Throughout this work, we assume that $E$ is a real Banach space, $E^{*}$ is the dual space of $E$ and $J:E \rightarrow 2^{E^{*}}$ is the normalized duality mapping defined by

$J(x)=\{f\in E^{*}:\langle x, f\rangle =\|x\|\|f\|, \|f\|=\|x\|\}, \quad \forall x\in E, $

where $\langle\cdot, \cdot\rangle$ denotes duality pairing between $E$ and $E^{*}$. A single-valued normalized duality mapping is denoted by $j$.

Let $C$ be a nonempty subset of $E$ and $T:C\rightarrow C$ be a mapping. We denote the set of fixed points of $T$ by $F(T)$, i.e., $F(T)=\{x\in C : Tx = x\}.$

Definition 1.1 (1) (see [1]) $T$ is said to be pseudocontractive, if for all $x, y \in C, $ there exists $j(x-y)\in J(x-y)$ such that $\langle Tx-Ty, j(x-y)\rangle \leq \|x-y\|^{2}.$

(2) (see [2]) $T$ is said to be uniformly L-Lipshitzian if there exists $L > 0$ such that

$\|T^{n}x-T^{n}y\| \leq L\|x-y\|$

for all $x, y \in C$ and $n\geq 1$.

(3) (see [3]) $T$ is said to be asymptotically pseudocontractive if there exists a sequence $\{k_n\} \subset [1, \infty) $ with $\lim\limits_{n\rightarrow \infty} k_n =1$, for any $x, y\in C$, there exists $j(x-y)\in J(x-y)$ such that

$\langle T^{n}x-T^{n}y, j(x-y)\rangle \leq k_{n}\|x-y\|^{2}, \quad n \geq 1.$

Recently, Guo Weiping and Guo Qi [4] introduced a new mapping as follows:

Definition 1.2 $T$ is said to be asymptotically hemi-pseudocontractive if $F(T)\neq \emptyset$ and there exists a sequence $\{k_n\} \subset [1, \infty) $ with $\lim_{n\rightarrow \infty} k_n =1$ such that, for any $x\in C$ and $p \in F(T)$, there exists $j(x-p)\in J(x-p)$ such that

$\langle T^{n}x-p, j(x-p)\rangle \leq k_{n}\|x-p\|^{2}, \quad n \geq 1.$

Remark 1.1 It is easy to see that if $T$ is an asymptotically pseudocontractive mapping with $F(T)\neq\emptyset$, then $T$ is an asymptotically hemi-pseudocontractive mapping. Conversely, Guo Weiping and Guo Qi [4] give an example to show that the converse is not true in general.

Let $C$ be a nonempty convex subset of $E$ with $C+C\subset C$ and $T: C\rightarrow C$ be a mapping. For any given $x_{1}\in C$, the sequence $\{x_{n}\}$ defined by

$\left \{\begin{array}{ll} x_{n+1} = (1-\alpha_{n})x_{n}+\alpha_{n}T^{n}y_{n}+u_{n}, \\ y_{n} = (1-\beta_{n})x_{n}+\beta_{n}T^{n}x_{n}+v_{n},\quad\forall n\geq 1,\end{array}\right.$

(1.1)

where $\{\alpha_{n}\}$, $\{\beta_{n}\}$ are two real sequences in [0, 1] and $\{u_{n}\}$, $\{v_{n}\}$ are two bounded sequences in $C$.

Letting $\beta_{n}=0$, $v_{n}=0$, $\forall n\geq 1$ in (1.1), then the sequence $\{x_{n}\}$ defined by (1.1) is reduced to the following iterative scheme:

$x_{n+1} = (1-\alpha_{n})x_{n}+\alpha_{n}T^{n}x_{n}+u_{n}, \quad \forall n\geq 1. $

(1.2)

For any $z\in C$ and a set $K$ in $E$, we denote the distance between $z$ and $K$ by $d(z, K)=\inf\limits_{y\in K}\|z-y\|.$

Recently, Tang et al. [5] proved some sufficient and necessary conditions for strong convergence of Lipschitzian pseudoncontractive mappings in Banach spaces.

In this work, we give a new method and prove some sufficient and necessary conditions for the strong convergence of the iterations sequences (1.1) and (1.2) to a fixed point of asymptotically hemi-pseudocontractive in Banach spaces.

Lemma 1.1 (see [6]) Let $X$ be a Banach space and $x, y\in X.$ Then $\|x\|\leq\|x+ry\|$ for all $r>0$ if and only if there exists $j(x)\in J(x)$ such that $\langle y, j(x)\rangle\geq 0.$

Lemma 1.2 (see [7]) Let $\{a_{n}\}$, $\{\lambda_{n}\}$, $\{\sigma_{n}\}$ be three nonnegative sequences satisfying the following inequality:

$a_{n+1}\leq(1+\lambda_{n})a_{n}+\sigma_{n}, \forall n\geq n_{_{0}}, $

where $n_{0}$ is some nonnegative integer, $\sum\limits_{n=1}^{\infty}\lambda_{n}<\infty$ and $\sum\limits_{n=1}^{\infty}\sigma_{n}<\infty$. Then $\lim\limits_{n\rightarrow\infty}a_{n}$ exists. In particular, if $\liminf\limits_{n\rightarrow\infty}a_{n}=0$, then $\lim\limits_{n\rightarrow\infty}a_{n}=0$.

First, we prove the following lemmas.

Lemma 2.1 Let $C$ be a nonempty subset of a Banach space $E$ and $T:C\rightarrow C$ be an asymptotically hemi-pseudocontractive mapping with the sequence $\{k_{n}\}\subset [1, \infty)$, $\lim\limits_{n\rightarrow\infty}k_{n}=1$. Then

$\|x-p\|\leq\|x-p+r[(k_{n}I-T^{n})x-(k_{n}I-T^{n})p]\|$

(2.1)

for all $x\in C$, $p\in F(T)$, $r > 0$ and $n\geq 1$, where $I$ is identity mapping.

Proof Since $T$ is an asymptotically hemi-pseudocontractive mapping with the sequence $\{k_{n}\}$, for all $x\in C$ and $p\in F(T)$, there exists $j(x-p)\in J(x-p)$, such that

$\langle T^{n}x-p, j(x-p)\rangle\leq k_{n}\|x-p\|^{2}=k_{n}\langle x-p, j(x-p)\rangle, n\geq 1,$

and so

$\langle(k_{n}I-T^{n})x-(k_{n}I-T^{n})p, j(x-p)\rangle \geq 0.$

Therefore, (2.1) holds by Lemma 1.1. This completes the proof.

Lemma 2.2  Let $E$ be a real Banach space and $C$ be a nonempty convex subset of $E$ with $C+C\subset C$ and $T:C\rightarrow C$ be a uniformly $L$-Lipschitzian asymptotically hemi-pseudocontractive mapping with the sequence $\{k_{n}\}\subset [1, \infty)$, $\lim\limits_{n\rightarrow\infty}k_{n}=1$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ be two real sequences in [0, 1] and $\{u_n\}$, $\{v_n\}$ be two bounded sequences in $C$. Suppose that the sequence $\{x_n\}$ is defined by (1.1) satisfying the following conditions:

(ⅰ) $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2}<\infty$, $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$;

(ⅱ)$\sum\limits_{n=1}^{\infty}\|u_{n}\|<\infty$, $\sum\limits_{n=1}^{\infty}\|v_{n}\|<\infty$;

(ⅱ) $\sum\limits_{n=1}^{\infty}\alpha_{n}(k_{n}-1)<\infty$.

Then (1) there exist two sequences $\{r_{n}\}, \{s_{n}\}\subset[0, \infty)$, such that $\sum\limits_{n=1}^{\infty}r_{n}<\infty$, $\sum\limits_{n=1}^{\infty}s_{n}<\infty$ and

$\|x_{n+1}-p\|\leq(1+r_{n})\|x_{n}-p\|+s_{n}$

(2.2)

for all $p\in F(T)$ and $n\geq 1$.

(2) The limit $\lim\limits_{n\rightarrow\infty}d(x_{n}, F(T))$ exists.

Proof (1) Let $p\in F(T)$, by (1.1), we have

$\begin{align} x_{n}&=x_{n+1}+\alpha_{n}x_{n}-\alpha_{n}T^{n}y_{n}-u_{n}\\ &=(1+\alpha_{n})x_{n+1}+\alpha_{n}(k_{n}I-T^{n})x_{n+1}-(1+k_{n})\alpha_{n}x_{n+1}\\ &\quad +\alpha_{n}x_{n}+\alpha_{n}(T^{n}x_{n+1}-T^{n}y_{n})-u_{n}\\ &=(1+\alpha_{n})x_{n+1}+\alpha_{n}(k_{n}I-T^{n})x_{n+1}-(1+k_{n})\alpha_{n}[x_{n}+\alpha_{n}(T^{n}y_{n}-x_{n})+u_{n}]\\ &\quad +\alpha_{n}x_{n}+\alpha_{n}(T^{n}x_{n+1}-T^{n}y_{n})-u_n\\ &=(1+\alpha_{n})x_{n+1}+\alpha_{n}(k_{n}I-T^{n})x_{n+1}-(1+k_{n})\alpha_{n}x_{n}+(1+k_{n})\alpha_{n}^{2}(x_{n}-T^{n}y_{n})\\ &\quad +\alpha_{n}x_{n}+\alpha_{n}(T^{n}x_{n+1}-T^{n}y_{n})-[(1+k_n)\alpha_{n}+1]u_n\\ &=(1+\alpha_{n})x_{n+1}+\alpha_{n}(k_{n}I-T^{n})x_{n+1}-k_{n}\alpha_{n}x_{n}+(1+k_{n})\alpha_{n}^{2}(x_{n}-T^{n}y_{n})\\ &\quad +\alpha_{n}(T^{n}x_{n+1}-T^{n}y_{n})-[(1+k_n)\alpha_{n}+1]u_n \end{align}$

(2.3)

and

$p=(1+\alpha_{n})p+\alpha_{n}(k_{n}I-T^{n})p-k_{n}\alpha_{n}p.$

(2.4)

Together with (2.3) and (2.4), we can obtain

$\begin{align} x_{n}-p=&(1+\alpha_{n})(x_{n+1}-p)+\alpha_{n}[(k_{n}I-T^{n})x_{n+1}-(k_{n}I-T^{n})p]-k_{n}\alpha_{n}(x_{n}-p)\\ &+(1+k_{n})\alpha_{n}^{2}(x_{n}-T^{n}y_{n})+\alpha_{n}(T^{n}x_{n+1}-T^{n}y_{n})-[(1+k_n)\alpha_{n}+1]u_n. \end{align}$

(2.5)

Notice that

$\begin{eqnarray*} && (1+\alpha_{n})(x_{n+1}-p)+\alpha_{n}[(k_{n}I-T^{n})x_{n+1}-(k_{n}I-T^{n})p]\\ &=&(1+\alpha_{n})[(x_{n+1}-p)+\frac{\alpha_{n}}{1+\alpha_{n}}((k_{n}I-T^{n})x_{n+1}-(k_{n}I-T^{n})p)].\end{eqnarray*}$

Using Lemma 2.1, we obtain that

$\|(1+\alpha_{n})(x_{n+1}-p)+\alpha_{n}(k_{n}I-T^{n})(x_{n+1}-p)\| \geq(1+\alpha_{n})\|x_{n+1}-p\|.$

(2.6)

It follows from (2.5) and (2.6) that

$\begin{align} \|x_{n}-p\|&\geq(1+\alpha_{n})\|x_{n+1}-p\|-k_{n}\alpha_{n}\|x_{n}-p\|-(1+k_{n})\alpha_{n}^{2}\|x_{n}-T^{n}y_{n}\|\\ &\quad -\alpha_{n}\|T^{n}x_{n+1}-T^{n}y_{n}\|-[(1+k_n)\alpha_{n}+1]\|u_n\|. \end{align}$

This implies that

$\begin{align} (1+\alpha_{n})\|x_{n+1}-p\|&\leq(1+k_{n}\alpha_{n})\|x_{n}-p\|+(1+k_{n})\alpha_{n}^{2}\|x_{n}-T^{n}y_{n}\|\\ &\quad +\alpha_{n}\|T^{n}x_{n+1}-T^{n}y_{n}\|+[(1+k_n)\alpha_{n}+1]\|u_n\|. \end{align}$

(2.7)

Next, we make the following estimations:

$\begin{align} \|y_{n}-p\|&=\|(1-\beta_{n})(x_{n}-p)+\beta_{n}(T^{n}x_{n}-p)+v_{n}\|\\ &\leq(1-\beta_{n})\|x_{n}-p\|+\beta_{n}\|T^{n}x_{n}-p\|+\|v_{n}\|\\ &\leq(1-\beta_{n})\|x_{n}-p\|+L\beta_{n}\|x_{n}-p\|+\|v_{n}\|\\ &\leq(1-\beta_{n}+L\beta_{n})\|x_{n}-p\|+\|v_{n}\|, \\ \|x_{n}-T^{n}y_{n}\| &\leq\|x_{n}-p\|+\|p-T^{n}y_{n}\|\\ &\leq\|x_{n}-p\|+L\|y_{n}-p\|\\&\leq[1+L(1-\beta_{n}+L\beta_{n})]\|x_{n}-p\|+L\|v_{n}\|, \end{align}$

(2.8)

$\begin{align} \|x_{n}-y_{n}\|&\leq\beta_{n}\|x_{n}-T^{n}x_{n}\|+\|v_{n}\|\\ &\leq\beta_{n}\|x_{n}-p\|+\beta_{n}\|T^{n}x_{n}-p\|+\|v_{n}\|\\ &\leq\beta_{n}(1+L)\|x_{n}-p\|+\|v_{n}\|,\end{align}$

(2.9)

$\begin{align}\|T^{n}x_{n+1}-T^{n}y_{n}\|&\leq L\|x_{n+1}-y_{n}\|\\ &=L\|x_{n}-y_{n}+\alpha_n(T^{n}y_{n}-x_{n})+u_{n}\|\\ &\leq L\|x_{n}-y_{n}\|+\alpha_{n}L\|T^{n}y_{n}-x_{n}\|+L\|u_{n}\|. \end{align}$

(2.10)

Substituting (2.8) and (2.9) into (2.10), we have

$\begin{align}\label{E:2.11} \|T^{n}x_{n+1}-T^{n}y_{n}\|&\leq[\alpha_{n}L+\alpha_{n}L^{2}(1-\beta_{n}+L\beta_{n})+\beta_{n}L(1+L)]\|x_{n}-p\|\\ &\quad+L(1+\alpha_{n}L)\|v_{n}\|+L\|u_{n}\|. \end{align}$

(2.11)

Substituting (2.8) and (2.11) into (2.7), we have

$\begin{align} (1+\alpha_{n})\|x_{n+1}-p\|&\leq(1+k_{n}\alpha_{n})\|x_{n}-p\|\\ &\quad+(1+k_{n})\alpha_{n}^{2}[(1+L(1-\beta_{n}+L\beta_{n}))\|x_{n}-p\|+L\|v_{n}\|]\\ &\quad+\alpha_{n}[\alpha_{n}L+\alpha_{n}L^{2}(1-\beta_{n}+L\beta_{n})+\beta_{n}L(1+L)] \|x_{n}-p\|\\ &\quad+\alpha_{n}L(1+\alpha_{n}L)\|v_{n}\|+\alpha_{n}L\|u_{n}\|+ [(1+k_n)\alpha_{n}+1]\|u_n\|. \end{align}$

By $1+\alpha_{n}\geq 1$, this implies that

$\begin{align} \|x_{n+1}-p\|&\leq(1+(k_{n}-1)\alpha_{n})\|x_{n}-p\|\\ &\quad+(1+k_{n})\alpha_{n}^{2}[(1+L(1-\beta_{n}+L\beta_{n}))\|x_{n}-p\|+L\|v_{n}\|]\\ &\quad+\alpha_{n}[\alpha_{n}L+\alpha_{n}L^{2}(1-\beta_{n}+L\beta_{n})+\beta_{n}L(1+L)]\|x_{n}-p\|\\ &\quad+\alpha_{n}L(1+\alpha_{n}L)\|v_{n}\|+\alpha_{n}L\|u_{n}\|+[(1+k_n)\alpha_{n}+1]\|u_n\|\\ &=[1+(k_{n}-1)\alpha_{n}+(1+k_{n})(1+L(1-\beta_{n}+L\beta_{n}))\alpha_{n}^{2}\\ &\quad+\alpha_{n}^{2}L(1+L(1-\beta_{n}+L\beta_{n}))\\ &\quad+\alpha_{n}\beta_{n}L(1+L)]\|x_{n}-p\|+L[(1+k_{n})\alpha_{n}^{2}+\alpha_{n}(1+\alpha_{n}L)]\|v_{n}\|\\ &\quad+[1+(1+k_{n})\alpha_{n}+\alpha_{n}L]\|u_{n}\|\\ &=(1+r_{n})\|x_{n}-p\|+s_{n}, \end{align}$

where

$\begin{eqnarray*}r_{n}&=&(k_{n}-1)\alpha_{n}+(1+k_{n})(1+L(1-\beta_{n}+L\beta_{n}))\alpha_{n}^{2}\\ && +\alpha_{n}^{2}L(1+L(1-\beta_{n}+L\beta_{n}))+\alpha_{n}\beta_{n}L(1+L), \\ s_{n}&=& L[(1+k_{n})\alpha_{n}^{2}+\alpha_{n}(1+\alpha_{n}L)]\|v_{n}\|+[1+(1+k_{n})\alpha_{n}+\alpha_{n}L]\|u_{n}\|, \end{eqnarray*}$

and $\sum\limits_{n=1}^{\infty}r_{n}< \infty$, $\sum\limits_{n=1}^{\infty}s_{n}< \infty$ by conditions (ⅰ)-(ⅲ). This completes the proof of (1).

(2) Taking the infimum over all $p\in F(T)$ on both sides in (2.2), we get

$d(x_{n+1}, F(T))\leq (1+r_{n})d(x_{n}, F(T))+s_{n}.$

It follows from Lemma 1.2 that the limit $\lim\limits_{n\rightarrow\infty}d(x_{n}, F(T))$ exists.

Theorem 2.1  Let $E$ be a real Banach space and $C$ be a nonempty closed convex subset of $E$ with $C+C\subset C$ and $T:C\rightarrow C$ be a uniformly $L$-Lipschitzian asymptotically hemi-pseudocontractive mapping with the sequence $k_{n}\subset [1, \infty)$, $\lim\limits_{n\rightarrow\infty}k_{n}=1$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ be two real sequences in [0, 1] and $\{u_n\}$, $\{v_n\}$ be two bounded sequences in $C$. Suppose that the sequence $\{x_n\}$ is defined by (1.1) satisfying the following conditions:

(ⅰ) $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2}<\infty$, $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$;

(ⅱ) $\sum\limits_{n=1}^{\infty}\|u_{n}\|<\infty$, $\sum\limits_{n=1}^{\infty}\|v_{n}\|<\infty$;

(ⅲ) $\sum\limits_{n=1}^{\infty}\alpha_{n}(k_{n}-1)<\infty$.

Then $\{x_{n}\}$ converges strongly to a fixed point of $T$ if and only if $\liminf\limits_{n\rightarrow\infty}d(x_{n}, F(T))=0.$

Proof The necessary of Theorem 2.1 is obvious. We just need to prove the sufficiency. Assume that $\liminf\limits_{n\rightarrow\infty}d(x_{n}, F(T))=0$, by Lemma 1.2, then $\lim\limits_{n\rightarrow\infty}d(x_{n}, F(T))=0$.

Next, we show that $\{x_{n}\}$ is a Cauchy sequence. In fact, for any $p\in F(T)$ and any positive integers $m, n$, $m>n$, from inequality $1+x\leq e^{x}$, $x\geq 0$ and Lemma 2.2, we have

$\begin{align} \|x_{m}-p\|&\leq \prod\limits_{j=n}^{m-1}(1+r_{j})\|x_{n}-p\|+\sum\limits_{j=n}^{m-1}s_{j}\prod\limits_{j=n}^{m-1}(1+r_{j})\\ &\leq e^{\sum\limits_{j=n}^{m-1}r_{j}}\|x_{n}-p\|+e^{\sum\limits_{j=n}^{m-1}r_{j}}\sum\limits_{j=n}^{m-1}s_{j} \leq M\|x_{n}-p\|+M\sum\limits_{j=n}^{m-1}s_{j}, \end{align}$

where $M=e^{\sum\limits_{j=1}^{\infty}r_{j}}$. Thus, we have

$ \|x_{n}-x_{m}\|\leq\|x_{n}-p\|+\|x_{m}-p\| \leq(1+M)\|x_{n}-p\|+M\sum\limits_{j=n}^{\infty}s_{j}. $

Taking the infimum over all $p\in F(T)$, we have

$\|x_{n}-x_{m}\|\leq (1+M) d(x_{n}, F(T))+ M\sum\limits_{j=n}^{\infty}s_{j}.$

It follows from $\sum\limits_{j=1}^{\infty}s_{j}<\infty$ and $\lim\limits_{n\rightarrow\infty}d(x_{n}, F)=0$ that $\{x_{n}\}$ is a Cauchy sequence. Since $C$ is a nonempty closed convex subset of $E$, so there exists a $p_{0}\in C$ such that $x_{n}\rightarrow p_{0}$ as $n\rightarrow \infty.$ Further, since $T$ is uniformly $L$-Lipschitzian, it is easy to prove that $F(T)$ is closed. Again since $\lim\limits_{n\rightarrow\infty}d(x_{n}, F(T))=0$ and so $p_{0}\in F(T)$. This shows that $\{x_{n}\}$ converges strongly to a fixed point of $T$. This completes the proof.

Corollary 2.1  Under the assumptions of Theorem 2.1. Then $\{x_{n}\}$ converges strongly to a fixed point $p$ of $T$ if and only if there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ which converges strongly to $p$.

Proof It follows from $\liminf\limits_{n\rightarrow\infty}d(x_{n}, F)\leq\liminf\limits_{k\rightarrow\infty}d(x_{n_{k}}, F) \leq \lim\limits_{k\rightarrow\infty}\|x_{n_{k}}-p\|=0$ and Theorem 2.1 that Corollary 2.1 holds. This completes the proof.

Letting $\beta_{n}=0$ and $v_{n}=0$ for all $n\geq 1$ in Theorem 2.1, we obtain the following results:

Theorem 2.2  Let $E$ be a real Banach space and $C$ be a nonempty closed convex subset of $E$ with $C+C\subset C$ and $T:C\rightarrow C$ be a uniformly $L$-Lipschitzian asymptotically hemi-pseudocontractive mapping with the sequence $\{k_{n}\}\subset [1, \infty)$, $\lim\limits_{n\rightarrow\infty}k_{n}=1$. Let $\{\alpha_{n}\}$ be a real sequence in [0, 1] and $\{u_n\}$ be a bounded sequence in $C$. Suppose that the sequence $\{x_n\}$ is defined by (1.2) satisfying the following conditions:

(ⅰ)$\sum\limits_{n=1}^{\infty}\alpha_{n}^{2}<\infty$; (ⅱ)$\sum\limits_{n=1}^{\infty}\|u_{n}\|<\infty$; (ⅲ)$\sum\limits_{n=1}^{\infty}\alpha_{n}(k_{n}-1)<\infty$.

Then $\{x_{n}\}$ converges strongly to a fixed point of $T$ if and only if $\liminf\limits_{n\rightarrow\infty}d(x_{n}, F(T))=0.$

Corollary 2.2  Under the assumptions of Theorem 2.2. Then $\{x_{n}\}$ converges strongly to a fixed point $p$ of $T$ if and only if there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ which converges strongly to $p$.

Remark 2.1 By Remark 1.1, clearly Theorems 2.1, 2.2 and Corollaries 2.1, 2.2 hold for uniformly $L$-Lipschitzian and asymptotically pseudoncontractive mappings with $F(T)\neq\emptyset$.