An inertial parallel iterative method for solving generalized mixed equilibrium problems and common fixed point problem in reflexive Banach spaces

Abstract

By combining the shrinking projection method with the parallel splitting-up technique and the inertial term, we introduce a new inertial parallel iterative method for finding common solutions of a finite system of generalized mixed equilibrium problems and common fixed points of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings. After that, we prove a strong convergence result for the proposed iteration in reflexive Banach spaces. By this theorem, we obtain some convergence results for generalized mixed equilibrium problems in reflexive Banach spaces. In addition, we give a numerical example to illustrate the proposed iterations. The obtained results are improvements and extensions to some known results in this area.

1 Introduction

The equilibrium problem (EP) was introduced by Muu and Oettli [1] in 1992. Later, some sufficient condition for the existence of a the solution for (EP) was studied by Blum and Oettli [2], Noor and Oettli [3]. The equilibrium problem consists of finding $u \in U$ such that

$$\begin{aligned} f(u,v) \ge 0, \forall v\in U, \end{aligned}$$

where U is a nonempty, closed, convex subset of a Banach space W, and ${f: U \times U \longrightarrow \mathbb {R}}$ is a bifunctional mapping satisfying $f(u,u)=0$ for all $u \in U$. The set

$$\begin{aligned} EP(f) =\big \{ u \in U: f(u,v) \ge 0, \forall v \in U \big \} \end{aligned}$$

denotes the set of solutions of (EP). The equilibrium problem had a great influence in the development of some branches of pure and applied sciences. The equilibrium problem theory provides a natural and novel approach for some problems arising in nonlinear analysis, physics and engineering, image reconstruction, economics, finance, game theory and optimization. In 2008, Peng and Yao [4] extended the equilibrium problem (EP) to the generalized mixed equilibrium problem (GMEP). Assume that $\langle u^*, v \rangle$ is the value of the function of $u^*$ at $v \in W$, $A: U \longrightarrow \mathbb {R}$ is a real valued function and $B: U \longrightarrow W^*$ is a nonlinear mapping. Then the generalized mixed equilibrium problem (GMEP) is to find $u \in U$ such that

$$\begin{aligned} f(u,v)+ \langle B(u), v-u \rangle + A(v) - A(u) \ge 0, \forall v\in U. \end{aligned}$$

The symbol $GMEP(f,A, B) =\big \{ u \in U: f(u,v)+ \langle B(u), v-u \rangle + A(v) - A(u) \ge 0, \forall v \in U \big \}$ denotes the set of solutions of (GMEP). In particular, if $B\equiv 0$, then (GMEP) is reduced into the mixed equilibrium problem (MEP) which is to find $u \in U$ such that

$$\begin{aligned} f(u,v)+ A(v) \ge A(u), \forall v\in U. \end{aligned}$$

If $A \equiv 0$, then (GMEP) is reduced into the generalized equilibrium problem (GEP) which is to find $u \in U$ such that

$$\begin{aligned} f(u,v)+ \langle B(u), v-u \rangle \ge 0, \forall v\in U. \end{aligned}$$

If $f\equiv 0$, then (GMEP) is reduced into the mixed variational inequality (MVI) of Browder type which is to find $u\in U$ such that

$$\begin{aligned} \langle B(u), v-u \rangle + A(v) \ge A(u), \forall v \in U. \end{aligned}$$

If $B\equiv 0$ and $A \equiv 0$, then (GMEP) is reduced into the equilibrium problem (EP).

In recent times, the authors have studied many iterative methods for solving the equilibrium problem and its generalizations in the setting of Hilbert spaces and Banach spaces. Furthermore, some authors proposed certain iterative methods for finding common solutions of the equilibrium problem or its generalizations and fixed point problem for nonexpansive mappings or generalized nonexpansive mappings in Hilbert spaces and Banach spaces. In 2007, using the hybrid projection method, Tada and Takahashi [5] proposed the following hybrid iterative method for finding common elements of an equilibrium problem and fixed point problem for a nonexpansive mapping S in Hilbert space W.

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1 \in U \\ u_n \in U \text { such that } f(u_n, v) + \frac{1}{r_n} \langle v- u_n, u_n-z_n \rangle \ge 0, \forall v \in U \\ v_n= b_nz_n + (1-b_n) Su_n \\ C_{n}= \big \{ u \in W: \Vert v_n-u\Vert \le \Vert z_n-u\Vert \big \} \\ Q_{n}= \big \{ u \in W: \langle z_n-u, z_n-z_1 \rangle \le 0 \big \} \\ z_{n+1}= P_{C_{n} \cap Q_n}(z_1), \forall n \ge 1, \end{array}\right. } \end{aligned}$$

(1)

where $\{b_n \} \subset [0,1]$ and $\{r_n \} \subset (0,\infty )$. In addition, under the suitable conditions, the authors proved that the sequence $\{z_n \}$ strongly converges to $p = P_{F(S) \cap EP(f)}(z_1)$, where F(S) denotes the set of fixed points of the mapping S. In 2016, Alizadeh and Moradlou [6] generalized the main results in [5] by proposing the following hybrid iterative method for solving an equilibrium problem and fixed point problem for a generalized hybrid mapping S in Hilbert space W.

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1 \in U \\ y_n=a_nz_n+(1-a_n)Sz_n \\ u_n \in U \text { such that } f(u_n, v) + \frac{1}{r_n} \langle v- u_n, u_n-z_n \rangle \ge 0, \forall v \in U \\ v_n=b_n y_n +(1-b_n)Su_n \\ C_{n}= \big \{ u \in U: \Vert v_n-u\Vert \le \Vert z_n-u\Vert \big \} \\ Q_{n}= \big \{ u \in U: \langle z_n-u, z_n-z_1 \rangle \le 0 \big \} \\ z_{n+1}= P_{C_{n} \cap Q_n}(z_1), \forall n \ge 1, \end{array}\right. } \end{aligned}$$

(2)

where $\{a_n \}, \{b_n \} \subset [0,1]$ and $\{r_n \} \subset (0,\infty )$. Furthermore, the authors proved that the sequence $\{z_n \}$ strongly converges to $p = P_{F(S) \cap EP(f)}(z_1)$. Note that if $a_n=1$ for all $n \in \mathbb {N}$, then the iteration (2) becomes the iteration (1).

An interesting work naturally raised is to extend and improve the convergence results of the iterative methods for equilibrium problems and fixed point problems from a Hilbert space to a Banach space. The fact that some characteristic properties and results in Hilbert spaces are not available in more general Banach spaces. To overcome these difficulties, the authors combined the normalized duality mapping, the Lyapunov functional and the generalized projection to construct some iterative methods for equilibrium problems and fixed point problems in smooth Banach spaces [7, 8]. In another approach, some authors used the Bregman distance and the Bregman projection in reflexive Banach spaces instead of the norm and the metric projection in Hilbert spaces. By these ways, some authors introduced many iterative methods for finding common elements of the solutions set of the equilibrium problems and the fixed point set of mappings with respect to the Bregman distance in reflexive Banach spaces [9, 10] and the references therein.

In 2014, Chang et al. [11] introduced the notion of a Bregman totally quasi-asymptotically nonexpansive mapping as a generalization of a Bregman strongly nonexpansive mapping. After that, some convergence results for the equilibrium problems and the fixed point problem for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces were established [12, 13].

In 2014, Anh and Chung [14] introduced a parallel splitting-up technique to construct two parallel hybrid methods for finding a common fixed point of a finite family of relatively nonexpansive mappings. By this idea, some authors proposed many parallel iterative methods for finite system of equilibrium problems and a finite family of generalized nonexpansive mappings [15, 16]. In 2017, Tuyen [17] proposed some parallel iterative methods for solving a system of generalized mixed equilibrium problems.

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1 \in W, U_1 =W \\ u_n^{(k)} =\textrm{Res}_{f_k, A_k, B_k}(z_n)\\ k_n= \textrm{argmax} \{ D_g(u_n^{(k)}, z_n): k =1,2,\dots ,M\}, \overline{u}_n= u^{(k_n)}_n \\ U_{n+1}= \big \{ u \in U_n: D_g(u, \overline{u}_n) \le D_g(u, z_n) \big \} \\ u_{n+1}= P^g_{U_{n+1}}(z_1), \forall n \ge 1. \end{array}\right. } \end{aligned}$$

(3)

Recently, there were many methods for constructing new iteration processes which generalize some previous ones. In 2008, Mainge [18] proposed the inertial Mann iteration by combining the Mann iteration process and the inertial extrapolation. In 2018, Chidume et al. [19] introduced an inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces.

Motivated by the mentioned works, we introduce a new inertial parallel iterative method for finding common solutions of a finite system of generalized mixed equilibrium problems and common fixed points of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings. After that, we prove a strong convergence result for the proposed iteration in reflexive Banach spaces. By this theorem, we obtain some convergence results for generalized mixed equilibrium problems in reflexive Banach spaces. In addition, we give a numerical example to illustrate the obtained results.

2 Preliminaries

Assume that W is a real reflexive Banach space, U is a nonempty, closed and convex subset of W, $W^*$ is the dual space of W. Throughout this paper, we suppose that $g: W \longrightarrow (-\infty , + \infty ]$ is a proper, lower semi-continuous and convex function. The set $\textrm{dom}g =\{u \in W: g(u) < + \infty \}$ denotes the domain of g. For any $u \in \mathrm {int(dom}g)$ and $v \in W$, we denote by $g'(u,v)$ the right-hand derivative of g at u in the direction v, that is

$$\begin{aligned} g'(u,v)= \lim \limits _{\lambda \downarrow 0}\frac{g(u+\lambda v)-g(u) }{\lambda }. \end{aligned}$$

(4)

The function g is called Gâteaux differentiable at u if the limit (4) exists for any v. In this case, the gradient of g at u is the function $\nabla g(u)$, which is defined by $\langle \nabla g(u), v \rangle =g'(u,v)$ for all $v \in W$. The function g is called Gâteaux differentiable on $\mathrm {int(dom}g)$ if it is Gâteaux differentiable at each $u \in \mathrm {int(dom}g)$. The function g is called Fréchet differentiable at u if the limt (4) is attained uniformly in $\Vert v\Vert =1$. The function g is called uniformly Fréchet differentiable on a subset U of W if the limit (4) is attained uniformly for $u \in U$ and $\Vert v\Vert =1$.

Proposition 1

[20, Proposition 1] Let W be a real reflexive Banach space, and $g: W \longrightarrow (-\infty , +\infty ]$ be uniformly Fréchet differentiable and bounded on bounded subsets of W. Then $\nabla g$ is uniformly continuous on bounded subsets of W from the strong topology of W to the strong topology of $W^*$.

Let $u \in \mathrm {int(dom}g),$ the subdifferential g at $u \in W$ is defined by

$$\begin{aligned} \partial g(u)= \{u^* \in W^*: g(u)+ \langle u^*, v-u \rangle \le g(v), \forall v \in W \}, \end{aligned}$$

and the Fenchel conjugate of g is the function $g^*: W^* \longrightarrow (-\infty , +\infty ]$ defined by

$$\begin{aligned} g^*(u^*) = \sup \{ \langle u^*, u \rangle -g(u): u \in W\}, \forall u^* \in W^*. \end{aligned}$$

Definition 1

[11, Definition 2.2] Suppose that W is a real reflexive Banach and $g: W \longrightarrow (-\infty , +\infty ]$ is a function. Then g is called Legendre if the following two conditions are satisfied.

• 1. $\mathrm {Int (dom} g )\ne \emptyset$, g is Gâteaux differentiable on $\mathrm { int (dom}g)$ and $\textrm{dom} (\nabla g)= \mathrm {int (dom} g).$

• 2. $\mathrm {Int (dom } g^*) \ne \emptyset$, $g^*$ is Gâteaux differentiable on $\mathrm {int (dom}g^*)$ and $\textrm{dom} (\nabla g^*) = \mathrm {int (dom } g^*).$

Remark 1

[21] Let W be a real reflexive Banach space and $g: W \longrightarrow (-\infty , +\infty ]$ be a Legendre function. Then

• 1. g is a Legendre function if and only if $g^*$ is a Legendre function.

• 2. $(\partial f)^{-1}= \partial g^*$.

• 3. $\nabla g= (\nabla g^*)^{-1}$, $\textrm{ran} (\nabla g)=\textrm{dom} (\nabla g^*)= \mathrm { int (dom}g^*)$ and $\textrm{ran} ( \nabla g^*) =\textrm{dom} (\nabla g)= \mathrm { int (dom}g),$ where $\textrm{ran} (\nabla g)$ denotes the range of $\nabla g$.

• 4. g and $g^*$ are strictly convex on the interior of their respective domains.

Definition 2

[22, p. 234] Assmue that W is a real reflexive Banach space and $g: W \longrightarrow (-\infty , +\infty ]$ is a Gâteaux differentiable function. Then the function $D_g: \textrm{dom} g \times \mathrm {int(dom}g) \longrightarrow [0,+ \infty )$, defined by $D_g(u, v)=g(u)-g(v) -\langle \nabla g(v), u-v \rangle$ is called the Bregman distance with respect to g.

Notice that the Bregman distance is not a distance in the usual sense of the term. In general, $D_g(u,u)=0$, but $D_g(u,v)=0$ may not imply $u=v$; $D_g$ is not symmetric and does not satisfy the triangle inequality. By the definition of the Bregman distance, we have $D_g(u,v) +D_g(v,w)-D_g(u,w)= \langle \nabla g(w)- \nabla g(v), u-v \rangle$ for all $u \in \textrm{dom}g$ and $v, w \in \mathrm {int (dom}g)$. Note that from [23, p.7], for all $u \in W$, we have

$$\begin{aligned} D_g \Big ( u, \nabla g^* \big ( \sum \limits _{n=1} ^{m} \lambda _n\nabla g(u_n) \big ) \Big ) \le \sum \limits _{ n=1} ^{m} \lambda _n D_g(u, u_n), \end{aligned}$$

(5)

where $\{u_n\}_{n=1}^m \subset W$ and $\{\lambda _n \}_{n=1}^m \subset [0,1]$ with $\sum \limits _{ n=1} ^{m} \lambda _n =1.$

Definition 3

[24, p. 69] Let W be a real reflexive Banach space, $g: W \longrightarrow (-\infty , +\infty ]$ is a convex and Gâteaux differentiable function, and U be a nonempty, closed and convex subset of $\mathrm {int(dom} g)$. The Bregman projection of $u \in \mathrm {int(dom} g)$ onto U is the unique vector $P^g_{U}(u) \in U$ such that $D_g \big (P^g_{U}(u) , u\big )= \inf \big \{ D_g(v, u): v \in U \big \}.$

Remark 2

[12, Remark 2.2] Let W be a smooth, strictly convex Banach space and $g(u)= \Vert u\Vert ^2$ for all $u\in W$. Then $\nabla g(u) = 2 Ju$ for all $u \in W$ and J is the normalized duality mapping which is defined by $J(u)= \{ u^* \in W^*: \langle u, u^* \rangle =\Vert u\Vert ^2 = \Vert u\Vert ^*\}$ for all $u \in W$. Therefore, Bregman distance $D_g(u,v)$ is reduced into $\phi (u,v)$, where $\phi (u,v)$ is a Lyapunov function which is defined by $\phi (u,v)=\Vert u\Vert ^2 -2 \langle u, Jv \rangle + \Vert v\Vert ^2.$ Thus, the Bregman projection $P^g_{U}(u)$ is reduced into the generalized projection $\Pi _{U}(u)$ in smooth Banach which is defined by $\phi \big (\Pi _{U}(u), u \big ) = \min \big \{ \phi (v, u): v \in U \}.$

If W is a Hilbert space and $g(u)=\Vert u\Vert ^2$ for all $u\in W$, then $U(u,v)=\Vert u-v\Vert ^2$ for all $u, v \in W$, and J is the identity mapping. Therefore, the Bregman projection $P^g_{U}(u)$ is reduced into the metric projection from W onto U.

Definition 4

[25, p. 1] Let W be a real reflexive Banach space, $g: W \longrightarrow (-\infty , +\infty ]$ be a convex and Gâteaux differentiable function. Then

• 1. g is called totally convex at $u \in \mathrm {int (dom} g)$ if any $t >0$, we have

  $$\begin{aligned} v_g(u, t) := \inf \big \{ D_g(v,u): v \in \textrm{dom} g, \Vert v-u\Vert =t\big \} >0. \end{aligned}$$

• 2. g is called totally convex if g is totally convex at every point $u \in \mathrm {int (dom} g)$.

• 3. g is called totally convex on bounded subsets of W if any nonempty bounded subset B of W and $t>0$, we have $v_g(B, t) := \inf \big \{ v_g(u,t): u \in B \cap \textrm{dom} g \big \} >0.$

Proposition 2

[24, Lemma 2.1.2] Let W be a real reflexive Banach space, $g: W \longrightarrow (-\infty , +\infty ]$ be a convex and Gâteaux differentiable function. Then g is totally convex on bounded subsets of W if and only if any sequence $\{u_n\} \subset \mathrm {int (dom} g)$ and $\{v_n\} \subset \textrm{dom} g$ such that $\{u_n\}$ is bounded and $\lim \limits _{n\rightarrow \infty }D_g(v_n, u_n)=0$, we have $\lim \limits _{n\rightarrow \infty } \Vert v_n-u_n\Vert =0.$

Proposition 3

[26, Proposition 2.3] Let W be a real Banach space, $g: W \longrightarrow \mathbb {R}$ be Legendre such that $\nabla g^*$ is bounded on bounded subsets of $\mathrm {int(dom} g^*)$, $u \in W$ and $\{u_n \} \subset W$ satisfying $\{D_g(u, u_n) \}$ is bounded. Then the sequence $\{u_n \}$ is bounded.

Proposition 4

[27, Corollary 4.4] Let W be a real reflexive Banach space, ${g: W \longrightarrow (-\infty , +\infty ]}$ be Gâteaux differentiable and totally convex on $\mathrm {int(dom} g)$, U be a nonempty, closed and convex subset of $\mathrm {int(dom} g)$ and $u \in \mathrm {int(dom}g)$. Then the following statements are equivalent.

• 1. $w = P^g_{U}(u)$.

• 2. w is the unique vector such that $\langle \nabla g(u) - \nabla g(w), w-v \rangle \ge 0$ for all $v \in U$.

• 3. w is the unique vector such that $D_g(v,w) + D_g(w, u) \le D_g(v,u)$ for all $v \in U$.

Definition 5

[28] Let W be a Banach space and denote by $S_1= \{u \in W: \Vert u\Vert < 1 \}$ and $B_{\varepsilon }= \{u \in W: \Vert u\Vert \le \varepsilon \}$ for some $\varepsilon >0$. Then $g: W \longrightarrow \mathbb {R}$ is called uniformly convex on bounded subsets of W if $\rho _{\varepsilon }(t) >0$ for all $t, \varepsilon >0$, where the function $\rho _{\varepsilon }: [0, \infty ) \longrightarrow [0, \infty )$ is defined by

$$ \rho _{\varepsilon }(t)= \inf \limits _{u, v \in B_{\varepsilon }, \Vert u-v\Vert =t, \eta \in (0,1)} \frac{\eta g(u) +(1-\eta ) g(v) -g(\eta u + (1-\eta )v) }{\eta (1-\eta )}. $$

Note that the notion of an uniformly smooth on bounded subset for a mapping, we can find in [28]. Furthermore if g is uniformly convex, then the function $\rho _{\varepsilon }$ is nondecreasing mapping. In addition, $\rho _{\varepsilon }(t) =0$ if and only if $t=0$ ([28, p. 203]).

Remark 3

[29, p. 6] The function g is totally convex on bounded subsets of W if and only if g is uniformly convex on bounded subsets of W.

Definition 6

[30, Definition 1.3.7] Let W be a Banach space and $g: W \longrightarrow (-\infty , +\infty ]$ be a function. Then g is called strongly coercive if $\lim \limits _{\Vert u\Vert \rightarrow +\infty }\frac{g(u)}{\Vert u\Vert }=+ \infty .$

By using [29, Lemma 2.2], we get the following lemma. The proof of this lemma is easy and is omitted.

Lemma 1

Let W be a real reflexive Banach space, $g: W \longrightarrow \mathbb {R}$ be a Legendre, strongly coercive function which is uniformly Fréchet differentiable and bounded on bounded subsets of W. Then

$$\begin{aligned} D_g \Big ( u, \nabla g^* \big ( \sum \limits _{ n=1} ^{m} a_n \nabla g(u_n) \big ) \Big ) \le \sum \limits _{ n=1} ^{m} a_n D_g(u, u_n) - a_i a_j \rho ^*_{\varepsilon }\big ( \Vert \nabla g(u_i)-\nabla g(u_j)\Vert \big ), \end{aligned}$$

where $i, j \in \{1,2, \dots , m\}$, $\nabla g(u_n) \in B^*_{\varepsilon } =\{ u \in X^*: \Vert u\Vert \le {\varepsilon }\}$ and $a_n \in [0,1]$ such that ${\sum \limits _{ n=1} ^{m} a_n =1,}$ and the $\rho ^*_{\varepsilon }$ is defined as in Definition 5.

Let us denote by $F(S) =\{u \in W: Su=u\}$ the set of fixed points of the mapping $S: W \longrightarrow W.$

Definition 7

Let W be a real reflexive Banach space, $g: W \longrightarrow \mathbb {R}$ be a Gâteaux differentiable function and $S: W \longrightarrow W$ be a mapping. Then

• 1. ([31], Definition 2) S is called a Bregman quasi-nonexpansive mapping if ${F(S) \ne \emptyset }$ and for all $u \in W$ and $p \in F(S)$, we have $D_g(p, Su) \le D_g(p,u)$.

• 2. ([32], Definition 2.10) S is called a Bregman quasi-asymptotically nonexpansive mapping if ${F(S) \ne \emptyset }$ and there exists a real sequence $\{k_n \} \subset [1, \infty )$ with ${\lim \limits _{n\rightarrow \infty }k_n=1}$ such that $D_g(p, S^nu) \le k_nD_g(p,u)$ for all $u \in W$ and $p \in F(S)$.

• 3. ([11], Definition 2.10) S is called a Bregman totally quasi-asymptotically nonexpansive mapping if ${F(S) \ne \emptyset }$ and there exist nonnegative real sequences $\{\eta _n\}$, $\{\mu _n\}$ with $\lim \limits _{n\rightarrow \infty }\eta _n=\lim \limits _{n\rightarrow \infty }\mu _n =0$ and a strictly increasing continuous function $\xi : [0, \infty ) \longrightarrow [0, \infty )$ with $\xi (0)=0$ such that

  $$ D_g(u, S^nx) \le D_g(u, x)+ \eta _n \xi (D_g(u, x)) + \mu _n $$

  for all $u \in W$ and $p \in F(S).$

• 4. ([31], Definition 2) S is called a Bregman firmly nonexpansive mapping if for all ${u, v \in W,}$ we have $\langle \nabla g(Su) - \nabla g(Sv), Su-Sv \rangle \le \langle \nabla g(u) - \nabla g(v), Su-Sv \rangle .$

• 5. S is called closed if any sequence $\{u_n \}$ in W such that ${\lim \limits _{n\rightarrow \infty }u_n=u \in W}$ and $\lim \limits _{n\rightarrow \infty }Su_n=v \in W$, we have $Su=v$.

• 6. ([33], p.3877) S is called uniformly asymptotically regular on W if for any bounded subset U of W, we have $\lim \limits _{n\rightarrow \infty } \sup \limits _{u \in U}{\Vert S^{n+1}u-S^nu\Vert }=0.$

Remark 4

• 1. Every Bregman quasi-asymptotically nonexpansive mapping is a Bregman totally quasi-asymptotically nonexpansive mapping with $\xi (t)=t$ for all $t \ge 0$, $\eta _n=k_n-1$ with $k_n \ge 1$ satisfying $\lim \limits _{n\rightarrow \infty }k_n=1$, and $\mu _n=0$, but the converse is not true.

• 2. Every Bregman firmly nonexpansive mapping is a Bregman quasi-nonexpansive mapping.

Lemma 2

[11, Lemma 2.16] Suppose that W is a real reflexive Banach space, $g: W \longrightarrow (-\infty , + \infty ]$ is a Legendre function which is totally convex on bounded subsets of W, and U is a nonempty, closed and convex subset of $\mathrm {int(dom} g)$. Let $S: U \longrightarrow U$ be a closed and Bregman totally quasi-asymptotically nonexpansive mapping. Then F(S) is a closed and convex subset of U.

For solving the problem (GMEP), let us assume that f, A, B satisfy the following conditions.

• (C1) $f(u, u)=0$ for all $u \in U$.

• (C2) f is monotone, that is, $f(u, v) +f(v, u) \le 0$ for all $u, v \in U$.

• (C3) For all $u, v, w \in U$, we have $\limsup \limits _{t \downarrow 0}f(tw +(1-t)u, v) \le f(u, v).$

• (C4) For each $u\in U$, $v \longmapsto f(u, v)$ is convex and lower semi-continuous.

• (C5) $A: U \longrightarrow \mathbb {R}$ is a lower semi-continuous and convex function.

• (C6) $B: U \longrightarrow W^*$ is a continuous monotone mapping.

In order to find the solution of the problem (GMEP), Darvish [9] introduced the notion of mixed resolvent of f. Later, this notion was studied in [17].

Definition 8

[9, Definition 2.4] Let W be a real reflexive Banach space, U be a nonempty, closed and convex subset of W, $g: W \longrightarrow (-\infty , + \infty ]$ be a Gâteaux differentiable function. Assume that $f: U \times U \longrightarrow \mathbb {R}$, $A: U \longrightarrow \mathbb {R}$ and $B: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$. The mixed resolvent of f is the operator $\textrm{Res}^g_{f, A, B}: W \longrightarrow 2^U$ defined by

$$\begin{aligned} \textrm{Res}^g_{f, A, B} (u)= & {} \Big \{ w \in U: f(w, v) +A(v) + \langle B(u), v-w \rangle \\{} & {} \quad + \langle \nabla g(w) - \nabla g(u), v-w \rangle \ge A(w) \ \text {for all} \ v \in U \Big \}. \end{aligned}$$

After that by using a similar idea of [10, Lemma 1], the author of [9] proved that if $g: W \longrightarrow (-\infty , + \infty ]$ is strongly coercive and Gâteaux differentiable, then ${\textrm{dom} \big (\textrm{Res}^g_{f, A, B}\big )=W.}$ We find that the formula of the function $\textrm{Res}_{f, A, B}^{g}$ contains the term B(u) for all $u \in W$. Since $\textrm{dom} B =U \subset W$, the value B(u) does not exist for all $u \in W \setminus U$. Motivated by this confusion, we revise the formula of the function $\textrm{Res}_{f, A, B}^{g}$ by replacing the term B(u), $u \in W$ by B(w), $w \in U$. Then the formula of the mixed resolvent $\textrm{Res}^g_{f, A, B}$ becomes the following formula.

$$\begin{aligned} \textrm{Res}^g_{f, A, B} (u)= & {} \Big \{ w \in U: f(w, v) +A(v) + \langle B(w), v-w \rangle \nonumber \\{} & {} \ \ \ \ \ \ \ \ + \langle \nabla g(w) - \nabla g(u), v-w \rangle \ge A(w) \ \text {for all} \ v \in U \Big \}. \end{aligned}$$

(6)

Note that the idea of the formula (6) was pointed out in [12, Lemm 2.5].Next,byusingtheideaof[10, Lemma1], we will prove that $\textrm{dom} \big (\textrm{Res}^g_{f, A, B}\big )=W$ under some suitable conditions, where the function $\textrm{Res}^g_{f, A, B}$ is defined by (6). The proof of following lemma is easy by using [2, Theorem 1 & p.130-131] and is omitted.

Lemma 3

Let W be a real reflexive Banach space, U be a nonempty, closed and convex subset of W, $g: W \longrightarrow (-\infty , + \infty ]$ be a strongly coercive and Gâteaux differentiable function. Assume that $f: U \times U \longrightarrow \mathbb {R}$, $A: U \longrightarrow \mathbb {R}$ and $B: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$. Then $\textrm{dom} \big (\textrm{Res}^g_{f, A, B}\big )=W.$

The following lemma presents some properties of the mixed resolvent $\textrm{Res}_{f, A, B}^{g}$ which is defined by (6). The proof of this lemma is similar to the proof [9, Lemma 2.8]. Furthermore, these results have been studied in [12, Lemm 2.5].

Lemma 4

[9, Lemma 2.8] Let W be a real reflexive Banach space, U be a nonempty, closed and convex subset of  W, $g: W \longrightarrow (-\infty , + \infty ]$ be a Legendre function. Assume that $f: U \times U \longrightarrow \mathbb {R}$, $A: U \longrightarrow \mathbb {R}$ and $B: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$. Then

• 1. $\textrm{Res}_{f, A, B}^{g}$ is a single-valued.

• 2. $\textrm{Res}_{f, A, B}^{g}$ is a Bregman firmly nonexpansive mapping.

• 3. $F\big ( \textrm{Res}_{f, A, B}^{g} \big ) = GMEP(f, A, B)$ with $F\big ( \textrm{Res}_{f, A, B}^{g} \big )=\{u\in U: \textrm{Res}_{f, A, B}^{g}(u)=u \}$.

• 4. GMEP(f, A, B) is a closed and convex subset of W.

• 5. For all $p \in F\big ( \textrm{Res}_{f, A, B}^{g} \big )$ and $u \in W$, we have

  $$D_g\big ( p, \textrm{Res}_{f, A, B}^{g}(u) \big )+ D_g\big ( \textrm{Res}_{f, A, B}^{g}(u), u\big ) \le D_g(p, u).$$

3 Main results

Let $S_i: W \longrightarrow W$ be Bregman totally quasi-asymptotically nonexpansive mappings with nonnegative real sequences $\{\eta ^{(i)}_n \}$ and $\{\mu ^{(i)}_n \}$ satisfying $\lim \limits _{n\rightarrow \infty }\eta ^{(i)}_n=\lim \limits _{n\rightarrow \infty }\mu ^{(i)}_n=0$ and strictly increasing continuous functions $\xi ^{(i)}: [0, \infty ) \longrightarrow [0, \infty )$ with $\xi ^{(i)} (0)=0$ for each $i \in I : = \{1, 2, ..., N\}$ with $N \in \mathbb {N}$. Put

$$\eta _n =\max \{\eta ^{(i)}_n: i \in I \}, \mu _n =\max \{\mu ^{(i)}_n: i \in I \}, \text {and} \ \xi (t) =\max \{ \xi ^{(i)}(t): i \in I \} $$

for all $t \ge 0$. Then $\lim \limits _{n\rightarrow \infty }\eta _n=\lim \limits _{n\rightarrow \infty }\mu _n=0,$ $\xi (0)=0$, and we have

$$D_g(p, S_i^nu) \le D_g(p, u)+ \eta _n \xi (D_g(p, u)) + \mu _n $$

for all $u \in W$, $p \in \bigcap \limits _{i\in I} F(S_i)$ and for all $i \in I$.

Theorem 5

Suppose that W is a real reflexive Banach space, and U is a nonempty, closed and convex subset of W. Let $g: W \longrightarrow \mathbb {R}$ be Legendre, strongly coercive on W, and g be bounded, totally convex, uniformly Fréchet differentiable on bounded subsets of W. For each $k \in K:= \{1,2, \dots , M\}$ with $M \in \mathbb {N}$, $f_k: U \times U \longrightarrow \mathbb {R}$, $A_k: U \longrightarrow \mathbb {R}$ and $B_k: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$. For each $i \in I$, $S_i: W \longrightarrow W$ is a closed, uniformly asymptotically regular and Bregman totally quasi-asymptotically nonexpansive mapping with nonnegative real sequences $\{\eta ^{(i)}_n \}$ and $\{\mu ^{(i)}_n \}$ satisfying ${\lim \limits _{n\rightarrow \infty }\eta ^{(i)}_n=\lim \limits _{n\rightarrow \infty }\mu ^{(i)}_n=0}$ and strictly increasing continuous function $\xi ^{(i)}: [0, \infty ) \longrightarrow [0, \infty )$ with $\xi ^{(i)} (0)=0$ such that $\mathcal {F} = \Big ( \bigcap \limits _{i \in I} F(S_i) \Big ) \bigcap \Big ( \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k)\Big )$ is nonempty and bounded. Let $\{z_n \}$ be a sequence generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1, z_2 \in U, U_1=U_2=U\\ x_n=z_n+ c_n(z_n-z_{n-1}) \ \text {for all} \ n \ge 2 \\ y_n^{(i)}= \nabla g^* \Big ( a_n \nabla g(x_n) + (1-a_n) \nabla g(S^n_i(x_n)) \Big ) \ \text {for all} \ i \in I \\ i_n=\textrm{argmax} \{ D_g(x_n, y_n^{(i)}): i \in I \}, \ \overline{y}_n = y_n^{(i_n)} \\ u_n^{(k)}= \textrm{Res}^g_{f_k, A_k, B_k}(x_n) \ \text {for all} \ k \in K \\ k_n=\textrm{argmax} \{ D_g(x_n, u_n^{(k)}):k \in K\}, \ \overline{u}_n = u_n^{(k_n)} \\ _n^{(j)}= \nabla g^* \Big ( b_n \nabla g(\overline{y}_n) + (1-b_n) \nabla g (S^n_j(\overline{u}_n)) \Big ) \ \text {for all} \ j \in I \\ j_n=\textrm{argmax} \{ D_g(x_n, v_n^{(j)}): j \in I \}, \ \overline{v}_n = v_n^{(j_n)} \\ U_{n+1}= \big \{ u \in U_n: D_g(u, \overline{v}_n) \le D_g(u, x_n) +\gamma _n \big \} \\ z_{n+1}= P^g_{U_{n+1}}(z_1), \end{array}\right. } \end{aligned}$$

(7)

where $\gamma _n= \eta _n \sup \big \{ \xi \big (D_g(u, x_n)\big ): u \in \mathcal {F}\big \}+ \mu _n$, and $\{a_n\}, \{b_n\}, \{c_n\} \subset [0,1]$ such that ${\lim \limits _{n\rightarrow \infty } a_n=1}$ and ${\liminf \limits _{n\rightarrow \infty } b_n(1-b_n)>0}$, and the function $\textrm{Res}_{f_k, A_k, B_k}^{g}$ is defined as in (6). Then the sequence $\{z_n\}$ strongly converges to $p=P^g_{\mathcal {F}}(z_1)$.

Proof

The proof of Theorem 5 is divided into following six steps.

Step 1. We claim that $P^g_{\mathcal {F}}(z_1)$ is well-defined. Indeed, we conclude from Lemma 2 and Lemma 4 that $F(S_i)$ and $GMEP(f_k, A_k, B_k)$ are closed and convex sets for all $i\in I$ and $k \in K$. This proves that

$$\begin{aligned} \mathcal {F} = \Bigg (\bigcap \limits _{i\in I} F(S_i)\Big ) \bigcap \Big (\bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k)\Bigg ) \end{aligned}$$

is a closed and convex subset of U. Since $\mathcal {F}$ is a nonempty set, we find that $\mathcal {F}$ is a nonempty, closed and convex subset of U. This fact ensures that $P^g_{\mathcal {F}}(z_1)$ is well-defined.

Step 2. We claim that $P^g_{U_{n+1}} (z_1)$ is well-defined. Indeed, we first show that $U_{n}$ is closed and convex for all $n \ge 2$ by mathematical induction. Obviously, we have $U_2=U$ is closed and convex. Now, we assume that $U_m$ is closed and convex for some $m \ge 2$. It follows from the definition of $U_{m+1}$, we get that

$$\begin{aligned} U_{m+1}= & {} \big \{ u \in U_m: \langle \nabla g(x_m), u-x_m \rangle - \langle \nabla g( \overline{v}_m), u- \overline{v}_m \rangle \nonumber \\{} & {} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le g( \overline{v}_m)-g(x_m)+\gamma _m \big \}. \end{aligned}$$

(8)

Then by directly checking, we conclude that $U_{m+1}$ is convex. Furthermore, it follows from (8) and the continuity of $\nabla g(.)$ that $U_{m+1}$ is closed. Therefore, $U_{m+1}$ is closed and convex, and hence $U_{n}$ is closed and convex for all $n \ge 2$. Combining this with $U_1=U_2$ is closed and convex, we get that $U_{n}$ is closed and convex for all $n \in \mathbb {N}$.

Next, we claim that $\mathcal {F} \subset U_n$ for all $n\ge 2$ by mathematical induction. Obviously, we obtain $\mathcal {F} \subset U=U_2$. Suppose that $\mathcal {F} \subset U_m$ for some $m\ge 2$. Now, we prove that $\mathcal {F} \subset U_{m+1}.$ Assume that $u \in \mathcal {F}$. It follows from $\mathcal {F} \subset U_m$ that $u \in U_m$. By using (5) and the fact that $S_{i_m}$ is a Bregman totally quasi-asymptotically nonexpansive mapping, we get

$$\begin{aligned} D_g(u, \overline{y}_m)= & {} D_g(u, y^{(i_m)}_m) \nonumber \\= & {} D_g\Big (u, \nabla g^*\big ( a_{m} \nabla g(x_m) + (1-a_m) \nabla g(S^m_{i_m}x_m) \big ) \Big ) \nonumber \\\le & {} a_mD_g(u, x_m) + (1-a_m)D_g(u, S^m_{i_m}x_m) \nonumber \\\le & {} a_{m} D_g(u, x_m) +(1-a_m)[ D_g(u,x_m) + \eta _n \xi (D_g(u, x_m)) +\mu _m]\nonumber \\\le & {} D_g(u, x_m) +\gamma _m. \end{aligned}$$

(9)

From Lemma 4, we find that $\textrm{Res}^g_{f_{k_m}, A_{k_m}, B_{k_m}}$ is a Bregman firmly nonexpansive mapping and hence it is a Bregman quasi-nonexpansive mapping for each $k_m \in K$. Then, by Remark 4(2), we conclude that $\textrm{Res}^g_{f_{k_m}, A_{k_m}, B_{k_m}}$ is a Bregman quasi nonexpansive mapping. Therefore, we have

$$\begin{aligned} D_g(u, \overline{u}_m) = D_g(u, u^{(k_m)}_m) = D_g\big (u, \textrm{Res}^g_{f_{k_m}, A_{k_m}, B_{k_m}}(x_m ) \big ) \le D_g(u,x_m ). \end{aligned}$$

(10)

It follows from the strictly increasing property of $\xi$ and (10) that

$$\begin{aligned} \xi \big ( D_g(u, \overline{u}_m)\big ) \le \xi \big ( D_g(u, x_m) \big ). \end{aligned}$$

(11)

By combining (9), (10) and (11), we find that

$$\begin{aligned} D_g(u, \overline{v}_m)= & {} D_g(u, v^{(j_m)}_m) \nonumber \\= & {} D_g\Big (u, \nabla g^*\big ( b_m \nabla g(\overline{y}_m) + (1-b_m) \nabla g (S^m_{j_m}(\overline{u}_m)) \big ) \Big ) \nonumber \\\le & {} b_mD_g(u, \overline{y}_m) + (1-b_m)D_g(u, S^m_{j_m}(\overline{u}_m)) \nonumber \\\le & {} b_{m} D_g(u, \overline{y}_m) +(1-b_m)[ D_g(u,\overline{u}_m) + \eta _m \xi (D_g(u, \overline{u}_m)) +\mu _m]\nonumber \\\le & {} b_m [ D_g(u, x_m) +\gamma _m ]+ (1-b_m)[ D_g(u,x_m)+ \eta _m \xi (D_g(u, x_m)) +\mu _m]\nonumber \\\le & {} D_g(u, x_m) + \gamma _m. \end{aligned}$$

(12)

This leads to $u \in U_{m+1}$ and hence $\mathcal {F} \subset U_{m+1}$. This implies that $\mathcal {F} \subset U_{n}$ for all $n \ge 2$. It follows from $U_1=U_2$ that $\mathcal {F} \subset U_{n}$ for all $n \in \mathbb {N}$. Since $\mathcal {F}$ is nonempty, we obtain $U_{n}$ is nonempty.

By the above, we obtain that $U_{n}$ is nonempty, closed and convex. Therefore, we conclude that $P^g_{U_{n+1}} (z_1)$ is well-defined.

Step 3. We claim that $\{z_n\}$ is bounded and the limit $\lim \limits _{n\rightarrow \infty }D_g(z_n, z_1)$ exists. Indeed, we conclude from $z_{n}= P^g_{U_{n}}(z_1)$ and Proposition 4 that

$$\begin{aligned} D_g(y, z_n) +D_g(z_n,z_1) \le D_g(y, z_1) \end{aligned}$$

(13)

for all $y \in U_n$. Let $u \in \mathcal {F}$. It follows from $\mathcal {F} \subset U_n$ that $u \in U_n$. By choosing $y=u$ in (13), we get

$$\begin{aligned} D_g(u, z_n) +D_g(z_n,z_1) \le D_g(u, z_1). \end{aligned}$$

(14)

This proves that $D_g(z_n,z_1) \le D_g(u, z_1)- D_g(u, z_n) \le D_g(u, z_1)$. It means that $\{ D_g(u_n,u_1)\}$ is bounded. By [34, Lemma 1], we conclude that $\{u_n\}$ is bounded.

Next, from the definition of $U_n$, we get $z_{n+1}= P^g_{U_{n+1}}(z_1) \in U_{n+1} \subset U_n.$ By choosing ${y=z_{n+1}}$ in (13), we obtain $D_g(z_{n+1}, z_n) +D_g(z_n,z_1) \le D_g(z_{n+1}, z_1).$ This leads to $D_g(z_n,z_1) \le D_g(z_{n+1}, z_1)- D_g(z_{n+1}, z_n) \le D_g(z_{n+1}, z_1).$ Therefore, $\{ D_g(z_n,z_1)\}$ is a nondecreasing sequence. By using the boundedness of the sequence $\{ D_g(z_n,z_1)\}$, we find that the limit $\lim \limits _{n\rightarrow \infty }D_g(z_n, z_1)$ exists.

Step 4. We claim that $\lim \limits _{n\rightarrow \infty }z_n=p \in U$. Indeed, for all $m>n$, it follows from the definition of $z_m$ that $z_{m}= P^g_{U_{m}}(z_1) \in U_{m} \subset U_n.$ Therefore, by taking $y=z_m$ in (13), we obtain $D_g(z_{m}, z_n) +D_g(z_n,z_1) \le D_g(z_{m}, z_1)$. This leads to

$$\begin{aligned} 0 \le D_g(z_{m}, z_n) \le D_g(z_{m}, z_1) - D_g(z_n,z_1). \end{aligned}$$

(15)

Taking the limit (15) as $m,n \rightarrow \infty$, and using the existence of the limit $\lim \limits _{n\rightarrow \infty } D_g(z_n, z_1)$, we have

$$\begin{aligned} \lim \limits _{m, n \rightarrow \infty } D_g(z_{m}, z_n) =0. \end{aligned}$$

(16)

Then, it follows from (16), the boundedness of $\{z_n \}$ and Proposition 2 that

$$\begin{aligned} \lim \limits _{m, n\rightarrow \infty } \Vert z_{m} - z_n \Vert =0. \end{aligned}$$

(17)

This proves that $\{z_n \}$ is a Cauchy sequence in U. Since W is a Banach space and U is a closed subset of W, there exists $p \in U$ such that $\lim \limits _{n\rightarrow \infty }z_n=p$.

Step 5. We claim that $p \in \mathcal {F}$. First, we will prove that $p \in \bigcap \limits _{i\in I} F(S_i).$ Indeed, by choosing $m=n+1$ in (16) and (17), we get

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } D_g(z_{n+1}, z_n) = \lim \limits _{n\rightarrow \infty } \Vert z_{n+1} - z_n \Vert =0. \end{aligned}$$

(18)

It follows from $z_{n+1}= P^g_{U_{n+1}}(z_1) \in U_{n+1} \subset U_n$ and the definition of $U_n$ that

$$\begin{aligned} D_g(z_{n+1}, \overline{v}_n) \le D_g(z_{n+1}, x_n) +\gamma _n. \end{aligned}$$

(19)

Furthermore, we have $\Vert x_n-z_n\Vert = c_n \Vert z_n-z_{n-1}\Vert$. By combining this with (18) and the boundedness of $\{c_n \}$, we find that $\lim \limits _{n\rightarrow \infty } \Vert x_n-z_n\Vert =0$. Since $\lim \limits _{n\rightarrow \infty }z_n=p$, we get that $\lim \limits _{n\rightarrow \infty }x_n=p.$ Therefore, from (18) and $\lim \limits _{n\rightarrow \infty } \Vert x_n-z_n\Vert =0,$ we get $\lim \limits _{n\rightarrow \infty }\Vert z_{n+1} -x_n\Vert =0.$ By using the definition of $D_g$, we find that

$$\begin{aligned} |D_g(z_{n+1}, x_n)|= & {} |g(z_{n+1})-g(x_n)- \langle \nabla g(x_n),z_{n+1}-x_n \rangle | \nonumber \\\le & {} |g(z_{n+1})-g(x_n)| + \Vert z_{n+1}-x_n\Vert .\Vert \nabla g(x_n)\Vert . \end{aligned}$$

(20)

Furthermore, by [24, Proposition 1.1.10 & Proposition 1.1.11], we find that $\nabla g$ is bounded on bounded subsets of W. Then, by combining this with the boundedness of $\{x_n \}$, $\lim \limits _{n\rightarrow \infty }\Vert z_{n+1} -x_n\Vert =0$ and (20), we obtain

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }D_g(z_{n+1},x_n)=0. \end{aligned}$$

(21)

Let $u \in \mathcal {F}$. By using the definition of $D_g$, we get

$$\begin{aligned} |D_g(u, x_n)|= & {} |g(u)-g(x_n)- \langle \nabla g(x_n),u-x_n \rangle | \nonumber \\\le & {} |g(u)-g(x_n)| + \Vert u-x_n\Vert .\Vert \nabla g(x_n)\Vert \nonumber \\\le & {} |g(u)|+ |g(x_n)| + (\Vert u\Vert +\Vert x_n\Vert ).\Vert \nabla g(x_n)\Vert . \end{aligned}$$

(22)

It follows from (22), the boundedness of $\mathcal {F}$ and $\{x_n \}$, the uniform continuity of g and the boundedness on bounded subsets of $\nabla g$ that $|D_g(u, x_n)| <\infty$. This proves that the sequence $\{D_g(u, x_n)\}$ is bounded. By using $\lim \limits _{n\rightarrow \infty } \eta _n = \lim \limits _{n\rightarrow \infty }\mu _n=0$, we find that

$$ \lim \limits _{n\rightarrow \infty }\gamma _n= \lim \limits _{n\rightarrow \infty } \Big ( \eta _n \sup \big \{ \xi \big (D_g(u, x_n)\big ): u \in \mathcal {F}\big \}+ \mu _n \Big )=0. $$

By combining (19), (21) and $\lim \limits _{n\rightarrow \infty }\gamma _n= 0$, we find that $\lim \limits _{n\rightarrow \infty } D_g(z_{n+1}, \overline{v}_n)=0$. Furthermore, by (12) and the boundedness of $\{D_g(u, x_n)\}$, we obtain that $\{D_g(u, v^{(j)}_n)\}$ is bounded. Now, by [28, Proposition 3.6.4], we find that $g^*$ is bounded on bounded subsets of $W^*$. This implies that $\nabla g^*$ is bounded on bounded subsets of $W^*$. By combining this with the boundedness of $\{D_g(u, v^{(j)}_n)\}$ and using Proposition 3, we find that $\{v^{(j)}_n\}$ is bounded. This implies that $\{ \overline{v}_n\}$ is bounded. By combining this with $\lim \limits _{n\rightarrow \infty } D_g(z_{n+1}, \overline{v}_n)=0$ and using Proposition 2, we find that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert z_{n+1}- \overline{v}_n\Vert =0. \end{aligned}$$

(23)

By combining (23) and $\lim \limits _{n\rightarrow \infty } \Vert z_{n+1}- x_n\Vert =0$, we get $\lim \limits _{n\rightarrow \infty } \Vert x_{n}- \overline{v}_n\Vert =0.$ Then, by using the same proof as in that of (21), we find that $\lim \limits _{n\rightarrow \infty }D_g( x_{n}, \overline{v}_n )=0.$ By the definition $\overline{v}_n$, we get that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }D_g(x_{n}, v^{(j)}_n)=0. \end{aligned}$$

(24)

Next, by using the same proofs as in that of (23), we find that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert x_n-v^{(j)}_n\Vert =0. \end{aligned}$$

(25)

Since g is uniformly continuous and $\nabla g$ is uniformly continuous on bounded sets, from (25), we find that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert g(x_{n})- g(v^{(j)}_n)\Vert =\lim \limits _{n\rightarrow \infty } \Vert \nabla g(x_{n})- \nabla g(v^{(j)}_n)\Vert =0. \end{aligned}$$

(26)

Next, by using the same proofs as in that of (9) and (10), we conclude that

$$\begin{aligned} D_g(u, \overline{y}_n) \le D_g(u, x_n) +\gamma _n \end{aligned}$$

(27)

and

$$\begin{aligned} D_g(u, \overline{u}_n) \le D_g(u, x_n). \end{aligned}$$

(28)

Then, from (27), (28) and using the boundedness of $\{D_g(u, x_n) \}$, $\{\gamma _n \}$, we find that $\{D_g(u, \overline{y}_n) \}$ and $\{D_g(u, \overline{u}_n) \}$ are bounded. Note that $\nabla g^*$ is bounded on bounded subsets of $W^*$. By combining this with the boundedness of $\{D_g(u, \overline{y}_n) \}$, $\{D_g(u, \overline{u}_n) \}$ and using Proposition 3, we find that $\{ \overline{y}_n\}$ and $\overline{u}_n\}$ are bounded. Furthermore, for each $j \in I,$ we have

$$\begin{aligned} D_g(u, S_{j}^n( \overline{u}_n)) \le D_g(u, \overline{u}_n ) + \eta _n \xi \big (D_g(u, \overline{u}_n)\big ) +\mu _n. \end{aligned}$$

(29)

By (29) and the boundedness of $\{D_g(u, \overline{u}_n) \}$, we obtain that $\{D_g(u, S_{j}^n (\overline{u}_n))\}$ is bounded. By Proposition 3, we find that $\{S_{j}^n (\overline{u}_n)\}$ is bounded.

Since $\{\overline{y}_n\}$ and $\{S_{j}^n (\overline{u}_n)\}$ are bounded and $\nabla g$ is bounded on bounded subsets, we find that $\{ \nabla g(\overline{y}_n)\}$ and $\{ \nabla g (S_{j}^n (\overline{u}_n))\}$ are bounded. Put

$$ \varepsilon = \max \{ \sup \limits _{n\in \mathbb {N}} \Vert \nabla g(\overline{y}_n)\Vert , \sup \limits _{n\in \mathbb {N}} \Vert \nabla g(S_{j}^n (\overline{u}_n))\Vert \}. $$

This leads to $\nabla g(\overline{y}_n), \nabla g(S^n_{j} (\overline{u}_n)) \in B^*_{\varepsilon }$. Therefore, by using Lemma 1, we find that

$$\begin{aligned}{} & {} D_g(u, v^{(j)}_n)\nonumber \\= & {} D_g\Big (u, \nabla g^*\big ( b_n \nabla g(\overline{y}_n) + (1-b_n) \nabla g (S^n_{j}(\overline{u}_n)) \big )\Big )\nonumber \\\le & {} b_nD_g(u, \overline{y}_n) + (1-b_n) D_g(u, S^n_{j}(\overline{u}_n) ) - b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ) \nonumber \\\le & {} b_nD_g(u, \overline{y}_n) + (1-b_n) [D_g(u, \overline{u}_n) + \eta _n \xi (D_g(u, \overline{u}_n)) +\mu _n] \nonumber \\{} & {} \ \ - b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ). \end{aligned}$$

(30)

It follows from (28), (30) and the strictly increasing property of $\xi$, we find that

$$\begin{aligned} D_g(u, v^{(j)}_n)\le & {} b_nD_g(u, \overline{y}_n) + (1-b_n) [D_g(u, x_n) + \eta _n \xi (D_g(u, x_n)) +\mu _n]\nonumber \\{} & {} \ \ - b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ). \end{aligned}$$

(31)

By combining (27) and (31), we find that

$$\begin{aligned} D_g(u, v^{(j)}_n)\le & {} b_n[D_g(u, x_n) +\gamma _n ] + (1-b_n) [D_g(u, x_n) + \eta _n \xi (D_g(u, x_n)) +\mu _n] \\{} & {} - b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ) \\\le & {} D_g(u, x_n) + \gamma _n - b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ). \end{aligned}$$

This leads to

$$\begin{aligned} b_n(1-b_n) \rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big ) \le D_g(u, x_n) - D_g(u, v^{(j)}_n) + \gamma _n. \end{aligned}$$

(32)

Moreover, by using the property of the function $D_g$, we obtain

$$\begin{aligned}{} & {} |D_g(u, x_n) - D_g(u, v^{(j)}_n)|\nonumber \\= & {} |-D_g(x_n, v^{(j)}_n) + \langle \nabla g(v^{(j)}_n) - \nabla g(x_n), u- x_n \rangle | \nonumber \\\le & {} |D_g(x_n, v^{(j)}_n)|+\Vert \nabla g(v^{(j)}_n) - \nabla g(x_n)\Vert . \Vert u- x_n \Vert . \end{aligned}$$

(33)

It follows from (24), (26) and (33) that $\lim \limits _{n\rightarrow \infty } |D_g(u, x_n) - D_g(u,v^{(j)}_n)| =0$. By combining this with (32), $\lim \limits _{n\rightarrow \infty }\gamma _n=0$ and ${\liminf \limits _{n\rightarrow \infty } b_n(1-b_n)>0},$ we conclude that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\rho ^*_{\varepsilon } \big (\Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n)) \Vert \big )=0. \end{aligned}$$

(34)

Suppose that $\lim \limits _{n\rightarrow \infty } \Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n))\Vert >0.$ Then, there exist $r >0$ and a subsequence $\{k(n)\}$ of n such that $\Vert \nabla g (\overline{y}_{k(n)}) - \nabla g (S^{k(n)}_{j}(\overline{u}_{k(n)}))\Vert \ge r.$ It follows from the nondecreasing property of $\rho ^*_{\varepsilon }$ that

$$\begin{aligned} \rho ^*_{\varepsilon }( \Vert \nabla g (\overline{y}_{k(n)}) - \nabla g (S^{k(n)}_{j}(\overline{u}_{k(n)}))\Vert ) \ge \rho ^*_{\varepsilon }( r ) \end{aligned}$$

(35)

for all $n\in \mathbb {N}$. By taking the limit as $n \rightarrow \infty$ in (35) and using (34), we obtain that $0 \ge \rho ^*_{\varepsilon }(r )$. This contradicts the fact that $\rho ^*_{\varepsilon }(r) >0$. Therefore,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert \nabla g (\overline{y}_n) - \nabla g (S^n_{j}(\overline{u}_n))\Vert =0. \end{aligned}$$

(36)

Note that $\nabla g^*$ is uniformly continuous on bounded subsets. By combining this with $\nabla g = (\nabla g^*)^{-1}$ and (36), we conclude that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert \overline{y}_n - S^n_{j}(\overline{u}_n)\Vert =0. \end{aligned}$$

(37)

Now, by the definition of $\overline{y}_n$ and $\nabla g = (\nabla g^*)^{-1}$, we get

$$\begin{aligned} \nabla g( \overline{y}_n) = \nabla g\Big (\nabla g^* \big ( a_n \nabla g(x_n) + (1-a_n) \nabla g(S^n_i(x_n) )\big )\Big ) = a_n \nabla g(x_n) + (1-a_n) \nabla g(S^n_{i_n}(x_n)). \end{aligned}$$

This leads to

$$\begin{aligned} \Vert \nabla g(\overline{y}_n) - \nabla g( x_n) \Vert =(1- a_n) \Vert \nabla g(S^n_{i_n}(x_n))- \nabla g(x_n)\Vert . \end{aligned}$$

(38)

It follows from (37), (38), $\lim \limits _{n\rightarrow \infty }a_n=1$ and the boundedness of $\{x_n \}$ that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert \nabla g(\overline{y}_n) - \nabla g( x_n) |=0. \end{aligned}$$

(39)

Then, by using the uniform continuous on bounded subsets of $\nabla g^*$ , $\nabla g = (\nabla g^*)^{-1}$ and (39), we obtain

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert \overline{y}_n - x_n |=0. \end{aligned}$$

(40)

It follows from (37) and (40) that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert S^n_{j}(\overline{u}_n)- x_n \Vert =0. \end{aligned}$$

(41)

By combining (41) and $\lim \limits _{n\rightarrow \infty } x_n=p$, we have $\lim \limits _{n\rightarrow \infty } S^n_{j}(\overline{u}_n) =p$. Moreover, we have

$$\begin{aligned} \Vert S^{n+1}_{j}(\overline{u}_n) - p\Vert \le \Vert S^{n+1}_{j}(\overline{u}_n) - S^n_{j}(\overline{u}_n)\Vert + \Vert S^n_{j}(\overline{u}_n) -p\Vert . \end{aligned}$$

(42)

Then, we conclude from (42), $\lim \limits _{n\rightarrow \infty } S^n_{j}(\overline{u}_n) =p$ and the asymptotically regular property of $S_j$ that $\lim \limits _{n\rightarrow \infty } S^{n+1}_{j}(\overline{u}_n) =p$ for all $j \in I$. This proves that $\lim \limits _{n\rightarrow \infty } S(S^{n}_{j}(\overline{u}_n)) =p$. Since $S_j$ is closed, we find that $S_j(p)=p$ for all $j \in I$ and $p \in \bigcap \limits _{i\in I} F(S_i).$

Next, we prove that $p \in \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k).$ Indeed, for each $k \in K$, we have $u_n^{(k)} =\textrm{Res}^g_{f_k, A_k, B_k}(\overline{y}_n).$ It follows from (6) that

$$\begin{aligned} f_k(u_n^{(k)}, v) +A_k(v) + \langle B_k(u_n^{(k)}), v-u_n^{(k)} \rangle + \langle \nabla g(u_n^{(k)}) - \nabla g(\overline{y}_n), v-u_n^{(k)} \rangle \ge A_k(u_n^{(k)}) \ \text {for all} \ v \in U. \end{aligned}$$

By using the condition $(C_2)$, we get

$$\begin{aligned}{} & {} \langle B_k(u_n^{(k)}), v-u_n^{(k)} \rangle + \langle \nabla g(u_n^{(k)}) - \nabla g(\overline{y}_n), v-u_n^{(k)} \rangle +A_k(v) - A_k(u_n^{(k)}) \nonumber \\\ge & {} - f_k(u_n^{(k)}, v) \ge f_k(v,u_n^{(k)}). \end{aligned}$$

(43)

Since g is uniformly continuous and $\nabla g$ is uniformly continuous on bounded sets, by (41), we obtain

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert g(x_{n})- g(S^n_{j}(\overline{u}_n))\Vert =\lim \limits _{n\rightarrow \infty } \Vert \nabla g(x_{n})- \nabla g(S^n_{j}(\overline{u}_n))\Vert =0. \end{aligned}$$

(44)

Let $u \in \mathcal {F}$. By using the property of the Bregman distance, we have

$$\begin{aligned}{} & {} |D_g(u, x_n) - D_g(u, S^n_{j}(\overline{u}_n)) | \nonumber \\= & {} |-D_g(x_n, S^n_{j}(\overline{u}_n))+ \langle \nabla g(S^n_{j}(\overline{u}_n))- \nabla g(x_n), u-x_n \rangle | \nonumber \\\le & {} |D_g(x_n, S^n_{j}(\overline{u}_n))|+ \Vert \nabla g(S^n_{j}(\overline{u}_n))- \nabla g(x_n) \Vert . \Vert u-x_n\Vert \nonumber \\\le & {} |g(x_n)- g( S^n_{j}(\overline{u}_n))|+ \Vert \nabla g( S^n_{j}(\overline{u}_n))\Vert . \Vert x_n- S^n_{j}(\overline{u}_n)\Vert \nonumber \\{} & {} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \Vert \nabla g(S^n_{j}(\overline{u}_n))- \nabla g(x_n) \Vert . \Vert u-x_n\Vert . \end{aligned}$$

(45)

It follows from (41), (44) and (45) that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } |D_g(u, x_n) - D_g(u, S^n_{j}(\overline{u}_n)) | =0. \end{aligned}$$

(46)

Furthermore, by $\overline{u}_n =\textrm{Res}^g_{f_{k_n}, A_{k_n}, B_{k_n}}(x_n)$, Lemma 4 and (29), we find that

$$\begin{aligned} D_g(x_n, \overline{u}_n)\le & {} D_g(u, x_n)- D_g(u, \overline{u}_n ) \nonumber \\\le & {} D_g(u, x_n) - D_g(u,S^n_{j}(\overline{u}_n) ) + \eta _n \xi \big (D_g(u, \overline{u}_n)\big ) +\mu _n. \end{aligned}$$

(47)

Then, by (28) and the strictly increasing property of $\xi$, the inequality (47) becomes

$$\begin{aligned} D_g(x_n, \overline{u}_n)\le & {} D_g(u, x_n)- D_g(u,S^n_{j}(\overline{u}_n) ) + \eta _n \xi \big (D_g(u, x_n)\big ) +\mu _n \nonumber \\\le & {} D_g(u, x_n) - D_g(u,S^n_{j}(\overline{u}_n) ) +\gamma _n. \end{aligned}$$

(48)

It follows from (46), (48) and $\lim \limits _{n\rightarrow \infty }\gamma _n=0$ that $\lim \limits _{n\rightarrow \infty } D_g(x_n, \overline{u}_n) =0.$ By the definition $\overline{u}_n$, we get that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }D_g(x_{n}, u^{(k)}_n)=0. \end{aligned}$$

(49)

Then, by using the same proof as in that of (23), we find that $\lim \limits _{n\rightarrow \infty } \Vert x_n-u^{(k)}_n\Vert =0.$ By combining this with $\lim \limits _{n\rightarrow \infty } x_n=p$, we get that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } u^{(k)}_n=p. \end{aligned}$$

(50)

Moreover, it follows from (40) and $\lim \limits _{n\rightarrow \infty } \Vert x_n-u^{(k)}_n\Vert =0$ that $\lim \limits _{n\rightarrow \infty } \Vert u^{(k)}_n- \overline{y}_n\Vert =0$. Since $\nabla g$ is uniformly continuous on bounded subsets, we obtain $\lim \limits _{n\rightarrow \infty } \Vert \nabla g(u_n^{(k)}) - \nabla g(\overline{y}_n)\Vert =0.$ This implies that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }| \langle \nabla g(u_n^{(k)}) - \nabla g(\overline{y}_n), v-u_n^{(k)} \rangle |=0. \end{aligned}$$

(51)

Since $A_k$ is lower semi-continuous and (50), we find that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty } A_k(u_n^{(k)}) \ge A_k(p). \end{aligned}$$

(52)

By the condition $(C_4)$, we get that $f_k$ is lower semi-continuous in the second variable for each $k \in K$. It follows from (50) that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty } f_k(v, u_n^{(k)}) \ge f_k(v,p). \end{aligned}$$

(53)

We also have

$$\begin{aligned}{} & {} |\langle B_k(u_n^{(k)}), v-u_n^{(k)} \rangle - \langle B_k(p), v-p \rangle | \nonumber \\= & {} | \langle B_k(u_n^{(k)}) -B_k(p), v \rangle - \langle B_k(u_n^{(k)}), u_n^{(k)} \rangle + \langle B_k(p), p \rangle | \nonumber \\\le & {} | \langle B_k(u_n^{(k)}) -B_k(p), v \rangle | +| \langle B_k(u_n^{(k)}), u_n^{(k)} -p \rangle | + |\langle B_k(u_n^{(k)})- B_k(p), p \rangle |\nonumber \\\le & {} | \langle B_k(u_n^{(k)}) -B_k(p), v \rangle | +\Vert B_k(u_n^{(k)})\Vert .\Vert u_n^{(k)} -p \Vert + |\langle B_k(u_n^{(k)})- B_k(p), p \rangle |. \end{aligned}$$

(54)

It follows from (50), (54), the continuity of $B_k$ and $B_k(u_n^{(k)}) \in W^*$ that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \langle B_k(u_n^{(k)} ), v-u_n^{(k)} \rangle = \langle B_k(p), v-p \rangle . \end{aligned}$$

(55)

Then, by (43), (51), (52), (53) and (55), we find that

$$\begin{aligned} \langle B_k(p), v-p \rangle +A_k(v) - A_k(p) \ge f_k(v,p) \end{aligned}$$

(56)

for all $v \in U$. For all $t \in (0,1]$, put $v_t =tv+(1-t)p$. Due to $y, p \in U$ and U is convex, we have $v_t \in U$. Then, by replacing y by $v_t$ in (56), we conclude that

$$\begin{aligned} f_k(v_t, p) + \langle B_k(p), p-v_t \rangle + A_k(p)- A_k(v_t) \le 0. \end{aligned}$$

(57)

By using the condition $(C_1)$, the convexity in the second variable of $f_k$ and the convexity of $A_k$ and (57), we conclude that

$$\begin{aligned} 0= & {} f_k(v_t, v_t)= f_k(v_t, v_t) + \langle B_k(p), v_t-v_t \rangle + A_k(v_t)- A_k(v_t)\nonumber \\\le & {} tf_k(v_t,y) +(1-t) f_k(v_t,p) +t \langle B_k(p), y-v_t \rangle + (1-t)\langle B_k(p), p-v_t \rangle \nonumber \\{} & {} \ \ \ \ \ \ \ \ \ + tA_k(y) +(1-t) A_k(p) -A_k(v_t)\nonumber \\= & {} t \big [ f_k(v_t,v)+ \langle B_k(p), v-v_t \rangle + A_k(v) - A_k(v_t) \big ]\nonumber \\{} & {} \ \ \ \ \ \ \ \ \ + (1-t) \big [ f_k(v_t,p)+ \langle B_k(p), p-v_t \rangle + A_k(p) - A_k(v_t) \big ]\nonumber \\\le & {} t \big [ f_k(v_t,y)+ \langle B_k(p), v-v_t \rangle + A_k(v) - A_k(v_t) \big ]. \end{aligned}$$

(58)

It follows from (58) and $t>0$ that

$$\begin{aligned} f_k(v_t,v)+ \langle B_k(p), v-v_t \rangle + A_k(v) - A_k(v_t) \ge 0. \end{aligned}$$

(59)

Therefore, by the condition $(C_3)$, we have

$$\begin{aligned} \limsup \limits _{t\downarrow 0} f_k(v_t,v) = \limsup \limits _{t\downarrow 0} f_k(tv+(1-t)p,v) \le f_k(p,v). \end{aligned}$$

(60)

Since $A_k$ is lower semi-continuous, we get that $-A_k$ is upper semi-continuous. From $\lim \limits _{t\rightarrow 0}v_t=\lim \limits _{t\rightarrow 0}(tv+(1-t)p)= p$, we find that

$$\begin{aligned} \limsup \limits _{t \rightarrow 0}[ -A_k(v_t) ] \le - A_k(p). \end{aligned}$$

(61)

By (59), (60), (61) and $\lim \limits _{t\rightarrow 0}v_t= p$, we find that

$$f_k(p,v)+ \langle B_k(p), v-p \rangle + A_k(v) - A_k(p) \ge 0.$$

This implies that $p \in \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k).$ By the above, we conclude that

$$\begin{aligned} p \in \mathcal {F} = \Bigg (\bigcap \limits _{i\in I} F(S_i) \Big ) \bigcap \Big ( \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k)\Bigg ). \end{aligned}$$

Step 6. We claim that $p=P^g_{\mathcal {F}}(z_1)$. Indeed, we put $z=P^g_{\mathcal {F}}(z_1)$. We will show that $z=p$. From $z_{n}= P^g_{U_{n}}(z_1)$ and Definition 3, we find that

$$\begin{aligned} D_g(z_n, z_1) \le D_g(v, z_1) \end{aligned}$$

(62)

for all $v \in U_{n}$. It follows from $z=P^g_{\mathcal {F}}(z_1) \in \mathcal {F}$ and $\mathcal {F} \subset U_{n}$ that $z \in U_{n}$. Therefore, by taking $v= z$ in (62), we obtain

$$\begin{aligned} D_g(z_n, z_1) \le D_g(z, z_1). \end{aligned}$$

(63)

Furthermore, we have

$$\begin{aligned} | D_g(z_n, z_1)- D_g(p, z_1)|= & {} | g(z_n)-g(p) + \langle \nabla g(z_1), p-z_n \rangle | \nonumber \\\le & {} |g(z_n)-g(p)| + \Vert \nabla g(z_1)\Vert . \Vert p-z_n\Vert . \end{aligned}$$

(64)

Taking the limit as $n \longrightarrow \infty$ in (64) and using $\lim \limits _{n\rightarrow \infty }u_n=p$, the uniform continuity of g and the boundedness on bounded subsets of $\nabla g$ that $\lim \limits _{n\rightarrow \infty } D_g(z_n, z_1)=D_g(p, z_1).$ Then, it follows from (63) that $D_g(p, z_1) \le D_g(z, z_1)$. By the definition of z and $p \in \mathcal {F}$, we find that $p=z=P^g_{\mathcal {F}}(z_1)$. $\square$

In Theorem 5, by choosing $a_n=1$, we obtain the following result. Note that iteration (65) is an improvements to iteration (1) which was presented in [5].

Corollary 1

Suppose that W is a real reflexive Banach space, and U is a nonempty, closed and convex subset of W. Let $g: W \longrightarrow \mathbb {R}$ be Legendre, strongly coercive on W, and g be bounded, totally convex, uniformly Fréchet differentiable on bounded subsets of W. For each $k \in K:= \{1,2, \dots , M\}$ with $M \in \mathbb {N}$, $f_k: U \times U \longrightarrow \mathbb {R}$, $A_k: U \longrightarrow \mathbb {R}$ and $B_k: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$. For each $i \in I$, $S_i: W \longrightarrow W$ is a closed, uniformly asymptotically regular and Bregman totally quasi-asymptotically nonexpansive mapping with nonnegative real sequences $\{\eta ^{(i)}_n \}$ and $\{\mu ^{(i)}_n \}$ satisfying ${\lim \limits _{n\rightarrow \infty }\eta ^{(i)}_n=\lim \limits _{n\rightarrow \infty }\mu ^{(i)}_n=0}$ and strictly increasing continuous function $\xi ^{(i)}: [0, \infty ) \longrightarrow [0, \infty )$ with $\xi ^{(i)} (0)=0$ such that $\mathcal {F} = \Big ( \bigcap \limits _{i \in I} F(S_i) \Big ) \bigcap \Big ( \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k)\Big )$ is nonempty and bounded. Let $\{z_n \}$ be a sequence generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1, z_2 \in U, U_1=U_2=U\\ x_n=z_n+ c_n(z_n-z_{n-1}) \ \text {for all} \ n \ge 2 \\ u_n^{(k)}= \textrm{Res}^g_{f_k, A_k, B_k}(x_n) \ \text {for all} \ k \in K \\ k_n=\textrm{argmax} \{ D_g(x_n, u_n^{(k)}):k \in K\}, \ \overline{u}_n = u_n^{(k_n)} \\ v_n^{(j)}= \nabla g^* \Big ( b_n \nabla g(x_n) + (1-b_n) \nabla g (S^n_j(\overline{u}_n)) \Big ) \ \text {for all} \ j \in I \\ j_n=\textrm{argmax} \{ D_g(x_n, v_n^{(j)}): j \in I \}, \ \overline{v}_n = v_n^{(j_n)} \\ U_{n+1}= \big \{ u \in U_n: D_g(u, \overline{v}_n) \le D_g(u, x_n) +\gamma _n \big \} \\ z_{n+1}= P^g_{U_{n+1}}(z_1), \end{array}\right. } \end{aligned}$$

(65)

where $\gamma _n= \eta _n \sup \big \{ \xi \big (D_g(u, x_n)\big ): u \in \mathcal {F}\big \}+ \mu _n$, and $\{b_n\}, \{c_n\} \subset [0,1]$ such that ${\liminf \limits _{n\rightarrow \infty } b_n(1-b_n)>0}$, and the function $\textrm{Res}_{f_k, A_k, B_k}^{g}$ is defined as in (6). Then the sequence $\{z_n\}$ strongly converges to $p=P^g_{\mathcal {F}}(z_1)$.

Remark 5

In Theorem 5 and Corollary 1, by choosing $S_i=S$ for all $i \in I$, $f_k=f$, $A_k=A$ and $B_k=B$ for all $k \in K$, we get two convergence results for a generalized mixed equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces.

In Theorem 5, when $S_i$ is an identity mapping for all $i \in I$, we obtain the following corollary. Note that the iteration process (66) is an improvement of the the iteration process (3) in the sense of adding the inertial extrapolation. Therefore, Corollary 2 is a generalization of the main result in [17].

Corollary 2

Suppose that W is a real reflexive Banach space, and U is a nonempty, closed and convex subset of W. Let $g: W \longrightarrow \mathbb {R}$ be Legendre, strongly coercive on W, and g be bounded, totally convex, uniformly Fréchet differentiable on bounded subsets of W. For each $k \in K:= \{1,2, \dots , M\}$ with $M \in \mathbb {N}$, $f_k: U \times U \longrightarrow \mathbb {R}$, $A_k: U \longrightarrow \mathbb {R}$ and $B_k: U \longrightarrow W^*$ satisfy the conditions $(C_1)$ - $(C_6)$ such that $\mathcal {F}_1 = \bigcap \limits _{k \in K} GMEP(f_k, A_k, B_k)$ is nonempty and bounded. Let $\{z_n \}$ be a sequence generated by

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1, z_2 \in U, U_1 =U_2= U \\ x_n = z_n+c_n(z_n-z_{n-1}) \ \text {for all} \ n \ge 2\\ u_n^{(k)} =\textrm{Res}^g_{f_k, A_k, B_k}(x_n)\\ k_n= \textrm{argmax} \{ D_g(u_n^{(k)}, x_n):k \in K\}, \overline{u}_n= u^{(k_n)}_n\\ v_n= \nabla g^* \Big ( b_n \nabla g(x_n) + (1-b_n) \nabla g (\overline{u}_n)) \Big ) \\ U_{n+1}= \big \{ u \in U_n: D_g(u, v_n) \le D_g(u, x_n) +\gamma _n \big \} \\ z_{n+1}= P^g_{U_{n+1}}(z_1), \end{array}\right. } \end{aligned}$$

(66)

where $\gamma _n= \eta _n \sup \big \{ \xi \big (D_g(u, x_n)\big ): u \in \mathcal {F}\big \}+ \mu _n$, and $\{b_n\}, \{c_n\} \subset [0,1]$ such that ${\liminf \limits _{n\rightarrow \infty } b_n(1-b_n)>0}$, and the function $\textrm{Res}_{f_k, A_k, B_k}^{g}$ is defined as in (6). Then the sequence $\{z_n\}$ strongly converges to $p=P^g_{\mathcal {F}_1}(z_1)$.

Remark 6

• 1. Note that every Bregman quasi-asymptotically nonexpansive mapping is a Bregman totally quasi-asymptotically nonexpansive mapping with $\xi (t)=t$ for all $t \ge 0$, $\eta _n=k_n-1$ with $k_n \ge 1$ satisfying $\lim \limits _{n\rightarrow \infty }k_n=1$, and $\mu _n=0$ for all $n \in \mathbb {N}$. Therefore, the conclusions of Theorem 5, Corollary 1 and Corollary 2 hold when $S_i$ is a Bregman quasi-asymptotically nonexpansive mapping for all $i \in I$ and $\gamma _n= (k_n-1) \sup \big \{ D_g(u, x_n): u \in \mathcal {F}\big \}$ for all $n \in \mathbb {N}$.

• 2. The conclusions of Theorem 5, Corollary 1 and Corollary 2 are satisfied when (GMEP) is replaced by (MEP), (GEP), (MVI) and (EP).

Finally, we give a numerical example to illustrate for the convergence of iteration (66) and iteration (3).

Example 1

Let $W= \mathbb {R}$, $U=[0, 1]$, $g(u)=u^4$, $S_i(u)=\dfrac{u}{2^i}$ for all $u \in W$ and $i=1,2$. Let $B_k(u)=ku$, $A_k(u)=ku^2$ and $f_k(u, v)=k( -2u^2+uv+v^2)$ for all $u, v \in U$ and $k=1,2$. Then

• 1. By directly calculating, we have $\nabla g(u)= 4u^3$ for all $u \in W$, $g^*(w)= 3\root 3 \of {\Big (\frac{w}{4}\Big )^4}$ and $\nabla g^*(w) =\root 3 \of {\frac{w}{4}}$ for all $w \in W$.

• 2. For all $u, v \in W$, we have $D_g(u, v) =g(u)-g(v) -\langle \nabla g(v), u-v \rangle = u^4+3v^4-4uv^3.$

• 3. For each $i=1,2$, we obtain $F(S_i) =\{0\}$. Therefore, for all $p \in F(S_i)$ and $u \in U$, we find that $D_g(p, S_i^nu)=3 (S_i^nu)^4=3 \Big (\frac{u}{2^{ni}}\Big )^4 \le 3(u)^4 = D_g(0,u)= D_g(p,u).$ This proves that $S_i$ is a Bregman totally quasi-asymptotically nonexpansive mapping with $\eta ^{(i)}_n =\mu ^{(i)}_n=0$ for all $n \in \mathbb {N}$.

• 4. By directly checking, we find that $f_k, A_k, B_k$ satisfies the conditions $(C_1)$ - $(C_6)$.

• 5. Now, we will find the formula of $\textrm{Res}^g_{f_k,A_k, B_k}$ as in (6). Indeed, $w= \textrm{Res}^g_{f_k,A_k, B_k}(u)$ for all $u \in W$ if and only if

  $$\begin{aligned} f_k(w, v) +A_k(v) + \langle B_k(w), v-w \rangle + \langle \nabla g(w) - \nabla g(u), v-w \rangle \ge A_k(w) \end{aligned}$$

  (67)

  for all $v \in U$. By substituting $f_k, A_k, B_k$ into (67) and by directly calculating, we find that $kv^2 +(kw+2w^3-2u^3)v +2u^3w -2w^4-2kw^2\ge 0.$ Put

  $$\begin{aligned} h(v)= kv^2+(kw+2w^3-2u^3)v +2u^3w -2w^4-2kw^2. \end{aligned}$$

  Then, we have $\Delta = (3kw+2w^3-2u^3)^2.$ We consider the following two cases. Case 1. $\Delta >0$. Then the quadratic equation $h(v)=0$ have two solutions as follows.

  $$\begin{aligned} v_1=w \quad \text {and } \quad v_2= \frac{2u^3-2w^3-2kw}{k}. \end{aligned}$$

  In oder to $h(v) \ge 0$ for all $v \in \Omega$, we have the following cases. Case 1.1. $v_1=1$ and $v_2 > v_1$. Then $w=v_1=1$, and $v_2= \frac{2u^3-2k-2}{k}>1$ and hence $u > \root 3 \of {\frac{3k+2}{2}}$. Case 1.2. $v_1=0$ and $v_2 < v_1$. Then $w=v_1=0$, and $v_2= \frac{2u^3}{k}<0$ and hence $u < 0$. Case 2. $\Delta \le 0$. Then $3kw+2w^3=2u^3$ and $h(v) \ge 0$ for all $v \in U$. Note that $3kw+2w^3=2u^3$ if and only if $w= \frac{\Big (\root 3 \of {\sqrt{4u^6+2k^3}+2u^3}\Big )^2-\root 3 \of {2}k }{\root 3 \of {4} \root 3 \of {{\sqrt{4u^6+2k^3}+2u^3}}}.$ Since $w \in U$, we have $0 \le 3kw+2w^3 =2u^3 \le 3k+2$ and hence $0 \le u \le \root 3 \of {\frac{3k+2}{2}}$. Therefore,

  $$\begin{aligned} \textrm{Res}^g_{f_k, A_k, B_k}(u)=w= \left\{ \begin{array}{ll} 0 &{} \quad \hbox {if}\; u <0 \\ \frac{\Big (\root 3 \of {\sqrt{4u^6+2k^3}+2u^3}\Big )^2-\root 3 \of {2}k }{\root 3 \of {4} \root 3 \of { {\sqrt{4u^6+2k^3}+2u^3}}}&{}\quad \hbox {if}\;0 \le u \le \root 3 \of {\frac{3k+2}{2}} \\ \\ 1 &{}\quad \hbox {if}\; u>\root 3 \of {\frac{3k+2}{2}}.\end{array}\right. \end{aligned}$$

By the above, all assumptions in Theorem 5 are satisfied with the given functions $f_k, A_k, B_k, S_i$. Therefore, by Theorem 5, the sequence $\{z_n \}$ which is defined by (7) converges to $0 \in \Big ( \bigcap \limits _{i=1}^2 F(S_i) \Big ) \bigcap \Big ( \bigcap \limits _{ k =1}^{2} GMEP(f_k, A_k, B_k)\Big ).$

Next, we compare the rate of convergence of the iteration process (3) and the iteration process (66) to 0 which is a solution of a finite system of (GMEP). Numerical results of the mentioned iteration processes with the initial point $z_1=1, z_2=0.8$, $b_n=\frac{1}{10000n}$ and the different choices of $c_n$ are presented in the following table.

Table 1 Number of iterations of the processes (3) and (66)

Table 1 shows that for given mappings, the iteration process (66) has a better convergence rate and requires a small number of iterations than the iteration process (3).

4 Conclusions

In this paper, a new inertial parallel iterative method was proposed for finding common solutions of a finite system of generalized mixed equilibrium problems and common fixed points of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings. A strong convergence result for the proposed iteration in reflexive Banach spaces was established and proved. From this theorem, some convergence results for generalized mixed equilibrium problems in reflexive Banach spaces were given. In addition, a numerical example was provided to demonstrate the proposed iterations. On comparing our results with the main result of [5, 6, 17], we find that

• 1. Theorem 5 is a generalization of [6, Theorem 3.1] from an equilibrium problem and a generalized hybrid mapping in Hilbert spaces to a finite system of generalized mixed equilibrium problems and a finite family of Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces.

• 2. Corollary 1 is an improvement of [5, Theorem 3.1] from an equilibrium problem and a nonexpansive mapping in Hilbert spaces to a generalized mixed equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces.

• 3. Corollary 2 is an extension of [17, Theorem 3.6] in the sense of adding the inertial extrapolation to the iteration process (3). Furthermore, Example 1 was given to prove the efficiency of iteration (66) which has a better convergence rate and requires a small number of iterations than the iteration process  (3) which was presented in [17].