A Mean Convergence Theorem without Convexity for Finite Commutative Nonlinear Mappings in Reflexive Banach Spaces

Abstract

This paper investigates the Bregman version of the Takahashi-type generic 2-generalized nonspreading mapping which includes the generic 2-generalized Bregman nonspreading mapping as a special case. Relative to the attractive points of nonlinear mapping, the Baillon-type nonlinear mean convergence theorem for finite commutative generic 2-generalized Bregman nonspreading mappings without the convexity assumption is proved in the setting of reflexive Banach spaces. Using this result, some new and well-known nonlinear mean convergence theorems for the finite generic generalized Bregman nonspreading mapping, the 2-generalized Bregman nonspreading mapping and the normally 2-generalized hybrid mapping, among others, are established. Our results extend and generalize many corresponding ones announced in the literature.

1. Introduction

Let H be a real Hilbert space and C be a nonempty subset of H. Let $M:C\to H$ be a nonlinear mapping and denote the sets of the fixed and attractive points of M by $F\left(M\right)$ and $A\left(M\right)$, respectively, i.e., $F\left(M\right)=\{x\in C:Mx=x\}$ and $A\left(M\right)=\{x\in H:\parallel x-My\parallel \le \parallel x-y\parallel ,\forall y\in C\}$. A mapping M of C onto H is called an $(\alpha ,\beta )$-generalized hybrid mapping [1] if there exist $\alpha ,\beta \in \mathbb{R}$ such that

$${\alpha \parallel Mx-My\parallel }^2{+(1-\alpha )\parallel x-My\parallel }^2\le {\beta \parallel Mx-y\parallel }^2+(1-\beta ){\parallel x-y\parallel }^2,\forall x,y\in C.$$

We call the mapping M nonexpansive if $\alpha =1$ and $\beta =0$. We call the mapping M hybrid [2,3] if $\alpha =\frac{3}{2}$ and $\beta =\frac{1}{2}$. In addition, if $\alpha =2$ and $\beta =1$, the mapping M reduces to nonspreading [2,4], i.e., ${2\parallel Mx-My\parallel }^2\le {\parallel Mx-y\parallel }^2+{\parallel My-x\parallel }^2\forall x,y\in C$. In 1975, Bailon [5] proved the first nonlinear mean convergence theorem. He proved that a sequence $\left\{S_{n}x\right\}$ of the Cesaro mean defined for all $x\in C$ by

$$S_{n}x=\frac{1}{n}\sum _{k=0}^n{M}^{k}x$$

converges weakly to an element $u\in F\left(M\right)$, where $M:C\to H$ is known to be nonexpansive with $F\left(M\right)\ne \varnothing$. Kocourek et al. [1] extended the work of Baillon by considering a larger class of mappings M more general than that of nonexpansive. They proved that for any $x\in C$, a sequence $\left\{S_{n}x\right\}$ defined by

$$S_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}{M}^{k}x$$

converges weakly to an element $v\in F\left(M\right)$, where M is a generalized hybrid mapping with $F\left(M\right)$ nonempty. It is worth mentioning that the nonempty subset C of H is assumed to be closed and convex in the works of both Baillon [5] and Kocourek et al. [1]. However, not all the cases are true in respect to C, for example, when C is a star-shaped (see Definition 1 below) subset of H.

Takahashi and Takeuchi [6] introduced the concept of an attractive point of a nonlinear mapping in the setting of Hilbert spaces. They proved the attractive point and nonlinear mean convergence theorem without a convexity assumption for a generalized hybrid mapping $M:C\to H$ in the space. In fact, they defined sequences $\left\{v_n\right\}$ and $\left\{b_n\right\}$ by

$$v_1\in C,v_{n+1}=M{v}_n,b_n=\frac{1}{n}\sum _{k=1}^n{v}_k,$$

for all $n\in \mathbb{N}$ and proved that if $\left\{v_n\right\}$ is bounded, then $\left\{b_n\right\}$ converges weakly to an element $u\in A\left(M\right)$. Takahashi et al. [7] defined a sequence $\left\{S_{n}x\right\}$ for all $n\in \mathbb{N}$ by

$$S_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}{M}^{k}x$$

and proved that $\left\{S_{n}x\right\}$ converges weakly to $q\in A\left(M\right)$, where $q=\underset{n\to \infty }{\lim }P{M}^{n}x$ and P is a metric projection. Another class of mappings which is said to include a special case, that of the generalized hybrid, was introduced. By considering two commutative 2-generalized hybrid mappings $M,N:C\to H$, Hojo et al. [8] defined a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^2}\sum _{k=0}^n\sum _{l=0}^n{M}^k{N}^{l}x$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. They proved that the sequence $\left\{S_{n}x\right\}$ converges weakly to an element $p\in A\left(M\right)\cap A\left(N\right)$. By considering two commutative normally 2-generalized hybrid mappings M and N and a bounded sequence $\left\{x_n\right\}$, Hojo et al. [9] defined a sequence $\left\{S_n{x}_n\right\}$ by

$$S_n{x}_n=\frac{1}{{(n+1)}^2}\sum _{k=0}^n\sum _{l=0}^n{M}^k{N}^l{x}_n$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$ and proved that every cluster point of $\left\{S_n{x}_n\right\}$ is a point in $A\left(M\right)\cap A\left(M\right)$.

In 2013, Lin and Takahashi [10] extended the concept of the attractive point to smooth Banach spaces. By considering a generalized nonspreading mapping M of a nonempty subset C of a smooth and reflexive Banach space E onto itself, Lin et al. [11] defined a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}{M}^{k}x$$

for all $n\in \mathbb{N}$ and proved that if a subsequence $\left\{S_{n_i}x\right\}$ of $\left\{S_{n}x\right\}$ converges weakly to p then $p\in A\left(M\right)$. Takahashi et al. [12] defined a sequence $\left\{S_{n}x\right\}$ for all $n\in \mathbb{N}$ by

$$S_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}{M}^{k}x$$

and proved that $\left\{S_{n}x\right\}$ converges weakly to $q\in A\left(M\right)$, where $q=\underset{n\to \infty }{lim}R{M}^{n}x$ and R is a sunny generalized nonexpansive retraction. By considering two commutative 2-generalized nonspreading mappings M and N of a nonempty subset C of a smooth, strictly convex and reflexive Banach space E into itself, Takahashi et al. [13] (see also Alsulami et al. [14]) defined a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^2}\sum _{k=0}^n\sum _{l=0}^n{M}^k{N}^l{x}_n$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$ and proved that $\left\{S_{n}x\right\}$ converges weakly to $p\in A\left(M\right)\cap A\left(N\right)$.

For the Bregman version of the generalized nonspreading mapping, the generic generalized nonspreading mapping and the 2-generalized nonspreading mapping, see [15,16]. By considering two commutative generic 2-generalized nonspreading mappings M and N of a nonempty subset C of a smooth, strictly convex and reflexive Banach space E into itself, Hojo and Takahashi [17] defined a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^2}\sum _{k=0}^n\sum _{l=0}^n{M}^k{N}^l{x}_n$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$ and proved that $\left\{S_{n}x\right\}$ converges weakly to a point in $A\left(M\right)\cap A\left(N\right)$.

1.1. Our Contributions

Motivated and inspired by the corresponding results in [1,5,7,8,9,11,13,14,15,17,18,19], our contributions in this paper are:

• We first study the Bregman version of the Takahashi-type generic 2-generalized nonspreading mapping, which includes as a special case the Ali and Haruna-type [15] generic 2-generalized Bregman nonspreading mapping in reflexive Banach spaces.
• We then prove a nonlinear mean convergence theorem for finite commutative generic 2-generalized Bregman nonspreading mappings without convexity assumptions in the space.
• As an application of our main results, we establish some new and well-known mean convergence theorems for the finite generic generalized Bregman nonspreading mapping [16], the 2-generalized Bregman nonspreading mapping [15] and the normally 2-generalized hybrid mapping.
• Our results extend and generalize the corresponding ones in Ali and Haruna [15], Alsulami et al. [14], Baillon [5], Hojo and Takahashi [9,17], Hojo et al. [8], Kocourek et al. [1], Lin et al. [11] and Takahashi et al. [13].

1.2. Organization

We organize the rest of our paper as follows: Section 2 contains some basic definitions and related results which are needed in other subsequent sections. In Section 3, we present and discuss our main results.

2. Preliminaries

Definition 1.

Let C be a nonempty subset of H. Then C is called star-shaped if there exists a $z\in C$ such that for any $x\in C$ and $\lambda \in (0,1)$,

$$\lambda z+(1-\lambda )x\in C.$$

Such a $z\in C$ is called a center of the star-shaped set C.

A mapping $M:C\to H$ is called a normally generalized hybrid [7] if there exist $\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}$ such that

$$\begin{array}{c}{\alpha \parallel Mx-My\parallel }^2{+\beta \parallel x-My\parallel }^2{+\gamma \parallel Mx-y\parallel }^2+\delta {\parallel x-y\parallel }^2\le 0,\forall x,y\in C,\hfill \end{array}$$

where (a) $\alpha +\beta +\gamma +\delta \ge 0$ and (b) $\alpha +\beta >0$ or $\alpha +\gamma >0$. Observe that if $\alpha +\beta =-\gamma -\delta =1$, then M reduces to a generalized hybrid mapping.

A mapping M of C into H is called 2-generalized hybrid [20] if there exist ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in \mathbb{R}$ such that

$$\begin{array}{ccc}{\alpha }_1{\parallel M^{2}x-My\parallel }^2\hfill & +& {\alpha }_2{\parallel Mx-My\parallel }^2+(1-{\alpha }_1-{\alpha }_2){\parallel x-My\parallel }^2\hfill \\ & \le & {\beta }_1\parallel M^2{x-y\parallel }^2+{\beta }_2{\parallel Mx-y\parallel }^2+(1-{\beta }_1-{\beta }_2){\parallel x-y\parallel }^2,\forall x,y\in C.\hfill \end{array}$$

Observe that if ${\alpha }_1={\beta }_1=0$, then the mapping reduces to a generalized hybrid.

As a unification of the normally generalized hybrid mapping and the 2-generalized hybrid mapping, a new nonlinear mapping is introduced. A mapping $M:C\to H$ is called a normally 2-generalized hybrid [21] if there exist ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\beta }_1,{\beta }_2,{\beta }_3\in \mathbb{R}$ such that

$$\begin{array}{ccc}{\alpha }_1{\parallel M^{2}x-My\parallel }^2\hfill & +& {\alpha }_2{\parallel Mx-My\parallel }^2+{\alpha }_3{\parallel x-My\parallel }^2\hfill \\ & +& {\beta }_1\parallel M^2{x-y\parallel }^2+{\beta }_2{\parallel Mx-y\parallel }^2+{\beta }_3{\parallel x-y\parallel }^2\le 0,\forall x,y\in C,\hfill \end{array}$$

where ${\sum }_{i=1}^3({\alpha }_i+{\beta }_i)\ge 0$ and ${\sum }_{i=1}^3{\alpha }_i>0$.

In another development, the class of generalized hybrid mappings was extended to that of generalized nonspreading mappings in Banach spaces more general than Hilbert. Let E be a smooth Banach space. A mapping $M:C\to E$ is called generalized nonspreading [22] if there exist $\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}$ such that

$$\begin{array}{ccc}\alpha \varphi (Mx,My)\hfill & +& (1-\alpha )\varphi (x,My)+\gamma \left\{\varphi \right(My,Mx)-\varphi (My,x\left)\right\}\hfill \\ & \le & \beta \varphi (Mx,y)+(1-\beta )\varphi (x,y)+\delta \left\{\varphi \right(y,Mx)-\varphi (y,x\left)\right\},\forall x,y\in C,\hfill \end{array}$$

where a map $\varphi :E\times E\to \mathbb{R}$ is a function defined by $\varphi (x,y)={\parallel x\parallel }^2-2\langle x,Jy\rangle +{\parallel y\parallel }^2,$ for all $x,y\in E$ and $J:E\to 2^{E^{*}}$ is a duality map defined by $Jx=\{f\in E^{*}:\langle x,f\rangle =\parallel x{\parallel }^2=\parallel f{\parallel }^2\}.$ Observe that if $E=H$, then we have $\varphi (x,y)={\parallel x-y\parallel }^2$, and consequently, an $(\alpha ,\beta ,\gamma ,\delta )$-generalized nonspreading mapping reduces to an $(\alpha +\gamma ,\beta +\delta )$-generalized hybrid mapping.

A mapping $M:C\to E$ is called generic generalized nonspreading [12] if there exist $\alpha ,\beta ,\gamma ,\delta ,ϵ$ and $\zeta \in \mathbb{R}$ such that

$$\begin{array}{ccc}\alpha \varphi (Mx,My)\hfill & +& \beta \varphi (x,My)+\gamma \varphi (Mx,y)+\delta \varphi (x,y)\hfill \\ & \le & ϵ\left\{\varphi \right(My,Mx)-\varphi (My,x\left)\right\}+\zeta \left\{\varphi \right(y,Mx)-\varphi (y,x\left)\right\},\forall x,y\in C,\hfill \end{array}$$

(1)

where (i) $\alpha +\beta +\gamma +\delta \ge 0$ and (ii) $\alpha +\beta >0$. Observe that a generic generalized nonspreading mapping reduces to a generalized nonspreading mapping if $\alpha +\beta =-\gamma -\delta =1$.

A mapping $M:C\to E$ is called 2-generalized nonspreading [19] if there exist ${\alpha }_1,{\alpha }_2,{\beta }_1,$${\beta }_2,{\gamma }_1,{\gamma }_2$, ${\delta }_1,{\delta }_2\in \mathbb{R}$ such that

$$\begin{array}{ccc}{\alpha }_1\varphi (M^{2}x,My)\hfill & +& {\alpha }_2\varphi (Mx,My)+(1-{\alpha }_1-{\alpha }_2)\varphi (x,My)\hfill \\ & +& {\gamma }_1\{\varphi (My,M^{2}x)-\varphi (My,x)\}+{\gamma }_2\{\varphi (My,Mx)-\varphi (My,x)\}\hfill \\ & \le & {\beta }_1\varphi (M^{2}x,y)+{\beta }_2\varphi (Mx,y)+(1-{\beta }_1-{\beta }_2)\varphi (x,y)\hfill \\ & +& {\delta }_1\{\varphi (y,M^{2}x)-\varphi (y,x)\}+{\delta }_2\{\varphi (y,Mx)-\varphi (y,x)\},\forall x,y\in C.\hfill \end{array}$$

Observe that if ${\alpha }_1={\beta }_1={\gamma }_1={\delta }_1=0$, then a 2-generalized nonspreading mapping reduces to a generalized nonspreading.

A mapping $M:C\to E$ is called generic 2-generalized nonspreading [23] if there exist ${\alpha }_0,{\alpha }_1,{\alpha }_2,{\beta }_0,{\beta }_1,{\beta }_2,{\gamma }_0,{\gamma }_1,{\gamma }_2$, ${\delta }_0,{\delta }_1,{\delta }_2\in \mathbb{R}$ such that

$$\begin{array}{c}{\alpha }_2\varphi (M^{2}x,My)+{\alpha }_1\varphi (Mx,My)+{\alpha }_0\varphi (x,My)+{\beta }_2\varphi (M^{2}x,y)+{\beta }_1\varphi (Mx,y)+{\beta }_0\varphi (x,y)\hfill \\ \hspace{1em}\le {\gamma }_2\{\varphi (My,M^{2}x)-\varphi (My,Mx)\}+{\gamma }_1\{\varphi (My,Mx)-\varphi (My,x)\}+{\gamma }_0\{\varphi (My,x)-\varphi (My,M^{2}x)\}\hfill \\ \hspace{1em}+{\delta }_2\{\varphi (y,M^{2}x)-\varphi (y,Mx)\}+{\delta }_1\{\varphi (y,Mx)-\varphi (y,x)\}+{\delta }_0\{\varphi (y,x)-\varphi (y,M^{2}x)\},\forall x,y\in C,\hfill \end{array}$$

where ${\alpha }_0+{\alpha }_1+{\alpha }_2+{\beta }_0+{\beta }_1+{\beta }_2\ge 0$ and ${\alpha }_0+{\alpha }_1+{\alpha }_2>0$.

Let E be a real Banach space and $f:E\to \mathbb{R}$ be a function. The gradient of f at x is the function $\nabla f\left(x\right):E\to (-\infty ,+\infty ]$ defined by $\langle \nabla f\left(x\right),y\rangle$=$f^{\circ }(x,y)$, for any $x\in$int(dom(f)) and $y\in E$, where $f^{\circ }(x,y)$ is the derivative of f at x in the direction y which is defined as

$$\begin{array}{c}{f}^{\circ }(x,y):=\underset{t\to 0}{lim}\frac{f(x+ty)-f\left(x\right)}{t}.\hfill \end{array}$$

(2)

The function f is said to be G$\widehat{a}$teaux-differentiable at x if the limit in (2) exists for any y. In addition, f is said to be G$\widehat{a}$teaux-differentiable if it is G$\widehat{a}$teaux-differentiable at every $x\in$int(dom(f)). The function f is said to be Fr$\acute{e}$chet-differentiable at x if the limit in (2) is attained uniformly in y with $\parallel y\parallel =1$. In addition, f is said to be Fr$\acute{e}$chet-differentiable on a subset C of X if the limit (2) is attained uniformly for $x\in X$ and $\parallel y\parallel =1$. It is known from [24] that if a continuous convex function f is Fr$\acute{e}$chet-differentiable (resp. G$\widehat{a}$teaux-differentiable) in int(dom(f)), then $\nabla f$ is continuous (resp. norm-to-weak${}^{*}$ continuous) in int(dom(f)).

Let $f:E\to (-\infty ,+\infty ]$ be a convex and G$\widehat{a}$teaux-differentiable function. The Bregman distance with respect to f [25,26] denoted by $D_f$ is a function $D_f:$ dom$f\times$ int(dom$\left(f\right))\to [0,+\infty )$, defined by

$$D_f(x,y):=f\left(x\right)-f\left(y\right)-\langle \nabla f\left(y\right),x-y\rangle .$$

(3)

Remark 1.

If E is a smooth Banach space and $f\left(x\right)={\parallel x\parallel }^2$ for all $x\in E$, then the gradient $\nabla f\left(x\right)$ of f reduces to $2Jx$ for all $x\in E$, and subsequently, $D_f(x,y)=\varphi (x,y)$. In addition, if $E=H$ is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel }^2,\forall x,y\in E$.

For any $x\in domf$ and $y,z\in int\left(dom\right(f\left)\right)$, the three-point identity can easily be obtained from (3) and is given by

$$D_f(x,z)=D_f(x,y)+D_f(y,z)+\langle \nabla f\left(y\right)-\nabla f\left(z\right),x-y\rangle .$$

(4)

We now define the Bregman version of a Takahashi-type generic 2-generalized nonspreading mapping [23] in a reflexive Banach space E.

Definition 2.

A mapping $M:C\to E$ is called generic 2-generalized Bregman nonspreading if there exist ${\alpha }_0,{\alpha }_1,{\alpha }_2,{\beta }_0,{\beta }_1,{\beta }_2,{\gamma }_0,{\gamma }_1,{\gamma }_2$, ${\delta }_0,{\delta }_1,{\delta }_2\in \mathbb{R}$ such that ${\alpha }_0+{\alpha }_1+{\alpha }_2+{\beta }_0+{\beta }_1+{\beta }_2\ge 0$, ${\alpha }_0+{\alpha }_1+{\alpha }_2>0$ and

$$\begin{array}{c}{\alpha }_2{D}_f(M^{2}x,My)+{\alpha }_1{D}_f(Mx,My)+{\alpha }_0{D}_f(x,My)+{\beta }_2{D}_f(M^{2}x,y)+{\beta }_1{D}_f(Mx,y)+{\beta }_0{D}_f(x,y)\hfill \\ \hspace{1em}\le {\gamma }_2\{D_f(My,M^{2}x)-D_f(My,Mx)\}+{\gamma }_1\{D_f(My,Mx)-D_f(My,x)\}+{\gamma }_0\{D_f(My,x)-D_f(My,M^{2}x)\}\hfill \\ \hspace{1em}+{\delta }_2\{D_f(y,M^{2}x)-D_f(y,Mx)\}+{\delta }_1\{D_f(y,Mx)-D_f(y,x)\}+{\delta }_0\{D_f(y,x)-D_f(y,M^{2}x)\},\forall x,y\in C.\hfill \end{array}$$

Remark 2.

Observe that by setting ${\gamma }_2={\delta }_2=0$, the mapping in Definition 2 reduces to a generic 2-generalized Bregman nonspeading mapping in the sense of Ali and Haruna [16]. In addition, if E is smooth and $f\left(x\right)={\left|\right|x\left|\right|}^2$, then the mapping reduces to generic 2-generalized nonspreading in the sense of Takahashi [23]. Furthermore, if $E=H$ is a real Hilbert space, the mapping reduces to a normally 2-generalized hybrid mapping in the sense of Kondo and Takahashi [21].

Example 1.

Let $E=\mathbb{R}$ and $C=[0,2]$. Let $f\left(x\right)=x^2$ and $M:C\to C$ be defined by

$$Mx=\left\{\begin{array}{c}0,x\in [0,2)\hfill \\ 1,x=2.\hfill \end{array}\right.$$

Observe that for the choice of real numbers ${\alpha }_2={\alpha }_1={\alpha }_0=1$, ${\beta }_2={\beta }_1={\beta }_0={\gamma }_1={\gamma }_0={\delta }_1={\delta }_0=-1$ and ${\gamma }_2={\delta }_2=0$, we see that ${\sum }_{i=1}^3{\alpha }_i>0$; ${\sum }_{i=1}^3({\alpha }_i+{\beta }_i)\ge 0$ and $h(x,y)\le 0$, for all $x,y\in C$, where

$$\begin{array}{cc}h(x,y)\hfill & ={\alpha }_2{(M^{2}x-My)}^2+{\alpha }_1{(Mx-My)}^2+{\alpha }_0{(x-My)}^2+{\beta }_2{(M^{2}x-y)}^2+{\beta }_1{(Mx-y)}^2+{\beta }_0{(x-y)}^2\hfill \\ \hfill & -{\gamma }_2\{{(My-M^{2}x)}^2-{(My-Mx)}^2\}-{\gamma }_1\{{(My-Mx)}^2-{(My-x)}^2\}-{\gamma }_0\{{(My-x)}^2-{(My-M^{2}x)}^2\}\hfill \\ \hfill & -{\delta }_2\{{(y-M^{2}x)}^2-{(y-Mx)}^2\}-{\delta }_1\{{(y-Mx)}^2-{(y-x)}^2\}-{\delta }_0\{{(y-x)}^2-{(y-M^{2}x)}^2\}.\hfill \end{array}$$

Therefore, M is a generic 2-generalized Bregman nonspreading mapping.

Let E be a reflexive Banach space and T be a mapping of a nonempty subset C of $int\left(domf\right)$ into E. We denote the set of Bregman attractive points of T by $A_f\left(T\right)$ and that of Bregman skew-attractive points [27] by $B_{f}T$, i.e., $A_f\left(T\right)=\{x\in E:D_f(x,Ty)\le D_f(x,y),\forall y\in C\}$ and $B_f\left(T\right)=\{x\in E:D_f(Ty,x)\le D_f(y,x),\forall y\in C\}$.

Lemma 1

([27]). Let E be a reflexive Banach space and $g:E\to (-\infty ,+\infty ]$ a convex, continuous, strongly coercive and G$\widehat{a}$teaux-differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty subset of E and $T:C\to E$ be a mapping. Then $B_f\left(T\right)$ is closed and convex.

Lemma 2

([27]). Let E be a reflexive Banach space and $g:E\to (-\infty ,+\infty ]$ a convex, continuous, strongly coercive and G$\widehat{a}$teaux-differentiable function which is bounded on bounded sets, uniformly convex and uniformly smooth on bounded sets. Let C be a nonempty subset of E and $T:C\to E$ be a mapping. Let $T^{*}:\nabla fC\to E^{*}$ be the duality mapping of T. Then the following assertions hold:

 (1) 

$\nabla g{B}_f\left(T\right)=A_f\left(T^{*}\right)$;

 (2) 

$\nabla g{A}_f\left(T\right)=B_f\left(T^{*}\right)$.

In particular, $\nabla g{B}_f\left(T\right)$ is closed and convex.

Let C be a nonempty subset of a Banach space X. A mapping $\mathcal{R}$ of X into C is said to be sunny [28] if

$$\mathcal{R}(\mathcal{R}x+r(x-\mathcal{R}x\left)\right)=\mathcal{R}x,$$

for each $x\in X$ and $r\ge 0$. A mapping $\mathcal{R}:X\to C$ is said to be retraction [28] if $\mathcal{R}x=x$ for all $x\in C$. A nonempty subset C of X is said to be a sunny Bregman generalized nonexpansive retract (resp. a Bregman generalized nonexpansive retract) of X if there exists a sunny Bregman generalized nonexpansive retraction (resp. a Bregman generalized nonexpansive retraction) of X onto C, see [29] for details.

Lemma 3

([30]). Let E be a reflexive Banach space and $g:E\to (-\infty ,+\infty ]$ a convex, continuous, strongly coercive function which is bounded on bounded sets and uniformly convex and uniformly smooth on bounded sets. Let C be a nonempty closed subset of E. Then the following statements are equivalent

 (1) 

C is a sunny Bregman generalized nonexpansive retract of E;

 (2) 

C is a Bregman generalized nonexpansive retract of E;

 (3) 

$\nabla gC$ is closed and convex.

Using Lemma 3, the following result can be established.

Lemma 4.

Let E be a reflexive Banach space and let $\left\{C_i\right\}$ be a family of sunny Bregman generalized nonexpansive retracts of E such that ${\cap }_{i\in I}{C}_i$ is nonempty. Then ${\cap }_{i\in I}{C}_i$ is a sunny Bregman generalized nonexpansive retract of E.

Proof. 

It is easy to see $\nabla g\left({\cap }_{i\in I}{C}_i\right)={\cap }_{i\in I}\nabla g{C}_i$. Indeed,

$$\begin{array}{ccc}x\in \nabla g\left({\cap }_{i\in I}{C}_i\right)\hfill & \iff & {(\nabla g)}^{-1}x\in {\cap }_{i\in I}{C}_i\hfill \\ & \iff & {(\nabla g)}^{-1}x\in C_i,\forall i\in I\hfill \\ & \iff & x\in \nabla g{C}_i,\forall i\in I\hfill \\ & \iff & x\in {\cap }_{i\in I}\nabla g{C}_i.\hfill \end{array}$$

Thus, from Lemma 3 above, $\nabla g{C}_i$ is closed and closed for each $i\in I$. Therefore, ${\cap }_{i\in I}\nabla g{C}_i$ is closed and closed. Hence, we have that ${\cap }_{i\in I}{C}_i$ is a sunny Bregman generalized nonexpansive retract of E. □

Lemma 5

([30]). Let E be a reflexive Banach space and $g:E\to \mathbb{R}$ be a strongly coercive Bregman function. Let C be a nonempty closed subset of E and let $\mathcal{R}$ be a retraction from E onto C. Then the following assertions are equivalent:

 (1) 

$\mathcal{R}$ is sunny Bregman generalized nonexpansive;

 (2) 

$\langle x-\mathcal{R}x,\nabla g\left(y\right)-\nabla g\left(\mathcal{R}x\right)\rangle \le 0$, $\forall (x,y)\in E\times C$.

Lemma 6

([31]). Let E be a Banach space and $r>0$. Let ${\rho }_r$ be the gauge function of uniform convexity of g where $g:E\to \mathbb{R}$ is a convex function which is uniformly convex on bounded subsets of E. Then the following hold:

 (1) 

For any $x,y\in B_r$ and $\alpha \in (0,1)$, $g(\alpha x+(1-\alpha \left)y\right)\le \alpha g\left(x\right)+(1-\alpha )g\left(y\right)-\alpha (1-\alpha )$${\rho }_r\left(\right||x-y\left|\right|)$;

 (2) 

For any $x,y\in B_r$, ${\rho }_r\left(\right||x-y\left|\right|)\le D_g(x,y)$.

Lemma 7

([32], Theorem 7.3 (vi)). Suppose $u\in domf$ and $v\in int\left(dom\right(f\left)\right)$. If f is strictly convex, then $D_f(u,v)=0\iff u=v$.

3. Main Results

In this section, we prove a nonlinear mean convergence theorem without convexity for finite commutative generic 2-generalized Bregman nonspreading mappings. The following lemma will play a vital role.

Lemma 8.

Let $f:E\to \mathbb{R}$ be a convex and uniformly Fr$\acute{e}$chet-differentiable function which is bounded on bounded subsets of E. Let C be a nonempty subset of $int\left(dom\right(f\left)\right)$ and $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative generic 2-generalized Bregman nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and conve, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Proof. 

Since $M_1$ is a generic 2-generalized Bregman nonspreading mapping, then by Definition 2, we have

$$\begin{array}{ccc}{\alpha }_2{D}_f(M_1^{2}x,M_{1}y)\hfill & +& {\alpha }_1{D}_f(M_{1}x,M_{1}y)+{\alpha }_0{D}_f(x,M_{1}y)+{\beta }_2{D}_f(M_1^{2}x,y)+{\beta }_1{D}_f(M_{1}x,y)+{\beta }_0{D}_f(x,y)\hfill \\ \hfill & \le & {\gamma }_2\{D_f(M_{1}y,M_1^{2}x)-D_f(M_{1}y,M_{1}x)\}+{\gamma }_1\{D_f(M_{1}y,M_{1}x)-D_f(M_{1}y,x)\}\hfill \\ \hfill & +& {\gamma }_0\{D_f(M_{1}y,x)-D_f(M_{1}y,M_1^{2}x)\}+{\delta }_2\{D_f(y,M_1^{2}x)-D_f(y,M_{1}x)\}\hfill \\ \hfill & +& {\delta }_1\{D_f(y,M_{1}x)-D_f(y,x)\}+{\delta }_0\{D_f(y,x)-D_f(y,M_1^{2}x)\},\forall x,y\in C.\hfill \end{array}$$

Using the three-point identity (4), we obtain

$$\begin{array}{ccc}{\alpha }_2\{D_f(M_1^{2}x,y)\hfill & +& D_f(y,M_{1}y)+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),M_1^{2}x-y\rangle \}+{\beta }_2{D}_f(M_1^{2}x,y)\hfill \\ \hfill & +& {\alpha }_1\{D_f(M_{1}x,y)+D_f(y,M_{1}y)+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),M_{1}x-y\rangle \}+{\beta }_1{D}_f(M_{1}x,y)\hfill \\ \hfill & +& {\alpha }_0\{D_f(x,y)+D_f(y,M_{1}y)+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),x-y\rangle \}+{\beta }_0{D}_f(x,y)\hfill \\ \hfill & \le & {\gamma }_2\{D_f(M_{1}y,M_1^{2}x)-D_f(M_{1}y,M_{1}x)\}+{\gamma }_1\{D_f(M_{1}y,M_{1}x)-D_f(M_{1}y,x)\}\hfill \\ \hfill & +& {\gamma }_0\{D_f(M_{1}y,x)-D_f(M_{1}y,M_1^{2}x)\}+{\delta }_2\{D_f(y,M_1^{2}x)-D_f(y,M_{1}x)\}\hfill \\ \hfill & +& {\delta }_1\{D_f(y,M_{1}x)-D_f(y,x)\}+{\delta }_0\{D_f(y,x)-D_f(y,M_1^{2}x)\},\forall x,y\in C.\hfill \end{array}$$

This implies

$$\begin{array}{ccc}({\alpha }_2+{\beta }_2)D_f(M_1^{2}x,y)\hfill & +& ({\alpha }_1+{\beta }_1)D_f(M_{1}x,y)+({\alpha }_0+{\beta }_0)D_f(x,y)+({\alpha }_2+{\alpha }_1+{\alpha }_0)D_f(y,M_{1}y)\hfill \\ \hfill & +& \langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2(M_1^{2}x-y)+{\alpha }_1(M_{1}x-y)+{\alpha }_0(x-y)\rangle \hfill \\ & \le & {\gamma }_2\{D_f(M_{1}y,M_1^{2}x)-D_f(M_{1}y,M_{1}x)\}+{\gamma }_1\{D_f(M_{1}y,M_{1}x)-D_f(M_{1}y,x)\}\hfill \\ \hfill & +& {\gamma }_0\{D_f(M_{1}y,x)-D_f(M_{1}y,M_1^{2}x)\}+{\delta }_2\{D_f(y,M_1^{2}x)-D_f(y,M_{1}x)\}\hfill \\ \hfill & +& {\delta }_1\{D_f(y,M_{1}x)-D_f(y,x)\}+{\delta }_0\{D_f(y,x)-D_f(y,M_1^{2}x)\},\forall x,y\in C.\hfill \end{array}$$

(5)

Since $-({\alpha }_2+{\beta }_2+{\alpha }_1+{\beta }_1)\le {\alpha }_0+{\beta }_0$, we obtain from Inequality (5) that

$$\begin{array}{ccc}({\alpha }_2+{\beta }_2)(D_f(M_1^{2}x,y)\hfill & -& D_f(x,y))+({\alpha }_1+{\beta }_1)\left(D_f(M_{1}x,y)-D_f(x,y)\right)+({\alpha }_2+{\alpha }_1+{\alpha }_0)D_f(y,M_{1}y)\hfill \\ \hfill & +& \langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2(M_1^{2}x-x)+{\alpha }_1(M_{1}x-x)+({\alpha }_2+{\alpha }_1+{\alpha }_0)(x-y)\rangle \hfill \\ & \le & {\gamma }_2\{D_f(M_{1}y,M_1^{2}x)-D_f(M_{1}y,M_{1}x)\}+{\gamma }_1\{D_f(M_{1}y,M_{1}x)-D_f(M_{1}y,x)\}\hfill \\ \hfill & +& {\gamma }_0\{D_f(M_{1}y,x)-D_f(M_{1}y,M_1^{2}x)\}+{\delta }_2\{D_f(y,M_1^{2}x)-D_f(y,M_{1}x)\}\hfill \\ \hfill & +& {\delta }_1\{D_f(y,M_{1}x)-D_f(y,x)\}+{\delta }_0\{D_f(y,x)-D_f(y,M_1^{2}x)\},\forall x,y\in C.\hfill \end{array}$$

(6)

Following the hypothesis, we can take $x\in C$ such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded. Now, we replace x with $M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$ so that from Inequality (6) we obtain

$$\begin{array}{c}({\alpha }_2+{\alpha }_1+{\alpha }_0)D_f(y,M_{1}y)+({\alpha }_2+{\beta }_2)\left(D_f(M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+ ({\alpha }_1+{\beta }_1)\left(D_f(M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+ \langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2(M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}+ {\alpha }_1(M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+({\alpha }_2+{\alpha }_1+{\alpha }_0)(M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-y)\rangle \hfill \\ \hspace{1em}\le {\gamma }_2\{D_f(M_{1}y,M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_1\{D_f(M_{1}y,M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_0\{D_f(M_{1}y,M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_2\{D_f(y,M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_1\{D_f(y,M_1^{{\mu }_1+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_0\{D_f(y,M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{{\mu }_1+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\},\forall y\in C.\hfill \end{array}$$

(7)

Summing Inequality (7) with respect to ${\mu }_1=0,1,\cdots ,n$, we obtain

$$\begin{array}{c}({\alpha }_2+{\alpha }_1+{\alpha }_0)(n+1)D_f(y,M_{1}y)\hfill \\ \hspace{1em}+ ({\alpha }_2+{\beta }_2)(D_f(M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)+D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}u,y)-D_f(M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\hfill \\ \hspace{1em}- D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y))+({\alpha }_1+{\beta }_1)\left(D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+ \langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2((M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- (M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x))\hfill \\ \hspace{1em}+ {\alpha }_1(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}+ ({\alpha }_2+{\alpha }_1+{\alpha }_0)(\sum _{{\mu }_1=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-(n+1)y)\rangle \hfill \\ \hspace{1em}\le {\gamma }_2\{D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_1\{D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_0\{D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}+{\delta }_2\{D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_1\{D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_0\{D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\},\forall y\in C.\hfill \end{array}$$

(8)

Again, summing Inequality (8) with respect to ${\mu }_2=0,1,\cdots ,n$, we obtain

$$\begin{array}{c}({\alpha }_2+{\alpha }_1+{\alpha }_0){(n+1)}^2{D}_f(y,M_{1}y)\hfill \\ \hspace{1em}+({\alpha }_2+{\beta }_2)\sum _{{\mu }_2=0}^n(D_f(M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)+D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\hfill \\ \hspace{1em}-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y))+({\alpha }_1+{\beta }_1)\sum _{{\mu }_2=0}^n\left(D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2\sum _{{\mu }_2=0}^n((M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}-(M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x))\hfill \\ \hspace{1em}+{\alpha }_1\sum _{{\mu }_2=0}^n(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}+({\alpha }_2+{\alpha }_1+{\alpha }_0)(\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-{(n+1)}^{2}y)\rangle \hfill \\ \hspace{1em}\le {\gamma }_2\sum _{{\mu }_2=0}^n\{D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+{\gamma }_1\sum _{{\mu }_2=0}^n\{D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+{\gamma }_0\sum _{{\mu }_2=0}^n\{D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}-D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}+{\delta }_2\sum _{{\mu }_2=0}^n\{D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+{\delta }_1\sum _{{\mu }_2=0}^n\{D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+{\delta }_0\sum _{{\mu }_2=0}^n\{D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}-D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\},\forall y\in C.\hfill \end{array}$$

(9)

We continue summing Inequality (9) until with respect to ${\mu }_N=0,1,\cdots ,n$, and we obtain

$$\begin{array}{c}({\alpha }_2+{\alpha }_1+{\alpha }_0){(n+1)}^N{D}_f(y,M_{1}y)\hfill \\ \hspace{1em}+ ({\alpha }_2+{\beta }_2)\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n(D_f(M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)+D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\hfill \\ \hspace{1em}- D_f(M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y))\hfill \\ \hspace{1em}+ ({\alpha }_1+{\beta }_1)\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\left(D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+ \langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),{\alpha }_2\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n((M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- (M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x))+{\alpha }_1\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}+ ({\alpha }_2+{\alpha }_1+{\alpha }_0)(\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-{(n+1)}^{N}y)\rangle \hfill \\ \hspace{1em}\le {\gamma }_2\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_1\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\gamma }_0\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}+{\delta }_2\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_1\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+ {\delta }_0\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}- D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\},\forall y\in C.\hfill \end{array}$$

(10)

Dividing both sides of Inequality (10) by ${(n+1)}^N$ and letting $S_{n}x=\frac{1}{{(n+1)}^N}{\sum }_{{\mu }_1=0}^n{\sum }_{{\mu }_2=0}^n$$\cdots {\sum }_{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$, we obtain

$$\begin{array}{c}({\alpha }_2+{\alpha }_1+{\alpha }_0)D_f(y,M_{1}y)\hfill \\ \hspace{1em}+\frac{({\alpha }_2+{\beta }_2)}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n(D_f(M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)+D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\hfill \\ \hspace{1em}-D_f(M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y))\hfill \\ \hspace{1em}+\frac{({\alpha }_1+{\beta }_1)}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\left(D_f(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)-D_f(M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,y)\right)\hfill \\ \hspace{1em}+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),\frac{{\alpha }_2}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n(M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x+M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x\hfill \\ \hspace{1em}-M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}+\frac{{\alpha }_1}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n(M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-M_1^{}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+({\alpha }_2+{\alpha }_1+{\alpha }_0)(S_{n}x-y)\rangle \hfill \\ \hspace{1em}\le \frac{{\gamma }_2}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+\frac{{\gamma }_1}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+\frac{{\gamma }_0}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(M_{1}y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(M_{1}y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}-D_f(M_{1}y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_{1}y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \end{array}$$

(11)

$$\begin{array}{c}\hspace{1em}+\frac{{\delta }_2}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+\frac{{\delta }_1}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\}\hfill \\ \hspace{1em}+\frac{{\delta }_0}{{(n+1)}^N}\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\{D_f(y,M_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)+D_f(y,M_1{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(y,M_1^{n+1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}-D_f(y,M_1^{n+2}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\},\forall y\in C.\hfill \end{array}$$

(12)

Since E is reflexive and $\left\{S_{n}x\right\}$ is bounded, then there exists a subsequence $\left\{S_{n_j}x\right\}$ of $\left\{S_{n}x\right\}$ such that $\left\{S_{n_j}x\right\}$ converges weakly to some point $p\in E$. Now, replacing n with $n_j$ in Inequality (11) and allowing $j\to \infty$, we obtain

$$({\alpha }_2+{\alpha }_1+{\alpha }_0)\left(D_f(y,M_{1}y)+\langle \nabla f\left(y\right)-\nabla f\left(M_{1}y\right),p-y\rangle \right)\le 0.$$

(13)

Using Equation (4) and the fact that ${\alpha }_2+{\alpha }_1+{\alpha }_0\ge 0$, we obtain

$$D_f(y,M_{1}y)+D_f(p,M_{1}y)-D_f(y,M_{1}y)-D_f(p,y)\le 0,\forall y\in C.$$

Thus,

$$D_f(p,M_{1}y)\le D_f(p,y),\forall y\in C.$$

(14)

By the commutative nature of $M_1,M_2,\cdots ,M_N$, we can replace $M_1$ in (14) with any of the $M_2\cdots ,M_N$ so that for all $y\in C$ we obtain

$$\begin{array}{ccc}{D}_f(p,M_{2}y)\hfill & \le & D_f(p,y)\hfill \\ \hfill & ⋮& \end{array}$$

(15)

$$\begin{array}{ccc}{D}_f(p,M_{N}y)\hfill & \le & D_f(p,y).\hfill \end{array}$$

(16)

Therefore, from (14)–(16), we have $v\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$. Hence, every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then we put $y=v$ in (14)–(16) and we see that by Lemma 7, $v\in {\cap }_{l=1}^{N}F\left(M_l\right)$. Hence, every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$. This completes the proof. □

Following a similar argument as in the proof of Lemma 8, the following new results with respect to finite generic 2-generalized nonspreading mappings and normally 2-generalized hybrid mappings can be established.

Lemma 9.

Let E be a smooth, strictly convex and reflexive Banach space and C a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative generic 2-generalized nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}A\left(M_l\right)$. Additionally, if C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Proof. 

Let E be a smooth Banach space and $f\left(x\right)={\left|\right|x\left|\right|}^2$. Then by Remark 2, the mapping reduces to generic 2-generalized nonspreading in the sense of Takahashi [23]. Following a similar argument as in the proof of Lemma 8 with the use of $\varphi (x,y)=\varphi (x,z)+\varphi (z,y)+2\langle x-z,Jz-Jy\rangle$ in the place where Equation (4) is applied, we obtain the desired results. This completes the proof. □

Lemma 10.

Let E be a real Hilbert space and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative normally 2-generalized hybrid mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}A\left(M_l\right)$. Additionally, if C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Proof. 

Since E is a real Hilbert space, then the mapping reduces to normally 2-generalized hybrid [21], i.e., there exist $\overline{{\alpha }_2},\overline{{\alpha }_1},\overline{{\alpha }_0},\overline{{\beta }_2},\overline{{\beta }_1},\overline{{\beta }_0}\in \mathbb{R}$ such that

$$\begin{array}{ccc}\overline{{\alpha }_2}\left|\right|M^{2}x-My{\left|\right|}^2\hfill & +& \overline{{\alpha }_1}\left|\right|Mx-{My\left|\right|}^2+\overline{{\alpha }_0}\left|\right|x-{My\left|\right|}^2\hfill \\ & \le & \overline{{\beta }_2}\left|\right|M^{2}x-{y\left|\right|}^2+\overline{{\beta }_1}\left|\right|Mx-{y\left|\right|}^2+\overline{{\beta }_0}\left|\right|x-{y\left|\right|}^2,\forall x,y\in C,\hfill \end{array}$$

where $\overline{{\alpha }_2}={\alpha }_2-{\gamma }_2+{\gamma }_0$, $\overline{{\alpha }_1}={\alpha }_1+{\gamma }_2-{\gamma }_1$, $\overline{{\alpha }_0}={\alpha }_0+{\gamma }_1-{\gamma }_0$, $\overline{{\beta }_2}={\beta }_2-{\delta }_2+{\delta }_0$, $\overline{{\beta }_1}={\beta }_1+{\delta }_2-{\delta }_1$ and $\overline{{\beta }_0}={\beta }_0+{\delta }_1-{\delta }_0$ satisfying $\overline{{\alpha }_2}+\overline{{\alpha }_1}+\overline{{\alpha }_0}+\overline{{\beta }_2}+\overline{{\beta }_1}+\overline{{\beta }_0}\ge 0$ and $\overline{{\alpha }_2}+\overline{{\alpha }_1}+\overline{{\alpha }_0}>0$. Following similar argument as in the proof of Lemma 8 with the use of $\left|\right|x-{y\left|\right|}^2=\left|\right|x-{z\left|\right|}^2+\left|\right|z-{y\left|\right|}^2+2\langle x-z,z-y\rangle$ in the place where Equation (4) is applied, we obtain the desired results. This completes the proof. □

As direct consequences of Lemmas 8–10, the following results corresponding to the ones in Ali and Haruna [15], Hojo and Takahashi [17] and Hojo et al. [9] can be obtained as corollaries.

Corollary 1

([15], Theorem 3.3). Let $f:E\to \mathbb{R}$ be a convex and uniformly Fr$\acute{e}$chet-differentiable function which is bounded on bounded subsets of E. Let C be a nonempty subset of $int\left(dom\right(f\left)\right)$ and $M_1,M_2:C\to C$ be two commutative generic generalized Bregman nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}x:{\mu }_1,{\mu }_2\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^2{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{2}F\left(M_l\right)$.

Corollary 2

([17], Lemma 3.1). Let E be a smooth, strictly convex and reflexive Banach space and C a nonempty subset of E. Let $M_1,M_2:C\to C$ be two commutative generic 2-generalized nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}x:{\mu }_1,{\mu }_2\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{2}A\left(M_l\right)$. Additionally, if C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{2}F\left(M_l\right)$.

Corollary 3

([9], Lemma 3.1). Let E be a real Hilbert space and C be a nonempty subset of E. Let $M_1,M_2:C\to C$ be two commutative normally 2-generalized hybrid mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}x:{\mu }_1,{\mu }_2\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{2}A\left(M_l\right)$. Additionally, if C is closed and convex then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{2}F\left(M_l\right)$.

In view of the fact that the generic 2-generalized Bregman nonspreading (simply nonspreading) mapping unifies the generic generalized Bregman nonspreading (simply nonspreading) mapping and the 2-generalized Bregman nonspreading (simply nonspreading) mapping, the following results can be obtained from Lemmas 8 and 9 as corollaries. These results correspond to the ones in Ali and Haruna [15], Alsulami et al. [14] Takahashi et al. [12], Takahashi et al. [13] and Lin et al. [11] when one or two mappings are considered.

Corollary 4

([15]). Let $f:E\to \mathbb{R}$ be a convex and uniformly Fr$\acute{e}$chet-differentiable function which is bounded on bounded subsets of E. Let C be a nonempty subset of $int\left(dom\right(f\left)\right)$ and $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative generic generalized Bregman nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 5

([15]). Let $f:E\to \mathbb{R}$ be a convex and uniformly Fr$\acute{e}$chet-differentiable function which is bounded on bounded subsets of X. Let C be a nonempty subset of $int\left(dom\right(f\left)\right)$ and $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative 2-generalized Bregman nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 6

([11,12]). Let E be a smooth, strictly convex and reflexive Banach space and C a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative generic generalized nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 7

([13,14]). Let E be a smooth, strictly convex and reflexive Banach space and C a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative 2-generalized nonspreading mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}x:{\mu }_1,{\mu }_2\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^N{A}_f\left(M_l\right)$. Additionally, if f is strictly convex and C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

In addition, in view of the fact that the normally 2-generalized hybrid mapping unifies the normally generalized hybrid mapping and 2-generalized hybrid mapping, we obtain the following results from Lemma 10 as corollaries. These results correspond to the ones in Hojo et al. [8], Takahashi et al. [7] and Takahashi et al. [33] when only one or two mappings are considered.

Corollary 8

([7]). Let E be a real Hilbert space and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,$$M_N:C\to C$ be finite commutative normally generalized hybrid mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}A\left(M_l\right)$. Additionally, if C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 9

([8,33]). Let E be a real Hilbert space and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N:C\to C$ be finite commutative 2-generalized hybrid mappings such that the set $\{M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x:{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$ is bounded for some $x\in C$. Define a sequence $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}A\left(M_l\right)$. Additionally, if C is closed and convex, then every weak cluster point of $\left\{S_{n}x\right\}$ is a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$.

We now prove a nonlinear mean convergence theorem for finite commutative generic 2-generalized Bregman nonspreading mappings in a reflexive Banach space E. Let $D=\{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N):{\mu }_1,{\mu }_2,\cdots ,{\mu }_N\in \mathbb{N}\cup \left\{0\right\}\}$. Then D is a directed set by the binary relation:

$$({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\le ({\nu }_1,{\nu }_2,\cdots ,{\nu }_N)\mathrm{if}{\mu }_1\le {\nu }_1,{\mu }_2\le {\nu }_2,\cdots ,{\mu }_N\le {\nu }_N.$$

Theorem 1.

Let E be a smooth, strictly convex and reflexive Banach space and $f:E\to \mathbb{R}$ a strongly coercive Bregman function which is bounded, uniformly convex and uniformly smooth on bounded sets. Let $M_1,M_2,\cdots ,M_N$ be finite commutative generic 2-generalized Bregman nonspreading mappings of a nonempty subset C of $int\left(dom\right(f\left)\right)$ into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny Bregman generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$, and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Proof. 

Using Lemmas 1 and 2, we see that ${\cap }_{l=1}^N{B}_f\left(S_l\right)$ and $\nabla f\left({\cap }_{l=1}^N{A}_f\left(S_l\right)\right)$ are closed and convex, respectively. Thus, by Lemma 4, there exists a sunny Bregman generalized nonexpansive retraction $\mathcal{R}$ of E on ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ which is characterized (see Lemma 5) by

$$0\le \langle v-\mathcal{R}v,\nabla f\left(\mathcal{R}v\right)-\nabla f\left(u\right))\rangle ,\forall u\in {\cap }_{l=1}^N{B}_f\left(S_l\right),v\in C.$$

(17)

Adding $D_f(\mathcal{R}v,u)$ on both sides of Inequality (17), we obtain

$$\begin{array}{ccc}{D}_f(\mathcal{R}v,u)\hfill & \le & D_f(\mathcal{R}v,u)+\langle v-\mathcal{R}v,\nabla f\left(\mathcal{R}v\right)-\nabla f\left(u\right))\rangle \hfill \\ \hfill & =& D_f(\mathcal{R}v,u)+D_f(v,u)-D_f(v,\mathcal{R}v)-D_f(\mathcal{R}v,u)\hfill \\ & =& D_f(v,u)-D_f(v,\mathcal{R}v).\hfill \end{array}$$

(18)

Since ${\cap }_{l=1}^N{B}_f\left(M_l\right)\ne \varnothing$, then for any $u\in {\cap }_{l=1}^N{B}_f\left(M_l\right)$, $D_f(M_{1}v,u)\le D_f(v,u),D_f$$(M_{2}v,u)\le D_f(v,u)\cdots D_f(M_{N}v,u)\le D_f(v,u)$. It follows that for any $({\mu }_1,{\mu }_2\cdots ,{\mu }_N),$$({\nu }_1,{\nu }_2\cdots ,{\nu }_N)\in D$ with $({\mu }_1,{\mu }_2\cdots ,{\mu }_N)\le ({\nu }_1,{\nu }_2\cdots ,{\nu }_N)$, we have

$$\begin{array}{ccc}{D}_f(M_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x,\mathcal{R}{M}_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x)\hfill & \le & D_f(M_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x,\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ & \le & D_f(M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x).\hfill \end{array}$$

Therefore, the net $D_f(M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)$ is nonincreasing. Putting $u=\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$ and $v=M_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x$ in (18), we obtain from (2) of Lemma 6 that

$$\begin{array}{c}{\rho }_r\left(\right||\mathcal{R}{M}_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots ,M_N^{{\mu }_N}x\left|\right|)\le D_f(\mathcal{R}{M}_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x,\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)\hfill \\ \hspace{1em}\le D_f(M_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x,\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-D_f(M_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x,\mathcal{R}{M}_1^{{\nu }_1}{M}_2^{{\nu }_2}\cdots M_N^{{\nu }_N}x),\hfill \end{array}$$

where ${\rho }_r$ is a gauge function of uniform convexity. From the properties of ${\rho }_r$, $\{\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}\}$ is a Cauchy net, see [34]. Hence, $\{\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}\}$ converges strongly to $q\in {\cap }_{l=1}^N{B}_f\left(M_l\right)$ since ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ is closed by Lemma 1.

Next, we consider a fixed $x\in C$ and an arbitrary subsequence $\left\{S_{n_j}x\right\}$ of $\left\{S_{n}x\right\}$ that converges weakly to v. We know from Lemma 8 that $v\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$. Rewriting the characterization of the retraction, we have that for any $u\in {\cap }_{l=1}^N{B}_f\left(M_l\right)$,

$$0\le \langle M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\nabla f(\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-\nabla f\left(u\right)\rangle .$$

Thus,

$$\begin{array}{c}\langle M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\nabla f\left(u\right)-\nabla f\left(q\right)\rangle \hfill \\ \hspace{1em}\le \langle M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\nabla f(\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-\nabla f\left(q\right)\rangle \hfill \\ \hspace{1em}\le \left|\right|M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x\left|\right|·\left|\right|\nabla f(\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-\nabla f\left(q\right)\left|\right|\hfill \\ \hspace{1em}\le M\left|\right|\nabla f(\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-\nabla f\left(q\right)\left|\right|,\hfill \end{array}$$

(19)

where M is an upper bound for $\left|\right|M_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x-\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x\left|\right|$. Summing Inequality (19) with respect to ${\mu }_1=0,1,2,\cdots ,n$, ${\mu }_2=0,1,2,\cdots ,n$ up to ${\mu }_N=0,1,2,\cdots ,n$, and dividing through by ${(n+1)}^N$, we obtain

$$\begin{array}{c}\langle S_{n}x-\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,\nabla f\left(u\right)-\nabla f\left(q\right)\rangle \hfill \\ \hspace{1em}\le M\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=0}^n\sum _{{\mu }_2=0}^n\cdots \sum _{{\mu }_N=0}^n\left|\right|\nabla f(\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x)-\nabla f\left(q\right)\left|\right|,\hfill \end{array}$$

(20)

where $S_{n}x=\frac{1}{{(n+1)}^N}{\sum }_{{\mu }_1=0}^n{\sum }_{{\mu }_2=0}^n\cdots {\sum }_{{\mu }_N=0}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Replacing n with $n_j$ in (20) and allowing $j\to \infty$, keeping in mind that $\nabla f$ is continuous, we obtain

$$\langle v-q,\nabla f\left(u\right)-\nabla f\left(q\right)\rangle \le 0.$$

This inequality holds for any $u\in {\cap }_{l=1}^N{B}_f\left(M_l\right)$. Thus, $\mathcal{R}v=q$. Since $v\in {\cap }_{l=1}^N{B}_f\left(M_l\right)$, then $v=q$. Therefore, the sequence $\left\{S_{n}x\right\}$ converges weakly to the point q. If, in addition, C is closed and convex, then $q\in C$. Hence, $\left\{S_{n}x\right\}$ converges weakly to a point of ${\cap }_{l=1}^{N}F\left(M_l\right)$. This completes the proof. □

Following a similar argument as in Theorem 1, we can establish the following new results for finite generic 2-generalized nonspreading mappings and normally 2-generalized hybrid mappings.

Corollary 10.

Let E be a uniformly convex Banach space with a Fr$\acute{e}$chet-differentiable norm and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N$ be finite commutative generic 2-generalized nonspreading mappings of the nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 11.

Let E be a real Hilbert space and $M_1,M_2,\cdots ,M_N$ be finite commutative normally 2-generalized hybrid mappings of a nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the metric projection of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

As direct consequences of Theorem 1, Corollary 10 and Theorem 11, the following results can be obtained as corollaries. These results correspond to the ones in Ali and Haruna [15], Hojo and Takahashi [17], Hojo et al. [9] and Kondo and Takahashi [21].

Corollary 12

([15]). Let $f:E\to \mathbb{R}$ be a strongly coercive Bregman function which is bounded, uniformly convex and uniformly smooth on bounded sets. Let $M_1,M_2$ be two commutative generic 2-generalized Bregman nonspreading mappings of a nonempty subset C of $int\left(dom\right(f\left)\right)$ into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2$. Let $\mathcal{R}$ be the sunny Bregman generalized nonexpansive retraction of E onto ${\cap }_{l=1}^2{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^2{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 13

([17], Theorem 4.4). Let E be a uniformly convex Banach space with a Fr$\acute{e}$chet-differentiable norm and C be a nonempty subset of E. Let $M_1,M_2$ be two commutative generic 2-generalized nonspreading mappings of the nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2$. Let $\mathcal{R}$ be the sunny generalized nonexpansive retraction of E onto ${\cap }_{l=1}^2{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^2{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x$.

Corollary 14

([9], Theorem 3.2). Let E be a real Hilbert space and $M_1,M_2$ be two commutative normally 2-generalized hybrid mappings of a nonempty subset C of E into itself such that $A_f\left(M_l\right)\ne \varnothing$, for $l=1,2$. Let $\mathcal{R}$ be the metric projection of E onto ${\cap }_{l=1}^2{A}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^2{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

In view of the fact that the generic 2-generalized Bregman nonspreading (simply nonspreading) mapping unifies the generic generalized Bregman nonspreading (simply nonspreading) mapping and the 2-generalized Bregman nonspreading (simply nonspreading) mapping, we can prove the following results as corollaries which correspond to the ones in Alsulami et al. [14], Lin et al. [11] and Takahashi et al. [12,13] when only one or two mappings are considered.

Corollary 15.

Let $f:E\to \mathbb{R}$ be a strongly coercive Bregman function which is bounded, uniformly convex and uniformly smooth on bounded sets. Let $M_1,M_2,\cdots ,M_N$ be finite commutative generic generalized Bregman nonspreading mappings of a nonempty subset C of $int\left(dom\right(f\left)\right)$ into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny Bregman generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 16.

Let $f:E\to \mathbb{R}$ be a strongly coercive Bregman function which is bounded, uniformly convex and uniformly smooth on bounded sets. Let $M_1,M_2,\cdots ,M_N$ be finite commutative 2-generalized Bregman nonspreading mappings of a nonempty subset C of $int\left(dom\right(f\left)\right)$ into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny Bregman generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 17

([11,12]). Let E be a uniformly convex Banach space with a Fr$\acute{e}$chet-differentiable norm and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N$ be finite commutative generic generalized nonspreading mappings of the nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 18

([13,14]). Let E be a uniformly convex Banach space with a Fr$\acute{e}$chet-differentiable norm and C be a nonempty subset of E. Let $M_1,M_2,\cdots ,M_N$ be finite commutative 2-generalized nonspreading mappings of the nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the sunny generalized nonexpansive retraction of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

In view of the fact that the class of normally 2-generalized hybrid mappings unifies those of normally generalized hybrid and 2-generalized hybrid mappings, these results correspond to the ones in [7,8] when only one or two mappings are considered.

Corollary 19

([7]). Let E be a real Hilbert space and $M_1,M_2,\cdots ,M_N$ be finite commutative normally generalized hybrid mappings of a nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the metric projection of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.

Corollary 20

([8]). Let E be a real Hilbert space and $M_1,M_2,\cdots ,M_N$ be finite commutative 2-generalized hybrid mappings of a nonempty subset C of E into itself such that $A_f\left(M_l\right)=B_f\left(M_l\right)\ne \varnothing$, for $l=1,2,\cdots ,N$. Let $\mathcal{R}$ be the metric projection of E onto ${\cap }_{l=1}^N{B}_f\left(M_l\right)$ and define $\left\{S_{n}x\right\}$ by

$$S_{n}x=\frac{1}{{(n+1)}^N}\sum _{{\mu }_1=1}^n\sum _{{\mu }_2=1}^n\cdots \sum _{{\mu }_N=1}^n{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x,$$

for all $x\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^N{A}_f\left(M_l\right)$, where $q={lim}_{({\mu }_1,{\mu }_2,\cdots ,{\mu }_N)\in D}\mathcal{R}{M}_1^{{\mu }_1}{M}_2^{{\mu }_2}\cdots M_N^{{\mu }_N}x$. Additionally, if C is closed and convex, then $\left\{S_{n}x\right\}$ converges weakly to an element $q\in {\cap }_{l=1}^{N}F\left(M_l\right)$.