A note on “Implicit iterative process with errors”

Abstract

In this paper, I introduce an implicit iterative process with mixed errors for two finite family of total asymptotically nonexpansive mappings in a uniformly convex Banach space and prove strong convergence theorems under some conditions. My results improved and extended many know results existing in the literature.

Background

Let E is a normed space and K be a nonempty subset of E. I also assume that $\phi : \mathbb {R} ^{+}\rightarrow \mathbb {R} ^{+}$strictly increasing continuous function with $\phi (0)=0$. Let $T:K\rightarrow K$ be a mapping. A point $x\in K$ is called a fixed point of T if and only if $Tx=x$. I will denote by nonexpansive if $F\left( T\right)$ the set of fixed points of T, that is, $F(T):=\left\{ x\in K:Tx=x\right\}$. T is said to be nonexpansive if

$$\begin{aligned} \left\| Tx-Ty\right\| \le \left\| x-y\right\| \end{aligned}$$

for all $x, y\in K$. T is called asymptotically nonexpansive if for a sequence $\left\{ k_{n}\right\} \subset \left[ 1,\infty \right)$ with $k_{n}\rightarrow 1$,

$$\begin{aligned} \left\| T^{n}x-T^{n}y\right\| \le k_{n}\left\| x-y\right\| \end{aligned}$$

for all $x, y\in K$ and $n\ge 1$. T is said to be total asymptotically nonexpansive (see, e.g., Albert et al. 2006) if

$$\begin{aligned} \left\| T^{n}x-T^{n}y\right\| \le \left\| x-y\right\| +\mu _{n}\phi (\left\| x-y\right\| )+l_{n} \end{aligned}$$

for all $x, y\in K, n\ge 1$ where $\left\{ \mu _{n}\right\}$ and $\left\{ l_{n}\right\}$ nonnegative real sequences such that $\mu _{n}, l_{n}\rightarrow 0$ as $n\rightarrow \infty$. From the definition, I see that the class of total asymptotically nonexpansive mappings include the class of asymptotically nonexpansive mappings as a special case; see also Albert et al. (2006) for more details.

Proposition 1

Let K be a nonempty subset of $E,\left\{ S_{i}\right\} _{i=1}^{N},\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N total asymptotically nonexpansive mappings. Then there exist nonnegative real sequences $\left\{ \mu _{n}\right\}$ and $\left\{ l_{n}\right\} , n\ge 1$ with $\mu _{n}, l_{n}\rightarrow 0$ as $n\rightarrow \infty$ and strictly increasing continuous function $\phi : \mathbb {R} ^{+}\rightarrow \mathbb {R} ^{+}$ with $\phi (0)=0$ such that for all $x, y\in K,$

$$\begin{aligned} \left\| T_{i}^{n}x-T_{i}^{n}y\right\|\le \left\| x-y\right\| +\mu _{n}\phi \left( \left\| x-y\right\| \right) +l_{n},\quad n\ge 1, \end{aligned}$$

(1)

$$\begin{aligned} \left\| S_{i}^{n}x-S_{i}^{n}y\right\|\le \left\| x-y\right\| +\mu _{n}\phi \left( \left\| x-y\right\| \right) +l_{n},\quad n\ge 1, \quad \hbox{for}\,i=1,2,\ldots ,N.\end{aligned}$$

(2)

Proof

Since $T_{i}:K\rightarrow K$ is a total asymptotically nonexpansive mappings for $i=1,2,\ldots ,N,$ there exist nonnegative real sequences $\left\{ \mu _{in}\right\}$, $\left\{ l_{in}\right\}$, $n\ge 1$ with $\mu _{in}, l_{in}\rightarrow 0$ as $n\rightarrow \infty$ and strictly increasing continuous function $\phi _{i}: \mathbb {R} ^{+}\rightarrow \mathbb {R} ^{+}$ with $\phi _{i}\left( 0\right) =0$ such that for all $x, y\in K,$

$$\begin{aligned} \left\| T_{i}^{n}x-T_{i}^{n}y\right\| \le \left\| x-y\right\| +\mu _{in}\phi _{i}\left( \left\| x-y\right\| \right) +l_{in},\quad n\ge 1. \end{aligned}$$

Setting

$$\begin{aligned} \mu _{n}&= \max \left\{ \mu _{1n},\mu _{2n},\ldots ,\mu _{Nn}\right\} , \quad l_{n}=\max \left\{ l_{1n},l_{2n},\ldots ,l_{Nn}\right\} ,\\ \phi \left( a\right)&= \max \left\{ \phi _{1}\left( a\right) ,\phi _{2}\left( a\right) ,\ldots ,\phi _{N}\left( a\right) \right\} \text { for }a\ge 0, \end{aligned}$$

then I get that there exist nonnegative real sequences $\left\{ \mu _{n}\right\}$ and $\left\{ l_{n}\right\} , n\ge 1$ with $\mu _{n}, l_{n}\rightarrow 0$ as $n\rightarrow \infty$ and strictly increasing continuous function $\phi : \mathbb {R} ^{+}\rightarrow \mathbb {R} ^{+}$ with $\phi \left( 0\right) =0$ such that

$$\begin{aligned} \left\| T_{i}^{n}x-T_{i}^{n}y\right\|&\le \left\| x-y\right\| +\mu _{in}\phi _{i}\left( \left\| x-y\right\| \right) +l_{in} \\ &\le {} \left\| x-y\right\| +\mu _{n}\phi \left( \left\| x-y\right\| \right) +l_{n},\quad n\ge 1, \end{aligned}$$

for all $x,y\in K,$ and each $i=1,2,\ldots ,N.$

In a way similar to the above, I can also prove (2).□

Recently, fixed point problems based on implicit iterative processes have been considered by many authors, (see, for example, Chang et al. 2006; Cianciaruso et al. 2010; Sun 2003; Gu 2006; Qin et al. 2008; Xu and Ori 2001). In Hao (2010) established weak and strong convergence theorems of the implicit iteration process for a finite family of uniformly Lipschitz total asymptotically nonexpansive mappings in a real Hilbert space. In Hao et al. (2012) studied weak and strong convergence theorems for common fixed points of two finite family of asymptotically nonexpansive mappings in a uniformly convex Banach space.

Note the convergence problems of an implicit (an explicit) iterative process to a common fixed point, for total asymptotically nonexpansive (or asymptotically nonexpansive) in Banach space have been obtained by a number of authors (see more details, Mukhamedov and Saburov 2010a, b, 2011, 2012a, b).

Inspired and motivated by this facts, I introduce an implicit iterative process with mixed errors for two finite family of total asymptotically nonexpansive mappings in a uniformly convex Banach space. The results of this paper can be viewed as an improvement and extension of the corresponding results of Chang et al. (2006), Cianciaruso et al. (2010), Sun (2003), Hao et al. (2012), Hao (2010) and others.

Definition 1

Let $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:E\rightarrow E$ be 2N total asymptotically nonexpansive mappings. Define the sequence $\left\{ x_{n}\right\}$ as follows: $x_{0}\in K,$ and

$$\begin{aligned} x_{n} &= \left( 1-\alpha _{n}-\gamma _{n}\right) x_{n-1}+\alpha _{n}T_{i\left( n\right) }^{k\left( n\right) }y_{n}+\gamma _{n}u_{n}, \nonumber \\ y_{n} & =\left( 1-\beta _{n}-\delta _{n}\right) x_{n}+\beta _{n}S_{i\left( n\right) }^{k\left( n\right) }x_{n}+\delta _{n}v_{n}, n\ge 1 \end{aligned}$$

(3)

where $n=\left( k\left( n\right) -1\right) N+i\left( n\right)$, $i\left( n\right) \in 1,2,\ldots ,N$, $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ are four real sequences in $\left[ 0,1\right]$ satisfying $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1,\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences.

The purpose of this paper is to study the strong convergence of implicit iterative process with mixed errors for two finite family of total asymptotically nonexpansive mappings in Banach spaces.

Preliminaries

Let E be a Banach space with dimension $E\ge 2$. The modulus of E is the function $\delta _{E}:\left( 0,2\right] \rightarrow \left[ 0,1\right]$ defined by

$$\begin{aligned} \delta _{E}\left( \varepsilon \right) =\inf \left\{ 1-\left\| \frac{1}{2} (x+y)\right\| :\mathrm {\;}\left\| x\right\| =\left\| y\right\| =1,\mathrm {\;}\varepsilon =\left\| x-y\right\| \right\} . \end{aligned}$$

A Banach space E is uniformly convex if and only if $\delta _{E}\left( \varepsilon \right) >0$ for all $\varepsilon \in \left( 0,2\right]$.

Recall that a mapping $T:K\rightarrow K$ is $semi-compact$ ($or hemi-compact$) if any sequence $\left\{ x_{n}\right\}$ in K such that $\left\{ x_{n}-Tx_{n}\right\} \rightarrow 0$ as $n\rightarrow \infty$ has a convergent subsequence.

Lemma 1

(Tan and Xu 1993) Let $\left\{ a_{n}\right\},$ $\left\{ b_{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be sequences of nonnegative real numbers satisfying the inequality

$$\begin{aligned} a_{n+1}\le \left( 1+\delta _{n}\right) a_{n}+b_{n},\quad n\ge 1, \end{aligned}$$

if $\sum _{n=1}^{\infty }b_{n}<\infty$ and $\sum _{n=1}^{\infty }\delta _{n}<\infty$, then

• (i) $\lim \limits _{n\rightarrow \infty }a_{n}$ exists;

• (ii) In particular, if $\left\{ a_{n}\right\}$ has a subsequence $\left\{ a_{n_{k}}\right\}$ converging to 0, then $\lim \limits _{n\rightarrow \infty }a_{n}=0.$

Lemma 2

(Schu 1991) Let E be a uniformly convex Banach space, $\left\{ t_{n}\right\} _{n\ge 1}\subseteq \left[ b,c\right] \subset \left( 0,1\right)$, $\left\{ x_{n}\right\} _{n\ge 1}$ and $\left\{ y_{n}\right\} _{n\ge 1}$ be sequences in E. If $\lim \sup _{n\rightarrow \infty }\left\| x_{n}\right\| \le a$, $\lim \sup _{n\rightarrow \infty }\left\| y_{n}\right\| \le a$ and $\lim _{n\rightarrow \infty }\left\| t_{n}x_{n}+\left( 1-t_{n}\right) y_{n}\right\| =a$ for some constant $a\ge 0,$ then $\lim _{n\rightarrow \infty }\left\| x_{n}-y_{n}\right\| =0.$

Main results

Lemma 3

Let E be a real Banach space, let K be a nonempty,closed and convex subset of E and $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) and (2) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\quad \sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(4)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \cap F\left( S_{\dot{I}}\right) \ne \varnothing$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K. Let $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be four real sequences in $\left[ 0,1\right]$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{ \infty }\delta _{n}<\infty .$

  Starting from an arbitrary $x_{0}\in K$, define the sequence $\left\{ x_{n}\right\}$ by recursion (3). Then, $\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\|$ exists for all $p\in {\mathcal {F}}.$

Proof

Let $p\in {\mathcal {F}}$. Since $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K, I have

$$\begin{aligned} K=\max \left\{ \sup _{n\ge 1}\left\| u_{n}-p\right\| ,\sup _{n\ge 1}\left\| v_{n}-p\right\| \right\} . \end{aligned}$$

Since $S_{1},S_{2},\ldots ,S_{N}$ are total asymptotically nonexpansive mappings, it follows from (2) that

$$\begin{aligned} \left\| y_{n}-p\right\| & = \left\| \left( 1-\beta _{n}-\delta _{n}\right) \left( x_{n}-p\right) +\beta _{n}\left( S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p\right) +\delta _{n}\left( v_{n}-p\right) \right\| \nonumber \\ &\le \left( 1-\beta _{n}-\delta _{n}\right) \left\| x_{n}-p\right\| +\beta _{n}\left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p\right\| +\delta _{n}\left\| v_{n}-p\right\| \\ &\le \left( 1-\beta _{n}-\delta _{n}\right) \left\| x_{n}-p\right\| +\beta _{n}\left[ \left\| x_{n}-p\right\| +\mu _{n}\phi \left( \left\| x_{n}-p\right\| \right) +l_{n}\right] +\delta _{n}K \nonumber \\ &\le \left\| x_{n}-p\right\| +\beta _{n}\mu _{n}\phi \left( \left\| x_{n}-p\right\| \right) +\beta _{n}l_{n}+\varphi _{\left( 1\right) }^{n}, \end{aligned}$$

(5)

where $\varphi _{\left( 1\right) }^{n}=\delta _{n}K$. Since $\sum \nolimits _{n=1}^{\infty }\delta _{n}<\infty$, I can see that $\sum \nolimits _{n=1}^{\infty }\varphi _{\left( 1\right) }^{n}<\infty$. Note that $\phi$ is an increasing function, it follows that $\phi \left( \lambda \right) \le \phi \left( M\right)$ whenever $\lambda \le M$ and ( by hypothesis) $\phi \left( \lambda \right) \le M^{*}\lambda$ if $\lambda \ge M$. In either case, I have

$$\begin{aligned} \phi \left( \lambda \right) \le \phi \left( M\right) +M^{*}\lambda , \end{aligned}$$

(6)

for some $M,M^{*}>0$. Thus, from (5) and (6), I have

$$\begin{aligned} \left\| y_{n}-p\right\|&\le \left\| x_{n}-p\right\| +\beta _{n}\mu _{n}\left[ \phi \left( M\right) +M^{*}\left\| x_{n}-p\right\| \right]+\beta _{n}l_{n}+\varphi _{\left( 1\right) }^{n} \nonumber \\ &\le \left( 1+M^{*}\mu _{n}\right) \left\| x_{n}-p\right\| +R_{1}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 1\right) }^{n}, \end{aligned}$$

(7)

for some constant $R_{1}>0$. It follows from (6) and (7)

$$\begin{aligned} \left\| x_{n}-p\right\| &= \left\| \left( 1-\alpha _{n}-\gamma _{n}\right) \left( x_{n-1}-p\right) +\alpha _{n}\left( T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p\right) +\gamma _{n}\left( u_{n}-p\right) \right\| \nonumber \\ &\le \left( 1-\alpha _{n}-\gamma _{n}\right) \left\| x_{n-1}-p\right\| +\alpha _{n}\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p\right\| +\gamma _{n}\left\| u_{n}-p\right\| \nonumber \\ &\le \left( 1-\alpha _{n}-\gamma _{n}\right) \left\| x_{n-1}-p\right\| +\alpha _{n}\left[ \left\| y_{n}-p\right\| +\mu _{n}\phi \left( \left\| y_{n}-p\right\| \right) +l_{n}\right] \nonumber \\ &\quad +\gamma _{n}\left\| u_{n}-p\right\| \nonumber \\ &\le \left( 1-\alpha _{n}-\gamma _{n}\right) \left\| x_{n-1}-p\right\| \nonumber \\ &\quad +\alpha _{n}\left[ \left( 1+M^{*}\mu _{n}\right) \left\| x_{n}-p\right\| +R_{1}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 1\right) }^{n}\right] \nonumber \\ &\quad +\alpha _{n}\mu _{n}\left[ \phi \left( M\right) +M^{*}\left\| y_{n}-p\right\| \right] +\alpha _{n}l_{n}+\gamma _{n}K \nonumber \\ &\le \left( 1-\alpha _{n}-\gamma _{n}\right) \left\| x_{n-1}-p\right\| +\alpha _{n}\left\| x_{n}-p\right\| \nonumber \\ &\quad +\alpha _{n}M^{*}\left( 2+M^{*}\mu _{n}\right) \mu _{n}\left\| x_{n}-p\right\| \nonumber \\ &\quad +\alpha _{n}R_{1}\left( \mu _{n}+l_{n}\right) +\alpha _{n}\varphi _{\left( 1\right) }^{n}+\alpha _{n}\mu _{n}\phi \left( M\right) +\alpha _{n}l_{n}+\gamma _{n}K \nonumber \\ &\quad +\alpha _{n}\mu _{n}M^{*}R_{1}\left( \mu _{n}+l_{n}\right) +\alpha _{n}\mu _{n}M^{*}\varphi _{\left( 1\right) }^{n} \nonumber \\ &\le \left( 1-\alpha _{n}\right) \left\| x_{n-1}-p\right\| +\alpha _{n}\left( 1+M_{2}\mu _{n}\right) \left\| x_{n}-p\right\| \nonumber \\ &R_{2}\left( \mu _{n}+l_{n}\right) +\alpha _{n}\varphi _{\left( 1\right) }^{n}+\gamma _{n}K+\alpha _{n}\mu _{n}M^{*}\varphi _{\left( 1\right) }^{n}, \end{aligned}$$

(8)

for some constants $M_{2},R_{2}>0$. I note that

$$\begin{aligned} \left\| x_{n}-p\right\| &\le \left( 1-\alpha _{n}\right) \left\| x_{n-1}-p\right\| +\alpha _{n}\left( 1+M_{2}\mu _{n}\right) \left\| x_{n}-p\right\| \nonumber \\ &\quad +R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}, \end{aligned}$$

(9)

where $\varphi _{\left( 2\right) }^{n}=\alpha _{n}\varphi _{\left( 1\right) }^{n}+\gamma _{n}K+\alpha _{n}\mu _{n}M^{*}\varphi _{\left( 1\right) }^{n}$. Since $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{\infty }\varphi _{\left( 1\right) }^{n}<\infty$ , I can see that $\sum \nolimits _{n=1}^{\infty }\varphi _{\left( 2\right) }^{n}<\infty$. This implies that

$$\begin{aligned} \left\| x_{n}-p\right\| &\le \frac{1-\alpha _{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }\left\| x_{n-1}-p\right\| +\frac{ R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) } \nonumber \\ &\le \left( 1+\frac{\alpha _{n}M_{2}\mu _{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }\right) \left\| x_{n-1}-p\right\| +\frac{ R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }. \end{aligned}$$

(10)

By hypothesis (ii), it follows that there exists $\lambda <1$, such that $\alpha _{n}\le \lambda$ for big n. It follows that

$$\begin{aligned} \frac{\alpha _{n}M_{2}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }\le \frac{\lambda M_{2}}{1-\lambda \left( 1+M_{2}\mu _{n}\right) }. \end{aligned}$$

From $\lim _{n\rightarrow \infty }\mu _{n}=0$, it derives that $\lim _{n\rightarrow \infty }\frac{\lambda M_{2}}{1-\lambda \left( 1+M_{2}\mu _{n}\right) }=\frac{\lambda M_{2}}{1-\lambda }$. Then there exists a real constant $L_{1}>0$ such that

$$\begin{aligned} \frac{\lambda M_{2}}{1-\lambda \left( 1+M_{2}\mu _{n}\right) }\le L_{1},\quad \forall n\ge 1. \end{aligned}$$

It follows from the hypothesis that $\sum _{n\ge 1}\frac{\alpha _{n}M_{2}}{ 1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }<\infty$. Similarly, I can prove that

$$\begin{aligned} \sum _{n\ge 1}\frac{R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }<\infty . \end{aligned}$$

(11)

Besides, I can write

$$\begin{aligned} \left\| x_{n}-p\right\| \le \left( 1+L_{1}\mu _{n}\right) \left\| x_{n-1}-p\right\| +\frac{R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }, \end{aligned}$$

where a real constant $L_{1}>0$. Thus, I obtain from it that

$$\begin{aligned} \left\| x_{n}-p\right\| \le \left( 1+\delta _{n}\right) \left\| x_{n-1}-p\right\| +b_{n},\quad \forall n\ge n_{0}, \end{aligned}$$

(12)

where $\delta _{n}=L_{1}\mu _{n}$ and $b_{n}=\frac{R_{2}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 2\right) }^{n}}{1-\alpha _{n}\left( 1+M_{2}\mu _{n}\right) }$ and using the condition (iii) and (11), it is easy to see that $\sum _{n=1}^{\infty }\delta _{n}<\infty$ and $\sum _{n=1}^{\infty }b_{n}<\infty$. In view of Lemma 1, I find that $\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\|$ exist for all $p\in {\mathcal {F}}$. $\square$

Theorem 1

Let E be a real uniformly convex Banach space, let K be a nonempty,closed and convex subset of E and $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) and (2) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\quad \sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(13)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \cap F\left( S_{\dot{I}}\right) \ne \varnothing$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K. Let $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be four real sequences in $\left[ 0,1\right]$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{ \infty }\delta _{n}<\infty .$

Then the implicit iterative sequence $\left\{ x_{n}\right\}$ by (3) converges strongly to a common fixed point in ${\mathcal {F}}$ if and only if $\lim \inf _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$, where $d\left( x,{\mathcal {F}}\right)$ denotes the distance of x to set ${\mathcal {F}}$, i.e., $d\left( x,{\mathcal {F}}\right) =\inf _{y\in {\mathcal {F}}}d\left( x,y\right)$.

Proof

It suffices to show that $\lim \inf _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$ implies that $\left\{ x_{n}\right\}$ converges to a common fixed point of ${\mathcal {F}}.$

Necessity. Since (12) holds for all $p\in {\mathcal {F}}$, I obtain from it that

$$\begin{aligned} d\left( x_{n},{\mathcal {F}}\right) \le \left( 1+\delta _{n}\right) d\left( x_{n-1},{\mathcal {F}}\right) +b_{n},\quad \forall n\ge n_{0}, \end{aligned}$$

Lemma 1 that $\lim _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right)$ exists and so $\lim _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$.

Sufficiency. Now, I show that $\left\{ x_{n}\right\}$ is a Cauchy sequence in E. For any positive integers $m,n>n\ge n_{0}$, from $1+t\le e^{t}$ for all $t>0$ and (12), I have

$$\begin{aligned} \left\| x_{m}-p\right\| &\le \left( 1+\delta _{m}\right) \left\| x_{m-1}-p\right\| +b_{m} \\ & \le e^{\delta _{m}}\left\| x_{m-1}-p\right\| +b_{m} \\ &\le e^{\delta _{m}}\left( e^{\delta _{m-1}}\left\| x_{m-2}-p\right\| +b_{m-1}\right) +b_{m} \\ &\quad \vdots \\ &\le e^{\sum _{i=n+1}^{m}\delta _{i}}\left\| x_{n}-p\right\| +\sum _{k=n+1}^{m-1}b_{k}e^{\sum _{i=k+1}^{m}\delta _{i}}+b_{m} \\ &\le Q\left\| x_{n}-p\right\| +Q\sum _{k=n+1}^{\infty }b_{k}+b_{m}, \end{aligned}$$

where $Q=e^{\sum _{n=1}^{\infty }\delta _{n}}$. Thus for any $p\in {\mathcal {F}}$, I have

$$\begin{aligned} \left\| x_{n}-x_{m}\right\| &\le \left\| x_{n}-p\right\| +\left\| x_{m}-p\right\| \\ &\le \left( 1+Q\right) \left\| x_{n}-p\right\| +Q\sum _{k=n+1}^{\infty }b_{k}+b_{m}. \end{aligned}$$

Taking the infimum over all $p\in {\mathcal {F}}$, I obtain that

$$\begin{aligned} \left\| x_{n}-x_{m}\right\| \le \left( 1+Q\right) d\left( x_{n},{\mathcal {F}}\right) +Q\sum _{k=n+1}^{\infty }b_{k}+b_{m}. \end{aligned}$$

It follows from $\sum \nolimits _{n=1}^{\infty }b_{n}<\infty$ and $\lim _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$ that $\left\{ x_{n}\right\}$ is a Cauchy sequence. Since K is a closed subset of E and so it is complete. Hence, there exists a $p\in K$ such that $x_{n}\rightarrow p$ as $n\rightarrow \infty$.

Finally, I have to prove that $p\in {\mathcal {F}}$. By contradiction, i assume that p is not in ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{ \dot{I}}\right) \cap F\left( S_{\dot{I}}\right) \ne \varnothing$. Since ${\mathcal {F}}$ is a closed set, $d\left( p,{\mathcal {F}}\right) >0$. Thus for all $p\in {\mathcal {F}}$, I have that

$$\begin{aligned} \left\| p-p_{1}\right\| \le \left\| p-x_{n}\right\| +\left\| x_{n}-p_{1}\right\| . \end{aligned}$$

(14)

This implies that

$$\begin{aligned} d\left( p,{\mathcal {F}}\right) \le \left\| p-x_{n}\right\| +d\left( x_{n},{\mathcal {F}}\right) . \end{aligned}$$

(15)

From (14) and (15) $\left( n\rightarrow \infty \right)$, I have that $d\left( p,{\mathcal {F}}\right) \le 0$. This is a contradiction. Thus, $p\in {\mathcal {F}}$. This completes the proof. $\square$

Lemma 4

Let E be a uniformly convex Banach space, let K be a nonempty,closed and convex subset of E and $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N uniformly $L_{i}-Lipschitz$ total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) and (2) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\quad \sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(16)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \cap F\left( S_{\dot{I}}\right) \ne \varnothing$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K. Let $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be four real sequences in $\left[ \frac{L-1}{L},a\right]$, where $L=\max _{1\le i\le N}\left\{ L_{i}\right\} >1$ and a is some constant in $\left( 0,1\right)$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{ \infty }\delta _{n}<\infty .$

Let the sequence $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ be defined by (3). Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{l}x_{n}\right\| &= 0, \\ \lim _{n\rightarrow \infty }\left\| x_{n}-S_{l}x_{n}\right\| &= 0, \\ \lim _{n\rightarrow \infty }\left\| S_{l}x_{n}-T_{l}x_{n}\right\| &= 0, \quad \forall l=1,2,\ldots ,N. \end{aligned}$$

Proof

For all $p\in {\mathcal {F}}$, it follows from Lemma 3 that $\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\|$ exists. Let $\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\| =r$ for some $r\ge 0$. It follows from (7) that

$$\begin{aligned} \left\| y_{n}-p\right\| \le \left( 1+M^{*}\mu _{n}\right) \left\| x_{n}-p\right\| +R_{1}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 1\right) }^{n}, \end{aligned}$$

(17)

where $\sum \nolimits _{n=1}^{\infty }\mu _{n}<\infty ,\sum \nolimits _{n=1}^{\infty }l_{n}<\infty$ and $\sum \nolimits _{n=1}^{\infty }\varphi _{\left( 1\right) }^{n}<\infty$. Taking $\lim \sup _{n\rightarrow \infty }$ in both sides, I obtain

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| y_{n}-p\right\|& \le \lim \sup _{n\rightarrow \infty }\left[ \begin{array}{c} \left( 1+M^{*}\mu _{n}\right) \left\| x_{n}-p\right\| \\ +R_{1}\left( \mu _{n}+l_{n}\right) +\varphi _{\left( 1\right) }^{n} \end{array} \right] \nonumber \\ & \le r, \end{aligned}$$

(18)

and by (16) and (18)

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p\right\|&\le \lim \sup _{n\rightarrow \infty }\left[ \begin{array}{c} \left\| y_{n}-p\right\| \\ +\mu _{n}\phi \left( \left\| y_{n}-p\right\| \right) +l_{n} \end{array} \right] \nonumber \\ &\le r. \end{aligned}$$

(19)

Notice that

$$\begin{aligned} \left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right\| \le \left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p\right\| +\gamma _{n}\left\| u_{n}-x_{n-1}\right\| . \end{aligned}$$

(20)

It follows that

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right\| \le r, \end{aligned}$$

(21)

and

$$\begin{aligned} \left\| x_{n-1}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right\| \le \left\| x_{n-1}-p\right\| +\gamma _{n}\left\| u_{n}-x_{n-1}\right\| . \end{aligned}$$

(22)

These imply that

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| x_{n-1}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right\| \le r, \end{aligned}$$

(23)

and I have that

$$\begin{aligned} x_{n}-p&= \alpha _{n}\left( T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right) \nonumber \\ &+\left( 1-\alpha _{n}\right) \left( x_{n-1}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right) . \end{aligned}$$

(24)

Hence,

$$\begin{aligned} r &= \lim _{n\rightarrow \infty }\left\| x_{n}-p\right\| \nonumber \\ &= \lim _{n\rightarrow \infty }\left\| \begin{array}{c} \alpha _{n}\left( T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right) \\ +\left( 1-\alpha _{n}\right) \left( x_{n-1}-p+\gamma _{n}\left( u_{n}-x_{n-1}\right) \right) \end{array} \right\| . \end{aligned}$$

(25)

Using (19), (23), (25) and Lemma 2, I find

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-x_{n-1}\right\| =0. \end{aligned}$$

(26)

Notice that

$$\begin{aligned} \left\| x_{n}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\|&\le \left\| x_{n}-x_{n-1}\right\| +\left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| \\ & \le \left( 1+\alpha _{n}\right) \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| +\gamma _{n}\left\| u_{n}-x_{n-1}\right\| . \end{aligned}$$

It follows from (26) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| =0. \end{aligned}$$

(27)

Notice that

$$\begin{aligned} \left\| x_{n}-p\right\|&\le \left\| x_{n}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| +\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-p\right\| \\ &\le \left\| x_{n}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| \\ &\quad+\left\| y_{n}-p\right\| +\mu _{n}\phi \left( \left\| y_{n}-p\right\| \right) +l_{n}. \end{aligned}$$

By using (16) and (27), I obtain

$$\begin{aligned} r=\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\| \le \lim \inf _{n\rightarrow \infty }\left\| y_{n}-p\right\| . \end{aligned}$$

It follows that

$$\begin{aligned} r\le \lim \inf _{n\rightarrow \infty }\left\| y_{n}-p\right\| \le \lim \sup _{n\rightarrow \infty }\left\| y_{n}-p\right\| \le r. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| y_{n}-p\right\| =r. \end{aligned}$$

(28)

Since $\lim _{n\rightarrow \infty }\left\| x_{n}-p\right\| =r$ and (16), I see that

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p\right\|&\le \lim \sup _{n\rightarrow \infty }\left[ \begin{array}{c} \left\| x_{n}-p\right\| \\ +\mu _{n}\phi \left( \left\| x_{n}-p\right\| \right) +l_{n} \end{array} \right] \nonumber \\ & \le r. \end{aligned}$$

(29)

Notice that

$$\begin{aligned} \left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right\| \le \left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p\right\| +\delta _{n}\left\| v_{n}-x_{n}\right\| . \end{aligned}$$

It follows that

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right\| \le r, \end{aligned}$$

(30)

and

$$\begin{aligned} \left\| x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right\| \le \left\| x_{n}-p\right\| +\delta _{n}\left\| v_{n}-x_{n}\right\| , \end{aligned}$$

(31)

which implies that

$$\begin{aligned} \lim \sup _{n\rightarrow \infty }\left\| x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right\| \le r, \end{aligned}$$

(32)

and I have that

$$\begin{aligned} y_{n}-p&= \beta _{n}\left( S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right) +\left( 1-\beta _{n}\right) \left( x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right) . \end{aligned}$$

(33)

Hence,

$$\begin{aligned} r= & {} \lim _{n\rightarrow \infty }\left\| y_{n}-p\right\| \nonumber \\= & {} \lim _{n\rightarrow \infty }\left\| \begin{array}{c} \beta _{n}\left( S_{i\left( n\right) }^{k\left( n\right) }x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right) \\ +\left( 1-\beta _{n}\right) \left( x_{n}-p+\delta _{n}\left( v_{n}-x_{n}\right) \right) \end{array} \right\| . \end{aligned}$$

(34)

Using (30), (32), (34) and Lemma 2, I find

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-S_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| =0. \end{aligned}$$

(35)

Notice that

$$\begin{aligned} \left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-x_{n}\right\|&\le \left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-y_{n}\right\| +\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-x_{n}\right\| \\ &\le {} \left\| x_{n}-y_{n}\right\| +\mu _{n}\phi \left( \left\| x_{n}-y_{n}\right\| \right) +l_{n} +\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-x_{n}\right\| \\ &\le \beta _{n}\left\| x_{n}-S_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| +\delta _{n}\left\| v_{n}-x_{n}\right\| \\ &\quad +\mu _{n}\phi \left( \left\| x_{n}-y_{n}\right\| \right) +l_{n}+\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-x_{n}\right\| . \end{aligned}$$

It follows from (16), (27) and (35) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| =0. \end{aligned}$$

(36)

Notice that

$$\begin{aligned} \left\| y_{n}-x_{n}\right\| &= \left\| \beta _{n}\left( S_{i\left( n\right) }^{k\left( n\right) }x_{n}-x_{n}\right) +\delta _{n}\left( v_{n}-x_{n}\right) \right\| \\ &\le \beta _{n}\left\| S_{i\left( n\right) }^{k\left( n\right) }x_{n}-x_{n}\right\| +\delta _{n}\left\| v_{n}-x_{n}\right\| . \end{aligned}$$

In view of (35), I see from the restriction (iii) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| y_{n}-x_{n}\right\| =0. \end{aligned}$$

(37)

Notice that

$$\begin{aligned} \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| &\le \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| +\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| \\ &\le \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }y_{n}\right\| +\left\| y_{n}-x_{n}\right\| +\mu _{n}\phi \left( \left\| y_{n}-x_{n}\right\| \right) +l_{n}. \end{aligned}$$

It follows that (16), (26) and (37) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| =0. \end{aligned}$$

(38)

Notice that

$$\begin{aligned} \left\| x_{n}-x_{n-1}\right\| \le \alpha _{n}\left\| T_{i\left( n\right) }^{k\left( n\right) }y_{n}-x_{n-1}\right\| +\gamma _{n}\left\| u_{n}-x_{n-1}\right\| . \end{aligned}$$

In view of (26), I see from the restriction (iii) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-x_{n-1}\right\| =0. \end{aligned}$$

(39)

Notice that

$$\begin{aligned} \left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-S_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| \le \left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-x_{n}\right\| +\left\| x_{n}-S_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| . \end{aligned}$$

It follows from (35) and (36) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-S_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| =0. \end{aligned}$$

From (39), I also have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-x_{n+j}\right\| =0,\quad \forall j=1,2,\ldots ,N. \end{aligned}$$

(40)

For any positive $n>N$, it can be rewritten as $n=\left( k\left( n\right) -1\right) N+i\left( n\right)$, $i\left( n\right) \in \left\{ 1,2,\ldots ,N\right\}$. Note that

$$\begin{aligned} \left\| x_{n-1}-T_{n}x_{n}\right\| &\le \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| +\left\| T_{i\left( n\right) }^{k\left( n\right) }x_{n}-T_{n}x_{n}\right\| \nonumber \\ &\le \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| +L\left\| T_{i\left( n\right) }^{k\left( n\right) -1}x_{n}-x_{n}\right\| \nonumber \\ &\le \left\| x_{n-1}-T_{i\left( n\right) }^{k\left( n\right) }x_{n}\right\| +L\left\{ \left\| T_{i\left( n\right) }^{k\left( n\right) -1}x_{n}-T_{i\left( n-N\right) }^{k\left( n\right) -1}x_{n-N}\right\| \right. \nonumber \\ &\left. \quad +\left\| T_{i\left( n-N\right) }^{k\left( n\right) -1}x_{n-N}-x_{\left( n-N\right) -1}\right\| +\left\| x_{\left( n-N\right) -1}-x_{n}\right\| \right\} . \end{aligned}$$

(41)

On the other hand, I have $n-N=\left( \left( k\left( n\right) -1\right) -1\right) N+i\left( n\right) =\left( \left( k\left( n\right) -1\right) -1\right) N+i\left( n-N\right) ,$ i.e.,

$$\begin{aligned} k\left( n-N\right) =k\left( n\right) -1\text { and }i\left( n-N\right) =i\left( n\right) . \end{aligned}$$

It follows that

$$\begin{aligned} \left\| T_{i\left( n\right) }^{k\left( n\right) -1}x_{n}-T_{i\left( n-N\right) }^{k\left( n\right) -1}x_{n-N}\right\| \le L\left\| x_{n}-x_{n-N}\right\| , \end{aligned}$$

(42)

and

$$\begin{aligned} \left\| T_{i\left( n-N\right) }^{k\left( n\right) -1}x_{n-N}-x_{\left( n-N\right) -1}\right\| =\left\| T_{i\left( n-N\right) }^{k\left( n-N\right) }x_{n-N}-x_{\left( n-N\right) -1}\right\| . \end{aligned}$$

(43)

Combining (38), (40) with (41), I see that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n-1}-T_{n}x_{n}\right\| =0. \end{aligned}$$

(44)

On the other hand, I have that

$$\begin{aligned} \left\| x_{n}-T_{n}x_{n}\right\| \le \left\| x_{n}-x_{n-1}\right\| +\left\| x_{n-1}-T_{n}x_{n}\right\| . \end{aligned}$$

From (39) and (44), I obtain that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{n}x_{n}\right\| =0. \end{aligned}$$

(45)

Consequently, for any $j=1,2,\ldots ,N$, I see that

$$\begin{aligned} \left\| x_{n}-T_{n+j}x_{n}\right\| &\le \left\| x_{n}-x_{n+j}\right\| +\left\| x_{n+j}-T_{n+j}x_{n+j}\right\| \\ &\quad +\left\| T_{n+j}x_{n+j}-T_{n+j}x_{n}\right\| \\ &\le \left( 1+L\right) \left\| x_{n}-x_{n+j}\right\| +\left\| x_{n+j}-T_{n+j}x_{n+j}\right\| . \end{aligned}$$

From (40) and (45), I arrive at

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{n+j}x_{n}\right\| =0. \end{aligned}$$

Therefore, for $\forall i\in \left\{ 1,2,\ldots ,N\right\}$, there exists some $e\in \left\{ 1,2,\ldots ,N\right\}$ such that $n+e=i\left( \hbox {mod} N\right)$. It follows that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{i}x_{n}\right\| =\lim _{n\rightarrow \infty }\left\| x_{n}-T_{n+e}x_{n}\right\| =0. \end{aligned}$$

(46)

Similarly, by using the same argument as in the proof above, I have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-S_{i}x_{n}\right\| =\lim _{n\rightarrow \infty }\left\| x_{n}-S_{n+e}x_{n}\right\| =0. \end{aligned}$$

(47)

Since

$$\begin{aligned} \left\| S_{i}x_{n}-T_{i}x_{n}\right\| \le \left\| S_{i}x_{n}-x_{n}\right\| +\left\| T_{i}x_{n}-x_{n}\right\| , \end{aligned}$$

I find from (46) and (47) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| S_{i}x_{n}-T_{i}x_{n}\right\| =0, \text { \ \ }\forall i\in \left\{ 1,2,\ldots ,N\right\} . \end{aligned}$$

(48)

This completes the proof. $\square$

Theorem 2

Let E be a real uniformly convex Banach space, K be a nonempty,closed and convex subset of E. Let $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N uniformly $L_{i}\hbox{-}Lipschitz$ total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) and (2) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\quad \sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(49)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \cap F\left( S_{\dot{I}}\right) \ne \varnothing$. Assume that there exist at least a $T_{i} \left( i\in I\right)$ which is $semi-compact$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K. Let $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be four real sequences in $\left[ \frac{L-1}{L},a\right]$, where $L=\max _{1\le i\le N}\left\{ L_{i}\right\} >1$ and a is some constant in $\left( 0,1\right)$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{ \infty }\delta _{n}<\infty .$

Then the sequence $\left\{ x_{n}\right\}$ be defined by (3) converges strongly to a common fixed point of $\left\{ T_{1},T_{2},\ldots ,T_{N},S_{1},S_{2},\ldots ,S_{N}\right\}$ in E.

Proof

Without loss of generality, I may assume that $T_{1}$ is semicompact. It follows from (46) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n}-T_{1}x_{n}\right\| =0. \end{aligned}$$

(50)

By the semicompactness of $T_{1}$, I have there exists a subsequence $\left\{ x_{n_{i}}\right\}$ of $\left\{ x_{n}\right\}$ such that $x_{n_{i}}\rightarrow q\in K$ strongly. From (46), I have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n_{i}}-T_{l}x_{n_{i}}\right\| =\left\| q-T_{l}q\right\| =0, \end{aligned}$$

for all $l=1,2,\ldots ,N$. Also, it follows from (48) and (50), I have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| x_{n_{i}}-S_{l}x_{n_{i}}\right\| =\left\| q-S_{l}q\right\| =0, \end{aligned}$$

for any $l\in I$. This implies that $q\in {\mathcal {F}}$. From Lemma 3 , I know that $\lim _{n\rightarrow \infty }\left\| x_{n}-q\right\|$ exists for all $q\in {\mathcal {F}}$. It follows from $x_{n_{i}}\rightarrow q$ that $\lim _{n\rightarrow \infty }\left\| x_{n}-q\right\| =0$. Thus, the iterative sequence $\left\{ x_{n}\right\}$ defined by (3) converges strongly to a common fixed point of $\left\{ T_{1},T_{2},\ldots ,T_{N},S_{1},S_{2},\ldots ,S_{N}\right\}$ in E. This completes the proof. $\square$

Corollary 1

Let E be a real uniformly convex Banach space, K be a nonempty,closed and convex subset of E. Let $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be N uniformly $L_{i}-Lipschitz$ total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(51)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \ne \varnothing$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ x_{n}\right\}$ be the sequence defined by

$$\begin{aligned} \left\{ \begin{array}{l} n=\left( k\left( n\right) -1\right) +i\left( n\right) \in \left\{ 1,\ldots ,N\right\} , \\ x_{n}=\left( 1-\alpha _{n}-\gamma _{n}\right) x_{n-1}+\alpha _{n}T_{i\left( n\right) }^{k\left( n\right) }y_{n}+\gamma _{n}u_{n}, \\ y_{n}=\left( 1-\beta _{n}-\delta _{n}\right) x_{n}+\beta _{n}T_{i\left( n\right) }^{k\left( n\right) }x_{n}+\delta _{n}v_{n},\text { }n\ge 1, \end{array} \right. \end{aligned}$$

where $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences in K. Let $\left\{ \alpha _{n}\right\} ,\left\{ \beta _{n}\right\} ,\left\{ \gamma _{n}\right\}$ and $\left\{ \delta _{n}\right\}$ be four real sequences in $\left[ \frac{L-1}{L},a\right]$, where $L=\max _{1\le i\le N}\left\{ L_{i}\right\} >1$ and a is some constant in $\left( 0,1\right)$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$, $\sum \nolimits _{n=1}^{ \infty }\delta _{n}<\infty .$

Then

• (a) The implicit iterative sequence $\left\{ x_{n}\right\}$ converges strongly to a common fixed point in ${\mathcal {F}}$ if and only if $\lim \inf _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$, where $d\left( x,{\mathcal {F}}\right)$ denotes the distance of x to set ${\mathcal {F}}$, i.e., $d\left( x,{\mathcal {F}}\right) =\inf _{y\in {\mathcal {F}}}d\left( x,y\right) .$

• (b) If one of the mappings in $\left\{ T_{1},\ldots ,T_{N}\right\}$ is $semi-compact$, then the sequence $\left\{ x_{n}\right\}$ converges strongly to a common fixed point of $\left\{ T_{1},T_{2},\ldots ,T_{N}\right\} .$

Corollary 2

Let E be a real uniformly convex Banach space, K be a nonempty,closed and convex subset of E. Let $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be N uniformly $L_{i}-Lipschitz$ total asymptotically nonexpansive mappings $\left\{ \mu _{n}\right\}$, $\left\{ l_{n}\right\}$ defined by (1) such that

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\mu _{n}<\infty ,\quad \sum \limits _{n=1}^{\infty }l_{n}<\infty \end{aligned}$$

(52)

and ${\mathcal {F}}:\bigcap \nolimits _{i=1}^{N}F\left( T_{\dot{I}}\right) \ne \varnothing$. Assume that there exist M, $M^{*}>0$ such that $\phi (\lambda )\le M^{*}\lambda$ for all $\lambda \ge M$, $i\in \left\{ i=1,2,\ldots ,N\right\} .$ Let $\left\{ x_{n}\right\}$ be the sequence defined by

$$\begin{aligned} \left\{ \begin{array}{l} n=\left( k\left( n\right) -1\right) +i\left( n\right) \in \left\{ 1,\ldots ,N\right\} , \\ x_{n}=\left( 1-\alpha _{n}-\gamma _{n}\right) x_{n-1}+\alpha _{n}T_{i\left( n\right) }^{k\left( n\right) }x_{n}+\gamma _{n}u_{n},\text { }n\ge 1, \end{array} \right. \end{aligned}$$

where $\left\{ u_{n}\right\}$ is a bounded sequence in K. Let $\left\{ \alpha _{n}\right\}$ and $\left\{ \gamma _{n}\right\}$ be two real sequences in $\left[ \frac{L-1}{L},a\right]$, where $L=\max _{1\le i\le N}\left\{ L_{i}\right\} >1$ and a is some constant in $\left( 0,1\right)$ satisfying the following conditions:

• (i) $\alpha _{n}+\gamma _{n}\le 1$ and $\beta _{n}+\delta _{n}\le 1$ for all $n\ge 1;$

• (ii) $0<\lim \inf _{n\rightarrow \infty }\alpha _{n}<\lim \sup _{n\rightarrow \infty }\alpha _{n}<1;$

• (iii) $\sum \nolimits _{n=1}^{\infty }\gamma _{n}<\infty$.

Then

• (a) The implicit iterative sequence $\left\{ x_{n}\right\}$ converges strongly to a common fixed point in ${\mathcal {F}}$ if and only if $\lim \inf _{n\rightarrow \infty }d\left( x_{n},{\mathcal {F}}\right) =0$, where $d\left( x,{\mathcal {F}}\right)$ denotes the distance of x to set ${\mathcal {F}}$, i.e., $d\left( x,{\mathcal {F}}\right) =\inf _{y\in {\mathcal {F}}}d\left( x,y\right) .$

• (b) If one of the mappings in $\left\{ T_{1},\ldots ,T_{N}\right\}$ is $semi-compact$, then the sequence $\left\{ x_{n}\right\}$ converges strongly to a common fixed point of $\left\{ T_{1},T_{2},\ldots ,T_{N}\right\} .$

Remark 1

Since total asymptotically nonexpansive mappings include asymptotically nonexpansive mappings, Theorem 2 improves and generalizes Theorem 3.7 in Cianciaruso et al. (2010) and Theorem 3.7 in Hao et al. (2012).

Conclusion 1

• 1. My theorems and corolaries which include the corresponding results announced in Xu and Ori (2001), Sun (2003), Chang et al. (2006), Gu (2006) as special cases fundamentally improve and generalize the results of Cianciaruso et al. (2010) and Hao et al. (2012) in the following sense.

  • (i) Extend the mappings from the class of asimptotically nonexpansive mappings to the class of total asimptotically nonexpansive mappings.

  • (ii) Extend the mappings from $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be N mappings to $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:K\rightarrow K$ be 2N mappings.

• 2. I considered an Ishikawa-type iterative algorithm for the class of total asimptotically nonexpansive mappings. But, my results are also available in the Mann-type iterative algorithm for the class of total asimptotically nonexpansive mappings.

Example 1

Let E is the real line with the usual norm $\left| .\right|$, $K=\left( -1,1\right)$. Assume that $Tx=\sin x$ and $Sx=\sin \left( -x\right)$ for $x\in K$. Let $\phi$ be a strictly increasing continuous function such that $\phi : \mathbb {R} ^{+}\rightarrow \mathbb {R}^{+}$ with $\phi (0)=0$. Let $\{\mu _{n}\}_{n\ge 1}$ and $\{l_{n}\}_{n\ge 1}$ in $\mathbb {R}$ be two sequences defined by $\mu _{n}=\frac{1}{n}$ and $l_{n}=\frac{1}{n+1}$, for all $n\ge 1$ ($\lim _{n\rightarrow \infty }\mu _{n}=\lim _{n\rightarrow \infty }\frac{1}{n}=0$, $\lim _{n\rightarrow \infty }l_{n}=\frac{1}{n+1}=0$). Since $Tx=\sin x$ for $x\in K$, I have

$$\begin{aligned} \left| T^{n}x-T^{n}y\right| \le \left| x-y\right| . \end{aligned}$$

For all x, $y\in K,$ I obtain

$$\begin{aligned}&\left| T^{n}x-T^{n}y\right| -\left| x-y\right| -\mu _{n}\phi (\left| x-y\right| )-l_{n} \\&\quad \le \left| x-y\right| -\left| x-y\right| -\mu _{n}\phi (\left| x-y\right| )-l_{n} \\&\quad \le 0 \end{aligned}$$

for all $n=1,2,\ldots$, $\{\mu _{n}\}_{n\ge 1}$ and $\{l_{n}\}_{n\ge 1}$ with $\mu _{n}$, $l_{n}\rightarrow 0$ as $n\rightarrow \infty$ and so T is a total asymptoticaly nonexpansive mapping. Also, $Sx=\sin \left( -x\right)$ for $x\in K$, I have

$$\begin{aligned} \left| S^{n}x-S^{n}y\right| \le \left| x-y\right| . \end{aligned}$$

For all x, $y\in K,$ I obtain

$$\begin{aligned}&\left| S^{n}x-S^{n}y\right| -\left| x-y\right| -\mu _{n}\phi (\left| x-y\right| )-l_{n} \\&\quad \le \left| x-y\right| -\left| x-y\right| -\mu _{n}\phi (\left| x-y\right| )-l_{n} \\&\quad \le 0 \end{aligned}$$

for all $n=1,2,\ldots$, $\{\mu _{n}\}_{n\ge 1}$ and $\{l_{n}\}_{n\ge 1}$ with $\mu _{n}$, $l_{n}\rightarrow 0$ as $n\rightarrow \infty$ and so S is a total asymptoticaly nonexpansive mapping. Clearly, $F:=F\left( T\right) \cap F\left( S\right) =\left\{ 0\right\}$. Set

$$\begin{aligned} \alpha _{n}=\beta _{n}=\frac{n}{2n+1},\text { }\delta _{n}=\gamma _{n}=\frac{ n^{3}}{6n^{3}+1}\text { and }u_{n}=v_{n}=\frac{1}{n+1} \end{aligned}$$

for $n\ge 1$. In order to easily calculate, I modifed my iteration scheme for $n=1$. This scheme (53) is defined as follows:

$$\begin{aligned} x_{1} &= \left( 1-\alpha _{1}-\gamma _{1}\right) x_{0}+\alpha _{1}T_{i\left( 1\right) }^{k\left( 1\right) }y_{1}+\gamma _{1}u_{1}, \nonumber \\ y_{1}&= {} \left( 1-\beta _{1}-\delta _{1}\right) x_{1}+\beta _{1}S_{i\left( 1\right) }^{k\left( 1\right) }x_{1}+\delta _{1}v_{1}. \end{aligned}$$

(53)

The numerical experiment outcome obtained by using Scientific WorkPlace 5.5 show that as $x_{0}=0$, the computation of $y_{1}=9.0853\times 10^{-2}$. This example illustrates the efficiency of approximation of common fixed points of total asymptotically nonexpansive mappings.

Let $\left\{ S_{i}\right\} _{i=1}^{N}$, $\left\{ T_{i}\right\} _{i=1}^{N}:E\rightarrow E$ be 2N total asymptotically nonexpansive mappings; assuming existence of common fixed points of these operators, our theorems and method of proof easily carry over to this class of mappings using the following implicit iterative scheme $\left\{ x_{n}\right\}$ with errors:

$$\begin{aligned} \left\{ \begin{array}{l} x_{0}\in K, \\ x_{n}=a_{n}x_{n-1}+b_{n}T_{i\left( n\right) }^{k\left( n\right) }y_{n}+c_{n}u_{n}, \\ y_{n}=a_{n}^{^{\prime }}x_{n}+b_{n}^{^{\prime }}S_{i\left( n\right) }^{k\left( n\right) }x_{n}+c_{n}^{^{\prime }}v_{n},\text { }n\ge 1 \end{array} \right. \end{aligned}$$

(54)

where $n=\left( k\left( n\right) -1\right) N+i\left( n\right)$, $i\left( n\right) \in 1,2,\ldots ,N$, $\left\{ a_{n}\right\}$, $\left\{ b_{n}\right\}$, $\left\{ c_{n}\right\}$, $\left\{ a_{n}^{^{\prime }}\right\}$, $\left\{ b_{n}^{^{\prime }}\right\}$, $\left\{ c_{n}^{^{\prime }}\right\}$ are six real sequences in $\left[ 0,1\right]$ satisfying $a_{n}= b_{n}=c_{n}=1=a_{n}^{^{\prime }}= b_{n}^{^{\prime }}=c_{n}^{^{\prime }}$ for all $n\ge 1$, $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ are two bounded sequences.

In order not to enlarge this note unnecessarily, I only include total asymptotically nonexpansive mappings. But, in accordance with the above proof of theorem, one can easily prove in total asymptotically quasi nonexpansive mappings.

If $S=I$, then (54) transform to implicit iterative scheme defined by Mukhamedov and Saburov (see, more details Mukhamedov and Saburov 2012a). My theorems and corolaries also improve and generalize the mappings from the class of a finite family of quasi-asimptotically nonexpansive mappings to the class of a finite family of total quasi-asimptotically nonexpansive mappings.

If S, $T:E\rightarrow E$ be two total asymptotically nonexpansive mappings and take $S=I$, then (54) reduce to implicit iterative scheme defined by Mukhamedov and Saburov (see, more details Mukhamedov and Saburov 2011). My theorems and corolaries also improve and generalize the mappings from the class of quasi-asimptotically nonexpansive mappings to the class of total quasi-asimptotically nonexpansive mappings.