# Convergence theorems of a three-step iteration method for a countable family of pseudocontractive mappings

- Qingqing Cheng
^{1}, - Yongfu Su
^{1}Email author and - Jingling Zhang
^{1}

**2013**:100

https://doi.org/10.1186/1687-1812-2013-100

© Cheng et al.; licensee Springer 2013

**Received: **14 January 2013

**Accepted: **3 April 2013

**Published: **17 April 2013

## Abstract

The purpose of this paper is to construct a three-step iteration method (as follows) and obtain the convergence theorem for a countable family of Lipschitz pseudocontractive mappings in Hilbert space *H*. For the iteration format,

under suitable conditions, we prove that the sequence $\{{x}_{n}\}$ generated from above converges strongly to a common fixed point of ${\{{T}_{n}\}}_{n\ge 1}$. The results obtained in this paper improve and extend previous results that have been proved for this class of nonlinear mappings.

**MSC:**47H05, 47H09, 47H10.

## Keywords

## 1 Introduction

*C*be a nonempty subset of

*H*. A mapping $T:C\to H$ is said to be nonexpansive, if

*α*-strictly pseudocontractive in the terminology of Browder and Petryshyn [1] if for all $x,y\in C$ there exists $\alpha >0$ such that

*I*denotes the identity operator, then (1.2) can be rewritten as

*T*is called pseudocontractive if

*α*-strict pseudocontractive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinear monotone mappings, where a mapping

*A*with domain $D(A)$ and range $R(A)$ in

*H*is called monotone if the inequality

holds for every $x,y\in D(A)$ and for all $s>0$. We observe that *A* is monotone if and only if $T:=I-A$ is pseudocontractive, and thus a zero of *A*, $N(A):=\{x\in D(A):Ax=0\}$, is a fixed point of *T*, $F(T):=\{x\in D(T):Tx=x\}$. It is now well known (see, *e.g.*, [2]) that if *A* is monotone then the solutions of the equation $Ax=0$ correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 20 years or so, have been devoted to iterative methods for approximating fixed points of *T* when *T* is pseudocontractive (see, for example, [3–5] and the references contained therein).

where ${\{{\alpha}_{n}\}}_{n\ge 0}\subset (0,1)$ and satisfies the following additional assumptions: (i) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$; (ii) ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, the sequence ${\{{x}_{n}\}}_{n\ge 1}$ generated by (1.5) is generally referred to as the Mann iteration scheme in the light of Mann [6].

The Mann iteration process does not generally converge to a fixed point of *T* even when the fixed point exists. If, for example, *C* is a nonempty, closed, convex and bounded subset of a real Hilbert space, $T:C\to C$ is nonexpansive, and the Mann iteration process is defined by (1.5) with (i) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$; (ii) ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, one can only prove that the sequence is an approximate fixed point sequence, that is, $\parallel {x}_{n}-T{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$. To get the sequence ${\{{x}_{n}\}}_{n\ge 1}$ to converge to a fixed point of *T* (when such a fixed point exists), some type of compactness condition must be additionally imposed either on *C* (*e.g.*, *C* is compact) or on *T*.

Later, some authors tried to prove convergence of Mann iteration scheme to a fixed point of a much more general and important class of Lipschitz pseudocontractive mappings. But, in 2001, Chidume and Mutangadura [7] gave an example of a Lipschitz pseudocontractive self-map of a compact convex subset of a Hilbert space with a unique fixed point for which no Mann sequence converges. Consequently, for this class of maps, the Mann sequence may not converge to a fixed point of Lipschitz pseudocontractive mappings even when *C* is a compact convex subset of *H*.

In 1974, Ishikawa [8] introduced an iteration process, which in some sense is more general than that of Mann and which converges to a fixed point of a Lipschitz pseudocontractive self-map *T* of *C*. The following theorem is proved.

**Theorem IS** [8]

*If*

*C*

*is a compact convex subset of a Hilbert space*

*H*, $T:C\to C$

*is a Lipschitz pseudocontractive mapping and*${x}_{0}$

*is any point of*

*C*,

*then the sequence*$\{{x}_{n}\}$

*converges strongly to a fixed point of*

*T*,

*where*$\{{x}_{n}\}$

*is defined iteratively for each integer*$n\ge 0$

*by*

*where*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$

*are sequences of positive numbers satisfying the conditions*:

The iteration method of Theorem IS, which is now referred to as the Ishikawa iterative method has been studied extensively by various authors. But it is still an open question whether or not this method can be employed to approximate fixed points of Lipschitz pseudocontractive mappings without the compactness assumption on *C* or *T* (see, *e.g.*, [4, 9, 10]).

He proved that the sequence $\{{x}_{n}\}$ defined by (1.8) converges strongly to ${P}_{F(T)}{x}_{0}$, where ${P}_{C}$ is the metric projection from *H* into *C*. We observe that the iterative algorithm (1.7) generates a sequence $\{{x}_{n}\}$ by projecting ${x}_{0}$ onto the intersection of closed convex sets ${C}_{n}$ and ${Q}_{n}$ for each $n\ge 0$.

*et al.*[12] introduced the hybrid Mann algorithm as follows. Let

*C*be a nonempty, closed and convex subset of a real Hilbert space

*H*. Let $T:C\to C$ be a

*L*-Lipschitz pseudocontractive mapping such that $F(T)\ne \mathrm{\varnothing}$. Assume that the sequence $\{{\alpha}_{n}\}\subset [a,b]$ for some $a,b\in (0,\frac{1}{1+L})$. Then for ${C}_{1}=C$ and ${x}_{1}={P}_{{C}_{1}}{x}_{0}$, they proved that the sequence $\{{x}_{n}\}$ defined by

converges strongly to ${P}_{F(T)}{x}_{0}$.

*et al.*[13] generalized algorithm (1.8) to the hybrid Ishikawa iterative process. Let

*C*be a nonempty, closed and convex subset of a real Hilbert space

*H*. Let $T:C\to C$ be a Lipschitz pseudocontractive mapping. Let $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ be a sequence in $[0,1]$. Suppose that ${x}_{0}\in H$. For ${C}_{1}=C$ and ${x}_{1}={P}_{{C}_{1}}{x}_{0}$, define a sequence $\{{x}_{n}\}$ of

*C*as follows:

Then they proved that the hybrid algorithm (1.9) strongly converges to a fixed point of Lipschitz pseudocontractive mappings. It is worth mentioning that the schemes in (1.7)-(1.9) are not easy to compute. They involve computation of the intersection of ${C}_{n}$ and ${Q}_{n}$ for each $n\ge 1$.

*et al.*[18] generalized algorithm (1.9) to Ishikawa iterative process (not hybrid) as follows. Let

*C*be a nonempty, closed and convex subset of a real Hilbert space

*H*. Let ${T}_{i}:C\to C$, $i=1,2,\dots ,N$, be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constants ${L}_{i}$, for $i=1,2,\dots ,N$, respectively. Assume that the interior of $F:={\bigcap}_{i=1}^{N}F({T}_{i})$ is nonempty. Let $\{{x}_{n}\}$ be a sequence generated from an arbitrary ${x}_{0}\in C$ by

Under some conditions, $\{{x}_{n}\}$ converges strongly to ${x}^{\ast}\in F$.

Our concern now is the following: *Is it possible to construct a three-step iteration method and obtain a convergence theorem for a countable family of pseudocontractive mappings?*

It is our purpose in this paper to construct a three-step iteration method and obtain the convergence theorem for a countable family of pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on one of the mappings or on *C*. The results obtained in this paper improve and extend the results of Theorem IS, Zhou [14], Yao *et al.* [12], Tang *et al.* [13] and Habtu Zegeye *et al.* [18].

## 2 Preliminaries

*C*be a nonempty subset of a real Hilbert space

*H*. The mapping $T:C\to H$ is called Lipschitz or Lipschitz continuous if there exists $L>0$ such that

If $L=1$, then *T* is called nonexpansive; and if $L<1$ then *T* is called a contraction. It is easy to see from Eq. (2.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.

A countable family of mapping ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to H$ is called uniformly closed if ${x}_{n}\to {x}^{\ast}$ and $\parallel {x}_{n}-{T}_{n}{x}_{n}\parallel \to 0$ imply ${x}^{\ast}\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$.

In the sequel, we also need the following definition and lemma.

*H*be a real Hilbert space. The function $\varphi :H\times H\to R$ defined by

is studied by Alber [15], Kamimula and Takahashi [16] and Riech [17].

*ϕ*that

*ϕ*also has the following property:

**Lemma 2.1**

*Let*

*H*

*be a real Hilbert space*.

*Then for all*$x,y\in H$

*and*$\alpha \in [0,1]$

*the following inequality holds*:

**Remark 2.2**We now give an example of a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with the interior of the common fixed points nonempty. Suppose that $X:=R$ and $C:=[-1,1]\subset R$. Let ${\{{T}_{n}\}}_{n\ge 1}:C\to C$ be defined by

Then we observe that $F:={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=[-1,0]$, and hence the interior of the common fixed points is nonempty.

Therefore, from (2.5), (2.6) and (2.7) we obtain that ${\{{T}_{n}\}}_{n\ge 1}$ is a countable family of pseudocontractive mappings.

Therefore, from (2.8), (2.9) and (2.10) we obtain that ${\{{T}_{n}\}}_{n\ge 1}$ is uniformly Lipschitz.

Finally, we show that ${\{{T}_{n}\}}_{n\ge 1}$ is uniformly closed.

If there exists $\{{x}_{n}\}\subset {C}_{1}$ such that ${x}_{n}\to {x}^{\ast}\in [-1,0]$, and $|{x}_{n}-{T}_{n}{x}_{n}|=0$, we observe that $[-1,0]\subset F$;

If there exists $\{{x}_{n}\}\subset {C}_{2}$ such that ${x}_{n}\to {x}^{\ast}\in [0,1]$, if and only if ${x}^{\ast}=0$, we have that $|{x}_{n}-{T}_{n}{x}_{n}|=0$, it is obvious that $0\in F$;

- (i)
∃

*N*, as $n>N$, ${x}_{n}\in {C}_{2}$. The proof is the same as the proof of the second situation; - (ii)
∃

*N*, as $n>N$, ${x}_{n}\in {C}_{1}$. The proof is the same as the proof of the first situation; - (iii)
$\{{x}_{{n}_{k}}\}\subset {C}_{1}$, $\{{x}_{{n}_{j}}\}\subset {C}_{2}$. If there exists ${x}_{n}\to {x}^{\ast}$, then we have that ${x}^{\ast}=0$. The proof is the same as the proof of the second situation. So, we can obtain that ${\{{T}_{n}\}}_{n\ge 1}$ is uniformly closed.

## 3 Main results

**Theorem 3.1**

*Let*

*C*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*,

*let*${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to C$

*be a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitzian constants*${L}_{n}$,

*let*$L:={sup}_{n\ge 1}{L}_{n}$.

*Assume that the interior of*$F:={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}=\alpha >0$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in F$.

It is obviously that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $ exists, then $\{\parallel {x}_{n}-p\parallel \}$ is bounded. This implies that $\{{x}_{n}\}$, $\{{T}_{n}{x}_{n}\}$, $\{{z}_{n}\}$, $\{{T}_{n}{z}_{n}\}$, $\{{y}_{n}\}$ and $\{{T}_{n}{y}_{n}\}$ are also bounded.

*F*is nonempty, there exists ${p}^{\ast}\in F$ and $r>0$ such that ${p}^{\ast}+rh\in F$ whenever $\parallel h\parallel \le 1$. Thus, from the fact that $\varphi (x,y)={\parallel x-y\parallel}^{2}$, and (3.12) and (3.13), we get that

*h*with $\parallel h\parallel \le 1$ is arbitrary, we have

*C*is closed subset of

*H*, there exists ${x}^{\ast}\in C$ such that

Since ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are uniformly closed, then from (3.15) and (3.16), we obtain that ${x}^{\ast}\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=F$. The proof is complete. □

**Theorem 3.2**

*Let*

*C*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*,

*let*${T}_{n}:C\to C$

*be a finite family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitzian constants*${L}_{n}$, $n=1,2,\dots ,N$.

*Assume that the interior of*$F:={\bigcap}_{n=1}^{N}F({T}_{n})$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* ${T}_{n}:={T}_{n(modN)}$ *and* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) ${lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$ *for* $L:=max\{{L}_{n}:n=1,2,\dots ,N\}$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in F$.

If in Theorem 3.1, we consider a single Lipschitz pseudocontractive mapping, then we may change the conditions of Theorem 3.1.

**Theorem 3.3**

*Let*

*C*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*,

*let*$T:C\to C$

*be a Lipschitz pseudocontractive mappings with Lipschitzian constants*

*L*.

*Assume that the interior of*$F(T)$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) $\sum {\alpha}_{n}{\beta}_{n}{\gamma}_{n}=\mathrm{\infty}$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in F(T)$.

*Proof* Following the method of proof of Theorem 3.1, we obtain that ${x}_{n}\to {x}^{\ast}\in C$.

Thus, ${x}_{{n}_{k}}\to {x}^{\ast}$ and the continuity of *T* imply that ${x}^{\ast}=T{x}^{\ast}$, and hence ${x}^{\ast}\in F(T)$. □

## 4 Applications

**Theorem 4.1**

*Let*

*H*

*be a real Hilbert space*,

*let*${\{{A}_{n}\}}_{n=1}^{\mathrm{\infty}}:H\to H$

*be a countable family of uniformly Lipschitz monotone mappings with Lipschitzian constants*${L}_{n}$,

*let*$L:={sup}_{n\ge 1}{L}_{n}$.

*And if*$\parallel {A}_{n}{x}_{n}\parallel \to 0$, ${x}_{n}\to x$,

*then*$x\in {\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n})$.

*Assume that the interior of*$F:={\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n})$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) ${lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in F$.

*Proof* Suppose that ${T}_{n}x:=(I-{A}_{n})x$ for $n\ge 1$. Then we get that ${\{{T}_{n}\}}_{n\ge 1}$ is a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})={\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n})\ne \mathrm{\varnothing}$. Moreover, when ${A}_{n}$ is replaced by $I-{T}_{n}$, Scheme (4.1) reduces to Scheme (3.1) and hence the conclusion follows from Theorem 3.1. □

**Corollary 4.2**

*Let*

*H*

*be a real Hilbert space*,

*let*${A}_{n}:H\to H$

*be a finite family of uniformly Lipschitz monotone mappings with Lipschitzian constants*${L}_{n}$, $n=1,2,\dots ,N$.

*And if*$\parallel {A}_{n}{x}_{n}\parallel \to 0$, ${x}_{n}\to x$,

*then*$x\in {\bigcap}_{n=1}^{N}N({A}_{n})$.

*Assume that the interior of*$F:={\bigcap}_{n=1}^{N}N({A}_{n})$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* ${A}_{n}:={A}_{n(modN)}$ *and* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) ${lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$ *for* $L:=max\{{L}_{n}:n=1,2,\dots ,N\}$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in F$.

**Corollary 4.3**

*Let*

*H*

*be a real Hilbert space*,

*let*$A:H\to H$

*be a Lipschitz monotone mappings with Lipschitzian constants*

*L*.

*Assume that the interior of*$N(A)$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated from an arbitrary*${x}_{0}\in C$

*by the following algorithm*:

*where* $\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1)$ *satisfying the following conditions*: (i) ${\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}$, $\mathrm{\forall}n\ge 0$; (ii) $\sum {\alpha}_{n}{\beta}_{n}{\gamma}_{n}=\mathrm{\infty}$; (iii) ${sup}_{n\ge 1}{\gamma}_{n}\le \gamma $ *with* ${\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}\in N(A)$.

**Remark 4.4**In the paper [18], Scheme (2.28) of Theorem 2.5 and Scheme (2.29) of Corollary 2.6 are not correct, they are replaced by the following iterative algorithms, respectively.

## Declarations

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

## Authors’ Affiliations

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