Modified block iterative procedure for solving the common solution of fixed point problems for two countable families of total quasi-ϕ-asymptotically nonexpansive mappings with applications

Abstract

In this paper, we introduce a new iterative procedure which is constructed by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Finally, we apply the problem of a strong convergence theorem concerning maximal monotone operators in Banach spaces.

MSC:47H09, 47H10, 47H20, 47J20.

1 Introduction

Throughout this paper, we assume that E is a real Banach space, $E^{\ast }$ is the dual space of E. Let C be a nonempty, closed, and convex subset of E and $〈\cdot ,\cdot 〉$ be the pairing between E and $E^{\ast }$. We denote the strong convergence and weak convergence of a sequence $\{x_n\}$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively, and $J:E\to 2^{E^{\ast }}$ is the normalized duality mapping defined by

$$Jx=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^2={\parallel f\parallel }^2\},\mathrm{\forall }x\in E.$$

(1.1)

In the sequel, we use $F(T)$ to denote the set of fixed points of a mapping T and use ℝ and ${\mathbb{R}}^{+}$ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively.

Definition 1.1 Let E be a Banach space.

• (1) E is said to be strictly convex if $\frac{\parallel x+y\parallel }{2}<1$ for all $x,y\in U_E=\{z\in E:\parallel z\parallel =1\}$ with $x\ne y$.

• (2) E is said to be uniformly convex if for each $\epsilon \in (0,2]$, there exists $\delta >0$ such that $\frac{\parallel x+y\parallel }{2}\le 1-\delta$ for all $x,y\in U_E$ with $\parallel x-y\parallel >\epsilon$.

• (3) E is said to be smooth if the limit (1.2)

  $$\underset{t⟶0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

  (1.2)

exists for each $x,y\in U_E$.

• (4) E is said to be uniformly smooth if the limit (1.2) is attained uniformly for all $x,y\in U_E$.

Remark 1.2 The basic properties below hold (see [1, 2]).

• (1) If E is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

• (2) If E is a strictly convex reflexive Banach space, then $J^{-1}$ is hemicontinuous, that is, $J^{-1}$ is norm-to-weak^∗-continuous.

• (3) If E is a smooth and strictly convex reflexive Banach space, then J is single-valued, one-to-one, and onto.

• (4) A Banach space E is uniformly smooth if and only if $E^{\ast }$ is uniformly convex.

• (5) Each uniformly convex Banach space E has the Kadec-Klee property; that is, for any sequence $\{x_n\}\subset E$, if $\{x_n\}\rightharpoonup x\in E$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $x_n\to x$.

• (6) A Banach space E is strictly convex if and only if J is strictly monotone; that is,

  $$〈x-y,x^{\ast }-y^{\ast }〉>0,\text{whenever }x,y\in E,x\ne y,\text{ and }{x}^{\ast }\in Jx,y^{\ast }\in Jy.$$

• (7) Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.

• (8) $E^{\ast }$ is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.

Let E be a smooth and strictly convex reflexive Banach space, and let C be a nonempty, closed, and convex subset of E. We assume that the Lyapunov functional $\varphi :E\times E\to {\mathbb{R}}^{+}$ is defined by [3, 4]

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,\mathrm{\forall }x,y\in E.$$

From the definition of ϕ, it is easy to see that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2.$$

(1.3)

Let C be a nonempty, closed, and convex subset of E. For each $x\in E$, the generalized projection [3]${\mathrm{\Pi }}_C:E\to C$ is defined by

$${\mathrm{\Pi }}_C(x)=arg\underset{y\in C}{min}\varphi (x,y).$$

Lemma 1.3 [3, 4]

If C is a nonempty, closed, and convex subset of a smooth and strictly convex reflexive real Banach space E, then

• (1) for $x\in E$ and $u\in C$, one has

  $$u={\mathrm{\Pi }}_C(x)\iff 〈u-y,Jx-Ju〉\ge 0,\mathrm{\forall }y\in C.$$

• (2) $\varphi (x,{\mathrm{\Pi }}_C(y))+\varphi ({\mathrm{\Pi }}_C(y),y)\le \varphi (x,y)$, $\mathrm{\forall }x\in C$, $y\in E$.

• (3) $\varphi (x,y)=0$ if and only if $x=y$, $\mathrm{\forall }x,y\in C$.

Remark 1.4 If E is a real Hilbert space H, then $\varphi (x,y)={\parallel x-y\parallel }^2$ and ${\mathrm{\Pi }}_C=P_C$ (the metric projection of H onto C).

Definition 1.5 Let E be a smooth, strictly convex, and reflexive real Banach space, C be a nonempty, closed, and convex subset of E, $T:C\to C$ be a mapping, and $Fix(T)$ be the set of fixed points of T.

• (1) A point $p\in C$ is said to be an asymptotic fixed point of T if there exists a sequence $\{x_n\}\subset C$ such that $x_n\rightharpoonup p$ and $\parallel x_n-T{x}_n\parallel \to 0$. We denote the set of all asymptotic fixed points of T by $\hat{F}(T)$.

• (2) A point $p\in C$ is said to be a strong asymptotic fixed point of T if there exists a sequence $\{x_n\}\subset C$ such that $x_n\to p$ and $\parallel x_n-T{x}_n\parallel \to 0$. We denote the set of all strong asymptotic fixed points of T by $\tilde{F}(T)$.

Definition 1.6 Let E be a smooth, strictly convex, and reflexive real Banach space, and let C be a nonempty, closed, and convex subset of E.

• (1) A mapping $T:C\to C$ is said to be closed if for each $\{x_n\}\subset C$ with $x_n\to x$ and $T{x}_n\to y$, then $Tx=y$.

• (2) A mapping $T:C\to C$ is said to be relatively nonexpansive [5, 6] if $Fix(T)\ne \mathrm{\varnothing }$, $Fix(T)=\hat{F}(T)$, and

  $$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in Fix(T).$$

• (3) A mapping $T:C\to C$ is said to be weak relatively nonexpansive [7] if $Fix(T)\ne \mathrm{\varnothing }$, $Fix(T)=\tilde{F}(T)$, and

  $$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in Fix(T).$$

• (4) A mapping $T:C\to C$ is said to be quasi-ϕ-nonexpansive (relatively quasi-nonexpansive) if $Fix(T)\ne \mathrm{\varnothing }$ and

  $$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in Fix(T).$$

• (5) A mapping $T:C\to C$ is said to be quasi-ϕ-asymptotically nonexpansive (asymptotically relatively nonexpansive) if $Fix(T)\ne \mathrm{\varnothing }$ and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that

  $$\varphi (p,T^{n}x)\le k_n\varphi (p,x),\mathrm{\forall }x\in C,p\in Fix(T),\text{ and }\mathrm{\forall }n\ge 1.$$

• (6) A mapping $T:C\to C$ is said to be total quasi-ϕ-asymptotically nonexpansive if $Fix(T)\ne \varphi$ and there exists nonnegative real sequences $\{{\nu }_n\}$ and $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\zeta (0)=0$ such that

  $$\varphi (p,T^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in Fix(T),\text{ and }\mathrm{\forall }n\ge 1.$$

Remark 1.7 From Definition 1.6, it is easy to know that

• (1) every relatively nonexpansive mapping is closed;

• (2) every quasi-ϕ-asymptotically nonexpansive mapping is a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;

• (3) every quasi-ϕ-nonexpansive mapping is a quasi-ϕ-asymptotically nonexpansive mapping with $\{k_n=1\}$, but the converse is not true;

• (4) every weak relatively nonexpansive mapping is a quasi-ϕ-nonexpansive mapping because it does not require the condition $Fix(T)=\tilde{F}(T)$, but the converse is not true;

• (5) every relatively nonexpansive mapping is a weak relatively nonexpansive mapping, but the converse is not true.

Regarding the iterative methods of nonlinear operator equations for relatively nonexpansive mappings, in 2005 Matsushita and Takahashi [5] and in 2008 Plubtieng and Ungchittrakool [6] proved the following result, respectively.

Theorem MT Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed and convex subset of E. Let $T:C\to C$ be a relatively nonexpansive mapping, and let $\{{\alpha }_n\}$ be a real sequence in $[0,1)$ with ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$. Let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)JT{x}_n),\hfill \\ C_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ Q_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_n\cap Q_n}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(1.4)

where J is the duality mapping on E. If $Fix(T)\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{Fix(T)}(x_0)$, where ${\mathrm{\Pi }}_{Fix(T)}(\cdot )$ is the generalized projection from C onto $Fix(T)$.

Theorem PU Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty, closed, and convex subset of E. Let $T,S:C\to C$ be two relatively nonexpansive mappings with $\mathrm{\Omega }:=Fix(T)\cap Fix(S)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)JT{z}_n),\hfill \\ z_n=J^{-1}({\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}JT{x}_n+{\beta }_n^{(3)}JS{x}_n),\hfill \\ H_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)+{\alpha }_n({\parallel x\parallel }^2+2〈z,J{x}_n-J{x}_0〉)\},\hfill \\ W_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{H_n\cap W_n}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(1.5)

with the following restrictions:

• (1) $0<{\alpha }_n<1$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (2) $0\le {\beta }_n^{(1)},{\beta }_n^{(2)},{\beta }_n^{(3)}\le 1$, ${lim}_{n\to \mathrm{\infty }}{\beta }_n^{(1)}=0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n^{(2)}{\beta }_n^{(3)}>0$.

Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$, where ${\mathrm{\Pi }}_{\mathrm{\Omega }}(\cdot )$ is the generalized projection from C onto Ω.

In 2010, Su et al. [7] introduced the concept of a countable family of weak relatively nonexpansive mappings and proved the following theorem which extends and improves Theorem MT and Theorem PU.

Theorem SXZ Let E be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty, closed, and convex subset of E. Let ${\{T_n\}}_{n=0}^{\mathrm{\infty }},{\{S_n\}}_{n=0}^{\mathrm{\infty }}:C\to C$ be two countable families of weak relatively nonexpansive mappings such that $\mathrm{\Omega }:=({\bigcap }_{n=0}^{\mathrm{\infty }}Fix(T_n))\cap ({\bigcap }_{n=0}^{\mathrm{\infty }}Fix(S_n))\ne \mathrm{\varnothing }$.

Let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}J{T}_n{x}_n+{\beta }_n^{(3)}J{S}_n{x}_n),\hfill \\ C_n=\{z\in C_{n-1}\cap Q_{n-1}:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ C_0=\{z\in C:\varphi (z,y_0)\le \varphi (z,x_0)\},\hfill \\ Q_n=\{z\in C_{n-1}\cap Q_{n-1}:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ Q_0=C,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_n\cap Q_n}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(1.6)

with the conditions:

• (1) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$;

• (2) ${\beta }_n^{(1)},{\beta }_n^{(2)},{\beta }_n^{(3)}\in (0,1)$ such that ${\beta }_n^{(1)}+{\beta }_n^{(2)}+{\beta }_n^{(3)}=1$ for each $n\ge 1$;

• (3) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n^{(1)}{\beta }_n^{(2)}>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n^{(2)}{\beta }_n^{(3)}>0$.

Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$, where ${\mathrm{\Pi }}_{\mathrm{\Omega }}(\cdot )$ is the generalized projection from C onto Ω.

In 2011, Chang et al. [8] proved some approximation theorems for common fixed points of countable families of total quasi-ϕ-asymptotically nonexpansive mappings which contain several kinds of mappings as their special cases in Banach spaces. Next, Chang et al. [9] modified the Halpern-type iteration algorithm for a total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. Recently, Chang, Lee, and Chan [10] introduced a block hybrid projection algorithm for solving the convex feasibility problem and the generalized equilibrium problems for an infinite family of total quasi-ϕ-asymptotically nonexpansive mappings and they proved strong convergence theorems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

Theorem CLCY Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_i$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\mu }_1=0$, ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and $\zeta (0)=0$. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},C_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{z}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(1.7)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (p,x_n))+{\mu }_n$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$. Let $\{{\beta }_{n,0}\}$, $\{{\beta }_{n,i}\}$, and $\{{\alpha }_n\}$ be sequences in $[0,1]$ satisfying the following conditions:

• (1) for each $n\ge 0$, ${\beta }_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$ for all $i\ge 1$;

• (3) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$.

If $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i)$ is a nonempty and bounded subset of C, then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Theorem CLCZ Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let ${\{T_m\}}_{m=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_m$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\mu }_1=0$, ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and $\zeta (0)=0$. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_1\in E\mathit{\text{chosen arbitrarily}},C_1=C,\hfill \\ y_{n,m}=J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{T}_m^n{x}_n),m\ge 1,\hfill \\ C_{n+1}=\{z\in C_n:{sup}_{m\ge 1}\varphi (z,y_{n,m})\le {\alpha }_n\varphi (z,x_1)+(1-\alpha )\varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_1),\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(1.8)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (p,x_n))+{\mu }_n$, and ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$. If $\mathrm{\Omega }:={\bigcap }_{m=1}^{\mathrm{\infty }}Fix(T_m)$ is a nonempty and bounded subset of C, then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_1)$.

Theorem CLC Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_i$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\mu }_1=0$, ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and $\zeta (0)=0$. Let $B_i:C\to E^{\ast }$ ($i=1,2,3,\dots ,N$) be a finite family of continuous and monotone mappings. Let ${\psi }_i:C\to \mathbb{R}$ ($i=1,2,3,\dots ,N$) be a finite family of lower semi-continuous and convex functions, and let $F_i:C\times C\to \mathbb{R}$ ($i=1,2,3,\dots ,N$) be a finite family of bifunctions satisfying the conditions (A1)-(A4). Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^N{\mathfrak{F}}_i)\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is a nonempty and bounded subset of C, where ${\mathfrak{F}}_i$ ($i=1,2,3,\dots ,N$) is the set of the following generalized mixed quasi-equilibrium problems:

$$F_i(u,y)+〈B_{i}u,y-u〉+{\psi }_i(y)-{\psi }_i(u)\ge 0,\mathrm{\forall }y\in C,i=1,2,3,\dots ,N.$$

Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{x}_n),\hfill \\ u_n^{(i)}=K_{f_i,r_i}{K}_{f_{i-1},r_{i-1}}\cdot \cdot \cdot K_{f_2,r_2}{K}_{f_1,r_1}{y}_n,i=1,2,3,\dots ,N,\hfill \\ C_{n+1}=\{v\in C_n:{max}_{i=1,2,3,\dots ,N}\varphi (v,u_n^{(i)})\le \varphi (v,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(1.9)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (u,x_n))+{\mu }_n$, $\mathrm{\forall }n\ge 1$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$. Let $\{{\alpha }_{n,i}\}$, $\{{\beta }_n\}$ be sequences in $[0,1]$ satisfying the following conditions:

• (1) for each $n\ge 0$, ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\beta }_n){\alpha }_{n,0}{\alpha }_{n,i}>0$ for all $i\ge 1$.

Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

In this paper, motivated and inspired by the previously mentioned results, we introduce a new iterative procedure by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove a strong convergence theorem of the iterative procedure generated by these conditions. The results obtained in this paper extend and improve several recent results in this area.

2 Preliminaries

Definition 2.1 Let C be a nonempty, closed, and convex subset of a real Banach space E.

• (1) A mapping $T:C\to C$ is said to be nonexpansive if

  $$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

• (2) A mapping $T:C\to C$ is said to be uniformly L-Lipschitz continuous if there exists a constant $L>0$ such that

  $$\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,\mathrm{\forall }n\ge 1.$$

Definition 2.2 [11]

Let C be a nonempty, closed, and convex subset of a real Banach space E.

• (1) A countable family of mappings ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ is said to be a uniformly quasi-ϕ-asymptotically nonexpansive mapping if ${\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i)\ne \mathrm{\varnothing }$ and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that for each $i\ge 1$,

  $$\varphi (p,T_i^{n}x)\le k_n\varphi (p,x),\mathrm{\forall }x\in C,p\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(T_i),\text{ and }\mathrm{\forall }n\ge 1.$$

• (2) A countable family of mappings ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ is said to be a uniformly total quasi-ϕ-asymptotically nonexpansive mapping if ${\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i)\ne \mathrm{\varnothing }$ and there exist nonnegative real sequences $\{{\nu }_n\}$ and $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\zeta (0)=0$ such that for each $i\ge 1$,

  $$\varphi (p,T_i^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(T_i),\text{ and }\mathrm{\forall }n\ge 1.$$

Lemma 2.3 [4]

Let E be a uniformly smooth and strictly convex real Banach space, and let $\{x_n\}$ and $\{y_n\}$ be two sequences of E. If $\varphi (x_n,y_n)\to 0$ and either $\{x_n\}$ or $\{y_n\}$ is bounded, then $\parallel x_n-y_n\parallel \to 0$.

Lemma 2.4 [8]

Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. Let $\{x_n\}$ and $\{y_n\}$ be two sequences in C and $p\in E$. If $x_n\to p$ and $\varphi (x_n,y_n)\to 0$, then $y_n\to p$.

Lemma 2.5 [8]

Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. Let $T:C\to C$ be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing continuous function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), and $\zeta (0)=0$. If ${\mu }_1=0$, then the fixed point set $Fix(T)$ is a closed and convex subset of C.

Lemma 2.6 [11]

Let E be a uniformly convex Banach space, $r>0$ be a positive number, and $B_r(0)$ be a closed ball of E. Then for any given sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}\subset B_r(0)$ and for any given ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}\subset (0,1)$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n=1$, there exists a continuous, strictly increasing, and convex function g : $[0,2r)\to [0,\mathrm{\infty })$ with $g(0)=0$ such that for any positive integers i, j with $i<j$,

$${\parallel \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{\parallel x_n\parallel }^2-{\lambda }_i{\lambda }_{j}g(\parallel x_i-x_j\parallel ).$$

3 Main results

In this section, we shall use the modified block hybrid projection method to study a common solution of fixed point problems for two countable families of closed and $L_i,{\ell }_i$-Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. For the purpose, we assume the following hypotheses.

(A1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_i$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ and a strictly increasing function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), ${\mu }_1=0$, and $\zeta (0)=0$, and for each $i\ge 1$,

$$\varphi (p,T_i^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(T_i),\text{ and }\mathrm{\forall }n\ge 1.$$

(A2) Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly ${\ell }_i$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\kappa }_n\}$, $\{{\omega }_n\}$ and a strictly increasing function $\rho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\kappa }_n\to 0$, ${\omega }_n\to 0$ (as $n\to \mathrm{\infty }$), ${\omega }_1=0$, and $\rho (0)=0$, and for each $i\ge 1$,

$$\varphi (p,S_i^{n}x)\le \varphi (p,x)+{\kappa }_n\rho (\varphi (p,x))+{\omega }_n,\mathrm{\forall }x\in C,p\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(S_i),\text{ and }\mathrm{\forall }n\ge 1.$$

Theorem 3.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ and ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ satisfy the above conditions (A1)-(A2), respectively.

Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.1)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (p,x_n))+{\mu }_n$ and ${\theta }_n={\kappa }_n{sup}_{p\in \mathrm{\Omega }}\rho (\varphi (p,x_n))+{\omega }_n$.

Let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$ and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$, and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof We shall complete this proof of Theorem 3.1 in seven steps.

Step 1. We will show that Ω and $C_{n+1}$ are closed and convex for each $n\ge 0$.

In fact, it follows from Lemma 2.5 that $Fix(T_i)$ and $Fix(S_i)$, for any $i\ge 1$, are closed and convex subsets of C. Therefore, Ω is closed and convex in C.

Clearly, $C_0=C$ is closed and convex. Suppose that $C_n$ is closed and convex for some $n\ge 1$.

By the assumption of $C_{n+1}$,

$$\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n$$

is equivalent to

$${\parallel v\parallel }^2-2〈v,J{y}_n〉+{\parallel y_n\parallel }^2\le {\parallel v\parallel }^2-2〈v,J{x}_n〉+{\parallel x_n\parallel }^2+{\xi }_n.$$

So that $2〈v,J{x}_n〉-2〈v,J{y}_n〉=2〈v,J{x}_n-J{y}_n〉\le {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2+{\xi }_n$.

Again, by the assumption of $C_{n+1}$,

$$\varphi (v,z_n)\le \varphi (v,x_n)+{\theta }_n$$

is equivalent to

$${\parallel v\parallel }^2-2〈v,J{z}_n〉+{\parallel z_n\parallel }^2\le {\parallel v\parallel }^2-2〈v,J{x}_n〉+{\parallel x_n\parallel }^2+{\theta }_n.$$

So that $2〈v,J{x}_n〉-2〈v,J{z}_n〉=2〈v,J{x}_n-J{z}_n〉\le {\parallel x_n\parallel }^2-{\parallel z_n\parallel }^2+{\theta }_n$.

Hence, $C_{n+1}=\{v\in C_n:2〈v,J{x}_n-J{y}_n〉\le {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2+{\xi }_n\text{ and }2〈v,J{x}_n-J{z}_n〉\le {\parallel x_n\parallel }^2-{\parallel z_n\parallel }^2+{\theta }_n\}$ is closed and convex.

Step 2. We will show that $\{x_n\}$ is bounded and $\{\varphi (x_n,x_0)\}$ is a convergent sequence for all $n\ge 1$.

Indeed, it follows from (3.1) and Lemma 1.3(2) that

$$\begin{array}{rcl}\varphi (x_n,x_0)& =& \varphi ({\mathrm{\Pi }}_{C_n}(x_0),x_0)\\ \le & \varphi (p,x_0)-\varphi (p,{\mathrm{\Pi }}_{C_n}(x_0))\\ \le & \varphi (p,x_0),\mathrm{\forall }n\ge 0,p\in \mathrm{\Omega }.\end{array}$$

This implies that $\{\varphi (x_n,x_0)\}$ is bounded. By virtue of (1.3), the sequence $\{x_n\}$ is also bounded.

By the assumption of $C_n$, we have $C_{n+1}\subset C_n$, $x_n={\mathrm{\Pi }}_{C_n}(x_0)$ and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0)$.

This implies that $x_{n+1}\in C_{n+1}\subset C_n$ and

$$\varphi (x_n,x_0)\le \varphi (x_{n+1},x_0),\mathrm{\forall }n\ge 0.$$

Therefore, $\{\varphi (x_n,x_0)\}$ is a convergent sequence. Without loss of generality, we can assume that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_0)=d\ge 0.$$

(3.2)

Step 3. We will show that $\mathrm{\Omega }\subset C_n$ for all $n\ge 0$.

It is obvious that $\mathrm{\Omega }\subset C_0=C$. Suppose that $\mathrm{\Omega }\subset C_n$ for some $n\ge 1$. For any given $p\in \mathrm{\Omega }$, from (3.1) and Lemma 2.6, we compute

$$\begin{array}{rcl}\varphi (p,y_n)& =& \varphi (p,J^{-1}({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n))\\ =& {\parallel p\parallel }^2-2〈p,{\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n〉+{\parallel {\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\beta }_{n,0}〈p,J{x}_n〉-2\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}〈p,J{T}_i^n{x}_n〉\\ +{\beta }_{n,0}{\parallel x_n\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}{\parallel T_i^n{x}_n\parallel }^2\\ -{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ =& {\beta }_{n,0}\varphi (p,x_n)+(1-{\beta }_{n,0}){\parallel p\parallel }^2\\ -2\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}〈p,J{T}_i^n{x}_n〉+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}{\parallel T_i^n{x}_n\parallel }^2\\ -{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ =& {\beta }_{n,0}\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}\varphi (p,T_i^n{x}_n)-{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ \le & {\beta }_{n,0}\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}[\varphi (p,x_n)+{\nu }_n\zeta (\varphi (p,x_n))+{\mu }_n]\\ -{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ =& ({\beta }_{n,0}+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i})\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}[{\nu }_n\zeta (\varphi (p,x_n))+{\mu }_n]\\ -{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ \le & \varphi (p,x_n)+{\nu }_n\zeta (\varphi (p,x_n))+{\mu }_n-{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ \le & \varphi (p,x_n)+{\nu }_n\underset{p\in \mathrm{\Omega }}{sup}\zeta (\varphi (p,x_n))+{\mu }_n-{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ =& \varphi (p,x_n)+{\xi }_n-{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right).\end{array}$$

(3.3)

It follows that

$$\varphi (p,y_n)\le \varphi (p,x_n)+{\xi }_n,\text{where }{\xi }_n={\nu }_n\underset{p\in \mathrm{\Omega }}{sup}\zeta (\varphi (p,x_n))+{\mu }_n.$$

(3.4)

From (3.1) and Lemma 2.6, we compute

$$\begin{array}{rcl}\varphi (p,z_n)& =& \varphi (p,J^{-1}({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{y}_n))\\ =& {\parallel p\parallel }^2-2〈p,{\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{y}_n〉+{\parallel {\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{y}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\alpha }_{n,0}〈p,J{x}_n〉-2\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}〈p,J{S}_i^n{y}_n〉+{\alpha }_{n,0}{\parallel x_n\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}{\parallel S_i^n{y}_n\parallel }^2\\ -{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ =& {\alpha }_{n,0}\varphi (p,x_n)+(1-{\alpha }_{n,0}){\parallel p\parallel }^2-2\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}〈p,J{S}_i^n{y}_n〉+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}{\parallel S_i^n{y}_n\parallel }^2\\ -{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ =& {\alpha }_{n,0}\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}\varphi (p,S_i^n{y}_n)-{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ \le & {\alpha }_{n,0}\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}[\varphi (p,x)+{\kappa }_n\rho (\varphi (p,x))+{\omega }_n]\\ -{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ =& ({\alpha }_{n,0}+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i})\varphi (p,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}[{\kappa }_n\rho (\varphi (p,x))+{\omega }_n]\\ -{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ \le & \varphi (p,x_n)+{\kappa }_n\rho (\varphi (p,x))+{\omega }_n-{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ \le & \varphi (p,x_n)+{\kappa }_n\underset{p\in \mathrm{\Omega }}{sup}\rho (\varphi (p,x))+{\omega }_n-{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\\ =& \varphi (p,x_n)+{\theta }_n-{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right).\end{array}$$

(3.5)

It follows that

$$\varphi (p,z_n)\le \varphi (p,x_n)+{\theta }_n,\text{where }{\theta }_n={\kappa }_n\underset{p\in \mathrm{\Omega }}{sup}\rho (\varphi (p,x))+{\omega }_n.$$

(3.6)

By the assumptions of $\{{\nu }_n\}$, $\{{\mu }_n\}$, $\{{\kappa }_n\}$, and $\{{\omega }_n\}$, and from (3.3) and (3.5), we obtain

$${\xi }_n={\nu }_n\underset{p\in \mathrm{\Omega }}{sup}\zeta (\varphi (p,x_n))+{\mu }_n\to 0(\text{as }n\to \mathrm{\infty }),$$

(3.7)

and

$${\theta }_n={\kappa }_n\underset{p\in \mathrm{\Omega }}{sup}\rho (\varphi (p,x_n))+{\omega }_n\to 0(\text{as }n\to \mathrm{\infty }).$$

(3.8)

So, we get $p\in C_{n+1}$. This implies that $\mathrm{\Omega }\subset C_n$ for all $n\ge 0$, and the sequence $\{x_n\}$ is well defined.

Step 4. We will show that there exists some point $p^{\ast }\in C$ such that $x_n\to p^{\ast }$.

In fact, since $\{x_n\}$ is bounded and E is reflexive, then there exists a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $x_{n_i}\rightharpoonup p^{\ast }$ (some point in C).

Since $C_n$ is closed and convex and $C_{n+1}\subset C_n$, it follows that $C_n$ is weakly closed and $p^{\ast }\in C_n$ for each $n\ge 0$.

In view of $x_{n_i}={\mathrm{\Pi }}_{C_{n_i}}(x_0)$, we have

$$\varphi (x_{n_i},x_0)\le \varphi (p^{\ast },x_0),\mathrm{\forall }{n}_i\ge 0.$$

Since the norm $\parallel \cdot \parallel$ is weakly lower semi-continuous, we have

$$\begin{array}{rcl}\underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_{n_i},x_0)& =& \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\{{\parallel x_{n_i}\parallel }^2-2〈x_{n_i},J{x}_0〉+{\parallel x_0\parallel }^2\}\\ \ge & {\parallel p^{\ast }\parallel }^2-2〈p^{\ast },J{x}_0〉+{\parallel x_0\parallel }^2=\varphi (p^{\ast },x_0),\end{array}$$

and so

$$\varphi (p^{\ast },x_0)\le \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_{n_i},x_0)\le \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (x_{n_i},x_0)\le \varphi (p^{\ast },x_0).$$

This implies that ${lim}_{n_i\to \mathrm{\infty }}\varphi (x_{n_i},x_0)=\varphi (p^{\ast },x_0)$, and so $\parallel x_{n_i}\parallel \to \parallel p^{\ast }\parallel$. Since $x_{n_i}\rightharpoonup p^{\ast }$, and by virtue of the Kadec-Klee property of E, we obtain

$$\underset{n_i\to \mathrm{\infty }}{lim}{x}_{n_i}=p^{\ast },\text{as }{n}_i\to \mathrm{\infty }.$$

The sequence $\{\varphi (x_n,x_0)\}$ is convergent, and ${lim}_{n_i\to \mathrm{\infty }}\varphi (x_{n_i},x_0)=\varphi (p^{\ast },x_0)$, which implies that ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_0)=\varphi (p^{\ast },x_0)$. If there exists some subsequence $\{x_{n_j}\}\subset \{x_n\}$ such that $x_{n_j}\rightharpoonup q$, then from Lemma 1.3(2) we have

$$\begin{array}{rcl}\varphi (p^{\ast },q)& =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},x_{n_j})\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},{\mathrm{\Pi }}_{C_{n_j}}(x_0))\\ \le & \underset{n_i,n_j\to \mathrm{\infty }}{lim}[\varphi (x_{n_i},x_0)-\varphi ({\mathrm{\Pi }}_{C_{n_j}}(x_0),x_0)]\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}[\varphi (x_{n_i},x_0)-\varphi (x_{n_j},x_0)]\\ =& \varphi (p^{\ast },x_0)-\varphi (p^{\ast },x_0)=0.\end{array}$$

This implies that $p^{\ast }=q$, and so

$$\underset{n\to \mathrm{\infty }}{lim}{x}_n=p^{\ast }.$$

(3.9)

Step 5. We will show that ${lim}_{n\to \mathrm{\infty }}\parallel J{x}_n-J{y}_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel J{x}_n-J{z}_n\parallel =0$ as $n\to \mathrm{\infty }$.

Since $x_{n+1}\in C_{n+1}\subset C_n$, by the definition of $C_{n+1}$, we have

$$\begin{array}{rcl}\varphi (x_{n+1},x_n)& =& \varphi (x_{n+1},{\mathrm{\Pi }}_{C_n}(x_0))\\ \le & \varphi (x_{n+1},x_0)-\varphi ({\mathrm{\Pi }}_{C_n}(x_0),x_0)\\ =& \varphi (x_{n+1},x_0)-\varphi (x_n,x_0).\end{array}$$

(3.10)

Since ${lim}_{n⟶\mathrm{\infty }}\varphi (x_n,x_0)$ exists, and we are taking $n\to \mathrm{\infty }$ in (3.10), then $\varphi (x_{n+1},x_n)\to 0$.

It follows from Lemma 2.3 that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.11)

By the definition of $C_{n+1}$ and $x_{n+1}\in C_{n+1}$, we get

$$\begin{array}{r}\varphi (x_{n+1},y_n)\le \varphi (x_{n+1},x_n)+{\xi }_n,\text{and}\\ \varphi (x_{n+1},z_n)\le \varphi (x_{n+1},x_n)+{\theta }_n.\end{array}$$

(3.12)

From $\varphi (x_{n+1},x_n)\to 0$, ${\xi }_n\to 0$, and ${\theta }_n\to 0$, as $n\to \mathrm{\infty }$, we obtain

$$\varphi (x_{n+1},y_n)\to 0,\text{and}\varphi (x_{n+1},z_n)\to 0,\text{as }n\to \mathrm{\infty }.$$

(3.13)

Since ${lim}_{n⟶\mathrm{\infty }}{x}_n=p^{\ast }$, by virtue of Lemma 2.4, we get

$$\underset{n\to \mathrm{\infty }}{lim}{y}_n=p^{\ast },\text{and}\underset{n\to \mathrm{\infty }}{lim}{z}_n=p^{\ast }.$$

(3.14)

It follows from (3.9) and (3.14) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-y_n\parallel =0,\text{and}\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-z_n\parallel =0.$$

(3.15)

Since J is uniformly continuous on each bounded subset of E, then

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_n-J{y}_n\parallel =0,\text{and}\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_n-J{z}_n\parallel =0.$$

(3.16)

Step 6. We will show that $p^{\ast }\in \mathrm{\Omega }$, where $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$.

(6.1) First, we will show that $p^{\ast }\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i)$.

For any $i\ge 1$ and for any $p\in \mathrm{\Omega }$, it follows from (3.3), (3.7), (3.15), and (3.16) that

$$\begin{array}{rcl}{\beta }_{n,0}{\beta }_{n,i}g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)& \le & \varphi (p,x_n)-\varphi (p,y_n)+{\xi }_n\\ =& {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2-2〈p,J{x}_n-J{y}_n〉+{\xi }_n\\ =& (\parallel x_n\parallel +\parallel y_n\parallel )(\parallel x_n\parallel -\parallel y_n\parallel )-2〈p,J{x}_n-J{y}_n〉+{\xi }_n\\ \le & (\parallel x_n\parallel +\parallel y_n\parallel )(\parallel x_n-y_n\parallel )-2\parallel p\parallel \parallel J{x}_n-J{y}_n\parallel +{\xi }_n\\ \to & 0,\text{as }n\to \mathrm{\infty }.\end{array}$$

By the condition ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$, we obtain

$$g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\to 0,\text{as }n\to \mathrm{\infty }.$$

It follows from the property of g that

$$\parallel J{x}_n-J{T}_i^n{x}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

(3.17)

Since $x_n\to p^{\ast }$ and J is uniformly continuous on each bounded subset of E, it yields that $J{x}_n\to J{p}^{\ast }$.

Hence, from (3.17) we get

$$J{T}_i^n{x}_n\to J{p}^{\ast },\text{as }n\to \mathrm{\infty },\mathrm{\forall }i\ge 1.$$

(3.18)

Since $J^{-1}:E^{\ast }\to E$ is norm-to-weak^∗-continuous, we also have

$$T_i^n{x}_n\rightharpoonup p^{\ast },\text{as }n\to \mathrm{\infty },\mathrm{\forall }i\ge 1.$$

(3.19)

Again, since for any $i\ge 1$,

$$|\parallel T_i^n{x}_n\parallel -\parallel p^{\ast }\parallel |=|\parallel J\left(T_i^n{x}_n\right)\parallel -\parallel J{p}^{\ast }\parallel |\le \parallel J\left(T_i^n{x}_n\right)-J{p}^{\ast }\parallel \to 0,\text{as }n\to \mathrm{\infty },$$

from (3.19) and the Kadec-Klee property of E, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}{T}_i^n{x}_n=p^{\ast }.$$

(3.20)

On the other hand, by the assumption that for each $i\ge 1$, $T_i$ is uniformly $L_i$-Lipschitz continuous, we get

$$\begin{array}{rcl}\parallel T_i^{n+1}{x}_n-T_i^n{x}_n\parallel & \le & \parallel T_i^{n+1}{x}_n-T_i^{n+1}{x}_{n+1}\parallel +\parallel T_i^{n+1}{x}_{n+1}-x_{n+1}\parallel \\ +\parallel x_{n+1}-x_n\parallel +\parallel x_n-T_i^n{x}_n\parallel \\ \le & (L_i+1)\parallel x_{n+1}-x_n\parallel +\parallel T_i^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_n-T_i^n{x}_n\parallel .\end{array}$$

It follows from (3.9), (3.11), and (3.20) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T_i^{n+1}{x}_n-T_i^n{x}_n\parallel =0,\text{and}\underset{n\to \mathrm{\infty }}{lim}{T}_i^{n+1}{x}_n=p^{\ast },$$

and so

$$\underset{n\to \mathrm{\infty }}{lim}{T}_i^{n+1}{x}_n=\underset{n\to \mathrm{\infty }}{lim}{T}_i{T}_i^n{x}_n=\underset{n\to \mathrm{\infty }}{lim}{T}_i{p}^{\ast }=p^{\ast }.$$

In view of (3.20) and the closeness of $T_i$, it yields that $T_i{p}^{\ast }=p^{\ast }$ for all $i\ge 1$. This implies that

$$p^{\ast }\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(T_i).$$

(3.21)

(6.2) Next, we will show that $p^{\ast }\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)$.

For any $i\ge 1$ and for any $p\in \mathrm{\Omega }$, it follows from (3.5), (3.8), (3.15), and (3.16) that

$$\begin{array}{rcl}{\alpha }_{n,0}{\alpha }_{n,i}g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)& \le & \varphi (p,x_n)-\varphi (p,z_n)+{\theta }_n\\ =& {\parallel x_n\parallel }^2-{\parallel z_n\parallel }^2-2〈p,J{x}_n-J{z}_n〉+{\theta }_n\\ =& (\parallel x_n\parallel +\parallel z_n\parallel )(\parallel x_n\parallel -\parallel z_n\parallel )-2〈p,J{x}_n-J{z}_n〉+{\theta }_n\\ \le & (\parallel x_n\parallel +\parallel z_n\parallel )(\parallel x_n-z_n\parallel )-2\parallel p\parallel \parallel J{x}_n-J{z}_n\parallel +{\theta }_n\\ \to & 0,\text{as }n\to \mathrm{\infty }.\end{array}$$

By the condition ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$, $\mathrm{\forall }i\ge 1$, we obtain

$$g\left(\parallel J{y}_n-J{S}_i^n{y}_n\parallel \right)\to 0,\text{as }n\to \mathrm{\infty }.$$

It follows from the property of g that

$$\parallel J{y}_n-J{S}_i^n{y}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

(3.22)

Since $y_n\to p^{\ast }$ and J is uniformly continuous on each bounded subset of E, it yields that $J{y}_n\to J{p}^{\ast }$.

Hence, from (3.22) we have

$$J{S}_i^n{y}_n\to J{p}^{\ast },\text{as }n\to \mathrm{\infty },\mathrm{\forall }i\ge 1.$$

(3.23)

Since $J^{-1}:E^{\ast }\to E$ is norm-to-weak^∗-continuous, we also have

$$S_i^n{y}_n\rightharpoonup p^{\ast }\text{as }n\to \mathrm{\infty },\mathrm{\forall }i\ge 1.$$

(3.24)

Again, since for any $i\ge 1$,

$$|\parallel S_i^n{y}_n\parallel -\parallel p^{\ast }\parallel |=|\parallel J\left(S_i^n{y}_n\right)\parallel -\parallel J{p}^{\ast }\parallel |\le \parallel J\left(S_i^n{y}_n\right)-J{p}^{\ast }\parallel \to 0,\text{as }n\to \mathrm{\infty },$$

from (3.24) and the Kadec-Klee property of E, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}{S}_i^n{y}_n=p^{\ast }.$$

(3.25)

On the other hand, by the assumption that for each $i\ge 1$, $S_i$ is uniformly ${\ell }_i$-Lipschitz continuous, we have

$$\begin{array}{rcl}\parallel S_i^{n+1}{y}_n-S_i^n{y}_n\parallel & \le & \parallel S_i^{n+1}{y}_n-S_i^{n+1}{y}_{n+1}\parallel +\parallel S_i^{n+1}{y}_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \\ +\parallel x_n-y_n\parallel +\parallel y_n-S_i^n{y}_n\parallel \\ \le & {\ell }_i\parallel y_n-y_{n+1}\parallel +\parallel S_i^{n+1}{y}_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \\ +\parallel x_n-y_n\parallel +\parallel y_n-S_i^n{y}_n\parallel .\end{array}$$

From $x_n\to p^{\ast }$, $y_n\to p^{\ast }$ and $S_i^n{y}_n\to p^{\ast }$, as $n\to \mathrm{\infty }$, we obtain

$$\parallel y_n-y_{n+1}\parallel \to 0,\parallel S_i^{n+1}{y}_{n+1}-x_{n+1}\parallel \to 0,\text{and}\parallel y_n-S_i^n{y}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

And it follows from (3.11) and (3.15) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S_i^{n+1}{y}_n-S_i^n{y}_n\parallel =0,\text{and}\underset{n\to \mathrm{\infty }}{lim}{S}_i^{n+1}{y}_n=p^{\ast },$$

and so

$$\underset{n\to \mathrm{\infty }}{lim}{S}_i^{n+1}{y}_n=\underset{n\to \mathrm{\infty }}{lim}{S}_i{S}_i^n{y}_n=\underset{n\to \mathrm{\infty }}{lim}{S}_i{p}^{\ast }=p^{\ast }.$$

In view of (3.25) and the closeness of $S_i$, it yields that $S_i{p}^{\ast }=p^{\ast }$ for all $i\ge 1$. This implies that

$$p^{\ast }\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(S_i).$$

(3.26)

From (3.21) and (3.26), we can conclude that $p^{\ast }\in \mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$.

Step 7. Finally, we will show that $x_n\to p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Let $w={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$. Since $w\in \mathrm{\Omega }\subset C_n$ and $x_n={\mathrm{\Pi }}_{C_n}(x_0)$, we have

$$\varphi (x_n,x_0)\le \varphi (w,x_0),\mathrm{\forall }n\ge 0.$$

This implies that

$$\varphi (p^{\ast },x_0)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_0)\le \varphi (w,x_0).$$

(3.27)

In view of the definition of ${\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$, from (3.27), we have $p^{\ast }=w$. Therefore, $x_n\to p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

This completes the proof of Theorem 3.1. □

Theorem 3.2 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(B1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_i$-Lipschitz continuous, and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a real sequence $\{k_n\}\subset [1,\mathrm{\infty })$, $k_n\to 1$.

(B2) Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly ${\ell }_i$-Lipschitz continuous, and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a real sequence $\{l_n\}\subset [1,\mathrm{\infty })$, $l_n\to 1$.

Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is a nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i^n{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.28)

where ${\xi }_n={sup}_{p\in \mathrm{\Omega }}(k_n-1)(\varphi (p,x_n))$ and ${\theta }_n={sup}_{p\in \mathrm{\Omega }}(l_n-1)(\varphi (p,x_n))$.

Let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$ and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$ and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof Since ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings, by virtue of Remark 1.7(2), ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative sequences ${\nu }_n=k_n-1$, ${\mu }_n=0$, and ${\kappa }_n=l_n-1$, ${\omega }_n=0$, respectively, and a strictly increasing and continuous function $\zeta (t)=\rho (t)=t$, $t\ge 0$. Hence, ${\xi }_n={sup}_{p\in \mathrm{\Omega }}(k_n-1)(\varphi (p,x_n))\to 0$ and ${\theta }_n={sup}_{p\in \mathrm{\Omega }}(l_n-1)(\varphi (p,x_n))\to 0$ (as $n\to \mathrm{\infty }$). Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □

Theorem 3.3 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(C1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed and quasi-ϕ-nonexpansive mappings.

(C2) Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed and quasi-ϕ-nonexpansive mappings.

Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.29)

and let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$, and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$, and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof Since ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of closed and quasi-ϕ-nonexpansive mappings, by Remark 1.7(3), ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of closed and quasi-ϕ-asymptotically nonexpansive mappings with nonnegative sequences $k_n=1$ and $l_n=1$, respectively. Hence, ${\xi }_n={sup}_{p\in \mathrm{\Omega }}(k_n-1)(\varphi (p,x_n))=0$ and ${\theta }_n={sup}_{p\in \mathrm{\Omega }}(l_n-1)(\varphi (p,x_n))=0$. Therefore, all the conditions in Theorem 3.2 as ‘Ω is bounded in C’ and ‘for each $i\ge 1$, ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are uniformly $L_i,{\ell }_i$-Lipschitz continuous’ are of no use here. Thus, all the conditions in Theorem 3.2 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □

Theorem 3.4 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(D1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of weak relatively nonexpansive mappings.

(D2) Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of weak relatively nonexpansive mappings.

Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.30)

and let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$ and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$ and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof Since ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of weak relatively nonexpansive mappings, from Remark 1.7(4), ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of quasi-ϕ-nonexpansive mappings. Therefore, all the conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 3.4 can be obtained from Theorem 3.3 immediately. □

Theorem 3.5 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(E1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of relatively nonexpansive mappings.

(E2) Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of relatively nonexpansive mappings.

Suppose that $\mathrm{\Omega }:=({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i))$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{S}_i{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.31)

and let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$ and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$ and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof Since ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of relatively nonexpansive mappings, it follows from Remark 1.7(5) that ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ are countable families of weak relatively nonexpansive mappings. Therefore, all the conditions in Theorem 3.4 are satisfied. The conclusion of Theorem 3.5 can be obtained from Theorem 3.4 immediately. □

Remark 3.6 Theorems 3.1-3.5 generalize, improve, and extend the corresponding results in [5–8, 11–18], and [19] in the following aspects:

• (a) For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property. (Note that each uniformly convex Banach space must have the Kadec-Klee property.)

• (b) For the mappings, we extend the mappings from a nonexpansive mapping, a relatively nonexpansive mapping, a weakly relatively nonexpansive mapping, a quasi-ϕ-nonexpansive mapping or a quasi-ϕ-asymptotically nonexpansive mapping to a total quasi-ϕ-asymptotically nonexpansive mapping.

• (c) We extend a countable family of closed and uniformly $L_i$-Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings to two countable families of closed and uniformly $L_i$-Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings.

4 Deduced theorem

If we take $i=1$ in Theorem 3.1, then we obtain the following result.

Theorem 4.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(F1) Let $T:C\to C$ be a closed, uniformly L-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), $\zeta (0)=0$, and

$$\varphi (p,T^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in Fix(T),\mathit{\text{and}}\mathrm{\forall }n\ge 1.$$

(F2) Let $S:C\to C$ be a closed, uniformly ℓ-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences $\{{\kappa }_n\}$, $\{{\omega }_n\}$, and a strictly increasing function $\rho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\kappa }_n\to 0$, ${\omega }_n\to 0$ (as $n\to \mathrm{\infty }$), $\rho (0)=0$, and

$$\varphi (p,S^{n}x)\le \varphi (p,x)+{\kappa }_n\rho (\varphi (p,x))+{\omega }_n,\mathrm{\forall }x\in C,p\in Fix(S),\mathit{\text{and}}\mathrm{\forall }n\ge 1.$$

Suppose that $\mathrm{\Omega }:=Fix(T)\cap Fix(S)$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{S}^n{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(4.1)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (p,x_n))+{\mu }_n$ and ${\theta }_n={\kappa }_n{sup}_{p\in \mathrm{\Omega }}\rho (\varphi (p,x_n))+{\omega }_n$.

Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$;

• (2) $0\le {\beta }_n\le \beta <1$ for some $\beta \in (0,1)$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

If we set $S_i^n=I$ (identity mapping) for all $i=1,2,3,\dots$ in Theorem 3.1, then we obtain the following result.

Theorem 4.2 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(G1) Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a countable family of closed, uniformly $L_i$-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, and a strictly increasing function $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$), ${\mu }_1=0$, and $\zeta (0)=0$, and for each $i\ge 1$,

$$\varphi (p,T_i^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in \bigcap _{i=1}^{\mathrm{\infty }}Fix(T_i),\mathit{\text{and}}\mathrm{\forall }n\ge 1.$$

(G2) Suppose that $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i)$ is nonempty and bounded in C. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C,\hfill \\ z_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{y}_n),\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(4.2)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathrm{\Omega }}\zeta (\varphi (p,x_n))+{\mu }_n$. Let $\{{\alpha }_n\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$;

• (2) ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (3) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Remark 4.3 Theorem 4.2 contains the result of Chang et al. [8].

5 Applications

Now, we apply Theorem 3.5 to prove a strong convergence theorem concerning two maximal monotone operators in Banach spaces.

Let E be a smooth, strictly convex, and reflexive real Banach space, and let $A:E\to E^{\ast }$ be a maximal monotone operator. For each $r>0$, we can define a single value mapping $J_r:E\to D(A)$ by $J_r={(J+rA)}^{-1}J$, and such a mapping $J_r$ is called the resolvent of A. It is easy to prove that $A^{-1}(0)=F(J_r)$ for all $r>0$. Using Theorem 3.5, we can obtain the following strong convergence theorem for maximal monotone operators.

Theorem 5.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:

(H1) Let A, B be two maximal monotone operators from E to $E^{\ast }$, and let $J_r^A$, $J_r^B$ be the resolvent of A and B, respectively, where $r>0$.

(H2) Suppose that $\mathrm{\Omega }:=A^{-1}(0)\cap B^{-1}(0)$ is nonempty, and let ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{z_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}}\text{, }{C}_0=C\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{n,i}J{J}_{r_i}^B{y}_n)\hfill \\ y_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{J}_{r_i}^A{x}_n)\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,x_n)\mathit{\text{and}}\varphi (v,z_n)\le \varphi (v,x_n)\}\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}(x_0),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(5.1)

where $r_n>0$ with ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$, and let $\{{\alpha }_{n,i}\}$ and $\{{\beta }_{n,i}\}$ be coefficient sequences in $[0,1]$ satisfying the following conditions:

• (1) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{n,i}=1$, and ${\sum }_{i=0}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$, and ${lim\hspace{0.17em}inf}_{n⟶\mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$, $\mathrm{\forall }i\ge 1$.

Then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to some point $p^{\ast }$, where $p^{\ast }={\mathrm{\Pi }}_{\mathrm{\Omega }}(x_0)$.

Proof It is well known that for each $i\ge 1$, $J_{r_i}^A$ is a relatively nonexpansive mapping (see, for example, [5, 6, 19]). Therefore, for each $p\in F(J_{r_i}^A)$ and $w\in E$, we have

$$\varphi (p,J_{r_i}^{A}w)\le \varphi (p,w).$$

Again, by the same method, we can prove that the set of strong asymptotically fixed points

$$\tilde{F}\left({\left\{J_{r_i}^A\right\}}_{i=1}^{\mathrm{\infty }}\right)=\bigcap _{i=1}^{\mathrm{\infty }}F\left(J_{r_i}^A\right)=A^{-1}(0).$$

This implies that ${\{J_{r_i}^A\}}_{i=1}^{\mathrm{\infty }}$ is a countable family of weak relatively nonexpansive mappings with the common fixed point set ${\bigcap }_{n=1}^{\mathrm{\infty }}F(J_{r_i}^A)=A^{-1}(0)$. By a similar way, we can prove that ${\{J_{r_i}^B\}}_{i=1}^{\mathrm{\infty }}$ is a countable family of weak relatively nonexpansive mappings with the common fixed point set ${\bigcap }_{n=1}^{\mathrm{\infty }}F(J_{r_i}^B)=B^{-1}(0)$. Hence, the conclusion of Theorem 5.1 can be obtained from Theorem 3.5 immediately. □

Remark 5.2 Theorem 5.1 improves and extends Theorem 4.1 of Chang et al. [10] from one maximal monotone operator to two maximal monotone operators.