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# Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings

- H Zegeye
^{1}and - N Shahzad
^{2}Email author

**2013**:1

https://doi.org/10.1186/1687-1812-2013-1

© Zegeye and Shahzad; licensee Springer 2013

**Received:**21 September 2012**Accepted:**28 November 2012**Published:**2 January 2013

## Abstract

We introduce an iterative process which converges strongly to the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings. As a consequence, convergence result to a common minimum-norm fixed point of a finite family of nonexpansive mappings is proved.

**MSC:**47H09, 47H10, 47J05, 47J25.

## Keywords

- asymptotically nonexpansive mappings
- minimum-norm fixed point
- nonexpansive mappings
- split feasibility problem
- strong convergence

## 1 Introduction

*K*and

*D*be nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. The split feasibility problem is formulated as finding a point $\overline{x}$ satisfying

where *A* is bounded linear operator from ${H}_{1}$ into ${H}_{2}$. A split feasibility problem in finite dimensional Hilbert spaces was first studied by Censor and Elfving [1] for modeling inverse problems which arise in medical image reconstruction, image restoration and radiation therapy treatment planing (see, *e.g.*, [1–3]).

*D*. Set

*i.e.*, (1.1) has a solution), and let Ω denote the (closed convex) solution set of (1.1) (or equivalently, solution of (1.2)). Then, in this case, Ω has a unique element $\overline{x}$ if and only if it is a solution of the following variational inequality:

*A*. In addition, inequality (1.3) can be rewritten as

Recall that a point $\overline{x}\in K$ is said to be a fixed point of *T* if $T(\overline{x})=\overline{x}$. We denote the set of fixed points of *T* by $F(T)$, *i.e.*, $F(T):=\{\overline{x}\in K:T\overline{x}=\overline{x}\}$. Therefore, finding a solution to the split feasibility problem (1.1) is equivalent to finding the minimum-norm fixed point of the mapping $x\mapsto {P}_{K}(x-\gamma {A}^{\ast}(I-{P}_{D})Ax)$.

*asymptotically nonexpansive*self-mapping

*T*on

*K*; that is, we find a minimum-norm fixed point of

*T*which satisfies

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

*H*; a mapping $T:K\to K$ is said to be

*nonexpansive*if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in K$ and it is called

*asymptotically nonexpansive*if there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${k}_{n}\to 1$, as $n\to \mathrm{\infty}$, such that

The class of asymptotically nonexpansive mappings was introduced as a generalization of the class of nonexpansive mappings by Goebel and Kirk [4] who proved that if *K* is a nonempty closed convex bounded subset of a real uniformly convex Banach spaces which includes Hilbert spaces as a special case and *T* is an asymptotically nonexpansive self-mapping of *K*, then *T* has a fixed point.

*i.e.*, ${z}_{t}$ is the unique solution of the equation

In [5], Browder proved that, as $t\to 1$, ${z}_{t}$ converges strongly to the nearest point projection of *u* onto $F(T)$.

*Halpern iteration*) defined by

*T*that is closest to

*u*provided that $\{{\alpha}_{n}\}$ satisfies (i) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, (ii) $\sum {\alpha}_{n}=\mathrm{\infty}$ and (iii) ${lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}}{{\alpha}_{n+1}}=0$. Wittmann [7] also showed that the sequence $\{{x}_{n}\}$ defined by

converges strongly to the element of $F(T)$ which is nearest to *u* under certain conditions on $\{{a}_{n}\}\subset (0,1)$.

*T*be an asymptotically nonexpansive mapping from

*K*into itself with $F(T)$ nonempty. Then they proved that the sequence generated by

*u*. Shioji and Takahashi [8] also studied an explicit scheme for asymptotically nonexpansive mappings. They showed that the sequence $\{{x}_{n}\}$ defined by

where $\{{b}_{n}\}\subset (0,1)$ satisfies certain conditions, converges strongly to the element of $F(T)$ which is nearest to *u*.

Several authors have extended the above results either to a more general Banach spaces or to a more general class of mappings (see, *e.g.*, [9–18]).

It is worth mentioning that the methods studied above are used to approximate the fixed point of *T* which is closest to the point $u\in K$. These methods can be used to find the minimum-norm fixed point ${x}^{\ast}$ of *T* if $0\in K$. If, however, $0\notin K$, any of the methods above fails to provide the minimum-norm fixed point of *T*.

*T*, Yang

*et al.*[19] introduced an explicit scheme given by

They proved that under appropriate conditions on $\{{\alpha}_{n}\}$ and *β*, the sequence $\{{x}_{n}\}$ converges strongly to the minimum-norm fixed point of *T* in real Hilbert spaces.

More recently, Yao and Xu [20] have also shown that the explicit scheme ${x}_{n+1}={P}_{K}((1-{\alpha}_{n})T{x}_{n})$, $n\ge 1$, converges strongly to the minimum-norm fixed point of a nonexpansive self-mapping *T* provided that $\{{\alpha}_{n}\}$ satisfies certain conditions.

*A natural question arises whether we can extend the results of Yang et al.* [19]*and Yao and Xu* [20]*to a class of mappings more general than nonexpansive mappings or not.*

Let *K* be a closed convex subset of a real Hilbert space *H* and let ${T}_{i}:K\to K$, $i=1,2,\dots ,N$ be a finite family of asymptotically nonexpansive mappings.

It is our purpose in this paper to introduce an explicit iteration process which converges strongly to the common minimum-norm fixed point of $\{{T}_{i}:i=1,2,\dots ,N\}$. Our theorems improve several results in this direction.

## 2 Preliminaries

In what follows, we shall make use of the following lemmas.

**Lemma 2.1**

*Let*

*H*

*be a real Hilbert space*.

*Then*,

*for any given*$x,y\in H$,

*the following inequality holds*:

**Lemma 2.2** [21]

*Let*

*E*

*be a real Hilbert space and*${B}_{R}(0)$

*be a closed ball of*

*H*.

*Then*,

*for any given subset*$\{{x}_{0},{x}_{1},\dots ,{x}_{N}\}\subset {B}_{r}(0)$

*and for any positive numbers*${\alpha}_{0},{\alpha}_{1},\dots ,{\alpha}_{N}$

*with*${\sum}_{i=0}^{N}{\alpha}_{i}=1$,

*we have that*

**Lemma 2.3** [22]

*Let*

*K*

*be a closed and convex subset of a real Hilbert space*

*H*.

*Let*$x\in H$.

*Then*${x}_{0}={P}_{K}x$

*if and only if*

**Lemma 2.4** [23]

*Let* *H* *be a real Hilbert space*, *K* *be a closed convex subset of* *H* *and* $T:K\to K$ *be an asymptotically nonexpansive mapping*, *then* $(I-T)$ *is demiclosed at zero*, *i*.*e*., *if* $\{{x}_{n}\}$ *is a sequence in* *K* *such that* ${x}_{n}\rightharpoonup x$ *and* $T{x}_{n}-{x}_{n}\to 0$, *as* $n\to \mathrm{\infty}$, *then* $x=T(x)$.

**Lemma 2.5** [24]

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

*be a sequence of nonnegative real numbers satisfying the following relation*:

*where* $\{{\alpha}_{n}\}\subset (0,1)$, *and* $\{{\delta}_{n}\}\subset R$ *satisfying the following conditions*: ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, *and* ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0$, *as* $n\to \mathrm{\infty}$. *Then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.

**Lemma 2.6** [25]

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

*be a sequence of real numbers such that there exists a subsequence*$\{{n}_{i}\}$

*of*$\{n\}$

*such that*${a}_{{n}_{i}}<{a}_{{n}_{i}+1}$

*for all*$i\in \mathbb{N}$.

*Then there exists a nondecreasing sequence*$\{{m}_{k}\}\subset \mathbb{N}$

*such that*${m}_{k}\to \mathrm{\infty}$

*and the following properties are satisfied by all*(

*sufficiently large*)

*numbers*$k\in \mathbb{N}$:

*In fact*, ${m}_{k}=max\{j\le k:{a}_{j}<{a}_{j+1}\}$.

**Proposition 2.7** *Let* *H* *be a real Hilbert space*, *let* *K* *be a closed convex subset of* *H*, *and let* *T* *be an asymptotically nonexpansive mapping from* *K* *into itself*. *Then* $F(T)$ *is closed and convex*.

*Proof*Clearly, the continuity of

*T*implies that $F(T)$ is closed. Now, we show that $F(T)$ is convex. For $x,y\in F(T)$ and $t\in (0,1)$, put $z=tx+(1-t)y$. Now, we show that $z=T(z)$. In fact, we have

and hence, since ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$, we get that ${lim}_{n\to \mathrm{\infty}}{\parallel z-{T}^{n}z\parallel}^{2}=0$, which implies that ${lim}_{n\to \mathrm{\infty}}{T}^{n}z=z$. Now, by the continuity of *T*, we obtain that $z={lim}_{n\to \mathrm{\infty}}{T}^{n}z={lim}_{n\to \mathrm{\infty}}T({T}^{n-1}z)=T({lim}_{n\to \mathrm{\infty}}{T}^{n-1}z)=T(z)$. Hence, $z\in F(T)$ and that $F(T)$ is convex. □

## 3 Main result

We now state and proof our main theorem.

**Theorem 3.1**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*.

*Let*${T}_{i}:K\to K$

*be asymptotically nonexpansive mappings with sequences*$\{{k}_{n,i}\}$

*for each*$i=1,2,\dots ,N$.

*Assume that*$F:={\bigcap}_{i=1}^{N}F({T}_{i})$

*is nonempty*.

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

*be a sequence generated by*

*where* ${\alpha}_{n}\in (0,1)$ *such that* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${lim}_{n\to \mathrm{\infty}}\frac{({k}_{n,i}^{2}-1)}{{\alpha}_{n}}=0$, *for each* $i\in \{1,2,\dots ,N\}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, $\{{\beta}_{n,i}\}\subset [a,b]\subset (0,1)$ *for* $i=1,2,\dots ,N$, *satisfying* ${\beta}_{n,0}+{\beta}_{n,1}+\cdots +{\beta}_{n,N}=1$ *for each* $n\ge 1$. *Then* $\{{x}_{n}\}$ *converges strongly to the common minimum*-*norm point of* *F*.

*Proof*Let ${x}^{\ast}\in {P}_{F}0$. Let ${k}_{n}:=max\{{k}_{n,i}:i=1,2,\dots ,N\}$. Then from (3.1) and asymptotical nonexpansiveness of ${T}_{i}$, for each $i\in \{1,2,\dots ,N\}$, we have that

for some $M>0$, where ${\theta}_{n}:={\alpha}_{n}(1-{\beta}_{n,0})$ for all $n\in N$.

Now, we consider the following two cases.

for some $M>0$. But note that ${\theta}_{n}$ satisfies ${lim}_{n}{\theta}_{n}=0$ and ${\sum}_{n=1}^{\mathrm{\infty}}{\theta}_{n}=\mathrm{\infty}$. Thus, it follows from (3.15) and Lemma 2.5 that $\parallel {x}_{n}-{x}^{\ast}\parallel \to 0$, as $n\to \mathrm{\infty}$. Consequently, ${x}_{n}\to {x}^{\ast}$.

Thus, from (3.16) and the fact that $\frac{({k}_{{m}_{k}}^{2}-1)}{{\theta}_{{m}_{k}}}\to 0$, we obtain that $\parallel {x}_{{m}_{k}}-{x}^{\ast}\parallel \to 0$ as $k\to \mathrm{\infty}$. This together with (3.17) gives $\parallel {x}_{{m}_{k}+1}-{x}^{\ast}\parallel \to 0$ as $k\to \mathrm{\infty}$. But $\parallel {x}_{k}-{x}^{\ast}\parallel \le \parallel {x}_{{m}_{k}+1}-{x}^{\ast}\parallel $ for all $k\in \mathbb{N}$, thus we obtain that ${x}_{k}\to {x}^{\ast}$. Therefore, from the above two cases, we can conclude that $\{{x}_{n}\}$ converges strongly to a point ${x}^{\ast}$ of *F* which is the common minimum-norm fixed point of the family $\{{T}_{i},i=1,2,\dots ,N\}$ and the proof is complete. □

If in Theorem 3.1 we assume that $N=1$, then we get the following corollary.

**Corollary 3.2**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*.

*Let*$T:K\to K$

*be an asymptotically nonexpansive mapping with a sequence*$\{{k}_{n}\}$.

*Assume that*$F(T)$

*is nonempty*.

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

*be a sequence generated by*

*where* ${\alpha}_{n}\in (0,1)$ *such that* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${lim}_{n\to \mathrm{\infty}}\frac{({k}_{n}^{2}-1)}{{\alpha}_{n}}=0$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, $\{{\beta}_{n}\}\subset [a,b]\subset (0,1)$ *for each* $n\ge 1$. *Then* $\{{x}_{n}\}$ *converges strongly to the minimum*-*norm fixed point of* *T*.

If in Theorem 3.1 we assume that each ${T}_{i}$ is nonexpansive for $i=1,2,\dots ,N$, then the method of proof of Theorem 3.1 provides the following corollary.

**Corollary 3.3**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*.

*Let*${T}_{i}:K\to K$

*be nonexpansive mappings with*$F:={\bigcap}_{i=1}^{N}F({T}_{i})$

*nonempty*.

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

*be a sequence generated by*

*where* ${\alpha}_{n}\in (0,1)$ *such that* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, $\{{\beta}_{n,i}\}\subset [a,b]\subset (0,1)$, *for* $i=1,2,\dots ,N$, *satisfying* ${\beta}_{n,0}+{\beta}_{n,1}+\cdots +{\beta}_{n,N}=1$ *for each* $n\ge 1$. *Then* $\{{x}_{n}\}$ *converges strongly to the common minimum*-*norm point of* *F*.

If in Corollary 3.3 we assume that $N=1$, then we have the following corollary.

**Corollary 3.4**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*.

*Let*$T:K\to K$

*be a nonexpansive mapping with*$F(T)$

*nonempty*.

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

*be a sequence generated by*

*where* ${\alpha}_{n}\in (0,1)$ *such that* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, $\{{\beta}_{n}\}\subset [a,b]\subset (0,1)$ *for each* $n\ge 1$. *Then* $\{{x}_{n}\}$ *converges strongly to the minimum*-*norm point of* $F(T)$.

## 4 Applications

In this section, we study the problem of finding a minimizer of a continuously Fréchet-differentiable convex functional which has the minimum norm in Hilbert spaces.

*K*be a closed convex subset of a real Hilbert space

*H*. Consider the minimization problem given by

*φ*at $x\in K$. It is also known that the optimality condition (4.3) is equivalent to the following fixed point problem:

for all $\gamma >0$.

Now, we have the following corollary deduced from Corollary 3.2.

**Corollary 4.1**

*Let*

*K*

*be a closed convex subset of a real Hilbert space*

*H*.

*Let*

*φ*

*be a continuously Fréchet*-

*differentiable convex functional on*

*K*

*such that*${T}_{\gamma}:={P}_{K}(I-\gamma \u25bd\phi )$

*is asymptotically nonexpansive with a sequence*$\{{k}_{n}\}$

*for some*$\gamma >0$.

*Assume that the solution of the minimization problem*(4.1)

*is nonempty*.

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

*be a sequence generated by*

*where* ${\alpha}_{n}\in (0,1)$ *such that* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${lim}_{n\to \mathrm{\infty}}\frac{({k}_{n}^{2}-1)}{{\alpha}_{n}}=0$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, $\{{\beta}_{n}\}\subset [a,b]\subset (0,1)$ *for each* $n\ge 1$. *Then* $\{{x}_{n}\}$ *converges strongly to the minimum*-*norm solution of the minimization problem* (4.1).

**Remark 4.2** Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem 3.1 improves Theorem 3.2 of Yang *et al.* [19] and of Yao and Xu [20] to a more general class of a finite family of asymptotically nonexpansive mappings.

## Declarations

### Acknowledgements

The second author gratefully acknowledges the sup- port provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research.

## Authors’ Affiliations

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