Three Convergence Results for Inexact Iterates of Uniformly Locally Nonexpansive Mappings

Abstract

In 2006, together with D. Butnariu, we showed that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. In a recent paper of ours, we have extended this result to uniformly locally nonexpansive self-mappings of a complete metric space. In the present paper, we establish analogous results for uniformly locally nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space. In the particular case of a Banach space, if the operator is symmetric, then the set of all limit points of its iterates is also symmetric.

1. Introduction

For more than sixty years now, considerable research activity has been devoted to the fixed point theory of nonexpansive mappings in Banach and complete metric spaces [1,2,3,4,5,6,7,8,9,10,11,12]. The starting point of this research was Banach’s classical theorem [13], which asserts that a strict contraction possesses a unique fixed point. This research activity also encompasses the study of the asymptotic behavior of (inexact) orbits of a nonexpansive mapping and their convergence to its fixed points. It includes studies of feasibility, common fixed points, iterative methods, and variational inequalities, which find important applications in engineering, medical science, and the natural sciences [12,14,15,16,17,18,19,20,21,22].

In 2006, together with D. Butnariu [3] we showed that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. In [23], we have extended this result to uniformly locally nonexpansive self-mappings of a complete metric space. In the present paper, we establish analogous results for uniformly locally nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space. In the particular case of a Banach space, if the operator is symmetric, then the set of all limit points of its iterates is also symmetric.

Let $(Z,d)$ be a complete metric space. For each point $\xi \in Z$ and each nonempty set $E\subset Z$, put

$$d(\xi ,E):=inf\left\{d\right(\xi ,\eta ):\eta \in E\}.$$

For each point $\xi \in Z$ and each number $r>0$, set

$$B(\xi ,r):=\{\eta \in X:d(\xi ,\eta )\le r\}.$$

For each mapping $S:Z\to Z$, let $S^0\xi =\xi$ for all $\xi \in Z$, $S^1=S$ and $S^{i+1}=S\circ S^i$ for each integer $i\ge 0$. We denote the set of all fixed points of S by $F\left(S\right)$.

The convergence of the inexact orbits of nonexpansive mappings in metric spaces in the presence of summable computational errors was studied in [3], and the following result has been obtained there (see also Theorem 2.72 on page 97 of [24]).

Theorem 1.

Let the mapping $S:Z\to Z$ satisfy

$$d\left(S\right(\xi ),S(\eta \left)\right)\le d(\xi ,\eta ) for all \xi ,\eta \in Z$$

and for each $\xi \in Z$, let the sequence ${\left\{S^n\xi \right\}}_{n=1}^{\infty }$ converge in $(Z,d)$.

Assume that ${\left\{{\xi }_n\right\}}_{n=0}^{\infty }\subset Z$; the sequence ${\left\{r_n\right\}}_{n=0}^{\infty }\subset (0,\infty )$ satisfies

$$\sum _{n=0}^{\infty }{r}_n<\infty$$

and that

$$d({\xi }_{n+1},S\left({\xi }_n\right))\le r_n,n=0,1,\cdots .$$

Then, the sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ converges to a fixed point of S in $(Z,d)$.

The result stated above has found important applications. It is, for instance, an essential ingredient in the study of superiorization and the perturbation resilience of algorithms. See [14,15,16,19] and the references mentioned therein.

In this paper, our goal is to extend this result to uniformly locally nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space.

At this juncture, we present the following example, which demonstrates one of the possible applications of the results of [3] and the results of the current paper. Assume that $(Z,\parallel ·\parallel )$ is a Banach space, $d(\xi ,\eta )=\parallel \xi -\eta \parallel$ for each pair of points $\xi ,\eta \in Z$, a mapping $S:Z\to Z$ satisfies

$$d\left(S\right(\xi ),S(\eta \left)\right)\le d(\xi ,\eta ) for all \xi ,\eta \in Z$$

and that for each point $\xi \in Z$, the sequence ${\left\{S^n\left(\xi \right)\right\}}_{n=1}^{\infty }$ converges in the norm topology. Let ${\xi }_0\in Z$, ${\left\{{\beta }_n\right\}}_{n=0}^{\infty }$ be a summable sequence of positive numbers, ${\left\{w_n\right\}}_{n=0}^{\infty }\subset Z$ be a norm bounded sequence, and let for any nonnegative integer n,

$${\xi }_{n+1}=S({\xi }_n+{\beta }_n{w}_n).$$

Then, it follows from Theorem 1 that the sequence ${\left\{{\xi }_n\right\}}_{n=0}^{\infty }$ converges in the norm topology of Z and that its limit is a fixed point of S. If we need to find an approximate fixed point of S, then we construct the sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ defined above. With an appropriate choice of the bounded sequence ${\left\{w_n\right\}}_{n=0}^{\infty }$, the sequence ${\left\{{\xi }_n\right\}}_{n=1}^{\infty }$ possesses some useful property. For example, the sequence ${\left\{g\left({\xi }_n\right)\right\}}_{n=1}^{\infty }$ can be decreasing, where g is a given function.

2. The Main Results

Assume that $(Z,d)$ is a complete metric space, $K\subset Z$ is a nonempty closed set, $\mathrm{\Delta }>0$, and that a mapping $S:K\to Z$ satisfies

$$d\left(S\right(\xi ),S(\eta \left)\right)\le d(\xi ,\eta )$$

(1)

for each pair of points $\xi ,\eta \in K$ satisfying $d(\xi ,\eta )\le \mathrm{\Delta }$.

In this paper, we establish the following three results.

Theorem 2.

Assume that if a point $\xi \in K$ and $S^i\left(\xi \right)\in K$ for each integer $i\ge 1$, then the sequence ${\left\{S^i\left(\xi \right)\right\}}_{i=1}^{\infty }$ converges. Assume further that a sequence ${\left\{y_i\right\}}_{i=0}^{\infty }\subset K$ satisfies

$$\sum _{i=0}^{\infty }d(y_{i+1},S\left(y_i\right))<\infty$$

(2)

and

$$\underset{i\to \infty }{lim\; inf}d(y_i,X\setminus K)>0.$$

(3)

Then, the sequence ${\left\{y_i\right\}}_{i=0}^{\infty }$ converges to a fixed point of S.

Theorem 2 is proven in Section 4 below.

Theorem 3.

Let $\mathcal{F}$ be a nonempty subset of K and assume that if $\xi \in K$ and $S^i\left(\xi \right)\in K$ for each integer $i\ge 1$, then

$$\underset{i\to \infty }{lim}d(S^i\left(\xi \right),\mathcal{F})=0.$$

Let a sequence ${\left\{y_i\right\}}_{i=0}^{\infty }$ satisfy relations (2) and (3). Then

$$\underset{i\to \infty }{lim}d(x_i,\mathcal{F})=0.$$

Theorem 3 is proved in Section 5.

Theorem 4.

Assume that if $\xi \in K$ and $S^i\left(\xi \right)\in K$ for each integer $i\ge 1$, then there exists a nonempty compact set $E\left(\xi \right)\subset K$ such that

$$\underset{i\to \infty }{lim}d(S^i\left(\xi \right),E\left(\xi \right))=0.$$

Let a sequence ${\left\{y_i\right\}}_{i=0}^{\infty }\subset K$ satisfy relations (2) and (3). Then, there exists a compact set $E\subset X$ such that ${lim}_{i\to \infty }d(y_i,E)=0$.

Theorem 4 is proven in Section 6.

3. An Auxiliary Result

Lemma 1.

Assume that a sequence ${\left\{{\xi }_i\right\}}_{i=0}^{\infty }\subset K$ satisfies

$$\sum _{i=0}^{\infty }d({\xi }_{i+1},S\left({\xi }_i\right))<\infty ,$$

$n_0\ge 1$ is an integer,

$$0<{\mathrm{\Delta }}_0<\mathrm{\Delta },$$

$$\sum _{i=n_0}^{\infty }d({\xi }_{i+1},S\left({\xi }_i\right))<{\mathrm{\Delta }}_0$$

(4)

and

$$d({\xi }_i,X\setminus K)>{\mathrm{\Delta }}_{0}forallintegersi\ge n_0.$$

(5)

Assume further that

$${\eta }_{n_0}={\xi }_{n_0},$$

(6)

and that if $i\ge n_0$ is an integer and ${\eta }_i\in K$ is defined, then

$${\eta }_{i+1}=S\left({\eta }_i\right).$$

(7)

Then, ${\eta }_i\in K$ for every integer $i>n_0$, and for each integer $n>n_0$, we have

$$d({\xi }_n,{\eta }_n)\le \sum _{i=n_0+1}^{n}d({\xi }_i,S\left({\xi }_{i-1}\right)).$$

(8)

Proof. 

In view of (4), (6), and (7), ${\eta }_{n_0+1}\in X$ is well defined and

$$d({\xi }_{n_0+1},{\eta }_{n_0+1})=d({\xi }_{n_0+1},S\left({\eta }_{n_0}\right))=d({\xi }_{n_0+1},S\left({\xi }_{n_0}\right))<{\mathrm{\Delta }}_0.$$

(9)

By (5) and (9),

$${\eta }_{n_0+1}\in K.$$

(10)

Assume that $q>n_0$ is an integer, ${\eta }_n\in K$ is defined for all integers $n=n_0,\dots ,q$, and that for all integers $n=n_0+1,\dots ,q$, inequality (8) holds. (In view of (9) and (10), our assumptions do hold for $q=n_0+1$.) It follows from (8) with $n=q$ that

$$d({\xi }_q,{\eta }_q)<{\mathrm{\Delta }}_0,{\eta }_q\in K.$$

(11)

This implies that

$${\eta }_{q+1}=S\left({\eta }_q\right)$$

is well defined. Relations (1), (4), (8) and (11) imply that

$$d({\xi }_{q+1},S\left({\eta }_q\right))\le d({\xi }_{q+1},S\left({\xi }_q\right))+d(S\left({\xi }_q\right),S\left({\eta }_q\right))$$

$$\le d({\xi }_{q+1},S\left({\xi }_q\right))+d({\xi }_q,{\eta }_q)$$

$$\le d({\xi }_{q+1},S\left({\xi }_q\right))+\sum _{n=n_0+1}^{q}d({\xi }_n,S\left({\xi }_{n-1}\right))$$

$$=\sum _{n=n_0+1}^{q+1}d({\xi }_n,S\left({\xi }_{n-1}\right)).$$

By (4) and the above relation, we have

$$d({\xi }_{q+1},{\eta }_{q+1})\le \sum _{n=n_0+1}^{q+1}d({\xi }_n,S\left({\xi }_{n-1}\right))<{\mathrm{\Delta }}_0.$$

When combined with (5), this implies that

$${\eta }_{q+1}\in K.$$

Thus, we see that the assumption made regarding q holds for $q+1$ too. Summing up, we have shown using mathematical induction that ${\eta }_n\in K$ is well defined for all integers $n>n_0$ and that (8) holds for all integers $n>n_0$. This completes the proof of Lemma 1. □

4. Proof of Theorem 2

We may assume without any loss of generality that there exists a number ${\mathrm{\Delta }}_0\in (0,\mathrm{\Delta })$, such that

$$d({\xi }_i,X\setminus K)>{\mathrm{\Delta }}_0 \mathrm{for} \mathrm{all} \mathrm{integers} i\ge 0.$$

Let $ϵ\in (0,{\mathrm{\Delta }}_0)$ be given. There exists an integer $n_0>1$ such that

$$\sum _{i=n_0}^{\infty }d(y_{i+1},S\left(y_i\right))<ϵ/2<{\mathrm{\Delta }}_0/2.$$

(12)

Set

$$z_{n_0}=y_{n_0}$$

and for each integer $i\ge n_0$, if $z_i\in K$ is defined, then set

$$z_{i+1}=S\left(z_i\right).$$

Lemma 1 and relation (12) imply that $z_n\in K$ for each integer $n>n_0$, and for every integer $n>n_0$, we have

$$d(z_n,y_n)\le \sum _{n=n_0+1}^{n}d(y_i,S\left(y_{i-1}\right))<ϵ/2.$$

(13)

By our assumptions, there exists

$$\xi =\underset{n\to \infty }{lim}{z}_n.$$

(14)

It follows from (13) and (14) that for all sufficiently large natural numbers n, we have

$$d(y_n,\xi )\le d(y_n,z_n)+d(z_n,\xi )<ϵ/2+d(z_n,\xi )\le ϵ.$$

Since $ϵ$ is an arbitrary number in the interval $(0,{\mathrm{\Delta }}_0)$, we find that ${\left\{y_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence, and therefore, it converges. It is not difficult to see that its limit is a fixed point of S. This completes the proof of Theorem 2.

5. Proof of Theorem 3

By (3), we may assume without any loss of generality that there exists a number ${\mathrm{\Delta }}_0\in (0,\mathrm{\Delta })$ such that

$$B(y_i,{\mathrm{\Delta }}_0)\subset K \mathrm{for} \mathrm{all} \mathrm{integers} i\ge 0.$$

Let $ϵ\in (0,{\mathrm{\Delta }}_0)$ be given. By (2), there exists an integer $n_0\ge 1$ such that

$$\sum _{i=n_0}^{\infty }d(y_{i+1},S\left(y_i\right))<ϵ/2.$$

Set

$$z_{n_0}=y_{n_0}$$

and for each integer $i\ge n_0$, if $z_i\in K$ is defined, then set

$$z_{i+1}=S\left(z_i\right).$$

Lemma 1 implies that for each integer $n>n_0$, $z_n\in K$ is well defined and

$$d(z_n,y_n)\le \sum _{i=n_0+1}^{\infty }d(y_i,S\left(y_{i-1}\right))<ϵ/2.$$

By our assumptions, we have

$$\underset{n\to \infty }{lim}d(z_n,\mathcal{F})=0.$$

Thus, we see that for all sufficiently large natural numbers n,

$$d(y_n,\mathcal{F})\le d(y_n,z_n)+d(z_n,\mathcal{F})<ϵ/2+d(z_n,\mathcal{F})<ϵ.$$

Theorem 3 is proven.

6. Proof of Theorem 4

By (3), we may assume without any loss of generality that there exists a number ${\mathrm{\Delta }}_0\in (0,\mathrm{\Delta })$ such that

$$B(y_i,{\mathrm{\Delta }}_0)\subset K \mathrm{for} \mathrm{all} \mathrm{integers} i\ge 0.$$

Let $ϵ\in (0,{\mathrm{\Delta }}_0)$ be given. By (3), there exists an integer $n_0\ge 1$ such that

$$\sum _{i=n_0}^{\infty }d(y_{i+1},S\left(y_i\right))<ϵ/2.$$

Set

$$z_{n_0}=y_{n_0}$$

and for each integer $n\ge n_0$, if $z_n\in K$ is defined, then set

$$z_{n+1}=S\left(z_n\right).$$

Lemma 1 implies that for each integer $n>n_0$, $z_n\in K$ is well defined and

$$d(z_n,y_n)\le \sum _{i=n_0+1}^{\infty }d(y_{i+1},S\left(y_{i-1}\right))<ϵ/2.$$

By our assumptions, there exists a nonempty compact set $E_0\subset K$ such that

$$\underset{n\to \infty }{lim}d(z_n,E_0)=0.$$

Clearly, for every sufficiently large natural number $n>n_0$,

$$d(y_n,E_0)\le d(y_n,z_n)+d(z_n,E_0)<ϵ.$$

Thus, we have shown that there exists a compact set $E_0$ such that

$$d(y_n,E_0)<ϵ$$

for every sufficiently large natural number n. We may assume that $E_0$ is finite. Thus for every $ϵ>0$, there exists a finite set $E_{ϵ}$ such that

$$d(y_n,E_{ϵ})<ϵ$$

for every sufficiently large natural number n. This implies that each subsequence of ${\left\{y_i\right\}}_{i=0}^{\infty }$ has a convergent subsequence. Denote by E the set of all limit points of the sequence ${\left\{y_i\right\}}_{i=0}^{\infty }$. It is not difficult to see that E is compact and that

$$\underset{i\to \infty }{lim}d(y_i,E)=0,$$

as asserted.

This completes the proof of Theorem 4.