# Fixed point problems of nonexpansive mappings for nononvex set in Hilbert spaces

## Xianbing Wu1

1Department of Mathematics Yangtze Normal University, Chongqing, China

1Corresponding author

Vibroengineering PROCEDIA, Vol. 28, 2019, p. 201-205. https://doi.org/10.21595/vp.2019.21034
Received 19 September 2019; accepted 1 October 2019; published 19 October 2019

Copyright © 2019 Xianbing Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

In this paper, we introduce a new concept of W-nonexpansive mappings and obtain fixed point theorems for nonexpansive mappings for non-convex set. Our results resolve fixed pointed problem that nonexpansive mappings be not on closed convex set, and it extends fixed point theorems for nonexpansive mappings.

Keywords: W-nonexpansive mapping, Hilbert space, fixed point theorem, non-convex set.

#### 1. Introduction and preliminaries

Fixed point theory is widely applied in engineering. Browder (1965) [1], Kirk (1965) [2] obtained fixed points theorem for nonexpansive mapping. Non-expansion fixed point theory has made great progress, large number of results are obtained by authors (e.g. See [3-11]). let’s come up with some definitions.

Definition 1.1 Let $X$ be a nonempty set, the function $W:X×X\to \left[0,\infty \right)$ is called triangular if for all $x\text{,}$$y\in X\text{,}$ if $W\left(x,y\right)\ge 1\text{,}$$W\left(y,z\right)\ge 1$ or $W\left(y,x\right)\ge 1\text{,}$$W\left(y,z\right)\ge 1\text{,}$ then $W\left(x,z\right)\ge 1.$

Definition 1.2 Let $\left(X,d\right)$ be a metric space and $T:X\to X$ be a given mapping, if there exists a function $W:X×X\to \left[0,\infty \right)$ such that $W\left(x,y\right)d\left(Tx,Ty\right)\le d\left(x,y\right)$, $\forall x,y\in X$, then we say that $T$ is a $W$-nonexpansive mapping.

Clearly, any nonexpansive mapping is a $W$-nonexpansive mapping with $W\left(x,y\right)=1$ for all $x,y\in X$.

Definition 1.3 Let $T:X\to X$ be a mapping and $W:X×X\to \left[0,\infty \right)$ be a function. We say that $T$ is a $W$-admissible if $W\left(x,y\right)\ge 1⇒W\left(Tx,Ty\right)\ge 1$, $\forall x,y\in X.$

Definition 1.4 [4] Let $H$ be a Hilbert space, $T:H\to H$ is called demicompact if whenever $\left\{{x}_{n}\right\}\subset H$ is bounded and $\left\{T{x}_{n}-T{x}_{n}\right\}$ strongly convergent, then there exists a subsequence $\left\{{x}_{nk}\right\}$ of $\left\{{x}_{n}\right\}$ which is strongly convergent.

Next our main results are presented.

#### 2. Main results

Theorem 2.1 Let $E$ be a bounded closed convex subset of a Hilbert space $H$, $W:E×E\to \left[0,\infty \right)$ is triangular function, $T:E\to E$ is a $W$-nonexpansive mapping and it is $W$-admissible. If the following conditions are satisfied:

(w1) there exists ${x}_{0}\in E$ such that $W\left({x}_{0},T{x}_{0}\right)\ge 1$;

(w2) there exists a sequence $\left\{{s}_{j}\right\}\subseteq \left[0,1\right)$ with ${lim}_{n\to \infty }{s}_{j}=1$ such that for all $x,y\in E$, if $W\left(x,y\right)\ge 1$, then $W\left(x,\left(1-{s}_{j}\right)x+{s}_{j}y\right)\ge 1$, $\forall {s}_{j}\in \left\{{s}_{j}\right\}$;

(w3) if $\left\{{x}_{n}\right\}\subseteq E$ is satisfied $W\left({x}_{0},{x}_{n}\right)\ge 1\text{,}$ moreover ${x}_{n}\to {x}^{*}$ or ${x}_{n}\to {x}^{*}\in E\text{,}$ then $W\left({x}_{n},x*\right)\ge 1.$

Then $T$ has a fixed point.

Proof. Let ${x}_{0}\in X$ such that $\alpha \left({x}_{0},T{x}_{0}\right)\ge 1$. Take ${x}_{n+1j}=\left(1-{s}_{j}\right){x}_{0}+{s}_{j}T{x}_{nj}$ for all $j$, $n\in N$, there ${x}_{0}={x}_{0j}$. Now we fix $j$, for each $j\in N$, from (w2), we may obtain $W\left({x}_{0j},{x}_{1j}\right)\ge 1.$

Also, for $T$ is $W$-admissible, then $W\left(T{x}_{0j},T{x}_{1j}\right)\ge 1$ is obtained. According to $W$ is a triangular function and (w1), then $W\left({x}_{0j},T{x}_{1j}\right)\ge 1.$

Once again use (w2), then $W\left({x}_{0j},{x}_{2j}\right)\ge 1$ is also obtained. Continuously, we easily obtain:

(1)

Based on that $W$ is triangular, we may get:

(2)

So from Eq. (2) and for $T$ is $W$-nonexpansive, we have:

(3)
$‖{x}_{nj}-{x}_{mj}‖={s}_{j}‖T{x}_{n-1j}-T{x}_{m-1j}‖\le {s}_{j}W\left({x}_{n-1j},{x}_{m-1j}\right)‖T{x}_{n-1j}-T{x}_{m-1j}‖$

Let $n\to \mathrm{\infty }$, for $E$ is bounded we may get $‖{x}_{nj}-{x}_{mj}‖\to 0$, hence $\left\{{x}_{nj}\right\}$ is Cauchy sequence, it means there exists ${x}_{j}^{\mathrm{*}}\in E$ such that $\left\{{x}_{nj}\right\}$ convergent to ${x}_{j}^{\mathrm{*}}$, that is:

(4)
$\mathrm{l}\mathrm{i}{\mathrm{m}}_{n\to \mathrm{\infty }}‖{x}_{nj}-{x}_{j}^{\mathrm{*}}‖=0.$

Also, from Eq. (1) and (w3), we have:

(5)
$W\left({x}_{nj},{x}_{j}^{\mathrm{*}}\right)\ge 1.$

Once again by Eq. (1), for $W$ is triangular, so we have:

(6)
$W\left({x}_{0j},{x}_{j}^{\mathrm{*}}\right)\ge 1.$

Since $E$ is bounded, closed and convex in Hilbert $H$, then it is weakly compact. Hence there exists a ${x}^{\mathrm{*}}\in E$ such that:

(7)

From Eqs. (6, 7), applying (w3) we have:

(8)
$W\left({x}_{j}^{\mathrm{*}},{x}^{\mathrm{*}}\right)\ge 1.$

Next, we show that ${x}_{j}^{\mathrm{*}}=\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}$.

Indeed, according to ${x}_{nj}=\left(1-{s}_{j}\right){x}_{0}+{s}_{j}T{x}_{n-1j}$, $T$ is $W$-nonexpiansive and Eq. (5), we have:

(9)
$‖{x}_{j}^{\mathrm{*}}-\left(\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}\right)‖=‖{x}_{j}^{\mathrm{*}}-{x}_{nj}+{x}_{nj}-\left(\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}\right)‖$

Let $n\to \mathrm{\infty }$ in Eq. (9), utilize Eq. (4) we obtain $‖{x}_{j}^{\mathrm{*}}-\left[\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}\right]‖\to 0$, it implies that ${x}_{j}^{\mathrm{*}}=\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}$.

Finally, we show that ${x}^{\mathrm{*}}$ is a fixed point of $T$. If $y$ is any arbitrary point in $H$, we have:

(10)
${‖{x}_{j}^{\mathrm{*}}-y‖}^{2}={‖\left({x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}\right)+\left({x}^{\mathrm{*}}-y\right)‖}^{2}={‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖}^{2}+{‖{x}^{\mathrm{*}}-y‖}^{2}+2〈{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}},{x}^{\mathrm{*}}-y〉.$

Since ${x}_{j}^{\mathrm{*}}\to {x}^{\mathrm{*}}$, then , $\left(j\to \mathrm{\infty }\right).$

So, based on the above inequality and Eq. (10), we get:

(11)
$\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left({‖{x}_{j}^{\mathrm{*}}-y‖}^{2}-{‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖}^{2}\right)={‖{x}^{\mathrm{*}}-y‖}^{2}.$

Setting $y=T{x}^{\mathrm{*}}$ in Eq. (11), we have:

(12)
$\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left({‖{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖}^{2}-{‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖}^{2}\right)={‖{x}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖}^{2}.$

Moreover, since ${x}_{j}^{\mathrm{*}}=\left(1-{s}_{j}\right){x}_{0j}+{s}_{j}T{x}_{j}^{\mathrm{*}}$, then:

(13)
$‖T{x}_{j}^{\mathrm{*}}-{x}_{j}^{\mathrm{*}}‖=‖T{x}_{j}^{\mathrm{*}}-\left(1-{s}_{j}\right){x}_{0j}-{s}_{j}T{x}_{j}^{\mathrm{*}}‖=\left(1-{s}_{j}\right)‖T{x}_{j}^{\mathrm{*}}-{x}_{0j}‖.$

So, in Eq. (13) as $j\to \mathrm{\infty }$, for ${\mathrm{l}\mathrm{i}\mathrm{m}}_{j\to \mathrm{\infty }}{s}_{j}=1$ we have:

(14)
$‖T{x}_{j}^{\mathrm{*}}-{x}_{j}^{\mathrm{*}}‖\to 0.$

On the other hand, from Eq. (8) and since $T$ is $W$-nonexpansive mapping, we have:

$‖T{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖\le W\left({x}_{j}^{\mathrm{*}},{x}^{\mathrm{*}}\right)‖T{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖\le ‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖.$

Thus:

(15)
$‖{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖\le ‖{x}_{j}^{\mathrm{*}}-T{x}_{j}^{\mathrm{*}}‖+‖T{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖\le ‖{x}_{j}^{\mathrm{*}}-T{x}_{j}^{\mathrm{*}}‖+‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖,$

in turn:

(16)
$‖{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖-‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖\le ‖{x}_{j}^{\mathrm{*}}-T{x}_{j}^{\mathrm{*}}‖.$

Hence by Eq. (14), we have:

(17)
$\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(‖{x}_{j}^{\mathrm{*}}-T{x}^{\mathrm{*}}‖-‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖\right)\le \underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}‖{x}_{j}^{\mathrm{*}}-T{x}_{j}^{\mathrm{*}}‖=0.$

And, due to $E$ is bounded, we have also:

(18)
$\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\left({‖{x}_{j}^{\mathrm{*}}-T{x}_{\mathrm{*}}‖}^{2}-{‖{x}_{j}^{\mathrm{*}}-{x}^{\mathrm{*}}‖}^{2}\right)$

So, by Eq. (12), we get ${‖{x}^{\mathrm{*}}-T{x}_{\mathrm{*}}‖}^{2}=0$, that is, ${x}^{\mathrm{*}}$ is fixed point of $T$.

Now, we provide a method for computation of that fixed point ${x}^{\mathrm{*}}$.

Theorem 2.2 Suppose all conditions of the Theorem 2.1 are satisfied. Then the Krasnoselskij iteration $\left\{{x}_{n}{\right\}}_{0}^{\infty }$ given by:

(19)

converges to a fixed point of $T$.

Proof. Take the same ${x}_{0}\in E$ as Theorem 2.1, and such that $W\left({x}_{0},T{x}_{0}\right)\ge 1$. From (w2) we get:

(20)
$W\left({x}_{0},\left(1-s\right){x}_{0}+sT{x}_{0}\right)=W\left({x}_{0},{x}_{1}\right)\ge 1.$

For $W$ is triangular, so:

(21)
$W\left(T{x}_{0},{x}_{1}\right)\ge 1.$

Since $T$ is a $W$-admissible, from Eq. (20) we have:

(22)
$W\left(T{x}_{0},T{x}_{1}\right)\ge 1.$

Once again for $W$ is triangular, by Eqs. (21) and (22) we have $W\left({x}_{1},T{x}_{1}\right)\ge 1.$

Also, from (w2) we have $W\left({x}_{1},\left(1-s\right){x}_{1}+sT{x}_{1}\right)=W\left({x}_{1},{x}_{2}\right)\ge 1.$

Continuously, we can obtain:

(23)

Hence:

(24)
$W\left({x}_{0},{x}_{n}\right)\ge 1.$

Also form Theorem 2.1, we know that ${x}_{\mathrm{*}}$ is fixed point of $T$, and Based on all conditions of Theorem 2.1 are satisfied in Theorem 2.2, similarly we have:

(25)
$W\left({x}_{0},{x}_{j}^{\mathrm{*}}\right)\ge 1,$
(26)
$W\left({x}_{j}^{\mathrm{*}},{x}^{\mathrm{*}}\right)\ge 1.$

From Eqs. (25) and (26), for $W$ is triangular, then:

(27)
$W\left({x}_{0},{x}^{\mathrm{*}}\right)\ge 1.$

Also, by Eqs. (24) and (27), use $W$ is triangular, we get:

(28)
$W\left({x}_{n},{x}^{\mathrm{*}}\right)\ge 1.$

Based on Eq. (28), since $T$ is $W$-nonexpansive mapping, then we have:

(29)
$‖T{x}_{n}-T{x}^{\mathrm{*}}‖\le W\left({x}_{n},{x}^{\mathrm{*}}\right)‖T{x}_{n}-T{x}^{\mathrm{*}}‖\le ‖{x}_{n}-{x}^{\mathrm{*}}‖.$

So:

(30)
$‖{x}_{n+1}-{x}^{\mathrm{*}}‖=‖\left(1-s\right)\left({x}_{n}-{x}^{\mathrm{*}}\right)+s\left(T{x}_{n}-T{x}^{\mathrm{*}}\right)‖$

Continuously, we have $‖{x}_{n+1}-{x}^{\mathrm{*}}‖\le ‖{x}_{0}-{x}^{\mathrm{*}}‖$, which implies that $\left\{‖{x}_{n+1}-{x}^{\mathrm{*}}‖\right\}$ is monotone decrease bounded sequence. So ${\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}‖{x}_{n+1}-{x}^{\mathrm{*}}‖$ exists.

Next, we prove that $‖{x}_{n}-T{x}_{n}‖\to 0$:

(31)
${‖{x}_{n+1}-{x}^{\mathrm{*}}‖}^{2}={‖\left(1-s\right)\left({x}_{n}-{x}^{\mathrm{*}}\right)-s\left(T{x}_{n}-T{x}^{\mathrm{*}}\right)‖}^{2}$

Also, on the other hand for any constant $\lambda$:

(32)
${\lambda }^{2}{‖{x}_{n}-T{x}_{n}‖}^{2}={‖\left({x}_{n}-{x}^{\mathrm{*}}\right)-\left(T{x}_{n}-T{x}^{\mathrm{*}}\right)‖}^{2}$

Adding Eq. (31) to Eq. (32) and let ${\lambda }^{2}\le \left(1-s\right)s$ , we may obtain:

(33)
${‖{x}_{n+1}-{x}^{\mathrm{*}}‖}^{2}+{\lambda }^{2}{‖{x}_{n}-T{x}_{n}‖}^{2}\le \left(\left(1-s{\right)}^{2}+{s}^{2}+2{\lambda }^{2}\right){‖{x}_{n}-{x}^{\mathrm{*}}‖}^{2}$

It implies ${\lambda }^{2}{‖{x}_{n}-T{x}_{n}‖}^{2}\le {‖{x}_{n}-{x}^{\mathrm{*}}‖}^{2}-{‖{x}_{n+1}-{x}^{\mathrm{*}}‖}^{2}.$

Since ${\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}‖{x}_{n+1}-{x}^{\mathrm{*}}‖$ exists, in the above inequality let $n\to \mathrm{\infty }\text{,}$ it results ${\lambda }^{2}{‖{x}_{n}-T{x}_{n}‖}^{2}\to 0.$

It means $‖{x}_{n}-T{x}_{n}‖\to 0.$

For $T$ is demicompact, it results that there exists a strongly convergent subsequence $\left\{{x}_{{n}_{i}}\right\}\subseteq \left\{{x}_{n}\right\}$ such that ${x}_{{n}_{i}}\to {x}^{\mathrm{*}}\in F\left(T\right)$, that is, $‖{x}_{{n}_{i}}\to {x}^{\mathrm{*}}‖\to 0$. Also $\left\{‖{x}_{n}\to {x}^{\mathrm{*}}‖\right\}$ is convergent, it implies that $‖{x}_{n}\to {x}^{\mathrm{*}}‖\to 0$. Hence that $\left\{{x}_{n}\right\}$ is convergent to ${x}^{\mathrm{*}}\in F\left(T\right)$.

#### Acknowledgements

This work was supported by the Educational Science Foundation of Chongqing.

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