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

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

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

Let $C$ be a nonempty subset of $E$, $T: C\rightarrow C$ be a mapping. We denote the set of fixed points of $T$ by $F(T)$.

A mapping $T$ with domain $D(T)$ and range $R(T)$ in $E$ is called pseudocontractive [1], if there exists some $j(x-y)\in J(x-y)$ such that

$\langle j(x-y),Tx-Ty\rangle \leq\|x-y\|^{2} $

(1.1)

for all $x,y\in D(T)$. It is well known that [2] (1.1) is equivalent to the following:

$\|x-y\|\leq\|x-y+s[(I-T)x-(I-T)y]\| $

(1.2)

for all $s>0$ and all $x,y \in D(T).$

Theorem 1.1 [3] Let $C$ be a convex compact subset of a Hilbert space $H$ and $T:C\rightarrow C$ be a Lipschitzian pseudocontractive mapping. For any $x_{1}\in C$, suppose the sequence $\{x_{n}\}$ is defined by

$\begin{equation*} \left \{\begin{array}{ll} x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}Ty_{n} ,\\ y_{n}=(1-\beta_{n})x_{n}+\beta_{n}Tx_{n},\quad n\geq 1,\\ \end{array}\right. \end{equation*}$

(1.3)

where $\{\alpha_{n}\}$, $\{\beta_{n}\}$ are two real sequences in $[0, 1]$ satisfying

(ⅰ) $\alpha_{n}\leq\beta_{n},n\geq1$;

(ⅱ) $\lim\limits_{n\rightarrow\infty}\beta_{n}=0;$

(ⅲ) $\Sigma_{n=1}^{\infty}\alpha_{n}\beta_{n}=\infty.$

Then $\{x_{n}\}$ converges strongly to a fixed point of $T$.

Remark 1.1 (1) Since $0\leq\alpha_{n}\leq\beta_{n}\leq1, n\geq1$ and $\Sigma_{n=1}^{\infty}\alpha_{n}\beta_{n}=\infty,$ the iterative sequence (1.3) couldn't be reduced to Mann iterative sequence by setting $\beta_{n}=0$. The Mann iterative sequence [4] is defined by the following

$x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}Tx_{n},\quad n\geq 1, $

(1.4)

where $\{\alpha_{n}\}$ is a appropriate sequence in [0, 1].

(2) Chidume and Mutangadura [5] gave an example to show that the Mann iterative sequence failed to be convergent to a fixed point point of Lipschitzian pseudocontractive mapping.

Let $C$ be a nonempty convex subset of a real Banach space and $T: C\rightarrow C$ be a Lipschitzian pseudocontractive mapping, we introduce a composite implicit iteration process $\{x_{n}\}$ as follows:

$\begin{equation*} \left \{\begin{array}{ll} x_{1}\in C, \\ x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}Ty_{n} ,\\ y_{n}=(1-\beta_{n})x_{n}+\beta_{n}Tx_{n+1},\quad n\geq 1, \end{array}\right. \end{equation*}$

(1.5)

where $\{\alpha_{n}\}$, $\{\beta_{n}\}\subset[0,1].$ When $\beta_{n}=0$, $\forall n\geq 1,$ (1.4) is the special form of (1.5).

In 1974, Deimling [6] proved the following fixed point theorem.

Theorem 1.2 Let $E$ be a real Banach space, $K$ a nonempty closed convex subset of $E$, and $T: K\rightarrow K$ a continuous strongly pseudocontractive mapping. Then, $T$ has a unique fixed point in $K$.

Observe that if $C$ is a nonempty closed convex subset of $E$ and $T:C\rightarrow C$ is a Lipschitz pseudocontractive mapping, then for every $u\in C$ and $t,s\in(0,1),$ the mapping $S_{t,s}: C\rightarrow C$ is defined by $S_{t,s}x=(1-t)u+tT((1-s)u+sTx)$ satisfies that

$\langle S_{t,s}x-S_{t,s}y , j(x-y)\rangle \leq tsL^{2}\|x-y\|^{2}$

for all $x,y \in C$. Thus $S_{t,s}$ is strongly pseudocontractive if $tsL^{2}<1$. Since $S_{t,s}$ is also Lipschitiz, it follows from Theorem 1.2 that there exists a unique fixed point $x_{t,s}\in C$ of $S_{t,s}$ such that

$x_{t,s}=(1-t)u+tT((1-s)u+sTx_{t,s}).$

This shows that the implicit iteration sequence (1.5) can be employed for the approximation of fixed points of a Lipschitz pseudocontractive mapping.

In this paper, we give necessary and sufficient conditions for the strong convergence of iterative sequence (1.5) and Mann iterative sequence to a fixed point of a Lipschitzian pseudocontractive mapping in Banach spaces.

In order to prove main results, the following lemma is needed.

Lemma 1.1 [7] Let $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ be sequences of nonnegative real numbers satisfying the inequality

$ a_{n+1}\leq (1+c_{n})a_{n}+b_{n},\quad n\geq1.$

If $\Sigma_{n=1}^{\infty}c_{n}<+\infty,$ $\Sigma_{n=1}^{\infty}b_{n}<+\infty,$ then $\lim\limits_{n\rightarrow\infty}a_{n}$ exists.

First, we prove the following lemma.

Lemma 2.1 Let $E$ be a real Banach space and $C$ a nonempty closed convex subset of $E$. Let $T :C\rightarrow C$ be a Lipschitzian pseudocontractive mapping with Lipschitz constant $L>1$ and $F(T)\neq\emptyset$. Suppose that the sequence $\{x_{n}\}$ is defined by (1.5) such that $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$, $ \alpha_{n}\beta_{n}L^{2}<1$ for all $n\geq 1$ and $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2} <\infty$. Then

(ⅰ) There exist a sequence $\{r_{n}\}\subseteq (0,\infty)$ and some positive integer, such that $\sum\limits_{n=1}^{\infty}r_{n}<\infty$ and

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

for all $p\in F(T)$ and $n\geq n_0$.

(ⅱ) There exists a constant $M>1$, for all integer $m\geq 1$ such that

$\|x_{n+m}-p\|\leq M\|x_{n}-p\|$

for all $p\in F(T)$.

Proof Let $p\in F(T).$ By (1.5), we have

$\begin{eqnarray}&& x_{n}=x_{n+1}+\alpha_{n}x_{n}-\alpha_{n}Ty_{n}\nonumber\\ &=& x_{n+1}+\alpha_{n}(I-T)x_{n+1}+\alpha_{n}^{2}(x_{n}-Ty_{n})+\alpha_{n}(Tx_{n+1}-Ty_{n}).\end{eqnarray}$

(2.1)

Observe that

$\begin{equation}p=p+\alpha_{n}(I-T)p. \end{equation}$

(2.2)

It follows from (2.1) and (2.2) that

$\begin{eqnarray} x_{n}-p&=&x_{n+1}-p+\alpha_{n}[(I-T)x_{n+1}-(I-T)p]\nonumber\\ && +\alpha_{n}^{2}(x_{n}-Ty_{n})+\alpha_{n}(Tx_{n+1}-Ty_{n}).\end{eqnarray}$

(2.3)

Together with (2.3) and (1.2), we have

$\begin{eqnarray*}\|x_{n}-p\|&\geq&\|x_{n+1}-p+\alpha_{n}[(I-T)x_{n+1}-(I-T)p]\|\\ && -\alpha_{n}^{2}\|x_{n}-Ty_{n}\|-\alpha_{n}\|Tx_{n+1}-Ty_{n}\|\\ &\geq& \|x_{n+1}-p\|-\alpha_{n}^{2}\|x_{n}-Ty_{n}\|-\alpha_{n}\|Tx_{n+1}-Ty_{n}\|.\end{eqnarray*}$

This implies that

$\begin{equation}\|x_{n+1}-p\|\leq\|x_{n}-p\|+\alpha_{n}^{2}\|x_{n}-Ty_{n}\|+\alpha_{n}\|Tx_{n+1}-Ty_{n}\|.\end{equation}$

(2.4)

Next, we make the following estimations:

$\begin{eqnarray}&& \|y_{n}-p\|\leq(1-\beta_{n})\|x_{n}-p\|+\beta_{n}\|Tx_{n+1}-p\|\nonumber\\ &\leq& (1-\beta_{n})\|x_{n}-p\|+\beta_{n}L\|x_{n+1}-p\|,\end{eqnarray}$

(2.5)

$\begin{eqnarray}&& && \|x_{n}-Ty_{n}\|\leq\|x_{n}-p\|+\|p-Ty_{n}\|\nonumber\\ &\leq&\|x_{n}-p\|+L\|y_{n}-p\|\end{eqnarray}$

(2.6)

and

$\begin{eqnarray}&&\|Tx_{n+1}-Ty_{n}\|\leq L\|x_{n+1}-y_{n}\| = L\|x_{n}-y_{n}+\alpha_{n}(Ty_{n}-x_{n})\|\nonumber\\ &\leq& L\beta_{n}\|Tx_{n+1}-x_{n}\|+L\alpha_{n}\|Ty_{n}-x_{n}\|\nonumber \\ &\leq& L^{2}\beta_{n}\|x_{n+1}-p\|+L\beta_{n}\|x_{n}-p\|+L\alpha_{n}(\|x_{n}-p\|+L\|y_{n}-p\|).\end{eqnarray}$

(2.7)

Substituting (2.5), (2.6) and (2.7) into (2.4) yields that

$\begin{eqnarray*}&&\|x_{n+1}-p\|\leq\|x_{n}-p\|+\alpha_{n}^{2}[\|x_{n}-p\|+L(1-\beta_{n})\|x_{n}-p\|+L^{2}\beta_{n}\|x_{n+1}-p\|]\\ && +\alpha_{n}[L^{2}\beta_{n}\|x_{n+1}-p\|+L\beta_{n}\|x_{n}-p\|+L\alpha_{n}(\|x_{n}-p\|\\ &&+L(1-\beta_{n})\|x_{n}-p\|+L^{2}\beta_{n}\|x_{n+1}-p\|)]. \end{eqnarray*}$

This implies that

$\begin{eqnarray*}&&(1-L^{2}\alpha_{n}^{2}\beta_{n}-L^{2}\alpha_{n}\beta_{n}-L^{3}\alpha_{n}^{2}\beta_{n})\|x_{n+1}-p\|\\ &\leq&(1+\alpha_{n}^{2}+2L\alpha_{n}^{2}+L\alpha_{n}\beta_{n}+L^{2}\alpha_{n}^{2})\|x_{n}-p\|.\end{eqnarray*}$

Since $\lim\limits_{n\rightarrow\infty}\alpha_{n}\beta_{n}=0$, there exists integer $n_{0}>0$ such that $\alpha_{n}\beta_{n}\leq \frac{1}{6L^{3}}$ for all $n\geq n_{0}$ and

$\begin{eqnarray*}&&1-L^{2}\alpha_{n}^{2}\beta_{n}-L^{2}\alpha_{n}\beta_{n}-L^{3}\alpha_{n}^{2}\beta_{n}\geq 1-L^{2}\cdot\frac{1}{6L^{3}}-L^{2}\cdot\frac{1}{6L^{3}}-L^{3}\cdot\frac{1}{6L^{3}}\\ &=&\frac{5L-2}{6L}\geq \frac{5L-2L}{6L}=\frac{1}{2}.\end{eqnarray*}$

Therefore, we have

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

where $r_{n}=2[(L^{3}+2L^{2}+L)\alpha_{n}\beta_{n}+(1+L)^{2}\alpha_{n}^{2}].$ Since $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$ and $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2}<\infty$. Then $\sum\limits_{n=1}^{\infty}r_{n} <\infty$. By Lemma 1.1, we obtain that $\lim\limits_{n\rightarrow\infty}\|x_{n}-p\|$ exists. This completes the proof of part (ⅰ).

(ⅱ) For any $m\geq 1$ and $n\geq n_{0}$, we have

$\begin{eqnarray*}&&\|x_{n+m}-p\|\leq (1+r_{n+m-1})\|x_{n+m-1}-p\| \leq e^{r_{n+m-1}}\|x_{n+m-1}-p\|\\ &\leq& e^{r_{n+m-1}}e^{r_{n+m-2}}\|x_{n+m-2}-p\| \\ && \vdots\\ & \leq& e^{\Sigma_{k=n}^{{n+m-1}}r_{k}}\|x_{n}-p\|\\ &\leq& e^{\Sigma_{k=n}^{\infty}r_{n}}\|x_{n}-p\|= M \|x_{n}-p\|,\end{eqnarray*}$

where $M=e^{\Sigma_{k=n}^{\infty}r_{n}}$. This completes the proof.

Theorem 2.1  Let $E$ be a real Banach space and $C$ a nonempty closed convex subset of $E$. Let $T:C\rightarrow C$ be a Lipschitzian pseudocontractive mapping with Lipschitz constant $L>1$ and $F(T)\neq\emptyset$. Suppose that the sequence $\{x_{n}\}$ is defined by (1.5) such that $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$, $ \alpha_{n}\beta_{n}L^{2}<1$ for all $n\geq 1$ and $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2} <\infty$. Then $\{x_{n}\}$ converges strongly to a fixed point of $T$ if and only if $\liminf\limits_{n\rightarrow\infty}d(x_{n},F(T))=0$, where $d(x_{n},F(T))=\inf\limits_{q\in F(T)}\|x_{n}-q\|$.

Proof The necessity of Theorems 2.1 is obvious. We just need to prove the sufficiency. By Lemma 2.1, we have

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

By Lemma 1.1 and the condition $\liminf\limits_{n\rightarrow\infty}d(x_{n},F(T))=0$, then $\lim\limits_{n\rightarrow\infty}d(x_{n},F(T))=0.$

Next, we show that $\{x_{n}\}$ is a Cauchy sequence. For any $\varepsilon>0,$ there exists an integer $n_{1}>n_{0}>0$ such that

$d(x_{n},F(T))< \frac{\varepsilon}{4M}$

for all $n\geq n_{1}$. In particular, there exists $p_{1}\in F(T)$ and a constant $n_{2}>n_{1}$ such that

$\begin{equation}\|x_{n_{2}}-p_{1}\|<\frac{\varepsilon}{2M}.\end{equation}$

(2.8)

Using Lemma 2.1 (ⅱ) and (2.8), for all $n_{2}>n_{1}$ and $m\geq 1$, we have

$\begin{eqnarray*}&& \|x_{n+m}-x_{n}\|\leq \|x_{n+m}-p_{1}\|+\|p_{1}-x_{n}\|\\ &\leq& 2M\|x_{n_{2}}-p_{1}\|<2M\cdot\frac{\varepsilon}{2M}=\varepsilon.\end{eqnarray*}$

Hence, $\{x_{n}\}$ is a Cauchy sequence. Since $C$ is closed subset of $E$, so $\{x_{n}\}$ converges strongly to $p_{0}\in C$. It follows from $F(T)$ is a closed set and $\lim\limits_{n\rightarrow\infty}d(x_{n},F(T))=0$ that $p_{0}\in F(T)$. This shows that $\{x_{n}\}$ converges strongly to a fixed point of $T$ in $C$. This completes the proof.

Remark 2.1  Let $\beta_{n}=0$ in iterative sequence (1.5), we can obtain strong convergence theorem of Mann iterative sequence from Theorem 2.1.

Theorem 2.2  Let $E$ be a real Banach space and $C$ a nonempty closed convex subset of $E$. Let $\{T_{i}\}_{i=1}^{N}:C\rightarrow C$ be $N$ Lipschitzian pseudocontractive mappings with Lipschitz constant $L_{i}>1$ and $F=\bigcap_{i=1}^{N}F(T_{i})\neq \emptyset$. Suppose that the sequence $\{x_{n}\}$ is defined as follows:

$\begin{equation*} \left \{\begin{array}{ll} x_{1}\in C,\\ x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}T_{n}y_{n} ,\\ y_{n}=(1-\beta_{n})x_{n}+\beta_{n}T_{n}x_{n+1},\quad n\geq 1, \end{array}\right. \end{equation*}$

where $T_{n}=T_{n{\rm mod} N}$, $\{\alpha_{n}\}$, $\{\beta_{n}\}\subset[0,1]$, $\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}<\infty$, $ \alpha_{n}\beta_{n}L_{i}^{2}<1$ for all $n\geq1$ and $i\in \{1,2\cdots N \}$ and $\sum\limits_{n=1}^{\infty}\alpha_{n}^{2} <\infty$. Then $\{x_{n}\}$ converges strongly to a common fixed point of $T_{1},T_{2},\cdots, T_{N}$ if and only if $\liminf\limits_{n\rightarrow\infty}d(x_{n},F)=0$, where $d(x_{n},F)=\inf\limits_{q\in F}\|x_{n}-q\|.$

Proof Using the same method as given Theorem 2.1, we can prove Theorem 2.2. This completes the proof.