdot ,\cdot )$ might not be symmetric and $D(\cdot ,\cdot )$ might not satisfy the triangular inequality.

Remark 2.1

[15] The Bregman distance has the following properties:

• (1) the three-point identity, for any $x\in dom\,g$ and $y,z\in int(dom\,g)$,

  $$\begin{aligned} D(x,z)=D(x,y)+D(y,z)+\langle \triangledown g(y)-\triangledown g(z),x-y\rangle ; \end{aligned}$$

• (2) the four-point identity, for any $y,w\in dom\,g$ and $x,z\in int(dom\,g)$,

  $$\begin{aligned} D(y,x)-D(y,z)-D(w,x)+D(w,z)=\langle \triangledown g(z)-\triangledown g(x),y-w\rangle . \end{aligned}$$

Definition 2.2

[17] Let $g:E\rightarrow \mathbb {R}$ be a G$\hat{a}$teaux differentiable and convex function. The Bregman projection of $x\in int(dom\,g)$ onto the nonempty, closed, and convex set $C\subset dom\,g$ is the necessarily unique vector $\textit{Proj}_C^g(x)\in C$ satisfying the following:

$$\begin{aligned} D(\textit{Proj}_C^g(x),x)=\inf \{D(y,x):y\in C\}. \end{aligned}$$

Remark 2.2

(1) If E is a smooth Banach space and $g(x)=||x||^2$ for all $x\in E$, then we have that $\triangledown g(x)=2Jx$ for all x in E. Hence, $D(\cdot ,\cdot )$ reduces to the usual map $\phi (\cdot ,\cdot )$ as $D(x,y)=||x||^2-2\langle x,Jy\rangle +||y||^2=\phi (x,y)$, $\forall x,y\in E$. The Bregman projection $\textit{Proj}_C^g(x)$ reduces to the generalized projection $\Pi _C(x)$ (see [6, 11]), which is defined by

$$\begin{aligned} \phi (\Pi _C(x),x)=\inf \{\phi (y,x):y\in C\}, \end{aligned}$$

where $\phi :E\times E\rightarrow \mathbb {R}^+$ denotes the Lyapunov functional defined by $\phi (x,y)=||x||^2-2\langle x,Jy\rangle +||y||^2$, $\forall x,y\in E$. It is obvious from the definition of $\phi$ that $(||x||-||y||)^2\le \phi (x,y)\le (||x||+||y||)^2$.

(2) If E is a Hilbert space and $g(x)=||x||^2$ for all $x\in E$, then $D(x,y)=||x-y||^2$, the Bregman projection $\textit{Proj}_C^g(x)$ is reduced to the metric projection $P_C(x)$ of x onto C. For more details we refer the readers to [8].

Definition 2.3

[23] Let B be the closed unit ball of a Banach space E. A function $g:E\rightarrow \mathbb {R}$ is said to be

(1) uniformly smooth on bounded subsets of E if the function $\sigma _r:[0,+\infty )\rightarrow [0,+\infty ]$, defined by

$$\begin{aligned} \sigma _r(t)= & {} \sup _{x\in rB,y\in E,||y||=1,\alpha \in (0,1)}[\alpha g(x+(1-\alpha )ty)+(1-\alpha )g(x-\alpha ty)-g(x)]\\&\times [\alpha (1-\alpha )]^{-1/2}, \end{aligned}$$

satisfies $\lim _{t\downarrow 0}\frac{\sigma _r(t)}{t}=0,\forall r>0$;

(2) uniformly convex on bounded subsets of E if the gauge $\rho _r:[0,+\infty )\rightarrow [0,+\infty ]$ of uniform convexity of g, defined by

$$\begin{aligned} \rho _r(t)= & {} \inf _{x,y\in rB,||x-y||=t,\alpha \in (0,1)}[\alpha g(x)+(1-\alpha )g(y)-g(\alpha x+(1-\alpha )y)]\\&\times [\alpha (1-\alpha )]^{-1/2}, \end{aligned}$$

satisfies $\rho _r(t)>0$, $\forall r,t>0$.

Definition 2.4

(1) A mapping $T:C\rightarrow C$ is said to be Bregman totally quasi-D-asymptotically nonexpansive [22], if $F(T)\ne \emptyset$ and there exist nonnegative real sequences $\{\nu _n\}$,$\{\mu _n\}$ with $\nu _n,\mu _n\rightarrow 0 (as\;n\rightarrow +\infty )$ and a strictly increasing continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$ such that

$$\begin{aligned} D(p,T^nx)\le D(p,x)+\nu _n\cdot \zeta [D(p,x)]+\mu _n,\forall n\ge 1,x\in C,p\in F(T).\qquad \end{aligned}$$

(2.1)

(2) A mapping $T:C\rightarrow C$ is said to be Bregman quasi-D-asymptotically nonexpansive [22], if $F(T)\ne \emptyset$ and there exists a sequence $\{k_n\}\subset [1,+\infty )$ with $\lim _{n\rightarrow +\infty }k_n=1$ such that

$$\begin{aligned} D(p,T^nx)\le k_nD(p,x)\;{\textit{for}}\;all\;x\in C,\,p\in F(T)\;and\;n\ge 1. \end{aligned}$$

(2.2)

(3) A mapping $T:C\rightarrow C$ is said to be Bregman quasi-D-asymptotically nonexpansive in the intermediate sense with sequence $\{\nu _n\}$, if $F(T)\ne \emptyset$ and there exists a sequence $\{\nu _n\}$ in $[0,+\infty )$ with $\lim _{n\rightarrow +\infty }\nu _n=0$ such that

$$\begin{aligned} \limsup _{n\rightarrow +\infty } \sup _{x\in C,p\in F(T)}[D(p,T^nx)-(1+\nu _n)D(p,x)]\le 0. \end{aligned}$$

(2.3)

(4) A mapping $T:C\rightarrow C$ is said to be totally quasi-$\phi$-asymptotically nonexpansive [29], if $F(T)\ne \emptyset$ and there exist nonnegative real sequences $\{\nu _n\}$,$\{\mu _n\}$ with $\nu _n,\mu _n\rightarrow 0(as\;n\rightarrow +\infty )$ and a strictly increasing continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$ such that

$$\begin{aligned} \phi (p,T^nx)\le \phi (p,x)+\nu _n\cdot \zeta [\phi (p,x)]+\mu _n,\forall n\ge 1,x\in C,p\in F(T).\qquad \end{aligned}$$

(2.4)

(5) A mapping $T:C\rightarrow C$ is said to be quasi-$\phi$-asymptotically nonexpansive [6–9, 29], if $F(T)\ne \emptyset$ and there exists a sequence $\{k_n\}\subset [1,+\infty )$ with $\lim _{n\rightarrow +\infty }k_n=1$ such that $\phi (p,T^nx)\le k_n\phi (p,x)$ for all $x\in C, p\in F(T)$ and $n\ge 1$.

(6) A mapping $T:C\rightarrow C$ is said to be quasi-$\phi$-asymptotically nonexpansive in the intermediate sense with sequence $\{\nu _n\}$, if $F(T)\ne \emptyset$ and there exists a sequence $\{\nu _n\}$ in $[0,+\infty )$ with $\lim _{n\rightarrow +\infty }\nu _n=0$ such that

$$\begin{aligned} \limsup _{n\rightarrow +\infty } \sup _{x\in C,p\in F(T)}[\phi (p,T^nx)-(1+\nu _n)\phi (p,x)]\le 0. \end{aligned}$$

(2.5)

The following Example 2.1 shows that there is a Bregman totally quasi-D-asymptotically nonexpansive mapping which is not a Bregman D-nonexpansive mapping.

Example 2.1

Let $E=l^2$, $C=\{x\in l^2\big |\,||x||\le 1\}$, and $g(x)=||x||^2$, where $l^2=\{\sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots )\big |\,\sum _{n=1}^{+\infty }|\sigma _n|^2<+\infty \}$. $||\sigma ||=\big (\sum _{n=1}^{+\infty }|\sigma _n|^2\big )^{\frac{1}{2}}$, $\forall \sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots )\in l^2$; $\langle \sigma ,\eta \rangle =\sum _{n=1}^{+\infty }\sigma _n\eta _n$, $\forall \sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots ),\,\eta =(\eta _1,\eta _2,\cdots ,\eta _n,\cdots )\in l^2$.

Let $T:C\rightarrow C$ be a mapping defined by

$$\begin{aligned} T(x_1,x_2,x_3,\cdots )=(0,x_1^2,a_2x_2,a_3x_3,\cdots ),\,\forall (x_1,x_2,x_3,\cdots )\in C, \end{aligned}$$

where $\{a_i\}$ is a sequence in (0,1) such that $\Pi _{i=2}^{+\infty }a_i=\frac{1}{2}$. Let $g:E\rightarrow R$ be defined by $g(x)=||x||^2$, $x\in E$, then the Bregman distance $D(x,y)=g(x)-g(y)-\langle \triangledown g(y),x-y\rangle =||x||^2-||y||^2-\langle 2y,x-y\rangle =||x-y||^2$, $\forall x,y\in C$, $F(T)=\{0\}(\ne \emptyset )$ and E is a Hilbert space.

It is proved in Goebel and Kirk [3] that

• (i) $||Tx-Ty||\le 2||x-y||,\,\forall x,y\in C;$

• (ii) $||T^nx-T^ny||\le (2\Pi _{j=2}^{n}a_j)||x-y||,\,\forall x,y\in C,\,\forall n\ge 2.$

Letting $\zeta (t)=t$, $\forall t\ge 0$, $\{\mu _n\}$ be a nonnegative real sequence with $\mu _n\rightarrow 0$ as $n\rightarrow +\infty$ and

$$\begin{aligned} \nu _n=\left\{ \begin{array}{l} 3,\quad {\text {if}}\,\,n=1,\\ (2\Pi _{j=2}^{n}a_j)^2-1,\quad {\text {if}}\,\,n\ge 2\,\,{\text {and}}\,\,n\in N, \end{array} \right. \end{aligned}$$

then from (i) and (ii), we have

$$\begin{aligned} ||T^nx-T^ny||^2\le ||x-y||^2+\nu _n\zeta (||x-y||^2)+\mu _n,\,\,\forall x,y\in C,\,\forall n\ge 1, \end{aligned}$$

i.e.,

$$\begin{aligned} D(T^nx,T^ny)\le D(x,y)+\nu _n\zeta (D(x,y))+\mu _n,\,\,\forall x,y\in C,\,\forall n\ge 1. \end{aligned}$$

Let $x_0=(1,0,0,\cdots )$, $y_0=(\frac{1}{2},0,0,\cdots )\in C$, then

$$\begin{aligned}&D(Tx_0,Ty_0)=||Tx_0-Ty_0||^2=||(0,1^2,0,\cdots )-(0,\frac{1}{4},0,\cdots )||^2=(1-\frac{1}{4})^2=\frac{9}{16}\\&\quad >||x_0-y_0||^2=||(1,0,0,\cdots )-(\frac{1}{2},0,0,\cdots )||^2=(1-\frac{1}{2})^2=\frac{4}{16}=D(x_0,y_0). \end{aligned}$$

These imply that T is a Bregman totally D-asymptotically nonexpansive mapping with the nonempty fixed point set which is not a Bregman D-nonexpansive mapping. Hence, T is a Bregman totally quasi-D-asymptotically nonexpansive mapping which is not a Bregman D-nonexpansive mapping.

Let C be a nonempty closed convex subset of E and T be a mapping from C to itself. A point $p\in C$ is said to be an asymptotic fixed point of T [15, 16] if C contains a sequence $\{x_n\}$ which converges weakly to p such that $\lim _{n\rightarrow \infty }||x_n-Tx_n||=0$. A point $p\in C$ is said to be a strong asymptotic fixed point of T [15, 16] if C contains a sequence which converges strongly to p such that $\lim _{n\rightarrow \infty }||x_n-Tx_n||=0$. We denote the sets of asymptotic fixed points and strong asymptotic fixed points of T by $\widehat{F}(T)$ and $\widetilde{F}(T)$, respectively.

Definition 2.5

• (1) A mapping T from C into itself is said to be Bregman relatively nonexpansive [13], if ${\widehat{F}}^{}(T)=F(T)\ne \emptyset$ and $D(p,Tx)\le D(p,x)$ for all $x\in C$ and $p\in F(T)$;

• (2) T is said to be Bregman weak relatively nonexpansive [16], if $\widetilde{F}(T)=F(T)\ne \emptyset$ and $D(p,Tx)\le D(p,x)$ for all $x\in C$ and $p\in F(T)$;

• (3) T is said to be Bregman quasi-D-nonexpansive [20], if $F(T)\ne \emptyset$ and $D(p,Tx)\le D(p,x)$ for all $x\in C$ and $p\in F(T)$;

• (4) T is said to be Bregman firmly nonexpansive [21], if $\langle \triangledown g(Tx)-\triangledown g(Ty),Tx-Ty\rangle \le \langle \triangledown g(x)-\triangledown g(y),Tx-Ty\rangle$,$\forall x,y\in C$, or, equivalently, $D(Tx,Ty)+D(Ty,Tx)+D(Tx,x)+D(Ty,y)\le D(Tx,y)+D(Ty,x)$,$\forall x,y\in C$;

• (5) T is said to be Bregman strongly nonexpansive [12, 15, 18, 24], if $\widehat{F}(T)\ne \emptyset$ and $D(p,Tx)\le D(p,x)$ for all $x\in C$ and $p\in \widehat{F}(T)$ and if whenever $\{x_n\}\subset E$ is bounded, $p\in \widehat{F}(T)$ and $\lim _{n\rightarrow +\infty }[D(p,x_n)-D(p,Tx_n)]=0$, it follows that $\lim _{n\rightarrow +\infty }D(Tx_n,x_n)=0$;

• (6) T is said to be relatively quasi-nonexpansive [9, 10], if $\widehat{F}(T)=F(T)\ne \emptyset$ and $\phi (p,Tx)\le \phi (p,x)$ for all $x\in C$ and $p\in F(T)$;

• (7) T is said to be weak relatively nonexpansive [9–11], if $\widetilde{F}(T)=F(T)\ne \emptyset$ and $\phi (p,Tx)\le \phi (p,x)$ for all $x\in C$ and $p\in F(T)$;

• (8) T is said to be quasi-$\phi$-nonexpansive [4, 10, 11], if $F(T)\ne \emptyset$ and $\phi (p,Tx)\le \phi (p,x)$ for all $x\in C$ and $p\in F(T)$.

Remark 2.3

(1) If $\zeta (t)=t,t\ge 0$, then (2.1) reduces to

$$\begin{aligned} D(p,T^nx)\le (1+\nu _n)\cdot D(p,x)+\mu _n,\forall n\ge 1,x\in C,p\in F(T). \end{aligned}$$

(2.6)

In addition, if $\mu _n\equiv 0$ for all $n\ge 1$, then Bregman totally quasi-D-asymptotically nonexpansive mappings coincide with Bregman quasi-D-asymptotically nonexpansive mappings. If $\mu _n\equiv 0$ and $\nu _n\equiv 0$ for all $n\ge 1$, we obtain from (2.6) the class of mappings that includes the class of Bregman quasi-nonexpansive mappings. If $\nu _n\equiv 0$ and $\mu _n=\sigma _n=\max \{0,\sup _{x\in E,p\in F(T)}(D(p,T^nx)-D(p,x))\}$, for all $n\ge 1$, then (2.6) reduces to (2.3) which has been studied as mappings—Bregman quasi-D-asymptotically nonexpansive in the intermediate sense.

(2) From the definitions, it is obvious that if $\widehat{F}(T)=F(T)\ne \emptyset$, then a Bregman strongly nonexpansive mapping is a Bregman relatively nonexpansive mapping; a Bregman relatively nonexpansive mapping is a Bregman quasi-D-nonexpansive mapping; a Bregman quasi-D-nonexpansive mapping is a Bregman quasi-D-asymptotically nonexpansive mapping, but the converse is not true.

If taking $\zeta (t)=t$,$t\ge 0$,$\nu _n=k_n-1$,$\mu _n=0$,$\lim _{n\rightarrow +\infty }k_n=1$, then (2.1) can be rewritten as (2.2). This implies that each Bregman quasi-D-asymptotically nonexpansive mapping must be a Bregman total quasi-D-asymptotically nonexpansive mapping, but the converse is not true. In [24], Chang et al. give an example of Bregman total quasi-D-asymptotically nonexpansive mapping. A Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse is not true in general. Indeed, for any mapping $T:C\rightarrow C$, we have $F(T)\subset \widetilde{F}(T)\subset \widehat{F}(T)$. If T is Bregman relatively nonexpansive, then $F(T)=\widetilde{F}(T)=\widehat{F}(T)$. In [30], Naraghirad and Yao have given two examples of a Bregman weak relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping, and a Bregman quasi-nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weak relatively nonexpansive mapping.

(3) The class of quasi-$\phi$-(asymptotically) nonexpansive mappings is more general than that of relatively nonexpansive mappings which requires the restriction: $\widehat{F}(T)=F(T)$. A quasi-$\phi$-nonexpansive mapping with a nonempty fixed point set F(T) is a quasi-$\phi$-asymptotically nonexpansive mapping, but the converse may not be true. In the framework of Hilbert spaces, quasi-$\phi$-(asymptotically) nonexpansive mappings are reduced to quasi-(asymptotically) nonexpansive mappings.

The idea of the definition of a total asymptotically nonexpansive mapping is to unify various definitions of classes of mappings associated with the class of asymptotically nonexpansive mappings and to prove a general convergence theorem applicable to all these classes of nonlinear mappings.

The theory of fixed points with respect to Bregman distances has been studied in the last ten years and much intensively in the last six years. In [14], Bauschke and Combettes introduced an iterative method to construct the Bregman projection of a point onto a countable intersection of closed and convex sets in reflexive Banach spaces. They proved strong convergence theorem of the sequence produced by their method; for more detail, see [14, Theorem 4.6]. For some recent articles on the existence of fixed points for Bregman nonexpansive type mappings, we refer the readers to [12–24, 26, 27, 30].

We need the following lemmas for our main results.

Lemma 2.1

[12, 19] Let E be a Banach space and $g:E\rightarrow \mathbb {R}$ a G$\hat{a}$teaux differentiable function which is locally uniformly convex on E. Let $\{y_n\}$ and $\{z_n\}$ be sequences in E such that either $\{y_n\}$ or $\{z_n\}$ is bounded. Then

• (1) $\lim _{n\rightarrow +\infty }D(y_n,z_n)=0$ $\Longrightarrow$ $\lim _{n\rightarrow +\infty }||y_n-z_n||=0$;

• (2) if $x,x_n\in E$ and the sequence $\{D(x_n,x)\}$ is bounded, then the sequence $\{x_n\}$ is also bounded.

Lemma 2.2

[30] Let C be a nonempty closed convex subset of a reflexive Banach space E, Let $g:E\rightarrow \mathbb {R}$ be a G$\hat{a}$teaux differentiable and totally convex function, and let $x\in E$. Then

• (1) $z=\textit{Proj}_C^g(x)$ if and only if $\langle y-z,\triangledown g(x)-\triangledown g(z)\rangle \le 0$,$\forall y\in C$;

• (2) $D(y,\textit{Proj}_C^g(x))+D(\textit{Proj}_C^g(x),x)\le D(y,x),\forall x\in E,y\in C$.

Lemma 2.3

(Lemma 1.2 in [31]) Let C be a nonempty closed convex subset of Banach space E and $g:E\rightarrow (-\infty ,+\infty ]$ be a G$\hat{a}$teaux differentiable function which is locally uniformly convex on E. Let $T:C\rightarrow C$ be a closed and Bregman totally quasi-D-asymptotically nonexpansive mapping with nonnegative real sequences $\{\nu _n\}$, $\{\mu _n\}$ and a strictly increasing and continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$. If $\nu _n,\mu _n\rightarrow 0(as\; n\rightarrow +\infty )$, then F(T) is a closed convex subset of C.

Lemma 2.4

[30] Let E be a Banach space, $r>0$ be a positive number, and $g:E\rightarrow \mathbb {R}$ be a continuous and convex function which is uniformly convex on bounded subsets of E. Then

$$\begin{aligned} g(\sum _{n=1}^m\lambda _nx_n)\le \sum _{n=1}^m\lambda _ng(x_n)-\lambda _i\lambda _j\rho _r(||x_i-x_j||) \end{aligned}$$

for any given infinite subset $\{x_n\}\subset B_r(0)=\{x\in E:||x||\le r\}$ and for any given sequence $\{\lambda _n\}$ of positive numbers with $\sum _{n=1}^m\lambda _n=1$, for any $i,j\in \{1,2,\cdots ,m\}$ with $i<j$, where $\rho _r$ is the gauge of uniform convexity of g.

For solving the equilibrium problem, let us assume that the bifunction $f:C\times C\rightarrow \mathbb {R}$ satisfies the following conditions:

• (C1) $f(x,x)=0,\forall x\in C$;

• (C2) f is monotone, i.e.,  $f(x,y)+f(y,x)\le 0,\forall x,y\in C$;

• (C3) for each $y\in C$, the function $x\mapsto f(x,y)$ is upper semicontinuous;

• (C4) $\forall x\in C,y\mapsto f(x,y)$ is a convex and lower semicontinuous.

Proposition 2.1

(Lemma 2.9 in [19], Lemma 11 in [12]) Let E be a reflexive Banach space and $g:E\rightarrow \mathbb {R}$ a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subset of E. Let C be a nonempty, closed, and convex subset of E and $f:C\times C\rightarrow \mathbb {R}$ a bifunction satisfying conditions $(C1)--(C4)$ and ${ EP}(f)\ne \emptyset$. For $r>0$ and $x\in E$, define a mapping $T_r^f:E\rightarrow C$ as follows:

$$\begin{aligned} T_r^fx=\{z\in C:f(z,y)+\frac{1}{r}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}. \end{aligned}$$

Then, the following statements hold:

• (1) $dom(T_r^f)=E$;

• (2) $T_r^f$ is single valued;

• (3) $T_r^f$ is a Bregman firmly nonexpansive mapping, i.e., for all $x,y\in E$,

  $$\begin{aligned} \langle T_r^fx-T_r^fy,\triangledown g(T_r^fx)-\triangledown g(T_r^fy)\rangle \le \langle T_r^fx-T_r^fy,\triangledown g(x)-\triangledown g(y)\rangle ; \end{aligned}$$

• (4) $F(T_r^f)={ EP}(f)$;

• (5) ${ EP}(f)$ is closed and convex of E;

• (6) $D(q,T_r^fx)+D(T_r^fx,x)\le D(q,x),\forall q\in F(T_r^f)$.

Lemma 2.5

Let E be a reflexive Banach space and $g:E\rightarrow \mathbb {R}$ a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subset of E. Let C be a nonempty, closed, and convex subset of E and $f:C\times C\rightarrow \mathbb {R}$ a bifunction satisfying conditions (C1)–(C4) and ${ EP}(G)\ne \emptyset$, $\varphi :C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, and $A:C\rightarrow E^*$ be a continuous and monotone mapping. For $r>0$ and $x\in E$, define a mapping $T_r^G:E\rightarrow C$ as follows:

$$\begin{aligned} T_r^Gx=\{z\in C:G(z,y)+\frac{1}{r}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}, \end{aligned}$$

where $G(x,y)=f(x,y)+\varphi (y)-\varphi (x)+\langle Ax,y-x\rangle ,\forall x,y\in E$. Then, the following statements hold:

• (1) $dom(T_r^G)=E$;

• (2) $T_r^G$ is single valued;

• (3) $T_r^G$ is a Bregman firmly nonexpansive mapping, i.e., for all $x,y\in E$,

  $$\begin{aligned} \langle T_r^Gx-T_r^Gy,\triangledown g(T_r^Gx)-\triangledown g(T_r^Gy)\rangle \le \langle T_r^Gx-T_r^Gy,\triangledown g(x)-\triangledown g(y)\rangle ; \end{aligned}$$

• (4) $F(T_r^G)=\textit{GMEP}(f,\varphi )$;

• (5) $\textit{GMEP}(f,\varphi )$ is closed and convex of E;

• (6) $D(q,T_r^Gx)+D(T_r^Gx,x)\le D(q,x),\forall q\in F(T_r^G)$.

Proof

By Proposition 2.1 (or also see Lemma 2.9 in ref. [19], Lemma 11 in ref. [12]), we only need to prove a bifunction $G: C\times C\rightarrow R$ also satisfies the conditions (C1)–(C4). Indeed, it follows from the conditions (C1) and (C2) of $f: C\times C\rightarrow R$, and monotonicity of the mapping $A: C\rightarrow E^*$ that

$$\begin{aligned}&G(x,x)=f(x,x)+\phi (x)-\phi (x)+\langle Ax,x-x\rangle =f(x,x)=0,\\&G(x,y)+G(y,x)=f(x,y)+f(y,x)-\langle Ax-Ay,x-y\rangle \le 0-0=0. \end{aligned}$$

Using the condition (C3) of $f: C\times C\rightarrow R$, lower semi-continuity of $\phi : C\rightarrow R$, and continuity of the mapping $A: C\rightarrow E^*$, we have

$$\begin{aligned}&\overline{\lim }_{x\rightarrow x_0}G(x,y)\le \overline{\lim }_{x\rightarrow x_0}f(x,y)+\phi (y)-\underline{\lim }_{x\rightarrow x_0}\phi (x)+\overline{\lim }_{x\rightarrow x_0}\langle Ax,y-x\rangle \\&\quad =\overline{\lim }_{x\rightarrow x_0}f(x,y)+\phi (y)-\underline{\lim }_{x\rightarrow x_0}\phi (x)+\overline{\lim }_{x\rightarrow x_0}\big [\langle Ax-Ax_0,y-x\rangle \\&\qquad +\langle Ax_0,x_0-x\rangle +\langle Ax_0,y-x_0\rangle \big ]\\&\quad \le f(x_0,y)+\phi (y)-\phi (x_0)+\langle Ax_0,y-x_0\rangle =G(x_0,y)\,\,for \,\,all\,\,x_0\in C. \end{aligned}$$

By the condition (C4) of $f: C\times C\rightarrow R$ and lower semi-continuity of $\phi : C\rightarrow R$, we obtain that

$$\begin{aligned}&\underline{\lim }_{y\rightarrow y_0}G(x,y)=\underline{\lim }_{y\rightarrow y_0}\big [f(x,y)+\phi (y)-\phi (x)+\langle Ax,y-x\rangle \big ]\\&\quad \ge \underline{\lim }_{y\rightarrow y_0}f(x,y)+\underline{\lim }_{y\rightarrow y_0}\phi (y)-\phi (x)+\underline{\lim }_{y\rightarrow y_0}\langle Ax,y-x\rangle \\&\quad \ge f(x,y_0)+\phi (y_0)-\phi (x)+\langle Ax,y_0-x\rangle =G(x,y_0)\,\,for \,\,all\,\,y_0\in C. \end{aligned}$$

Thus, a bifunction $G: C\times C\rightarrow R$ satisfies the conditions (C1)–(C4). This completes the proof of Lemma 2.5. $\square$

Lemma 2.6

[32] Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:

• (1) g is strongly coercive and uniformly convex on bounded subsets of E;

• (2) $dom\,g^*=E^*$, $g^*$ is bounded on bounded subsets and uniformly smooth on bounded subsets of $E^*$;

• (3) $dom\,g^*=E^*$, $g^*$ is Fr$\acute{e}$chet differentiable and $\triangledown g^*$ is uniformly norm-to-norm continuous on bounded subsets of $E^*$.

Lemma 2.7

[32] Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a continuous convex function which is strongly coercive. Then the following assertions are equivalent:

• (1) g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;

• (2) $g^*$ is Fr$\acute{e}$chet differentiable and $\triangledown g^*$ is uniformly norm-to-norm continuous on bounded subsets of $E^*$;

• (3) $dom\,g^*=E^*$, $g^*$ is strongly coercive and uniformly convex on bounded subsets of $E^*$.

Lemma 2.8

Let C be a nonempty closed and convex subset of reflexive Banach space E, $g:E\rightarrow \mathbb {R}$ be a G$\hat{a}$teaux differentiable and totally convex function, and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings. Then there exist nonnegative real sequences $\{\nu _n\},\{\mu _n\}$, satisfying $\nu _n,\mu _n\rightarrow 0$ as $n\rightarrow +\infty$ and a strictly increasing continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$, such that for all $x\in C,p\in F(T)$, $D(p,T_i^nx)\le D(p,x)+\nu _n\cdot \zeta [D(p,x)]+\mu _n,\forall n\ge 1$, for $i=1,2,\cdots ,m$.

Proof

Since $T_i:C\rightarrow C$ is a Bregman totally quasi-D-asymptotically nonexpansive mapping for $i=1,2,\cdots ,m$, there exist nonnegative real sequences $\{\nu _n^{(i)}\},\{\mu _n^{(i)}\}$ with $\nu _n^{(i)},\mu _n^{(i)}\rightarrow 0$ as $n\rightarrow +\infty$ and strictly increasing continuous function $\zeta _i:\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta _i(0)=0$ such that for all $x\in C$, $p\in F(T)$

$$\begin{aligned} D(p,T_i^nx)\le D(p,x)+\nu _n^{(i)}\cdot \zeta [D(p,x)]+\mu _n^{(i)},\forall n\ge 1. \end{aligned}$$

Setting

$$\begin{aligned} \nu _n=\max _{i\in \{1,2,\cdots ,m\}}\{\nu _n^{(i)}\}, \mu _n=\max _{i\in \{1,2,\cdots ,m\}}\{\mu _n^{(i)}\}, \zeta (a)=\max _{i\in \{1,2,\cdots ,m\}}\{\zeta _i(a)\}\;\textit{for}\;a\ge 0, \end{aligned}$$

we get nonnegative real sequences $\{\nu _n\},\{\mu _n\}$ with $\nu _n,\mu _n\rightarrow 0$ as $n\rightarrow +\infty$ and a strictly increasing continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$, such that

$$\begin{aligned} D(p,T_i^nx)\le D(p,x)+\nu _n\cdot \zeta [D(p,x)]+\mu _n,\forall n\ge 1, \end{aligned}$$

for all $x\in C,\;p\in F(T)$ and each $i=1,2,\cdots ,m$. $\square$

3 Main Results

We now state and prove the main results of this paper.

Theorem 3.1

Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of closed and Bregman totally quasi-D-asymptotically nonexpansive mappings with nonnegative real sequences $\{\nu _n^{(i)}\}$,$\{\mu _n^{(i)}\}$ and a strictly increasing and continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$ and $\nu _n^{(i)},\mu _n^{(i)}\rightarrow 0(as\;n\rightarrow +\infty \;and \;for\; each\;i=1,2,\cdots ,m)$. Assume that $T_i$ is asymptotically regular on C for all $i=1,2,\cdots ,m$, i.e., $\lim _{n\rightarrow +\infty }\sup _{x\in K}||T_i^{n+1}x-T_i^nx||=0$ holds for any bounded subset K of C and $F=[\bigcap _{i=1}^mF(T_i)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{1}{r_{k,n}}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in E$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=\textit{Proj}_{C_1}^g(x_0),\\ y&_n=\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)],\\ z&_n=\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)],\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:D(z,u_n)\le \alpha _n D(z,x_n)+(1-\alpha _n)D(z,z_n)\le D(z,x_n)\\&+(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n\},\\ x&_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0), \end{aligned} \right. \end{aligned}$$

(3.1)

where $\zeta _n=\nu _n\cdot \sup _{z\in F}\zeta [D(z,x_n)]+\mu _n,\nu _n\!=\!\max _{1\le i\le m}\{\nu _n^{(i)}\},\mu _n=\max _{1\le i\le m}\{\mu _n^{(i)}\}$. $\{\alpha _n\}$,$\{\beta _n^{(i)}\} (i=0,1,\cdots ,m)$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _{n}^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $\textit{Proj}_F^gx_0$.

Proof

We define a bifunction $G_k:C\times C\rightarrow \mathbb {R}$ by

$$\begin{aligned} G_k(x,y)=f_k(x,y)+\varphi _k(y)-\varphi _k(x)+\langle A_kx,y-x\rangle ,\forall x,y\in C. \end{aligned}$$

Then, we prove from Lemma 2.5 that the bifunction $G_k$ satisfies conditions (C1)–(C4) for each $k=1,2,\cdots ,h$. Therefore, the generalized mixed equilibrium problem (1.1) is equivalent to the following equilibrium problem: find $x\in C$ such that $G_k(x,y)\ge 0,\;\forall y\in C$. Hence, ${\textit{GMEP}}(f_k,\varphi _k)={ EP}(G_k)$. By taking $\theta _n^k=T_{r_{k,n}}^{G_k}T_{r_{k-1,n}}^{G_{k-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1},k=1,2,\cdots ,h$ and $\theta _n^0=I$ for all $n\ge 1,$ we obtain $u_n=\theta _n^my_n$.

In view of Lemmas 2.3 and 2.5, we find that F is closed and convex, so that $\textit{Proj}_F^g(x_0)$ is well defined for any $x_0\in E$.

We divide the proof of Theorem 3.1 into six steps:

Step 1 We first show that $C_n$ is closed and convex for each $n\ge 1$.

In fact, it is obvious that $C_1=C$ is closed and convex. Suppose that $C_n$ is closed and convex for some $n\ge 1$. For any $z\in C_n$, we know that

$$\begin{aligned} D(z,u_n)\le \alpha _n D(z,x_n)+(1-\alpha _n)D(z,z_n)\le D(z,x_n)+(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n \end{aligned}$$

is equivalent to the following:

$$\begin{aligned} \langle z-u_n,\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)-\triangledown g(u_n)\rangle \le \alpha _nD(u_n,x_n)+(1-\alpha _n)D(u_n,z_n) \end{aligned}$$

and

$$\begin{aligned} \langle z-z_n,\triangledown g(x_n)-\triangledown g(z_n)\rangle \le D(z_n,x_n)+(1-\beta _n^{(0)})\zeta _n,\forall i\ge 1; \end{aligned}$$

noting that the left-hand sides of the last two inequalities are affine with respect to z, $C_{n+1}$ is closed and convex. Then, for each $n\ge 1$, $C_n$ is closed and convex.

Step 2 Assume that $F\subset C_n$ for all $n\ge 1$. Then the sequence $\{x_n\}$ is bounded.

In fact, by $x_n=\textit{Proj}_{C_n}^g(x_0)$, it then follows from Lemma 2.2 that

$$\begin{aligned} D(x_n,x_0)=D(\textit{Proj}_{C_n}^g(x_0),x_0)\le D(p,x_0)-D(p,x_n)\le D(p,x_0) \end{aligned}$$

for each $p\in F\subset C_n,\forall n\ge 1$. Hence, the sequence $\{D(x_n,x_0)\}$ is bounded. From Lemma 2.1, we see that the sequence $\{x_n\}$ is also bounded and so are $\{T_i^nx_n\}$, $\{y_n\}$, $\{z_n\}$, and $\{u_n\}$.

Step 3 Next, we show, by induction, that $F\subset C_n$ for all $n\ge 1$.

In fact, it is obvious that $F\subset C_1=C$. Suppose that $F\subset C_n$ for some $n\ge 1$. Let $p\in F$. Since $T_i:E\rightarrow C(i=1,2,\cdots ,m)$ is a finite family of closed and Bregman totally quasi-D-asymptotically nonexpansive mappings, by the definitions of $D(\cdot ,\cdot )$ and the sequence $\{x_n\}$, Lemma 2.4, and the fact of $\langle x,x^*\rangle \in \partial g\iff g(x)+g^*(x^*)=\langle x,x^*\rangle$, for each $i\ge 1$, we have

$$\begin{aligned} D(p,z_n)= & {} D(p,\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)])\nonumber \\= & {} g(p)-g\{\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)]\}-\langle p-\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)\nonumber \\&+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)],\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)\rangle \nonumber \\= & {} g(p)-\langle p,\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)\rangle +g^*[\beta _n^{(0)}\triangledown g(x_n)\nonumber \\&+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)]\nonumber \\\le & {} \beta _n^{(0)}g(p)+\sum _{i=1}^m\beta _n^{(i)}g(p)+\beta _n^{(0)}g^*[\triangledown g(x_n)]+\sum _{i=1}^m\beta _n^{(i)}g^*[\triangledown g(T_i^nx_n)]\nonumber \\\le & {} \beta _n^{(0)}D(p,x_n)+\sum _{i=1}^m\beta _n^{(i)}D(p,T_i^nx_n)\nonumber \\\le & {} \beta _n^{(0)}D(p,x_n)+\sum _{i=1}^m\beta _n^{(i)} \{D(p,x_n)+\nu _n^{(i)}\cdot \zeta [D(p,x_n)]+\mu _n^{(i)}\}\nonumber \\\le & {} \beta _n^{(0)}D(p,x_n)+\sum _{i=1}^m\beta _n^{(i)}\{D(p,x_n)+\nu _n\cdot \zeta [D(p,x_n)]+\mu _n\}\nonumber \\= & {} D(p,x_n)+(1-\beta _n^{(0)})\{\nu _n\cdot \zeta [D(p,x_n)]+\mu _n\}\nonumber \\\le & {} D(p,x_n)+(1-\beta _n^{(0)})\cdot \zeta _n. \end{aligned}$$

(3.2)

Observe that $p\in F$ implies $p\in C$. Thus, by the definition of the sequence $\{x_n\}$, (3.2), Lemmas 2.4 and 2.5, and the fact that $T_{r_{k,n}}^{G_k}(k=1,2,\cdots ,m)$ is a Bregman quasi-D-nonexpansive mapping, for each $p\in F$, we have

$$\begin{aligned} D(p,u_n)= & {} D(p,\theta _n^my_n)\nonumber \\\le & {} D(p,y_n)\nonumber \\= & {} D(p,\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)])\nonumber \\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)D(p,z_n)\nonumber \\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)[D(p,x_n)+(1-\beta _n^{(0)})\cdot \zeta _n]\nonumber \\\le & {} D(p,x_n)+(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n. \end{aligned}$$

(3.3)

This shows that $p\in C_{n+1}$, which implies that $F\subset C_{n+1}$. Hence $F\subset C_n$ for all $n\ge 1$.

Step 4 We show that $\{x_n\}$ is a Cauchy sequence.

In fact, in view of the construction of the sets $C_n$, we find that $C_{n+1}\subset C_n$ and $x_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0)\in C_{n+1}$ and from Lemma 2.5, we obtain

$$\begin{aligned} 0\le D(x_n,x_{n+1})\le D(x_n,x_0)-D(x_{n+1},x_0) \end{aligned}$$

for all $n\ge 1$. Thus, the sequence $\{D(x_n,x_0)\}$ is nondecreasing. It follows from the boundedness of $\{D(x_n,x_1)\}$ that the limit of $\{D(x_n,x_1)\}$ exists.

For any positive integer m, it then follows from Lemma 2.2 that

$$\begin{aligned} D(x_{n+m},x_{n+1})= & {} D(x_{n+m},\textit{Proj}_{C_{n+1}}^g(x_0))\nonumber \\\le & {} D(x_{n+m},x_0)-D(\textit{Proj}_{C_{n+1}}^g(x_0),x_0)\nonumber \\= & {} D(x_{n+m},x_0)-D(x_{n+1},x_0); \end{aligned}$$

(3.4)

it follows from (3.4) that $D(x_{n+m},x_{n+1})\rightarrow 0$ as $n\rightarrow \infty$. We have from Lemma 2.1 and the boundedness of $\{x_n\}$ that

$$\begin{aligned} x_{n+m}-x_{n+1}\rightarrow 0,\;n\rightarrow \infty . \end{aligned}$$

Hence, the sequence $\{x_n\}$ is Cauchy in C. Since E is a Banach space and C is closed and convex, there exists $p\in C$ such that $x_n\rightarrow p$ as $n\rightarrow \infty$. Now, since $D(x_{n+m},x_{n+1})\rightarrow 0$ as $n\rightarrow \infty$, we have in particular that $\lim _{n\rightarrow \infty }D(x_{n+2},x_{n+1})=0$ and this further implies that $\lim _{n\rightarrow \infty }||x_{n+1}-x_{n+2}||=0$ from Lemma 2.1.

Since $x_{n+2}=\textit{Proj}_{C_{n+2}}^g(x_0)\in C_{n+2}\subset C_{n+1}(\subset C)$ and by the definition of $C_{n+2}$, it follows that we have

$$\begin{aligned} D(x_{n+2},u_{n+1})\le D(x_{n+2},x_{n+1})+(1-\alpha _{n+1})(1-\beta _{n+1}^{(0)})\zeta _{n+1}\rightarrow 0,n\rightarrow \infty . \end{aligned}$$

From Lemma 2.1, we obtain that $\lim _{n\rightarrow \infty }||x_{n+2}-u_{n+1}||=0$. Therefore

$$\begin{aligned} ||x_{n+1}-u_{n+1}||\le ||x_{n+1}-x_{n+2}||+||x_{n+2}-u_{n+1}||\rightarrow 0. \end{aligned}$$

(3.5)

It follows from $\lim _{n\rightarrow +\infty }||x_{n}-p||=0$ and (3.5) that

$$\begin{aligned} u_{n}\rightarrow p,n\rightarrow \infty . \end{aligned}$$

(3.6)

Step 5 We prove that $p\in [\bigcap _{i=0}^{m}F(T_i)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]$.

(a) First we prove that $p\in \bigcap _{i=0}^{+\infty }F(T_i)$.

Since g is uniformly smooth on bounded subsets of E, we have that $\triangledown g(\cdot )$ is uniformly norm-to-norm continuous on any bounded sets and (3.5), and we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }||\triangledown g(x_n)-\triangledown g(u_n)||=0. \end{aligned}$$

(3.7)

It follows from the boundedness of the sequences $\{x_n\}$, Lemma 2.9, and $D(p,T_i^nx_n)\le D(p,x_n)+\nu _n\cdot \zeta [D(p,x_n)]+\mu _n$ for each $p\in F$ and $i=1,2,\cdots ,m$ that the sequences $\{\triangledown g(x_n)\}$ and $\{\triangledown g(T_i^nx_n)\}$ are bounded. Thus, there exists $r>0$ such that $\{\triangledown g(x_n)\}\subset B_r(0)$ and $\{\triangledown g(T_i^nx_n)\}\subset B_r(0)$. For each $p\in F$, we have from Lemmas 2.2, 2.4, and (3.3) that

$$\begin{aligned} D(p,u_n)= & {} D(p,\theta _n^my_n)\\&\le D(p,y_n)\\= & {} D(p,\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)])\\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)D(p,z_n)\\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)\cdot [\beta _n^{(0)}D(p,x_n)+\sum _{i=1}^m\beta _n^{(i)}D(p,T_i^nx_n)\\&-\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)]\\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)\cdot \{\beta _n^{(0)}D(p,x_n)+\sum _{i=1}^m\beta _n^{(i)}[D(p,x_n)\\&+\nu _n^{(i)}\cdot \zeta (D(p,x_n))+\mu _n^{(i)}]-\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)\}\\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)\cdot [D(p,x_n)+(1-\beta _n^{(0)})\zeta _n\\&-\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)]\\\le & {} \alpha _nD(p,x_n)+(1-\alpha _n)D(p,x_n)+(1-\beta _n^{(0)})\zeta _n\\&-(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)\\\le & {} D(p,x_n)+\zeta _n-(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||). \end{aligned}$$

This implies that

$$\begin{aligned} 0\le (1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}\rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)\le D(p,x_n)-D(p,u_n)+\zeta _n.\nonumber \\ \end{aligned}$$

(3.8)

On the other hand, we have

$$\begin{aligned} \begin{aligned} |D(p,x_n)-D(p,u_n)|&=|-D(x_n,u_n)+\langle x_n-p,\triangledown g(u_n)-\triangledown g(x_n)\rangle |\\&\le D(x_n,u_n)+||x_n-p||\cdot ||\triangledown g(u_n)-\triangledown g(x_n)||. \end{aligned} \end{aligned}$$

In view of (3.5) and (3.7), we obtain

$$\begin{aligned} D(p,x_n)-D(p,u_n)\rightarrow 0,n\rightarrow \infty . \end{aligned}$$

(3.9)

Combining (3.8)–(3.9), $\lim _{n\rightarrow +\infty }\zeta _n=0$, and the assumption $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0$ for $i=1,2,\cdots ,m$, we have

$$\begin{aligned} \rho _r^*(||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||)\rightarrow 0,n\rightarrow \infty . \end{aligned}$$

It follows from the property of $\rho _r^*$ that

$$\begin{aligned} \lim _{n\rightarrow +\infty }||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||=0. \end{aligned}$$

(3.10)

Since $x_n\rightarrow p$ as $n\rightarrow \infty$ and $\triangledown g(\cdot )$ is uniformly norm-to-norm continuous on any bounded sets, we obtain that

$$\begin{aligned} ||\triangledown g(x_n)-\triangledown g(p)||\rightarrow 0\;as\;n\rightarrow \infty . \end{aligned}$$

(3.11)

Note that

$$\begin{aligned} ||\triangledown g(T_i^nx_n)-\triangledown g(p)||\le ||\triangledown g(p)-\triangledown g(x_n)||+||\triangledown g(x_n)-\triangledown g(T_i^nx_n)||. \end{aligned}$$

From (3.10) and (3.11), we see that

$$\begin{aligned} \lim _{n\rightarrow +\infty }||\triangledown g(T_i^nx_n)-\triangledown g(p)||=0. \end{aligned}$$

(3.12)

Since g is uniformly convex and uniformly smooth on bounded subsets of E, from Lemma 2.7, we have that $\triangledown g^*(\cdot )$ is also uniformly norm-to-norm continuous on any bounded sets. It follows from (3.12) that

$$\begin{aligned} \lim _{n\rightarrow +\infty }||T_i^nx_n-p||=0. \end{aligned}$$

(3.13)

Noting that $||T_i^{n+1}x_n-p||\le ||T_i^{n+1}x_n-T_i^nx_n||+||T_i^nx_n-p||$, the asymptotic regularity of $T_i$ and (3.13), we have $\lim _{n\rightarrow +\infty }||T_i^{n+1}x_n-p||=0$. That is, $T_i(T_i^nx_n)\rightarrow p$ as $n\rightarrow \infty$, it follows from the closeness of $T_i$ that $T_ip=p$ for $i=1,2,\cdots ,m$, i.e.,  $p\in \bigcap _{i=1}^{m}F(T_i)$.

(b) Now we prove that $p\in \bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)=\bigcap _{k=1}^{h}{} { EP}(G_k)$.

In fact, in view of $u_n=\theta _n^hy_n$, (3.3), and Lemma 2.5, for each $q\in F(\theta _n^k)$, we have

$$\begin{aligned} 0\le D(u_n,y_n)=D(\theta _n^hy_n,y_n)\le D(p,y_n)-D(p,\theta _n^hy_n)\le D(p,x_n)-D(p,u_n)+\zeta _n. \end{aligned}$$

It follows from (3.9) and $\lim _{n\rightarrow +\infty }\zeta _n=0$ that $D(u_n,y_n)\rightarrow 0$ as $n\rightarrow \infty$. Using Lemma 2.1, we see that $||u_n-y_n||\rightarrow 0$ as $n\rightarrow \infty$. Furthermore, $||x_n-y_n||\le ||x_n-u_n||+||u_n-y_n||\rightarrow 0$ as $n\rightarrow \infty$. Since $x_n\rightarrow p$, $n\rightarrow \infty$ and $||x_n-y_n||\rightarrow 0$, $n\rightarrow \infty$, then $y_n\rightarrow p$, $n\rightarrow \infty$. By the fact that $\theta _n^k(k=1,2,\cdots ,h)$ is Bregman relatively nonexpansive and using Lemma 2.5 again, we have that

$$\begin{aligned} 0\le D(\theta _n^ky_n,y_n)\le D(p,y_n)-D(p,\theta _n^ky_n)\le D(p,x_n)-D(p,\theta _n^ky_n)+\zeta _n.\nonumber \\ \end{aligned}$$

(3.14)

Observe that

$$\begin{aligned} \begin{aligned} D(p,u_n)&=D(p,\theta _n^hy_n)=D(p,T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n)\\&=D(p,T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots \theta _{n}^{k}y_n)\le D(p,\theta _n^ky_n). \end{aligned} \end{aligned}$$

(3.15)

Using (3.15) in (3.14), we obtain that $0\le D(\theta _n^ky_n,y_n)\le D(p,x_n)-D(p,u_n)+\zeta _n\rightarrow 0,n\rightarrow \infty$. Then Lemma 2.1 implies that $\lim _{n\rightarrow \infty }||\theta _n^ky_n-y_n||=0,k=1,2,\cdots ,h$. Now $||\theta _n^ky_n-p||\le ||\theta _n^ky_n-y_n||+||y_n-p||\rightarrow 0$,$n\rightarrow \infty$,$k=1,2,\cdots ,m$. Similarly, $\lim _{n\rightarrow +\infty }||\theta _n^{k-1}y_n-p||=0,k=1,2,\cdots ,m$. This further implies that

$$\begin{aligned} \lim _{n\rightarrow +\infty }||\theta _n^{k-1}y_n-\theta _n^ky_n||=0. \end{aligned}$$

(3.16)

Also, since $\triangledown g(\cdot )$ is uniformly norm-to-norm continuous on any bounded sets and using (3.16), we obtain that $\lim _{n\rightarrow +\infty }||\triangledown g(\theta _n^ky_n)-\triangledown g(\theta _n^{k-1}y_n)||=0$. From the assumption $\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$ for each $k=1,2,\cdots ,h,$ we see that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{||\triangledown g(\theta _n^ky_n)-\triangledown g(\theta _n^{k-1}y_n)||}{r_{k,n}}=0. \end{aligned}$$

(3.17)

By Lemma 2.5, we have that for each $k=1,2,\cdots ,h$,

$$\begin{aligned} G_k(\theta _n^ky_n,y)+\frac{1}{r_{k,n}}\langle y-\theta _n^ky_n,\triangledown g(\theta _n^ky_n)-\triangledown g(\theta _n^{k-1}y_n)\rangle \ge 0,\forall y\in C. \end{aligned}$$

Furthermore, replacing n by $n_j$ in the last inequality and using condition (C2), we obtain

$$\begin{aligned} \begin{aligned}&||y-\theta _{n_j}^ky_{n_j}||\cdot \frac{||\triangledown g(\theta _{n_j}^ky_{n_j})-\triangledown g(\theta _{n_j}^{k-1}y_{n_j})||}{r_{k,n_j}}\\&\quad \ge \frac{1}{r_{k,n_j}} \langle y-\theta _{n_j}^ky_{n_j},\triangledown g(\theta _{n_j}^ky_{n_j})-\triangledown g(\theta _{n_j}^{k-1}y_{n_j})\rangle \\&\quad \ge -G_k(\theta _{n_j}^ky_{n_j},y)\ge G_k(y,\theta _{n_j}^ky_{n_j}),\forall y\in C. \end{aligned} \end{aligned}$$

By taking the limit as $j\rightarrow +\infty$ in the above inequality, for each $k=1,2,\cdots ,h,$ we have from the condition (C4), (3.17), and $\theta _{n_j}^ky_{n_j}\rightarrow p$ that $G_k(y,p)\le 0,\forall y\in C$.

For $0<t\le 1$ and $y\in C$, define $y_t=ty+(1-t)p$. It follows from $y,p\in C$ that $y_t\in C$ which yields that $G_k(y_t,p)\le 0$. It follows from the conditions (C1) and (C4) that

$$\begin{aligned} 0=G_k(y_t,y_t)\le tG_k(y_t,y)+(1-t)G_k(y_t,p)\le tG_k(y_t,y). \end{aligned}$$

That is

$$\begin{aligned} G_k(y_t,y)\ge 0. \end{aligned}$$

Letting $t\rightarrow 0^+$, from the condition (C3), we obtain that $G_k(p,y)\ge 0,\forall y\in C$. This implies that $p\in { EP}(G_k)$,$k=1,2,\cdots ,h$, i.e., $p\in \bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)=\bigcap _{k=1}^{h}{} { EP}(G_k)$. Thus we have $p\in F$.

Step 6 Finally, we prove that $p=\textit{Proj}_F^g(x_0)$.

From Lemma 2.2 and the definition of $x_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0)$, we see that

$$\begin{aligned} \langle x_{n+1}-z,\triangledown g(x_0)-\triangledown g(x_{n+1})\rangle \ge 0,\forall z\in C_{n+1}. \end{aligned}$$

Since $F\subset C_n$ for each $n\ge 1$, we have

$$\begin{aligned} \langle x_{n+1}-w,\triangledown g(x_0)-\triangledown g(x_{n+1})\rangle \ge 0,\forall w\in F. \end{aligned}$$

Let $n\rightarrow +\infty$ in the last inequality, we see that $\langle p-w,\triangledown g(x_0)-\triangledown g(p)\rangle \ge 0,\forall w\in F$. In view of Lemma 2.2, we obtain that $p=\textit{Proj}_F^g(x_0)$. This completes the proof of Theorem 3.1. $\square$

Remark 3.1

(1) If we suppose that $T_i$ is uniformly $L_i$-Lipschitz continuous on C for each $i=1,2,\cdots ,m$, then the assumption that $T_i$ is closed and asymptotically regular on C can be removed in Theorem 3.1.

(2) Theorem 3.1 improves and extends Theorem 12— the main result of Zhu, Chang, and Liu [12] in the following aspects:

(a) from a Bregman strongly nonexpansive mapping to a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings;

(b) from an equilibrium problem to a system of generalized mixed equilibrium problem;

(c) if we set $m=h=1,A_1=\varphi _1=0,\nu _n=\mu _n=0$ , and $\beta _n^{(0)}=0$ for all $n\in N^+$, then (3.1) reduces to (1.7) which has been studied as mapping—Bregman strongly nonexpansive mapping.

The space in Theorem 3.1 can be applicable to $L^P$, $P>1$. Now, we give the following Example 3.1 in order to support Theorem 3.1. Meanwhile, Example 3.1 also shows that there is a finite family of closed, uniformly asymptotically regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings which are not Bregman D-nonexpansive mappings.

Example 3.1

Let $E=l^2$ and $C=\{x\in l^2\big |\,||x||\le 1\}$, where $l^2=\big \{\sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots )\,\big |\sum _{n=1}^{+\infty }|\sigma _n|^2<+\infty \big \}$.$||\sigma ||=\big (\sum _{n=1}^{+\infty }|\sigma _n|^2\big )^{\frac{1}{2}}$,$\forall \sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots )\in l^2$;$\langle \sigma ,\eta \rangle =\sum _{n=1}^{+\infty }\sigma _n\eta _n$, $\forall \sigma =(\sigma _1,\sigma _2,\cdots ,\sigma _n,\cdots )$, $\eta =(\eta _1,\eta _2,\cdots ,\eta _n,\cdots )\in l^2$. Let $x_0=(1,0,0,\cdots )$, then $x_0\in C$ and $||x_0||=1$. Define the following finite family of mappings $T_i:C\rightarrow C$ by

$$\begin{aligned}&T_i(x_1,x_2,x_3,\cdots )\\&\quad =\left\{ \begin{array}{l} (0,x_1^2,a_2x_2,a_3x_3,\cdots ),\,\,if\,\,x\in \{x=(x_1,x_2,x_3,\cdots )\,|\,x=\frac{x_0}{2^n}\in C\}\mathop {=}\limits ^{\bigtriangleup }Q,\\ -\frac{1}{i+1}(x_1,x_2,x_3,\cdots ),\,\,if\,\,x\in \{x=(x_1,x_2,x_3,\cdots )\,|\,x\in C\,\,and\,\,x\ne \frac{x_0}{2^n}\}, \end{array} \right. \end{aligned}$$

for all $i=1,2,\cdots ,m$ and $n\ge 1,\,n\in N$, where $\{a_j\}$ is a sequence in (0,1) such that $\Pi _{j=2}^{+\infty }a_j=\frac{1}{2}$.

It is proved in Goebel and Kirk [3] that

• (i) $||T_ix-T_iy||\le 2||x-y||,\,\forall x,y\in Q,\,i=1,2,\cdots ,m;$

• (ii) $||T_i^nx-T_i^ny||\le (2\Pi _{j=2}^{n}a_j)||x-y||,\,\forall x,y\in Q,\,\forall n\ge 2,\,i=1,2,\cdots ,m.$

It is clear that $F(T_i)=\{0\}$ for all $i=1,2,\cdots ,m$, and E is a Hilbert space. Let $g:E\rightarrow R$ be defined by $g(x)=||x||^2$, $x\in E$, then the Bregman distance $D(x,y)=||x-y||^2$ for all $x,y\in E$.

Let $\zeta (t)=t$, $\forall t\ge 0$, and $\{\mu _n^{(i)}\}$ be a nonnegative real sequence with $\mu _n^{(i)}\rightarrow 0$ as $n\rightarrow +\infty$ and for each $i=1,2,\cdots ,m.$ For any $p\in F(T_i)=\{0\}$ and $x\in C$, we consider the following two cases:

(1) If $x\in Q$, then from (i) and (ii), we have

$$\begin{aligned} D(p,T_i^nx)= & {} ||p-T_i^nx||^2=||0-T_i^nx||^2\\= & {} ||T_i^n0-T_i^nx||^2\le (2\Pi _{j=2}^{n}a_j)^2||0-x||^2\\= & {} ||x||^2+[(2\Pi _{j=2}^{n}a_j)^2-1]\cdot ||x||^2=D(0,x)\\&+\,[(2\Pi _{j=2}^{n}a_j)^2-1]\cdot D(0,x)\\\le & {} D(p,x)+[(2\Pi _{j=2}^{n}a_j)^2-1]\cdot \zeta (D(p,x))+\mu _n^{(i)}; \end{aligned}$$

(2) If $x\in C\backslash Q$, then $x\ne \frac{x_0}{2^n}$, $x\in C$ and $T_i^nx=\frac{(-1)^n}{(i+1)^n}x$, we have

$$\begin{aligned} D(p,T_i^nx)= & {} ||p-T_i^nx||^2=||0-T_i^nx||^2=||T_i^nx||^2=\frac{1}{(i+1)^{2n}}||x||^2\\\le & {} ||0-x||^2+\frac{1}{(i+1)^{2n}}||0-x||^2+\mu _n^{(i)}\\= & {} D(p,x)+\frac{1}{(i+1)^{2n}}D(p,x)+\mu _n^{(i)}\\= & {} D(p,x)+\frac{1}{(i+2)^{2n}}\zeta (D(p,x))+\mu _n^{(i)}. \end{aligned}$$

It follows from (1) and (2) that we obtain that

$$\begin{aligned} D(p,T_i^nx)\le & {} D(p,x)+\nu _n^{(i)}\cdot \zeta (D(p,x))+\mu _n^{(i)},\,\forall p\in F(T),\,x\in C,\,n\ge 1,\\&i=1,2,\cdots ,m, \end{aligned}$$

where $\nu _n^{(i)}=\max \left\{ (2\Pi _{j=2}^{n}a_j)^2-1,\frac{1}{(i+1)^{2n}}\right\}$.

Noting that $0\le \lim _{n\rightarrow +\infty }\nu _n^{(i)}\le \lim _{n\rightarrow +\infty }\max \{(2\Pi _{j=2}^{n}a_j)^2-1,\frac{1}{2^{2n}}\}=0$, we have $\lim _{n\rightarrow +\infty }\nu _n^{(i)}=0$. These imply that $T_i:C\rightarrow C$ is a finite family of uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings for every $i=1,2,\cdots ,m$. Next, we claim that $T_i$ is not a Bregman D-nonexpansive mapping for all $i=1,2,\cdots ,m$. Indeed, let $s=\frac{3x_0}{5}$, $t=\frac{x_0}{2}\in C$, then

$$\begin{aligned} D(T_is,T_it)= & {} ||T_is-T_it||^2=\big |\big |(-\frac{1}{i+1})(\frac{3}{5},0,0,\cdots )-(0,\frac{1}{4},0,\cdots )\big |\big |^2\\= & {} \frac{9}{25(i+1)^2}+\frac{1}{16}>(\frac{1}{10})^2=||\frac{3x_0}{5}-\frac{x_0}{2}||^2=||s-t||^2=D(s,t) \end{aligned}$$

for all $i=1,2,\cdots ,m$.

Now, if $x\in Q$, then $T_i^dx=\frac{1}{2^{2n}}(\Pi _{j=2}^{d}a_j)\cdot \big (\underbrace{ 0,\cdots ,0 }_{d},1,0,\cdots \big )$ for all $i=1,2,\cdots ,m$ and $d\ge 2$, $d\in N$. For any bounded subset K of C, we have

$$\begin{aligned}&0\le \lim _{n\rightarrow +\infty }\sup _{y\in K}||T_i^{n+1}y-T_i^ny||\\&\quad \le \lim _{n\rightarrow +\infty }\max \Big (\big |\big |\frac{1}{2^{2n}}(\Pi _{j=2}^{n+1}a_j)\cdot \big (\underbrace{ 0,\cdots ,0 }_{n+1},1,0,\cdots \big )\\&\qquad -\frac{1}{2^{2n}}(\Pi _{j=2}^{n}a_j)\cdot \big (\underbrace{ 0,\cdots ,0 }_{n},1,0,\cdots \big )\big |\big |,\\&\sup _{y\in K\backslash \{\frac{x_0}{2^n}\}}\big |\big |\frac{(-1)^{n+1}}{(i+1)^{n+1}}y-\frac{(-1)^n}{(i+1)^n}y\big |\big |\Big )\\&\quad \le \lim _{n\rightarrow +\infty }\max \Big (\frac{1}{2^{2n}}(\Pi _{j=2}^{n}a_j)\cdot \sqrt{a_{n+1}^2+1},\sup _{y\in C\backslash \{\frac{x_0}{2^n}\}}\frac{i+2}{(i+1)^{n+1}}||y||\Big )\\&\quad \le \lim _{n\rightarrow +\infty }\max \Big (\frac{\sqrt{1^2+1}}{2^{2n}}(\Pi _{j=2}^{n}a_j),\frac{i+2}{(i+1)^{n+1}}\cdot 1\Big )\\&\quad \le \lim _{n\rightarrow +\infty }\max \Big (\frac{\Pi _{j=2}^{n}a_j}{2^{2n-0.5}},\frac{2i+2}{(i+1)^{n+1}}\Big )\le \lim _{n\rightarrow +\infty }\max \Big (\frac{1}{2^{2n-0.5}},\frac{2}{(i+1)^{n}}\Big )\\&\quad \le \lim _{n\rightarrow +\infty }\max \Big (\frac{1}{2^{2n-0.5}},\frac{1}{2^{n-1}}\Big )= \lim _{n\rightarrow +\infty }\frac{1}{2^{n-1}}=0. \end{aligned}$$

This implies that $\lim _{n\rightarrow +\infty }\sup _{y\in K}||T_i^{n+1}y-T_i^ny||=0$, i.e., $T_i$ is uniformly asymptotically regular on C for all $i=1,2,\cdots ,m$.

For any sequence $\{y_n\}\subseteq C$ such that $\lim _{n\rightarrow +\infty }y_n=x^0$ and $\lim _{n\rightarrow +\infty }T_iy_n=y^0$, we consider the following two cases:

• (1) If the sequence $y_n=\frac{x_0}{2^n}$ and $\lim _{n\rightarrow +\infty }y_n=x^0$, then we have that $x^0=0$ and

  $$\begin{aligned} 0= & {} \lim _{n\rightarrow +\infty }||T_iy_n-y^0||=\lim _{n\rightarrow +\infty }||(0,\frac{1}{2^{2n}},0,\cdots )-y^0||\\\ge & {} \limsup _{n\rightarrow +\infty }\Big |||y^0||-\frac{1}{2^{2n}}\Big |=||y^0||\ge 0, \end{aligned}$$

  which implies that $y^0=0$ and $T_ix^0=-\frac{x^0}{i+1}=0=y^0$.

• (2) If $y_n\ne \frac{x_0}{2^n}$, $y_n\in C$ and $\lim _{n\rightarrow +\infty }y_n=x^0$, then it follows from

  $$\begin{aligned} 0= & {} \lim _{n\rightarrow +\infty }||T_iy_n-y^0||=\lim _{n\rightarrow +\infty }|| -\frac{1}{i+1}y_n-y^0||\\= & {} \lim _{n\rightarrow +\infty }||-\frac{1}{i+1}(y_n-x^0)-(y^0+\frac{x^0}{i+1})||\\\ge & {} \limsup _{n\rightarrow +\infty }\big |||y^0+\frac{x^0}{i+1}||-||\frac{y_n-x^0}{i+1}||\big |=||y^0+\frac{x^0}{i+1}||\ge 0 \end{aligned}$$

  that $y^0=-\frac{x^0}{i+1}$, hence $T_ix^0=-\frac{1}{i+1}x^0=y^0$.

In summary, we can obtain that the map $T_i$ is closed for every $i=1,2,\cdots ,m$.

Choose $i=1,2,\cdots ,m$, for any $n\ge 1$ and $n\in N$, we may set $x_n=\frac{x_0}{2^{n+1}}$, then $x_n\in C$ and $x_n\rightarrow 0\in F(T_i)=\{0\}$ as $n\rightarrow +\infty$.

Finally, it is obvious that the family $\{T_i\}$ satisfies all the aspects of the hypothesis of Theorem 3.1.

Setting $\zeta (t)=t,t\ge 0,\nu _n^{(i)}=k_n^{(i)}-1,\lim _{n\rightarrow +\infty }k_n^{(i)}=1,\mu _n^{(i)}\equiv 0$ for each $i=1,2,\cdots ,m$ in Theorem 3.1, we have the following:

Corollary 3.1

Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of closed and Bregman quasi-D-asymptotically nonexpansive mappings with real sequences $\{k_n^{(i)}\}\subseteq [1,+\infty )$ and $k_n^{(i)}\rightarrow 1(as\;n\rightarrow +\infty \;and\;for\; each\;i=1,2,\cdots ,m)$. Assume that $T_i$ is asymptotically regular on C for all $i=1,2,\cdots ,m$, i.e., $\lim _{n\rightarrow +\infty }\sup _{x\in K}||T_i^{n+1}x-T_i^nx||=0$ holds for any bounded subset K of C, and $F=[\bigcap _{i=1}^mF(T_i)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{1}{r_{k,n}}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in E$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=\textit{Proj}_{C_1}^g(x_0),\\ y&_n=\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)],\\ z&_n=\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_i^nx_n)],\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:D(z,u_n)\le \alpha _n D(z,x_n)+(1-\alpha _n)D(z,z_n)\le D(z,x_n)\\ +&(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n\},\\ x&_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0), \end{aligned} \right. \end{aligned}$$

where $\zeta _n\!=\!\max _{i\in \{1,2,\cdots ,m\}}\{k_n^{(i)}-1\}\cdot \sup _{z\in F}\{D(z,x_n)\}$. $\{\alpha _{n}\}, \{{\beta _{n}^{(i)}}\}{(i=0,1,\cdots ,m)}$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $\textit{Proj}_F^gx_0$.

Remark 3.2

Corollary 3.1 improves and extends Theorem 3.5 of Chang et al. [22] from modified Halpern-type iterative algorithms to modified Ishikawa-type iterative algorithms.

Setting $k_n^{(i)}\equiv 1$ for each $i=1,2,\cdots ,m$ and for all $n\ge 1$ in Corollary 3.1, we have the following.

Corollary 3.2

Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of closed and Bregman quasi-nonexpansive mappings. $F=[\bigcap _{i=1}^mF(T_i)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{1}{r_{k,n}}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in E$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=\textit{Proj}_{C_1}^g(x_0),\\ y&_n=\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)],\\ z&_n=\triangledown g^*[\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g(T_ix_n)],\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:D(z,u_n)\le \alpha _n D(z,x_n)+(1-\alpha _n)D(z,z_n)\le D(z,x_n)\},\\ x&_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0). \end{aligned} \right. \end{aligned}$$

(3.18)

$\{\alpha _n\}$,$\{\beta _n^{(i)}\}(i=0,1,\cdots ,m)$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $\textit{Proj}_F^gx_0$.

Remark 3.3

Corollary 3.2 improves Theorem 3.1 in [8] in the following aspects:

• (1) For the structure of Banach spaces, we extend the duality mapping to a more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.

• (2) For the mappings, we extend the mapping from two quasi-$\phi$-nonexpansive mappings to a finite family of Bregman quasi-nonexpansive mappings.

• (3) For equilibrium problems, we extend equilibrium problem from an equilibrium problem to a system of generalized mixed equilibrium problem.

• (4) If we set $m=2,h=1,A_1=\varphi _1=0,g(x)=||x||^2,\nu _n^{(1)}=\nu _n^{(2)}=\mu _n^{(1)}=\mu _n^{(2)}=0$, and $\alpha _n\equiv 0$ for all $n\in N^+$, then (3.18) reduces to (1.6) which has been studied as mapping—quasi-$\phi$-nonexpansive mapping.

Using Corollary 3.2, we obtain the following strong convergence theorem for maximal monotone operators.

Corollary 3.3

Let E be a reflexive Banach space and let $g:E\rightarrow \mathbb {R}$ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $B_i:E\rightarrow E^*(i=1,2,\cdots ,m)$ be a finite family of maximal monotone operators satisfying $dom(B_i)\subseteq C$, let $r_i>0$ and $Res_{r_i,B_i}^g=(\triangledown g+r_iB_i)^{-1}\triangledown g$ be the g-resolvent of $B_i$ for all $i=1,2,\cdots ,m$. $F=[\bigcap _{i=1}^mB_i^{-1}(0)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{1}{r_{k,n}}\langle y-z,\triangledown g(z)-\triangledown g(x)\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in E$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=\textit{Proj}_{C_1}^g(x_0),\\ y&_n=\triangledown g^*[\alpha _n\triangledown g(x_n)+(1-\alpha _n)\triangledown g(z_n)],\\ z&_n=\triangledown g^*\{\beta _n^{(0)}\triangledown g(x_n)+\sum _{i=1}^m\beta _n^{(i)}\triangledown g[Res_{r_i,B_i}^g(x_n)]\},\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:D(z,u_n)\le \alpha _n D(z,x_n)+(1-\alpha _n)D(z,z_n)\le D(z,x_n)\},\\ x&_{n+1}=\textit{Proj}_{C_{n+1}}^g(x_0). \end{aligned} \right. \end{aligned}$$

$\{\alpha _n\}$,$\{\beta _n^{(i)}\}(i=0,1,\cdots ,m)$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $\textit{Proj}_F^gx_0$.

Proof

Setting $T_i=Res_{r_i,B_i}^g$ for each $i=1,2,\cdots ,m$, from [15], we know that $T_i$ is closed, and Bregman quasi-nonexpansive mapping and $B_i^{-1}(0)=F(Res_{r_i,B_i}^g)$ for all $r_i>0$. Thus, $T_i$ is closed and Bregman quasi-nonexpansive mapping for all $i=1,2,\cdots ,m$. It follows from Corollary 3.2 that Corollary 3.3 holds. $\square$

Remark 3.4

Setting $m=h=1,A_k=\varphi _k=\alpha _n\equiv 0$ for all $n\ge 1$ in Corollary 3.3, we can obtain the main result—Theorem 2.1 in [33].

If $g(x)=||x||^2$ for all $x\in E$, then, by Theorem 3.1, the following results hold.

Corollary 3.4

Let C be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of closed and totally quasi-$\phi$-asymptotically nonexpansive mappings with nonnegative real sequences $\{\nu _n^{(i)}\}$,$\{\mu _n^{(i)}\}$ and a strictly increasing and continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$ and $\nu _n^{(i)},\mu _n^{(i)}\rightarrow 0(as\;n\rightarrow +\infty \;and \;for\; each\;i=1,2,\cdots ,m)$. Assume that $T_i$ is asymptotically regular on C for all $i=1,2,\cdots ,m$, i.e., $\lim _{n\rightarrow +\infty }\sup _{x\in K}||T_i^{n+1}x-T_i^nx||=0$ holds for any bounded subset K of C and $F=[\bigcap _{i=1}^mF(T_i)]\cap [\bigcap _{k=1}^h{\textit{GMEP}}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{2}{r_{k,n}}\langle y-z,Jz-Jx\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in E$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=\Pi _{C_1}(x_0),\\ y&_n=J^{-1}[\alpha _nJx_n+(1-\alpha _n)Jz_n],\\ z&_n=J^{-1}[\beta _n^{(0)}Jx_n+\sum _{i=1}^m\beta _n^{(i)}J(T_i^nx_n)],\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:\phi (z,u_n)\le \alpha _n \phi (z,x_n)+(1-\alpha _n)\phi (z,z_n)\le \phi (z,x_n)\\&+(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n\},\\ x&_{n+1}=\Pi _{C_{n+1}}(x_0), \end{aligned} \right. \end{aligned}$$

where $\zeta _n\!=\!\nu _n\cdot \sup _{z\in F}\zeta [\phi (z,x_n)]+\mu _n,\nu _n=\max _{1\le i\le m}\{\nu _n^{(i)}\},\mu _n=\max _{1\le i\le m}\{\mu _n^{(i)}\}$. $\{\alpha _n\}$,$\{\beta _n^{(i)}\} (i=0,1,\cdots ,m)$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\liminf _{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $\Pi _Fx_0$.

Remark 3.5

Corollary 3.4 improves and extends Theorems 3.1 and 3.2—the main result of Petrot, Wattanawitoon, and Kumam [6] in the following aspects:

(a) from two relatively quasi-nonexpansive mappings to a finite family of totally quasi-$\phi$-asymptotically nonexpansive mappings;

(b) from an equilibrium problem to a system of generalized mixed equilibrium problem;

(c) setting $m=2,h=1,\zeta (t)=t,\mu _n^{(i)}=\nu _n^{(i)}\equiv 0$ for each $i=1,2$ and for all $n\in N^+$ in Corollary 3.4, we can obtain Theorem 3.1 in [6].

Setting $E=$ Hilbert space H in Theorem 3.1, we have the following Corollary 3.5.

Corollary 3.5

Let C be a nonempty, closed, and convex subset of real Hilbert space H. For each $k=1,2,\cdots ,h$, let $A_k:C\rightarrow E^*$ be a continuous and monotone mapping, $\varphi _k:C\rightarrow \mathbb {R}$ be a lower semicontinuous and convex functional, let $f_k:C\times C\rightarrow \mathbb {R}$ be a bifunction satisfying (C1)–(C4) and $T_i:C\rightarrow C(i=1,2,\cdots ,m)$ be a finite family of closed and totally quasi-$\phi$-asymptotically nonexpansive mappings with nonnegative real sequences $\{\nu _n^{(i)}\}$,$\{\mu _n^{(i)}\}$ and a strictly increasing and continuous function $\zeta :\mathbb {R}^+\rightarrow \mathbb {R}^+$ with $\zeta (0)=0$ and $\nu _n^{(i)},\mu _n^{(i)}\rightarrow 0(as\;n\rightarrow +\infty \;and \;for\; each\;i=1,2,\cdots ,m)$. Assume that $T_i$ is asymptotically regular on C for all $i=1,2,\cdots ,m$, i.e., $\lim _{n\rightarrow +\infty }\sup _{x\in K}||T_i^{n+1}x-T_i^nx||=0$ holds for any bounded subset K of C and $F=[\bigcap _{i=1}^mF(T_i)]\cap [\bigcap _{k=1}^h\textit{GMEP}(f_k,\varphi _k)]\ne \emptyset$. For each $k=1,2,\cdots ,h,\{r_{k,n}\}_{n=1}^{+\infty }\subset (0,+\infty )$ satisfying $\liminf _{n\rightarrow +\infty }r_{k,n}>0$, for all $z,y\in C$, $G_k(z,y)=f_k(z,y)+\varphi _k(y)-\varphi _k(z)+\langle A_kz,y-z\rangle$, $T_{r_{k,n}}^{G_k}(x)=\{z\in C:G_k(z,y)+\frac{2}{r_{k,n}}\langle y-z,z-x\rangle \ge 0,\forall y\in C\}$. For an arbitrarily initial point $x_0\in H$, define the sequence $\{x_n\}$ by

$$\begin{aligned} \left\{ \begin{aligned} C&_1=C,\\ x&_1=P_{C_1}(x_0),\\ y&_n=\alpha _nx_n+(1-\alpha _n)z_n,\\ z&_n=\beta _n^{(0)}x_n+\sum _{i=1}^m\beta _n^{(i)}T_i^nx_n,\\ u&_n=T_{r_{h,n}}^{G_h}T_{r_{h-1,n}}^{G_{h-1}}\cdots T_{r_{2,n}}^{G_2}T_{r_{1,n}}^{G_1}y_n,\\ C&_{n+1}=\{z\in C_n:||z-u_n||^2\le \alpha _n ||z-x_n||^2+(1-\alpha _n)||z-z_n||^2\le ||z-x_n||^2\\&+(1-\alpha _n)(1-\beta _n^{(0)})\cdot \zeta _n\},\\ x&_{n+1}=P_{C_{n+1}}(x_0), \end{aligned} \right. \end{aligned}$$

where $\zeta _n=\nu _n\cdot \sup _{z\in F}\zeta (||z-x_n||^2)+\mu _n,\nu _n=\max _{1\le i\le m}\{\nu _n^{(i)}\},\mu _n=\max _{1\le i\le m}\{\mu _n^{(i)}\}$. $\{\alpha _n\}$,$\{\beta _n^{(i)}\}(i=0,1,\cdots ,m)$ are real sequences in [0, 1] which satisfy the conditions: $\sum _{i=0}^m\beta _n^{(i)}=1$ and $\hbox {lim inf}_{n\rightarrow \infty }(1-\alpha _n)\beta _n^{(0)}\beta _n^{(i)}>0(i=0,1,\cdots ,m)$. Then the sequence $\{x_n\}$ converges strongly to $P_Fx_0$.

Remark 3.6

Corollary 3.5 improves and extends Corollaries 3.2 and 3.3 of Qin et al. [8] in many aspects.