Modified Inertial-Type Multi-Choice CQ-Algorithm for Countable Weakly Relatively Non-Expansive Mappings in a Banach Space, Applications and Numerical Experiments

Abstract

In this paper, some new modified inertial-type multi-choice CQ-algorithms for approximating common fixed point of countable weakly relatively non-expansive mappings are presented in a real uniformly convex and uniformly smooth Banach space. New proof techniques are used to prove strong convergence theorems, which extend some previous work. The connection and application to maximal monotone operators are demonstrated. Numerical experiments are conducted to illustrate that the rate of convergence is accelerated compared to some previous ones for some special cases.

1. Introduction and Preliminaries

In this paper, suppose E is a real Banach space and $E^{*}$ is its dual space. $“u_n\to u”$ and $“u_n\rightharpoonup u”$ denote $\{u_n\}$ converges strongly or weakly to u, as $n\to \infty$, respectively.

A Banach space E is said to be uniformly convex [1] if, for any two sequences $\{u_n\}$ and $\{v_n\}$ in E such that $\parallel u_n\parallel =\parallel v_n\parallel =1$ and $li{m}_{n\to \infty }\parallel u_n+v_n\parallel =2,$ one has $li{m}_{n\to \infty }\parallel u_n-v_n\parallel =0$.

The function ${\delta }_E:[0,+\infty )\to [0,+\infty )$ is said to be the modulus of smoothness of a Banach space E [1] if it is defined as follows:

$${\delta }_E(t)=sup\{\frac{1}{2}(\parallel u+v\parallel +\parallel u-v\parallel )-1:u,v\in E,\parallel u\parallel =1,\parallel v\parallel \le t\}.$$

A Banach space E is said to be uniformly smooth [1] if $li{m}_{t\to 0}\frac{{\delta }_E(t)}{t}=0$. A Banach space E is said to have Property $(H)$: if for any sequence $\{u_n\}\subset E$ which satisfies both $u_n\rightharpoonup u$ and $\parallel u_n\parallel \to \parallel u\parallel$ as $n\to \infty$, then $u_n\to u$, as $n\to \infty$. The uniformly convex and uniformly smooth Banach space has Property $(H)$.

The normalized duality mapping $J_E:E\to 2^{E^{*}}$ is defined by

$$J_E(u)=\{g\in E^{*}:⟨u,g⟩={\parallel u\parallel }^2={\parallel g\parallel }^2\},u\in E,$$

where $⟨u,g⟩$ denotes the value of $g\in E^{*}$ at $u\in E$.

If E is a real uniformly convex and uniformly smooth Banach space, then $J_E$ is single-valued, surjective and $J_E(ku)=k{J}_E(u)$ for $u\in E$ and $k\in (0,+\infty )$. Moreover, $J_E^{-1}=J_{E^{*}}$ and both $J_E$ and $J_E^{-1}$ are uniformly continuous on each bounded subset of E or $E^{*}$, respectively [2].

The Lyapunov functional $\phi :E\times E\to (0,+\infty )$ is defined as follows [3]:

$$\phi (u,v)={\parallel u\parallel }^2-2⟨u,j_E(v)⟩+{\parallel v\parallel }^2,$$

for all $u,v\in E,j_E(v)\in J_E(v).$

Suppose K is a subset of $E.$ Let $T:K\to E$ be a single-valued mapping.

(1)

If $Tp=p,$ then p is called a fixed point of $T.$ The set of fixed points of T is denoted by $Fix(T)$;

(2)

if there exists a sequence $\{u_n\}\subset K$ with $u_n\rightharpoonup p\in K$ such that $u_n-T{u}_n\to 0,$ as $n\to \infty ,$ then p is called an asymptotic fixed point of $T$ [4]. The set of asymptotic fixed points of T is denoted by $\widehat{Fix}(T)$;

(3)

if there exists a sequence $\{u_n\}\subset K$ with $u_n\to p\in K$ such that $u_n-T{u}_n\to 0,$ as $n\to \infty ,$ then p is called a strong asymptotic fixed point of $T$ [4]. The set of strong asymptotic fixed points of T is denoted by $\widetilde{Fix}(T)$;

(4)

T is called strongly relatively non-expansive [5,6,7] if $\widehat{Fix}(T)=Fix(T)\ne \varnothing$ and $\phi (p,Tu)\le \phi (p,u)$ for $u\in K$ and $p\in Fix(T)$;

(5)

T is called weakly relatively non-expansive [4,8] if $\widetilde{Fix}(T)=Fix(T)\ne \varnothing$ and $\phi (p,Tu)\le \phi (p,u)$ for $u\in K$ and $p\in Fix(T)$.

It is obvious that strongly relatively non-expansive mappings are weakly relatively non-expansive mappings.

If E is a real reflexive, strictly convex and smooth Banach space with K being its non-empty closed and convex subset, then for all $u\in E,$ there exists a unique element $u_0\in K$ such that $\phi (u_0,u)=inf\{\phi (y,u):y\in K\}.$ In this case, the generalized projection mapping ${\Pi }_K:E\to K$ is defined by ${\Pi }_{K}u=u_0,$ for all $u\in E$ [3].

If E is a real reflexive, strictly convex Banach space with K being its non-empty closed and convex subset, then for all $u\in E,$ there exists a unique element $u_0\in K$ such that $\parallel u-u_0\parallel =inf\{\parallel u-y\parallel :y\in K\}.$ In this case, the metric projection mapping $P_K:E\to K$ is defined by $P_{K}u=u_0,$ for all $x\in E$ [9].

In a Hilbert space H, ${\Pi }_K=P_K$.

$A:E\to 2^{E^{*}}$ is called monotone [10] if for all $v_i\in A{u}_i,$$i=1,2,$ one has $⟨u_1-u_2,v_1-v_2⟩\ge 0.$A is called maximal monotone if A is monotone and $R(J_E+\lambda A)=E^{*},$ for all $\lambda >0.$$x\in D(A)$ is called a zero point of A if $Ax=0.$ The set of zero points of A is denoted by $A^{-1}0.$

In a real uniformly convex and uniformly smooth Banach E, the resolvent of the maximal monotone operator A, $Q_r^A,$ is defined by $Q_r^{A}x={(J_E+rA)}^{-1}{J}_{E}x$, for $x\in E$ and $r>0.$ Moreover, if $A^{-1}0\ne \varnothing ,$ then $Q_r^A$ is a strongly relatively non-expansive mapping [11], and $Fix(Q_r^A)=A^{-1}0$, for $r>0.$

Strongly relatively non-expansive mappings were introduced by Matsushita and Takahashi in 2004 (see [5,6,7]). The study on strongly or weakly relatively non-expansive mappings draw much attention from mathematicians until now since it includes important nonlinear mappings as special, e.g., the resolvent of a maximal monotone operator and generalized projection ([12,13,14,15,16,17,18,19,20]).

Recall that in 2003, Nakajo and Takahashi [21] presented the following CQ iterative algorithm to approximate the fixed point of a non-expansive mapping T in a real Hilbert space H:

$$\left\{\begin{array}{c}{u}_0\in K,\hfill \\ v_n={\alpha }_n{u}_n+(1-{\alpha }_n)T{u}_n,\hfill \\ C_n=\{p\in K:\parallel v_n-p\parallel \le \parallel u_n-p\parallel \},\hfill \\ Q_n=\{p\in K:⟨p-u_n,u_0-u_n⟩\le 0\},\hfill \\ u_{n+1}=P_{C_n\bigcap Q_n}(u_0),n\in N\bigcup \{0\}.\hfill \end{array}\right.$$

(1)

The result that $\{u_n\}$ converges strongly to a point in $Fix(T)$ is proved.

In 2005, Matsushita and Takahashi [7] extended the topic of non-expansive mappings to that of strongly relatively non-expansive mappings. They presented the following CQ iterative algorithm to approximate the fixed points of a strongly relatively non-expansive mapping T in a real uniformly convex and uniformly smooth Banach space E:

$$\left\{\begin{array}{c}{u}_0\in K,\hfill \\ v_n=J_E^{-1}[{\alpha }_n{J}_E{u}_n+(1-{\alpha }_n)J_{E}T{u}_n],\hfill \\ C_n=\{p\in K:\phi (p,v_n)\le \phi (p,u_n)\},\hfill \\ Q_n=\{p\in K:⟨p-u_n,J_E{u}_0-J_E{u}_n⟩\le 0\},\hfill \\ u_{n+1}={\Pi }_{C_n\bigcap Q_n}(u_0),n\in N\bigcup \{0\}.\hfill \end{array}\right.$$

(2)

The result that $\{u_n\}$ converges strongly to a point in $Fix(T)$ is proved.

In 2009, Wei et al. [14] presented the following hybrid iterative algorithm to approximate the common fixed point of two strongly relatively non-expansive mappings $T_1$ and $T_2$ in a real uniformly convex and uniformly smooth Banach space E:

$$\left\{\begin{array}{c}{u}_0\in K,\hfill \\ v_n=J_E^{-1}[{\alpha }_n{J}_E{u}_n+(1-{\alpha }_n)J_E{T}_1{u}_n],\hfill \\ w_n=J_E^{-1}[{\beta }_n{J}_E{u}_n+(1-{\beta }_n)J_E{T}_2{v}_n],\hfill \\ C_n=\{p\in K:\phi (p,w_n)\le {\beta }_n\phi (p,u_n)+(1-{\beta }_n)\phi (p,v_n)\le \phi (p,u_n)\},\hfill \\ Q_n=\{p\in K:⟨p-u_n,J_E{u}_0-J_E{u}_n⟩\le 0\},\hfill \\ u_{n+1}={\Pi }_{C_n\bigcap Q_n}(u_0),n\in N\bigcup \{0\}.\hfill \end{array}\right.$$

(3)

The result that $u_n\to {\Pi }_{Fix(T_1)\bigcap Fix(T_2)}(u_0)$ is proved, as $n\to \infty$.

In 2010, Su et al. [8] extended the topic of strongly relatively non-expansive mappings to that for weakly relatively non-expansive mappings. They presented the following iterative algorithm for two countable families of weakly relatively non-expansive mappings $\{T_n^{(1)}\}$ and $\{T_n^{(2)}\}$ in a real uniformly convex and uniformly smooth Banach space E:

$$\left\{\begin{array}{c}{u}_0\in K,\hfill \\ v_n=J_E^{-1}[{\beta }_n^{(1)}{J}_E{u}_n+{\beta }_n^{(2)}{J}_E{T}_n^{(1)}{u}_n+{\beta }_n^{(3)}{J}_E{T}_n^{(2)}{u}_n],\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{u}_n+(1-{\alpha }_n)J_E{v}_n],\hfill \\ C_n=\{p\in C_{n-1}\bigcap Q_{n-1}:\phi (p,w_n)\le \phi (p,u_n)\},\hfill \\ C_0=\{p\in C:\phi (p,w_0)\le \phi (p,u_0)\},\hfill \\ Q_n=\{p\in C_{n-1}\bigcap Q_{n-1}:⟨p-u_n,J_E{u}_0-J_E{u}_n⟩\le 0\},\hfill \\ Q_0=K,\hfill \\ u_{n+1}={\Pi }_{C_n\bigcap Q_n}(u_0),n\in N\bigcup \{0\}.\hfill \end{array}\right.$$

(4)

The result that $u_n\to {\Pi }_{({\bigcap }_{n=1}^{\infty }Fix(T_n^{(1)}))\bigcap ({\bigcap }_{n=1}^{\infty }Fix(T_n^{(2)}))}(u_0)$ is proved, as $n\to \infty$.

In 2019, Duan et al. [11] presented some new multi-choice iterative algorithms to approximate common fixed point of two groups of countable weakly relatively non-expansive mappings $\{T_n^{(1)}\}$ and $\{T_n^{(2)}\}$ in a real uniformly convex and uniformly smooth Banach space E:

$$\left\{\begin{array}{c}{u}_0\in E,\hfill \\ v_n=J_E^{-1}[{\alpha }_n{J}_E{u}_n+(1-{\alpha }_n){\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{u}_n],\hfill \\ w_n=J_E^{-1}[{\beta }_n{J}_E{u}_n+(1-{\beta }_n){\displaystyle \sum _{i=1}^{\infty }}{b}_i{J}_E{T}_i^{(2)}{v}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:\phi (p,v_n)\le \phi (p,u_n),\phi (p,w_n)\le {\beta }_n\phi (p,u_n)+(1-{\beta }_n)\phi (p,v_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel p-u_0{\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N\bigcup \{0\}.\hfill \end{array}\right.$$

(5)

The result that $\{u_n\}$ converges strongly to a point in $({\bigcap }_{n=1}^{\infty }Fix(T_n^{(1)}))\bigcap ({\bigcap }_{n=1}^{\infty }Fix(T_n^{(2)}))$ is proved.

Compared to the previous ones (e.g., (2)–(4)), the iterative element $u_{n+1}$ in (5) can be chosen arbitrarily in the set $Q_{n+1}$ and the metric projection $P_{C_{n+1}}(u_0)$ is involved instead the generalized projection.

Recall that the inertial-type algorithm was first proposed by Polyak [22] as an acceleration process in solving a smooth convex minimization problem. The inertial-type algorithm involves a two-step iterative method where the next iterate is defined by making use of the previous two iterates. Inserting an inertial term in an algorithm accelerates the rate of convergence of the iterative sequence [16]. Much devotion is contributed to the inertial-type algorithm (see [23,24,25] and the references therein).

In 2018, Dong et al. [26] presented the following inertial-type CQ iterative algorithm for a non-expansive mapping T in a real Hilbert space $H:$

$$\left\{\begin{array}{c}{u}_0,u_1\in H,\hfill \\ w_n=u_n+{\alpha }_n(u_n-u_{n-1}),\hfill \\ v_n=(1-{\beta }_n)w_n+{\beta }_{n}T{w}_n,\hfill \\ C_n=\{p\in H:\parallel v_n-p\parallel \le \parallel w_n-p\parallel \},\hfill \\ Q_n=\{p\in H:⟨p-u_n,u_0-u_n⟩\le 0\},\hfill \\ u_{n+1}=P_{C_n\bigcap Q_n}(u_0),n\in N.\hfill \end{array}\right.$$

(6)

The result that $u_n\to P_{Fix(T)}(u_0)$ is proved, as $n\to \infty$.

Motivated by Dong’s work, Chidume et al. [16] presented the following inertial-type algorithm for strongly relatively non-expansive mappings $\{T_i\}$ in a real uniformly convex and uniformly smooth Banach space E:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ C_0=E,\hfill \\ w_n=u_n+{\alpha }_n(u_n-u_{n-1}),\hfill \\ v_n=J_E^{-1}[(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n],\hfill \\ C_{n+1}=\{p\in C_n:\phi (p,v_n)\le \phi (p,w_n)\},\hfill \\ u_{n+1}={\Pi }_{C_{n+1}}(u_0),n\in N,\hfill \end{array}\right.$$

(7)

where $Tx=J_E^{-1}({\Sigma }_{i=1}^{\infty }{a}_i(b_i{J}_{E}x+(1-b_i)J_E{T}_{i}x)),$ for $x\in E.$ The result that $u_n\to {\Pi }_{({\bigcap }_{n=1}^{\infty }Fix(T_n))}(u_0)$ is proved, as $n\to \infty$.

Our paper is organized as follows: in Section 2, we shall improve a key result of Chidume [16] from strongly relatively non-expansive mappings to weakly relatively non-expansive mappings and present a new proof method. In Section 3, we shall construct some new iterative algorithms for a countable weakly relatively non-expansive mappings in a real uniformly convex and uniformly smooth Banach space and prove some strong convergence theorems. In Section 4, we shall demonstrate the applications of the new results in Section 3 to countable maximal monotone operators. In Section 5, we shall do numerical experiments to show the advantage of the newly obtained iterative algorithms in the sense that the convergence rate is accelerated compared to the previous ones for some special cases.

The following preliminaries are needed.

Lemma 1

([15]). Assume that E is a uniformly convex and also a uniformly smooth Banach space, K is its non-empty closed and convex subset, $T:K\to K$ is a weakly relatively non-expansive mapping. Then $Fix(T)$ is a convex and closed subset of E.

Lemma 2

([19]). Assume that E is a uniformly convex and also a uniformly smooth Banach space, $\{x_n\}$ and $\{y_n\}$ are two sequences in E. If one of $\{x_n\}$ and $\{y_n\}$ is bounded and also $\phi (x_n,y_n)\to 0,$ then $x_n-y_n\to 0,$ as $n\to \infty .$

Definition 1

([27]). Assume $\{K_n\}$ is a sequence of non-empty closed and convex subsets of E. Then $s-liminf{K}_n:=\{x\in E:\mathit{there}\mathit{exists}{x}_n\in K_n\mathit{for}\mathit{almost}\mathit{all}\mathrm{n}\mathit{such}\mathit{that}{x}_n\to x,\mathrm{as}n\to \infty \}$ is called the strong lower limit of $\{K_n\}$; $w-limsup{K}_n:=\{x\in E:\mathit{there}\mathit{exists}\mathrm{a}\mathit{subsequence}{K}_{n_m}\mathrm{of}{K}_n\mathit{such}\mathit{that}{x}_{n_m}\in K_{n_m}\mathrm{for}\mathrm{every}{n}_m\mathrm{and}{x}_{n_m}\rightharpoonup x,\mathrm{as}{n}_m\to \infty \}$ is called the weak upper limit of $\{K_n\}$; if $s-liminf{K}_n=w-limsup{K}_n$, the common value is denoted by $lim{K}_n$ and is called the limit of $\{K_n\}.$

Lemma 3

([27]). Assume $\{K_n\}$ is decreasing, closed and convex in E, then $lim{K}_n={\bigcap }_{n=1}^{\infty }{K}_n.$

Lemma 4

([28]). Let E be a real uniformly convex Banach space. If $lim{K}_n\ne \varnothing$, then $P_{K_n}x\rightharpoonup P_{lim{K}_n}x,$ for $\forall x\in E.$ Moreover, if E has Property $(H)$, then $P_{K_n}x\to P_{lim{K}_n}x,$ for for all $x\in E.$

Lemma 5

([13]). Assume E is a real uniformly convex Banach space and $r\in (0,+\infty )$. Then there exists a continuous, strictly increasing and convex function $g:[0,2r]\to [0,+\infty )$ such that $g(0)=0$ and $\parallel {\sum }_{i=1}^{\infty }{c}_i{x}_i{\parallel }^2\le {\sum }_{i=1}^{\infty }{c}_i\parallel x_i{\parallel }^2-c_1{c}_{m}g(\parallel x_1-x_m\parallel ),$ for all ${\{x_n\}}_{n=1}^{\infty }\subset \{x\in E:\parallel x\parallel \le r\}$, ${\{c_n\}}_{n=1}^{\infty }\subset (0,1)$ with ${\sum }_{n=1}^{\infty }{c}_n=1$ and $m\in N.$

Lemma 6

([10]). Assume $A:E\to 2^{E^{*}}$ is maximal monotone, then

(1) 

$A^{-1}0$ is closed and convex in E;

(2) 

if $x_n\to x$ and $y_n\in A{x}_n$ with $y_n\rightharpoonup y,$ or $x_n\rightharpoonup x$ and $y_n\in A{x}_n$ with $y_n\to y,$ then $x\in D(A)$ and $y\in Ax.$

2. Improvement and New Proof Techniques for Chidume’s Results

Lemma 7

([16]). Assume that E is a real uniformly convex and also a uniformly smooth Banach space, $T:E\to E$ is a strongly relatively non-expansive mapping with $Fix(T)\ne \varnothing .$ Let $\{u_n\}$ be generated by the following iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ C_0=E,\hfill \\ w_n=u_n+{\alpha }_n(u_n-u_{n-1}),\hfill \\ v_n=J_E^{-1}[(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n],\hfill \\ C_{n+1}=\{p\in C_n:\phi (p,v_n)\le \phi (p,w_n)\},\hfill \\ u_{n+1}={\Pi }_{C_{n+1}}(u_0),n\in N,\hfill \end{array}\right.$$

(8)

where ${\alpha }_n\in (0,1)$ and $\beta \in (0,1).$ Then $u_n\to {\Pi }_{Fix(T)}(u_0)$, as $n\to \infty$.

We shall improve Lemma 7 as follows:

Theorem 1.

Assume that E is a real uniformly convex and also a uniformly smooth Banach space, $T:E\to E$ is a weakly relatively non-expansive mapping with $Fix(T)\ne \varnothing .$ Let $\{u_n\}$ be generated by the following iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ C_0=E,\hfill \\ w_n=u_n+{\alpha }_n(u_n-u_{n-1}),\hfill \\ v_n=J_E^{-1}[(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n],\hfill \\ C_{n+1}=\{p\in C_n:\phi (p,v_n)\le \phi (p,w_n)\},\hfill \\ u_{n+1}=P_{C_{n+1}}(u_0),n\in N,\hfill \end{array}\right.$$

(9)

where ${\alpha }_n\in (0,1)$ and $\beta \in (0,1).$ Then $u_n\to P_{Fix(T)}(u_0)$, when $n\to \infty$.

Proof. 

We need three steps to finish the proof.

Step 1. $Fix(T)\subset C_n$, where $C_n$ is a convex and closed subset of E, for all $n\in N.$ This ensures $\{u_n\}$ is well-defined.

Since $\phi (p,v_n)\le \phi (p,w_n)$ is equivalent to $\parallel v_n{\parallel }^2-{\parallel w_n\parallel }^2\le 2⟨p,J_E{v}_n-J_E{w}_n⟩$, then $C_n$ is closed and convex, for $n\in N.$

For all $q\in Fix(T),$ since T is weakly relatively non-expansive, then

$$\begin{array}{l}\phi (q,v_n)={\parallel q\parallel }^2-2⟨q,(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n⟩+{\parallel (1-\beta )J_E{w}_n+\beta J_{E}T{w}_n\parallel }^2\\ \le (1-\beta )\phi (q,w_n)+\beta \phi (q,T{w}_n)\le \phi (q,w_n).\end{array}$$

This ensures that $Fix(T)\subset C_n$ for $n\in N.$

Step 2. $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

It follows from Step 1 and Lemma 3 that $lim{C}_n$ exists and $lim{C}_n={\bigcap }_{n=1}^{\infty }{C}_n\ne \varnothing$. Lemma 4 implies that $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$. From iterative algorithm (9), one has $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Note that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in C_{n+1},$ then $\phi (P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),v_n)\le \phi (P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),w_n)\to 0,$ as $n\to \infty$. Thus Lemma 2 implies that $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Step 3. $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)=P_{Fix(T)}(u_0)$.

Since $Fix(T)\subset {\bigcap }_{n=1}^{\infty }{C}_n$, then $\parallel P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)-u_0\parallel \le \parallel P_{Fix(T)}(u_0)-u_0\parallel .$

To show that $\parallel P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)-u_0\parallel \ge \parallel P_{Fix(T)}(u_0)-u_0\parallel ,$ it suffices to show that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T)$.

In fact, for all $q\in Fix(T),$ using Lemma 5, one has

$$\begin{array}{l}\phi (q,v_n)={\parallel q\parallel }^2-2⟨q,(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n⟩+\parallel (1-\beta )J_E{w}_n+\beta J_{E}T{w}_n{\parallel }^2\\ \le {\parallel q\parallel }^2-2(1-\beta )⟨q,J_E{w}_n⟩-2\beta ⟨q,J_{E}T{w}_n⟩+(1-\beta )\parallel w_n{\parallel }^2+\beta \parallel T{w}_n{\parallel }^2-\beta (1-\beta )g(\parallel w_n-T{w}_n\parallel )\\ =(1-\beta )\phi (q,w_n)+\beta \phi (q,T{w}_n)-\beta (1-\beta )g(\parallel w_n-T{w}_n\parallel )\\ \le \phi (q,w_n)-\beta (1-\beta )g(\parallel w_n-T{w}_n\parallel ).\end{array}$$

It follows from Step 2 that, $\beta (1-\beta )g(\parallel w_n-T{w}_n\parallel )\le \phi (q,w_n)-\phi (q,v_n)\to 0,$ as $n\to \infty .$ Thus $w_n-T{w}_n\to 0,$ as $n\to \infty .$ Combining with the facts that $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and Lemma 1, one has $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T).$

By now, we have showed that $\parallel P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)-u_0\parallel =\parallel P_{Fix(T)}(u_0)-u_0\parallel .$ Since the metric projection $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ is unique, then $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)=P_{Fix(T)}(u_0).$

This completes the proof. □

Remark 1.

Compared to Lemma 7, three novelties can be seen in Theorem 1. One is that the discussion is extended from strongly relatively non-expansive mapping to weakly relatively non-expansive mapping. The second is that the generalized projections ${\Pi }_{C_{n+1}}(u_0)$ and ${\Pi }_{Fix(T)}(u_0)$ are replaced by the metric projections $P_{C_{n+1}}(u_0)$ and $P_{Fix(T)}(u_0)$ even if in a Banach space, which means that the computation may be easily realized theoretically. The third is that different technique is employed to show the convergence of $\{u_n\}$.

Lemma 8

([12]). Assume that E is a real uniformly convex and also a uniformly smooth Banach space, $T_i:E\to E,$$(i\in N),$ is a sequence of countable strongly relatively non-expansive mappings with ${\bigcap }_{i=1}^{\infty }Fix(T_i)\ne \varnothing .$ Suppose $\{a_i\}\subset (0,1)$ and $\{b_i\}\subset (0,1)$ satisfy that ${\sum }_{i=1}^{\infty }{a}_i=1.$ Define $T:E\to E$ as follows:

$$Tx=J_E^{-1}({\Sigma }_{i=1}^{\infty }{a}_i(b_i{J}_{E}x+(1-b_i)J_E{T}_{i}x)),$$

for $\forall x\in E.$ The result that T is strongly relatively non-expansive is true. Moreover, $Fix(T)={\bigcap }_{i=1}^{\infty }Fix(T_i)$.

Based on Lemmas 7 and 8, a countable family of strongly relatively non-expansive mapping ${\{T_i\}}_{i=1}^{\infty }$ is studied as follows:

Lemma 9

([16]). Assume that E, $T_i:E\to E,$$\{a_i\}$, $\{b_i\}$ and $T:E\to E$ are the same as those in Lemma 8. Let $\{u_n\}$ be generated by the iterative algorithm (7). Then under the assumptions that ${\alpha }_n\in (0,1)$ and $\beta \in (0,1)$, one has $u_n\to {\Pi }_{{\bigcap }_{i=1}^{\infty }Fix(T_i)}(u_0)$, as $n\to \infty$.

Lemma 9 can be improved as follows, which is obtained directly in view of Theorem 1 and Lemma 8.

Theorem 2.

Under all of the assumptions of Lemma 9, suppose $\{u_n\}$ is generated by the following iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ C_0=E,\hfill \\ w_n=u_n+{\alpha }_n(u_n-u_{n-1}),\hfill \\ v_n=J_E^{-1}[(1-\beta )J_E{w}_n+\beta J_{E}T{w}_n],\hfill \\ C_{n+1}=\{p\in C_n:\phi (p,v_n)\le \phi (p,w_n)\},\hfill \\ u_{n+1}=P_{C_{n+1}}(u_0),n\in N.\hfill \end{array}\right.$$

(10)

Then $u_n\to P_{{\bigcap }_{i=1}^{\infty }Fix(T_i)}(u_0)$, as $n\to \infty$.

3. Inertial-Type Iterative Algorithms with New Set $C_n$

Definition 2

([29]). Let E be a real Banach space with K being its non-empty closed and convex subset. Define the function $G:K\times E^{*}\to (-\infty ,+\infty )$ as follows:

$$G(x,y)={\parallel x\parallel }^2-2⟨x,y⟩+{\parallel y\parallel }^2+2\rho f(x),$$

for all $x\in K,y\in E^{*},$ where $\rho >0$ and $f:K\to (-\infty ,+\infty )$ is a proper, convex and lower-semi-continuous mapping. Obviously, if $K\equiv E$ and $f(u)\equiv 0$, then $G(u,J_{E}v)\equiv \phi (u,v)$ for all $u,v\in E.$

Theorem 3.

Assume that E is a real uniformly convex and also a uniformly smooth Banach space. Let $T_i^{(1)},T_i^{(2)}:E\to E$ be weakly relatively non-expansive mappings, for each $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N,\hfill \end{array}\right.$$

(11)

where G is the function defined in Definition 2. Under the following assumptions

$(i)$

$({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\ne \varnothing ;$

$(ii)$

$\{a_i\}\subset (0,1)$ and $\{b_i\}\subset (0,1)$ satisfy ${\sum }_{i=1}^{\infty }{a}_i=1={\sum }_{i=1}^{\infty }{b}_i;$

$(iii)$

$\{{\lambda }_n\}$ is a real number sequence in $(0,+\infty )$ with ${\lambda }_n\to 0,$ as $n\to \infty ;$

$(iv)$

$\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two real number sequences in $[0,1)$ with $0\le su{p}_n{\alpha }_n<1$ and $0\le su{p}_n{\beta }_n<1$;

$(v)$

$\{t_n\}$ is a real number sequence in $(-\infty ,+\infty )$ with $t_n\nrightarrow \pm \infty$ as $n\to \infty$,

we have $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0),$ when $n\to \infty .$

Proof. 

We need nine steps to finish the proof.

Step 1. $({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\subset C_n,$ for $n\in N.$

For all $q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})),$ if $n=1,$ it is obvious that $q\in C_1=E$. Noticing the definitions of Lyapunov functional and weakly relatively non-expansive mappings, one has:

$$\begin{array}{c}G(q,J_E{w}_1)={\parallel q\parallel }^2-2⟨q,{\alpha }_1{J}_E{v}_1+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_1⟩\hfill \\ +\parallel {\alpha }_1{J}_E{v}_1+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_1{\parallel }^2+2\rho f(q)\hfill \\ \le {\alpha }_1\phi (q,v_1)+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i\phi (q,T_i^{(1)}{v}_1)+2\rho f(q)\hfill \\ \le {\alpha }_1\phi (q,v_1)+(1-{\alpha }_1)\phi (q,v_1)+2\rho f(q)=G(q,J_E{v}_1)\hfill \end{array}$$

and

$$\begin{array}{c}G(q,J_E{y}_1)\le {\parallel q\parallel }^2-2{\beta }_1⟨q,J_E{v}_1⟩-2(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i⟨q,J_E{T}_i^{(2)}{w}_1⟩\hfill \\ +{\beta }_1\parallel v_1{\parallel }^2+(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i{\parallel T_i^{(2)}{w}_1\parallel }^2+2\rho f(q)\hfill \\ ={\beta }_1\phi (q,v_1)+(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i\phi (q,T_i^{(2)}{w}_1)+2\rho f(q)\hfill \\ \le {\beta }_1\phi (q,v_1)+(1-{\beta }_1)\phi (q,w_1)+2\rho f(q)\hfill \\ ={\beta }_{1}G(q,J_E{v}_1)+(1-{\beta }_1)G(q,J_E{w}_1).\hfill \end{array}$$

Therefore, $q\in C_2.$ That is, $({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\subset C_2.$

Suppose it is true when $n\le k.$ Now, if $n=k+1$, compute the following:

$$\begin{array}{c}G(q,J_E{w}_{k+1})={\parallel q\parallel }^2-2⟨q,{\alpha }_{k+1}{J}_E{v}_{k+1}+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_{k+1}⟩\hfill \\ +\parallel {\alpha }_{k+1}{J}_E{v}_{k+1}+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_{k+1}{\parallel }^2+2\rho f(q)\hfill \\ \le {\alpha }_{k+1}\phi (q,v_{k+1})+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i\phi (q,T_i^{(1)}{v}_{k+1})+2\rho f(q)\hfill \\ \le {\alpha }_{k+1}\phi (q,v_{k+1})+(1-{\alpha }_{k+1})\phi (q,v_{k+1})+2\rho f(q)=G(q,J_E{v}_{k+1}).\hfill \end{array}$$

Moreover,

$$\begin{array}{c}G(q,J_E{y}_{k+1})\le {\parallel q\parallel }^2-2{\beta }_{k+1}⟨q,J_E{v}_{k+1}⟩-2(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i⟨q,J_E{T}_i^{(2)}{w}_{k+1}⟩\hfill \\ +{\beta }_{k+1}\parallel v_{k+1}{\parallel }^2+(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i{\parallel T_i^{(2)}{w}_{k+1}\parallel }^2+2\rho f(q)\hfill \\ ={\beta }_{k+1}\phi (q,v_{k+1})+(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i\phi (q,T_i^{(2)}{w}_{k+1})+2\rho f(q)\hfill \\ \le {\beta }_{k+1}\phi (q,v_{k+1})+(1-{\beta }_{k+1})\phi (q,w_{k+1})+2\rho f(q)\hfill \\ ={\beta }_{k+1}G(q,J_E{v}_{k+1})+(1-{\beta }_{k+1})G(q,J_E{w}_{k+1}).\hfill \end{array}$$

Therefore, $q\in C_{k+2}.$ By induction, $\varnothing \ne ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\subset C_n,$ for all $n\in N.$

Step 2. $C_n$ is a convex and closed subset of $E,$ for all $n\in N.$

If $n=1,$ the result is trivial.

If $n\in N\backslash \{1\},$ then noticing the following facts: $G(p,J_E{w}_n)\le G(p,J_E{v}_n)$ is equivalent to $2⟨p,J_E{v}_n-J_E{w}_n⟩\le \parallel v_n{\parallel }^2-{\parallel w_n\parallel }^2$ and $G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)$ is equivalent to $2{\beta }_n⟨p,J_E{v}_n⟩+2(1-{\beta }_n)⟨p,J_E{w}_n⟩-2⟨p,J_E{y}_n⟩\le {\beta }_n\parallel v_n{\parallel }^2+(1-{\beta }_n)\parallel w_n{\parallel }^2-{\parallel y_n\parallel }^2.$

Thus $C_n$ is closed and convex, for all $n\in N.$

Step 3. $Q_n$ is a non-empty subset of $E,$$\forall n\in N$. This guarantees that $\{u_n\}$ is well-defined.

In fact, if $n=1,$ the result is trivial.

If $n\in N\backslash \{1\},$ then we can see from the definition of metric projection that $\parallel P_{C_{n+1}}(u_0)-u_0\parallel =in{f}_{y\in C_{n+1}}\parallel y-u_0\parallel .$ Then from the definition of infimum, for ${\lambda }_{n+1},$ there exists ${\zeta }_{n+1}\in C_{n+1}$ such that $\parallel u_0-{\zeta }_{n+1}{\parallel }^2\le {\parallel P_{C_{n+1}}(u_0)-u_0\parallel }^2+{\lambda }_{n+1}.$ This ensures that $Q_{n+1}\ne \varnothing ,$ for $n\in N.$

Step 4. $P_{C_{n+1}}(u_0)\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

It follows from Steps 1 and 2 and Lemma 3 that $lim{C}_n$ exists and $lim{C}_n={\bigcap }_{n=1}^{\infty }{C}_n\ne \varnothing$. Since E has Property (H), then Lemma 4 ensures that $P_{C_{n+1}}(u_0)\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Step 5. Both $\{u_n\}$ and $\{P_{C_{n+1}}(u_0)\}$ are bounded.

It follows from Step4 immediately that $\{P_{C_{n+1}}(u_0)\}$ is bounded. Since $u_{n+1}\in Q_{n+1},$ then $\parallel u_0-u_{n+1}{\parallel }^2\le {\parallel P_{C_{n+1}}(u_0)-u_0\parallel }^2+{\lambda }_{n+1}$, $\forall n\in N$. Since ${\lambda }_n\to 0$ and $\{P_{C_{n+1}}(u_0)\}$ is bounded, then it is easy to see that $\{u_n\}$ is bounded.

Step 6. $u_{n+1}-P_{C_{n+1}}(u_0)\to 0,$ as $n\to \infty$.

Since $u_{n+1}\in Q_{n+1}\subset C_{n+1}$ and $C_n$ is convex, for $\forall t\in (0,1),$$t{P}_{C_{n+1}}(u_0)+(1-t)u_{n+1}\in C_{n+1}.$ Noticing the definition of metric projection,

$$\parallel P_{C_{n+1}}(u_0)-u_0\parallel \le \parallel t{P}_{C_{n+1}}(u_0)+(1-t)u_{n+1}-u_0\parallel .$$

Lemma 5 ensures that

$$\begin{array}{c}\parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2\le {\parallel t{P}_{C_{n+1}}(u_0)+(1-t)u_{n+1}-u_0\parallel }^2\hfill \\ \le t\parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+(1-t)\parallel u_{n+1}-u_0{\parallel }^2-t(1-t)g(\parallel P_{C_{n+1}}(u_0)-u_{n+1}\parallel ).\hfill \end{array}$$

Thus, $tg(\parallel P_{C_{n+1}}(u_0)-u_{n+1}\parallel )\le \parallel u_{n+1}-u_0{\parallel }^2-{\parallel P_{C_{n+1}}(u_0)-u_0\parallel }^2\le {\lambda }_{n+1}.$ Let $t\to 1$ and later $n\to \infty$, one can see $P_{C_{n+1}}(u_0)-u_{n+1}\to 0,$ as $n\to \infty .$

Step 7. $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and $y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

In fact, it follows from Step 4 and Step 6 that $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$. Since $v_n=u_n+t_n(u_n-u_{n-1}),$ then $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$. Thus $\phi (u_{n+1},v_n)\to 0,$ as $n\to \infty$.

Since $u_{n+1}\in Q_{n+1}\subset C_{n+1},$ then $G(u_{n+1},J_E{w}_n)\le G(u_{n+1},J_E{v}_n)$, which implies that $\phi (u_{n+1},w_n)\le \phi (u_{n+1},v_n)\to 0.$ Lemma 2 ensures that $u_{n+1}-w_n\to 0,$ and then $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Since $u_{n+1}\in Q_{n+1}\subset C_{n+1}$, then $G(u_{n+1},J_E{y}_n)\le {\beta }_{n}G(u_{n+1},J_E{v}_n)+(1-{\beta }_n)G(u_{n+1},J_E{w}_n)$, which implies that $\phi (u_{n+1},y_n)\le {\beta }_n\phi (u_{n+1},v_n)+(1-{\beta }_n)\phi (u_{n+1},w_n)\to 0,$ as $n\to \infty$. Thus $u_{n+1}-y_n\to 0,$ which implies that $y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Step 8. $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in ({\bigcap }_{i=1}^{\infty }Fix{(T_i)}^{(1)})\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})).$

First, we shall show that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(1)}).$

For all $q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})),$ by using Lemma 5, we have:

$$\begin{array}{l}\phi (q,J_E^{-1}({\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n))={\parallel q\parallel }^2-2⟨q,{\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n⟩+\parallel {\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n{\parallel }^2\\ \le {\parallel q\parallel }^2-2{\displaystyle \sum _{i=1}^{\infty }}{a}_i⟨q,J_E{T}_i^{(1)}{v}_n⟩+{\displaystyle \sum _{i=1}^{\infty }}{a}_i\parallel T_i^{(1)}{v}_n{\parallel }^2-a_1{a}_{m}g(\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\parallel )\\ ={\displaystyle \sum _{i=1}^{\infty }}{a}_i\phi (q,T_i^{(1)}{v}_n)-a_1{a}_{m}g(\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\parallel ).\end{array}$$

Then

$$\begin{array}{l}{a}_1{a}_{m}g(\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\parallel )\le {\displaystyle \sum _{i=1}^{\infty }}{a}_i\phi (q,T_i^{(1)}{v}_n)-\phi (q,J_E^{-1}({\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n))\\ \le \phi (q,v_n)-\phi (q,J_E^{-1}({\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n))\\ =\parallel v_n{\parallel }^2-2⟨q,J_E{v}_n⟩+2{\displaystyle \sum _{i=1}^{\infty }}{a}_i⟨q,J_E{T}_i^{(1)}{v}_n⟩-{\parallel {\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n\parallel }^2.\end{array}$$

(12)

Since $w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],$ then $J_E{w}_n-J_E{v}_n=(1-{\alpha }_n)({\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n-J_E{v}_n).$ Based on the properties of $J_E$ and $J_E^{-1}$, $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and $0\le su{p}_n{\alpha }_n<1,$ one has: ${\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n-J_E{v}_n\to 0,$$n\to \infty .$ Therefore,

$$J_E^{-1}({\displaystyle \sum _{i=1}^{\infty }}{a}_i{J}_E{T}_i^{(1)}{v}_n)\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$

(13)

as $n\to \infty .$ Moreover, from (12), we can also know that

$$J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\to 0,$$

as $n\to \infty ,$ for $m\ne 1.$

Note that for all $q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})),$ one has $(\parallel q\parallel -\parallel T_i^{(1)}{v}_n{\parallel )}^2\le \phi (q,T_i^{(1)}{v}_n)\le \phi (q,v_n)\le (\parallel q\parallel +\parallel v_n{\parallel )}^2.$ Then $\{T_i^{(1)}{v}_n\}$ is bounded for $\forall i\in N.$ We may assume that $M=sup\{\parallel T_i^{(1)}{v}_n\parallel :i,n\in N\}.$

Since ${\sum }_{i=1}^{\infty }{a}_i=1,$ then for all $\delta >0,$ there exists sufficiently large integer $N_0$ such that ${\sum }_{i=N_0+1}^{\infty }{a}_i<\frac{\delta }{4M}.$

From the fact that $J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\to 0,$ as $n\to \infty ,$ for all $m\in \{1,2,\cdots ,N_0\},$ we may know that there exists sufficiently large integer $M_0$ such that $\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_m^{(1)}{v}_n\parallel <\frac{\delta }{2}$ for all $n\ge M_0$ and $m\in \{2,\cdots ,N_0\}.$ Then if $n\ge M_0,$

$$\begin{array}{c}\parallel J_E{T}_1^{(1)}{v}_n-{\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n\parallel \le {\sum }_{i=2}^{N_0}{a}_i\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_i^{(1)}{v}_n\parallel +{\sum }_{i=N_0+1}^{\infty }{a}_i\parallel J_E{T}_1^{(1)}{v}_n-J_E{T}_i^{(1)}{v}_n\parallel \hfill \\ <({\sum }_{i=2}^{N_0}{a}_i)\frac{\delta }{2}+({\sum }_{i=N_0+1}^{\infty }{a}_i)2M<\frac{\delta }{2}+\frac{\delta }{2}=\delta .\hfill \end{array}$$

Therefore, $J_E{T}_1^{(1)}{v}_n-{\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n\to 0,$ as $n\to \infty .$ Then (13) implies that $T_1^{(1)}{v}_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty .$ Combining with the fact that $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ and by using Lemma 1, $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(1)}).$

Repeating the above process for showing $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(1)})$, we can also prove that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_m^{(1)}),$$\forall m\in N.$ Therefore, $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in {\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}).$

Next, we shall show that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(2)}).$

For all $q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})),$ similar to (12), we have:

$$\begin{array}{l}{b}_1{b}_{m}g(\parallel J_E{T}_1^{(2)}{w}_n-J_E{T}_m^{(2)}{w}_n\parallel )\\ \le \parallel w_n{\parallel }^2-2⟨q,J_E{w}_n⟩+2{\displaystyle \sum _{i=1}^{\infty }}{b}_i⟨q,J_E{T}_i^{(2)}{w}_n⟩-{\parallel {\displaystyle \sum _{i=1}^{\infty }}{b}_i{J}_E{T}_i^{(2)}{w}_n\parallel }^2.\end{array}$$

(14)

Since $y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],$ then $J_E{y}_n-J_E{v}_n=(1-{\beta }_n)({\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n-J_E{v}_n).$ From the facts that $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and $0\le su{p}_n{\beta }_n<1,$ we have: ${\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n-J_E{v}_n\to 0,$ which implies that

$$J_E^{-1}({\displaystyle \sum _{i=1}^{\infty }}{b}_i{J}_E{T}_i^{(2)}{w}_n)\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$

(15)

as $n\to \infty .$ Coming back to (14),

$$J_E{T}_1^{(2)}{w}_n-J_E{T}_m^{(2)}{w}_n\to 0,$$

as $n\to \infty ,$ for $m\ne 1.$

Similar to the above corresponding discussion, $\{T_i^{(2)}{w}_n\}$ is bounded for $\forall i\in N$. Then we may assume that $M^{\prime }=sup\{\parallel T_i^{(2)}{w}_n\parallel :i,n\in N\}.$

Since ${\sum }_{i=1}^{\infty }{b}_i=1,$ then for all $\delta >0,$ there exists sufficiently large integer $N_0$ such that ${\sum }_{i=N_0+1}^{\infty }{b}_i<\frac{\delta }{4M^{\prime }}.$

Since $J_E{T}_1^{(2)}{w}_n-J_E{T}_m^{(2)}{w}_n\to 0,$ as $n\to \infty ,$ for all $m\in \{1,2,\cdots ,N_0\},$ then there exists sufficiently large integer $M_0$ such that $\parallel J_E{T}_1^{(2)}{w}_n-J_E{T}_m^{(2)}{w}_n\parallel <\frac{\delta }{2}$ for all $n\ge M_0$ and $m\in \{2,\cdots ,N_0\}.$ Then if $n\ge M_0,$

$$\begin{array}{c}\parallel J_E{T}_1^{(2)}{w}_n-{\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n\parallel \le {\sum }_{i=2}^{N_0}{b}_i\parallel J_E{T}_1^{(2)}{w}_n-J_E{T}_i^{(2)}{w}_n\parallel +{\sum }_{i=N_0+1}^{\infty }{b}_i\parallel J_E{T}_1^{(2)}{w}_n-J_E{T}_i^{(2)}{w}_n\parallel \hfill \\ <({\sum }_{i=2}^{N_0}{b}_i)\frac{\delta }{2}+({\sum }_{i=N_0+1}^{\infty }{b}_i)2M^{\prime }<\frac{\delta }{2}+\frac{\delta }{2}=\delta .\hfill \end{array}$$

Therefore, $J_E{T}_1^{(2)}{w}_n-{\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n\to 0,$ and then (15) implies that $T_1^{(2)}{w}_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty .$ Combining with the fact that $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ we have $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(2)}).$

Repeating the above process for showing $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(2)})$, we can also prove that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_m^{(2)}),$ for all $m\in N.$ Therefore, $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in {\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}).$

Step 9. $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)=P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0)$.

From Steps 1 and 8, we can easily see that

$$\parallel P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0)-u_0\parallel =\parallel P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)-u_0\parallel .$$

Since the metric projection $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ is unique, then $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)=P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0)$.

This completes the proof. □

Remark 2.

Compare to [16], the restriction of $\{t_n\}$ is weaker. Compared to [11], the construction of $C_n$ is different, the inertial iterate is considered and the limit of the iterative sequence is precisely defined.

Theorem 4.

Assume that E is a real uniformly convex and also a uniformly smooth Banach space. Let $T_i^{(1)},T_i^{(2)}:E\to E$ be weakly relatively non-expansive mappings, for each $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(16)

Under the assumptions of (i)–(iii) and (v) in Theorem 3 and the following condition

${(iv)}^{\prime }$$\{{\alpha }_n\}\subset [0,1)$ and $\{{\beta }_n\}\subset [0,1)$ with ${\alpha }_n\to 0$ and ${\beta }_n\to 0$, one has: $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0),$ as $n\to \infty .$

Proof. 

Copy Steps 3, 4, 5, 6 and 9 in Theorem 3 and make some changes in Steps 1, 2, 7 and 8 as follows.

Step 1. $({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)})\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\subset C_n,$ for $n\in N.$

Now, for all $q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)})\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)})).$

For $n=1,$ the result that $q\in C_1=E$ is trivial. Note the property of weakly relatively non-expansive mapping, one can get the following:

$$\begin{array}{c}G(q,J_E{w}_1)={\parallel q\parallel }^2-2⟨q,{\alpha }_1{J}_E{v}_1+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_1⟩\hfill \\ +\parallel {\alpha }_1{J}_E{v}_1+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_1{\parallel }^2+2\rho f(q)\hfill \\ \le {\parallel q\parallel }^2-2{\alpha }_1⟨q,J_E{v}_1⟩-2(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i⟨q,J_E{T}_i^{(1)}{v}_1⟩\hfill \\ +{\alpha }_1\parallel v_1{\parallel }^2+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i{\parallel T_i^{(1)}{v}_1\parallel }^2+2\rho f(q)\hfill \\ ={\alpha }_1\phi (q,v_1)+(1-{\alpha }_1){\sum }_{i=1}^{\infty }{a}_i\phi (q,T_i^{(1)}{v}_1)+2\rho f(q)\hfill \\ \le {\alpha }_1\phi (q,v_1)+(1-{\alpha }_1)\phi (q,v_1)+2\rho f(q)={\alpha }_{1}G(q,J_E{v}_1)+(1-{\alpha }_1)G(q,J_E{v}_1)\hfill \end{array}$$

and

$$\begin{array}{c}G(q,J_E{y}_1)\le {\parallel q\parallel }^2-2{\beta }_1⟨q,J_E{v}_1⟩-2(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i⟨q,J_E{T}_i^{(2)}{w}_1⟩+{\beta }_1\parallel v_1{\parallel }^2+(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i{\parallel T_i^{(2)}{w}_1\parallel }^2+2\rho f(q)\hfill \\ ={\beta }_1\phi (q,v_1)+(1-{\beta }_1){\sum }_{i=1}^{\infty }{b}_i\phi (q,T_i^{(2)}{w}_1)+2\rho f(q)\hfill \\ \le {\beta }_1\phi (q,v_1)+(1-{\beta }_1)\phi (q,w_1)+2\rho f(q)={\beta }_{1}G(q,J_E{v}_1)+(1-{\beta }_1)G(q,J_E{w}_1).\hfill \end{array}$$

Thus $q\in C_2.$

Suppose it is true when $n\le k.$ For $n=k+1$, compute the following:

$$\begin{array}{c}G(q,J_E{w}_{k+1})={\parallel q\parallel }^2-2⟨q,{\alpha }_{k+1}{J}_E{v}_1+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_{k+1}⟩\hfill \\ +\parallel {\alpha }_{k+1}{J}_E{v}_1+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_{k+1}{\parallel }^2+2\rho f(q)\hfill \\ \le {\parallel q\parallel }^2-2{\alpha }_{k+1}⟨q,J_E{v}_1⟩-2(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i⟨q,J_E{T}_i^{(1)}{v}_{k+1}⟩\hfill \\ +{\alpha }_{k+1}\parallel v_1{\parallel }^2+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i{\parallel T_i^{(1)}{v}_{k+1}\parallel }^2+2\rho f(q)\hfill \\ ={\alpha }_{k+1}\phi (q,v_1)+(1-{\alpha }_{k+1}){\sum }_{i=1}^{\infty }{a}_i\phi (q,T_i^{(1)}{v}_{k+1})+2\rho f(q)\hfill \\ \le {\alpha }_{k+1}\phi (q,v_1)+(1-{\alpha }_{k+1})\phi (q,v_{k+1})+2\rho f(q)\hfill \\ ={\alpha }_{k+1}G(q,J_E{v}_1)+(1-{\alpha }_{k+1})G(q,J_E{v}_{k+1}).\hfill \end{array}$$

Moreover,

$$\begin{array}{c}G(q,J_E{y}_{k+1})\le {\parallel q\parallel }^2-2{\beta }_{k+1}⟨q,J_E{v}_1⟩-2(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i⟨q,J_E{T}_i^{(2)}{w}_{k+1}⟩\hfill \\ +{\beta }_{k+1}\parallel v_1{\parallel }^2+(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i{\parallel T_i^{(2)}{w}_{k+1}\parallel }^2+2\rho f(q)\hfill \\ ={\beta }_{k+1}\phi (q,v_1)+(1-{\beta }_{k+1}){\sum }_{i=1}^{\infty }{b}_i\phi (q,T_i^{(2)}{w}_{k+1})+2\rho f(q)\hfill \\ \le {\beta }_{k+1}\phi (q,v_1)+(1-{\beta }_{k+1})\phi (q,w_{k+1})+2\rho f(q)\hfill \\ ={\beta }_{k+1}G(q,J_E{v}_1)+(1-{\beta }_{k+1})G(q,J_E{w}_{k+1}).\hfill \end{array}$$

Then $q\in C_{k+2}.$ Therefore, by induction, $\varnothing \ne ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))\subset C_n,$ for all $n\in N.$

Step 2. $C_n$ is a convex and closed subset of $E.$

If $n=1$, the result is trivial. If $n\in N\backslash \{1\},$ then we notice that $G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n)$ is equivalent to $2{\alpha }_n⟨p,J_E{v}_1⟩+2(1-{\alpha }_n)⟨p,J_E{v}_n⟩-2⟨p,J_E{w}_n⟩\le {\alpha }_n\parallel v_1{\parallel }^2+(1-{\alpha }_n)\parallel v_n{\parallel }^2-{\parallel w_n\parallel }^2$ and $G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)$ is equivalent to $2{\beta }_n⟨p,J_E{v}_1⟩+2(1-{\beta }_n)⟨p,J_E{w}_n⟩-2⟨p,J_E{y}_n⟩\le {\beta }_n\parallel v_1{\parallel }^2+(1-{\beta }_n)\parallel w_n{\parallel }^2-{\parallel y_n\parallel }^2.$

Thus $C_n$ is closed and convex for $n\in N.$

Step 7. $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and $y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Steps 4 and 6 ensure that $u_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$. Then $v_n=u_n+t_n(u_n-u_{n-1})\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Since $u_{n+1}\in Q_{n+1}\subset C_{n+1}$, then $G(u_{n+1},J_E{w}_n)\le {\alpha }_{n}G(u_{n+1},J_E{v}_1)+(1-{\alpha }_n)G(u_{n+1},J_E{v}_n),$ which ensures that $0\le \phi (u_{n+1},w_n)\le {\alpha }_n\phi (u_{n+1},v_1)+(1-{\alpha }_n)\phi (u_{n+1},v_n)\to 0,$ since ${\alpha }_n\to 0,$ as $n\to \infty .$ Employing Lemma 2, $u_{n+1}-w_n\to 0$ and then $w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ when $n\to \infty$.

Since $u_{n+1}\in Q_{n+1}\subset C_{n+1}$, then $G(u_{n+1},J_E{y}_n)\le {\beta }_{n}G(u_{n+1},J_E{v}_1)+(1-{\beta }_n)G(u_{n+1},J_E{w}_n),$ which ensures that

$$\begin{array}{c}0\le \phi (u_{n+1},y_n)\le {\beta }_n\phi (u_{n+1},v_1)+(1-{\beta }_n)\phi (u_{n+1},w_n)\to 0,\hfill \end{array}$$

since ${\beta }_n\to 0,$ as $n\to \infty$. Thus $u_{n+1}-y_n\to 0$ and then $y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$ as $n\to \infty$.

Step 8. $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}).$

First, we shall show that $P_{{\bigcap }_{n=1}^{\infty }{U}_n}(u_1)\in Fix(T_1^{(1)}).$

For $\forall q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}),$ by using Lemma 5, we know that (12) in Theorem 3 is still true.

Since $w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],$ then $J_E{w}_n-J_E{v}_n={\alpha }_n(J_E{v}_1-J_E{v}_n)+(1-{\alpha }_n)({\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n-J_E{v}_n).$ Noticing the properties of $J_E$ and $J_E^{-1}$, $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$w_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and ${\alpha }_n\to 0,$ then ${\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n-J_E{v}_n\to 0,$ which implies that (13) is still true. Copy the corresponding part of Step 8 in Theorem 3, we have $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(1)}).$

Repeating the process for showing $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(1)})$, we have $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_m^{(1)}),$$\forall m\in N.$ Therefore, $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in {\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}).$

Next, we shall show that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in {\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}).$

For $\forall q\in ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}),$ (14) is still true.

Since $y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],$ then $J_E{y}_n-J_E{v}_n={\beta }_n(J_E{v}_1-J_E{v}_n)+(1-{\beta }_n)({\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n-J_E{v}_n).$ From the facts that $v_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0),$$y_n\to P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)$ and ${\beta }_n\to 0,$ we have: ${\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n-J_E{v}_n\to 0,$ which implies that (15) is still true. Copy the corresponding part of Step 8 in Theorem 3, we have $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(2)}).$

Repeating the process for showing $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_1^{(2)})$, we can also prove that $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in Fix(T_m^{(2)}),$ for all $m\in N.$ Therefore, $P_{{\bigcap }_{n=1}^{\infty }{C}_n}(u_0)\in {\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}).$

This completes the proof. □

Using similar methods, we can get the following theorems:

Theorem 5.

Assume that E is a real uniformly convex and also a uniformly smooth Banach space, $T_i^{(1)},T_i^{(2)}:E\to E$ are weakly relatively non-expansive mappings, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(17)

Under the assumptions of (i)–(iii) and (v) in Theorem 3 and the additional assumption that

${(iv)}^{″}$$\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two real number sequences in $[0,1)$ with $0\le su{p}_n{\beta }_n<1$ and ${\alpha }_n\to 0$, $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0),$ when $n\to \infty .$

Theorem 6.

Assume that E is a real uniformly convex and also a uniformly smooth Banach space, $T_i^{(1)},T_i^{(2)}:E\to E$ are weakly relatively non-expansive mappings, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(18)

Under the assumptions of (i)–(iii) and (v) in Theorem 3 and the additional assumption that

${(iv)}^{‴}$$\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two real number sequences in $[0,1)$ with $0\le su{p}_n{\alpha }_n<1$ and ${\beta }_n\to 0$, $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_0),$ as $n\to \infty .$

Remark 3.

Replace $u_0$ by $u_1$ in the set $Q_{n+1}$ in (11), (16)–(18), we can get the corresponding results of Theorems 3–6 that $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_1),$ as $n\to \infty .$

For example, modify (11) as follows:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{T}_i^{(1)}{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i^{(2)}{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_1{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_1)-u_1{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(19)

Under the assumptions of (i)–(v) in Theorem 3, one has: $u_n\to P_{({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))}(u_1),$ as $n\to \infty .$

4. Applications

Lemma 10

([10,11]). Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $A:E\to 2^{E^{*}}$ is a maximal monotone operator satisfying $A^{-1}0\ne \varnothing .$ Then, for all $x\in E,y\in A^{-1}0$ and $r>0,$ the result $\phi (y,{(J_E+rA)}^{-1}{J}_{E}x)+\phi ({(J_E+rA)}^{-1}{J}_{E}x,x)\le \phi (y,x)$ is true.

Lemma 11

([10,11]). If $A^{-1}0\ne \varnothing ,$ then under the assumptions of Lemma 10, one has ${(J_E+rA)}^{-1}{J}_E:E\to E$ is strongly relatively non-expansive, and $Fix({(J_E+rA)}^{-1}{J}_E)=A^{-1}0,$ for all $r>0.$

Remark 4.

Based on Lemma 11 and Theorems 3–6, we can get the following results.

Theorem 7.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $T_i:E\to E$ are weakly relatively non-expansive mappings and $A_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(20)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))\ne \varnothing ,$ under the assumptions (ii)–(v) in Theorem 3, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))}(u_0),$ as $n\to \infty .$

Theorem 8.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $T_i:E\to E$ are weakly relatively non-expansive mappings and $A_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(21)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))\ne \varnothing ,$ under the assumptions $(ii),(iii),{(iv)}^{\prime }$ and $(v)$ in Theorem 4, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))}(u_0),$ as $n\to \infty .$

Theorem 9.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $T_i:E\to E$ are weakly relatively non-expansive mappings and $A_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{C_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(22)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))\ne \varnothing ,$ under the assumptions $(ii),(iii),{(iv)}^{″}$ and $(v)$ in Theorem 5, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))}(u_0),$ as $n\to \infty .$

Theorem 10.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $T_i:E\to E$ are weakly relatively non-expansive mappings and $A_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{T}_i{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(23)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))\ne \varnothing ,$ under the assumptions $(ii),(iii),{(iv)}^{‴}$ and $(v)$ in Theorem 6, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i))}(u_0),$ as $n\to \infty .$

Remark 5.

Similar to the discussion of Theorems 7–10 and replace $T_i^{(2)}$ by ${(J_E+s{B}_i)}^{-1}{J}_E,$ where $B_i:E\to E^{*}$ is maximal monotone, we can get the following results:

Theorem 11.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $A_i,B_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{(J_E+s{B}_i)}^{-1}{J}_E{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(24)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)\ne \varnothing ,$ under the assumptions (ii)–(v) in Theorem 3, one has

$u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)}(u_0),$ as $n\to \infty .$

Theorem 12.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $A_i,B_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{(J_E+s{B}_i)}^{-1}{J}_E{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(25)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)\ne \varnothing ,$ under the assumptions $(ii),(iii),{(iv)}^{\prime }$ and $(v)$ in Theorem 4, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)}(u_0),$ as $n\to \infty .$

Theorem 13.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $A_i,B_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_1+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_n+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{(J_E+s{B}_i)}^{-1}{J}_E{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le {\alpha }_{n}G(p,J_E{v}_1)+(1-{\alpha }_n)G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_n)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(26)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)\ne \varnothing ,$$(ii),(iii),{(iv)}^{″}$ and $(v)$ in Theorem 5, one has $u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)}(u_0),$ as $n\to \infty .$

Theorem 14.

Assume that E is a real uniformly smooth and also a uniformly convex Banach space, $A_i,B_i:E\to E^{*}$ are maximal monotone operators, for all $i\in N$. Let $\{u_n\}$ be generated by the following inertial-type multi-choice iterative algorithm:

$$\left\{\begin{array}{c}{u}_0,u_1\in E,\hfill \\ v_n=u_n+t_n(u_n-u_{n-1}),\hfill \\ w_n=J_E^{-1}[{\alpha }_n{J}_E{v}_n+(1-{\alpha }_n){\sum }_{i=1}^{\infty }{a}_i{J}_E{(J_E+r{A}_i)}^{-1}{J}_E{v}_n],\hfill \\ y_n=J_E^{-1}[{\beta }_n{J}_E{v}_1+(1-{\beta }_n){\sum }_{i=1}^{\infty }{b}_i{J}_E{(J_E+r{B}_i)}^{-1}{J}_E{w}_n],\hfill \\ C_1=E=Q_1,\hfill \\ C_{n+1}=\{p\in C_n:G(p,J_E{w}_n)\le G(p,J_E{v}_n),\hfill \\ G(p,J_E{y}_n)\le {\beta }_{n}G(p,J_E{v}_1)+(1-{\beta }_n)G(p,J_E{w}_n)\},\hfill \\ Q_{n+1}=\{p\in C_{n+1}:\parallel u_0{-p\parallel }^2\le \parallel P_{U_{n+1}}(u_0)-u_0{\parallel }^2+{\lambda }_{n+1}\},\hfill \\ u_{n+1}\in Q_{n+1},n\in N.\hfill \end{array}\right.$$

(27)

If $({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)\ne \varnothing ,$$(ii),(iii),{(iv)}^{‴}$ and $(v)$ in Theorem 6, one has

$u_n\to P_{({\bigcap }_{i=1}^{\infty }{A}_i^{-1}0)\bigcap ({\bigcap }_{i=1}^{\infty }{B}_i^{-1}0)}(u_0),$ as $n\to \infty .$

Remark 6.

From Theorems 7–10, we can see that the main results of Theorems 3–6 in our paper are extensions of the corresponding results in [11,14,19].

Remark 7.

From Theorems 11–14, we can see that the main results of Theorems 3–6 in our paper can be further extended to the topic of designing iterative algorithms to approximate common zero points of two kinds of countable maximal monotone operators.

5. Numerical Experiments

In this part, some numerical experiments will be done to compare the performance of the new inertial-type iterative algorithms with non-inertial type algorithms in [11].

Example 1.

Let $E=(-\infty ,+\infty ).$ Suppose $T_i^{(1)},T_i^{(2)}:(-\infty ,+\infty )\to (-\infty ,+\infty )$ are defined as follows: $T_1^{(1)}x=x$ and $T_i^{(1)}x=\frac{5x}{3^i}$ for $i\in N\backslash \{1\}$, and $T_i^{(2)}x=\frac{x}{2^i}$ for $x\in (-\infty ,+\infty )$ and $i\in N.$ Then $\{T_i^{(1)}\}$ and $\{T_i^{(2)}\}$ are weakly relatively non-expansive mappings such that $({\bigcap }_{i=1}^{\infty }Fix(T_i^{(1)}))\bigcap ({\bigcap }_{i=1}^{\infty }Fix(T_i^{(2)}))=\{0\}.$ Let $a_i=\frac{1}{2^i}$, $b_i=\frac{2}{3^i}$ for $i\in N,$$t_n=\frac{2}{n}$, ${\alpha }_n={\beta }_n=\frac{1}{n}$ and ${\lambda }_{n+1}=\frac{1}{{(n+1)}^2}$ for $n\in N$. Then all of the conditions of Theorem 3 are satisfied for this special case.

Remark 8.

Take Example 1, we can choose the following iterative sequence $\{u_n\}$ from infinite choices based on iterative algorithm (19) in Remark 3:

$$\left\{\begin{array}{c}{u}_0=1,u_1=2=u_2,\hfill \\ v_n=u_n+\frac{2}{n}(u_n-u_{n-1}),n\in N,\hfill \\ w_n=\frac{2n+1}{3n}{v}_n,n\in N,\hfill \\ y_n=\frac{1}{n}{v}_n+\frac{2n-2}{5n}{w}_n,n\in N,\hfill \\ z_n=\frac{\frac{1}{n}{v}_n^2+\frac{n-1}{n}{w}_n^2-y_n^2}{\frac{2}{n}{v}_n+\frac{2n-2}{n}{w}_n-2y_n},n\in N\backslash \{1\},\hfill \\ wemaychoose{u}_{n+1}=\frac{u_1-\sqrt{{(u_1-z_n)}^2+\frac{1}{{(n+1)}^2}}+z_n}{2},n\in N\backslash \{1\}.\hfill \end{array}\right.(28)$$

With codes of Visual Basic Six, Figure 1 (see $u_{1(n)}$) is obtained.

Figure 1. Convergence of $\{u_n\}$ in (28) (denoted by $\{u_{1(n)}\}$) and $\{u_n\}$ in (29) (denoted by $\{u_{2(n)}\}$).

Remark 9.

Take Example 1, we can choose the following iterative sequence $\{u_n\}$ from infinite choices based on non-inertial type iterative algorithm (5):

$$\left\{\begin{array}{c}{u}_1=2,u_2=2,\hfill \\ v_n=\frac{2n+1}{3n}{u}_n,n\in N,\hfill \\ w_n=\frac{1}{n}{u}_n+\frac{2n-2}{5n}{v}_n,n\in N,\hfill \\ z_n=\frac{\frac{1}{n}{u}_n^2+\frac{n-1}{n}{v}_n^2-w_n^2}{\frac{2}{n}{u}_n+\frac{2n-2}{n}{v}_n-2w_n},n\in N\backslash \{1\},\hfill \\ wemaychoose{u}_{n+1}=\frac{u_1-\sqrt{{(u_1-z_n)}^2+\frac{1}{{(n+1)}^2}}+z_n}{2},n\in N\backslash \{1\}.\hfill \end{array}\right.(29)$$

With codes of Visual Basic Six, Figiure 1(see $u_{2(n)}$) is obtained.

Remark 10.

CFrom Figure 1, we may find that the inertial-type algorithm (28) converges faster than inertial-type algorithm (29) when n increases.