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# Fixed points for mappings satisfying some multi-valued contractions with w-distance

Fixed Point Theory and Applications20142014:246

https://doi.org/10.1186/1687-1812-2014-246

• Accepted: 4 December 2014
• Published:

## Abstract

The existence of fixed points and iterative approximations for some nonlinear multi-valued contraction mappings in complete metric spaces with w-distance are proved. Two examples are included. The results presented in this paper extend, improve and unify many known results in recent literature.

MSC:54H25, 47H10.

## Keywords

• multi-valued contractions
• w-distance
• fixed point theorems

## 1 Introduction and preliminaries

In 1996, Kada et al. [1] introduced the concept of w-distance and got some fixed point theorems for single-valued mappings under w-distance. In 2006, Feng and Liu [[2], Theorem 3.1] proved the following fixed point theorem for a multi-valued contractive mapping, which generalizes the nice fixed point theorem due to Nadler [[3], Theorem 5].

Theorem 1.1 ([2])

Let $\left(X,d\right)$ be a complete metric space and T be a multi-valued mapping from X into $CL\left(X\right)$, where $CL\left(X\right)$ is the family of all nonempty closed subsets of X. Assume that

(c1) the mapping $f:X\to {\mathbb{R}}^{+}$, defined by $f\left(x\right)=d\left(x,T\left(x\right)\right)$, $x\in X$, is lower semi-continuous;

(c2) there exist constants $b,c\in \left(0,1\right)$ with $c such that for any $x\in X$, there is $y\in T\left(x\right)$ satisfying
$bd\left(x,y\right)\le f\left(x\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}f\left(y\right)\le cd\left(x,y\right).$

Then T has a fixed point in X.

In 2007, Klim and Wardowski [[4], Theorem 2.1] extended Theorem 1.1 and proved the following result.

Theorem 1.2 ([4])

Let $\left(X,d\right)$ be a complete metric space and T be a multi-valued mapping from X into $CL\left(X\right)$ satisfying (c1). Assume that

(c3) there exist $b\in \left(0,1\right)$ and $\phi :{\mathbb{R}}^{+}\to \left[0,b\right)$ satisfying
$\underset{r\to {t}^{+}}{lim sup}\phi \left(r\right)
and for any $x\in X$, there is $y\in T\left(x\right)$ satisfying
$bd\left(x,y\right)\le d\left(x,T\left(x\right)\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}f\left(y\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right).$

Then T has a fixed point in X.

In 2009 and 2010, Ćirić [[5], Theorem 2.1] and Liu et al. [[6], Theorems 2.1 and 2.3] established a few fixed point theorems for some multi-valued nonlinear contractions, which include the multi-valued contraction in Theorem 1.1 as a special case.

Theorem 1.3 ([5])

Let $\left(X,d\right)$ be a complete metric space and T be a multi-valued mapping from X into $CL\left(X\right)$ satisfying (c1). Assume that

(c4) there exists a function $\phi :{\mathbb{R}}^{+}\to \left[a,1\right)$, $0, satisfying
$\underset{r\to {t}^{+}}{lim sup}\phi \left(r\right)<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {\mathbb{R}}^{+},$
and for any $x\in X$, there is $y\in T\left(x\right)$ satisfying
$\sqrt{\phi \left(f\left(x\right)\right)}d\left(x,y\right)\le f\left(x\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}f\left(y\right)\le \phi \left(f\left(x\right)\right)d\left(x,y\right).$

Then T has a fixed point in X.

Theorem 1.4 ([6])

Let T be a multi-valued mapping from a complete metric space $\left(X,d\right)$ into $CL\left(X\right)$ such that
where
$\alpha :B\to \left(0,1\right]$ and $\beta :B\to \left[0,1\right)$ satisfy that
$\underset{r\to {0}^{+}}{lim inf}\alpha \left(r\right)>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{r\to {t}^{+}}{lim sup}\frac{\beta \left(r\right)}{\alpha \left(r\right)}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,supf\left(X\right)\right).$

Then

(a1) for each ${x}_{0}\in X$, there exist an orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T and $z\in X$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=z$;

(a2) z is a fixed point of T in X if and only if the function $f\left(x\right)=d\left(x,T\left(x\right)\right)$, $x\in X$, is T-orbitally lower semi-continuous at z.

Theorem 1.5 ([6])

Let T be a multi-valued mapping from a complete metric space $\left(X,d\right)$ into $CL\left(X\right)$ such that
where
$\alpha :A\to \left(0,1\right]$ and $\beta :A\to \left[0,1\right)$ satisfy that
$\underset{r\to {t}^{+}}{lim inf}\alpha \left(r\right)>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{r\to {t}^{+}}{lim sup}\frac{\beta \left(r\right)}{\alpha \left(r\right)}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,diam\left(X\right)\right),$

and one of α and β is nondecreasing. Then

(a1) for each ${x}_{0}\in X$, there exist an orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T and $z\in X$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=z$;

(a2) z is a fixed point of T in X if and only if the function $f\left(x\right)=d\left(x,T\left(x\right)\right)$, $x\in X$, is T-orbitally lower semi-continuous at z.

In 2011, Latif and Abdou [[7], Theorem 2.1] generalized Theorem 1.3 and proved the following fixed point theorem for some multi-valued contractive mapping with w-distance.

Theorem 1.6 ([7])

Let $\left(X,d\right)$ be a complete metric space with a w-distance w, and let T be a multi-valued mapping from X into $CL\left(X\right)$. Assume that

(c5) the mapping $f:X\to {\mathbb{R}}^{+}$, defined by ${f}_{w}\left(x\right)=w\left(x,T\left(x\right)\right)$, $x\in X$, is lower semi-continuous;

(c6) there exists a function $\phi :{\mathbb{R}}^{+}\to \left[b,1\right)$, $0, satisfying
$\underset{r\to {t}^{+}}{lim sup}\phi \left(r\right)<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {\mathbb{R}}^{+}$
and for any $x\in X$, there is $y\in T\left(x\right)$ satisfying
$\sqrt{\phi \left({f}_{w}\left(x\right)\right)}w\left(x,y\right)\le {f}_{w}\left(x\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{f}_{w}\left(y\right)\le \phi \left({f}_{w}\left(x\right)\right)w\left(x,y\right).$

Then there exists ${v}_{0}\in X$ such that ${f}_{w}\left({v}_{0}\right)=0$. Further, if $w\left({v}_{0},{v}_{0}\right)=0$, then ${v}_{0}\in T\left({v}_{0}\right)$.

The purpose of this paper is to prove the existence of fixed points and iterative approximations for some multi-valued contractive mappings with w-distance. Two examples with uncountably many points are included. The results presented in this paper extend, improve and unify Theorem 3.1 in [2], Theorem 2.1 in [4], Theorems 2.1 and 2.2 in [5], Theorems 2.1 and 2.3 in [6], Theorems 2.1-2.3 and 2.5 in [7], Theorem 6 in [8], Theorems 2.2 and 2.4 in [9] and Theorems 3.1-3.4 in [10].

Throughout this paper, we assume that ${\mathbb{R}}^{+}=\left[0,\mathrm{\infty }\right)$, ${\mathbb{N}}_{0}=\mathbb{N}\cup \left\{0\right\}$, where denotes the set of all positive integers.

Definition 1.7 ([1])

A function $w:X×X\to {\mathbb{R}}^{+}$ is called a w-distance in X if it satisfies the following:

(w1) $w\left(x,z\right)\le w\left(x,y\right)+w\left(y,z\right)$, $\mathrm{\forall }x,y,z\in X$;

(w2) for each $x\in X$, a mapping $w\left(x,\cdot \right):X\to {\mathbb{R}}^{+}$ is lower semi-continuous, that is, if ${\left\{{y}_{n}\right\}}_{n\in \mathbb{N}}$ is a sequence in X with ${lim}_{n\to \mathrm{\infty }}{y}_{n}=y\in X$, then $w\left(x,y\right)\le {lim inf}_{n\to \mathrm{\infty }}w\left(x,{y}_{n}\right)$;

(w3) for any $\epsilon >0$, there exists $\delta >0$ such that $w\left(z,x\right)\le \delta$ and $w\left(z,y\right)\le \delta$ imply $d\left(x,y\right)\le \epsilon$.

For any $u\in X$, $D\subseteq X$, w-distance w and $T:X\to CL\left(X\right)$, put
and

A sequence ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ in X is called an orbit of T at ${x}_{0}\in X$ if ${x}_{n}\in T\left({x}_{n-1}\right)$ for all $n\in \mathbb{N}$. A function $g:X\to {\mathbb{R}}^{+}$ is said to be T-orbitally lower semi-continuous at $z\in X$ if $g\left(z\right)\le {lim inf}_{n\to \mathrm{\infty }}g\left({x}_{n}\right)$ for each orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}\subset X$ of T with ${lim}_{n\to \mathrm{\infty }}{x}_{n}=z$. A function $\phi :{A}_{w}\to {\mathbb{R}}^{+}$ is called subadditive in ${A}_{w}$ if $\phi \left(s+t\right)\le \phi \left(s\right)+\phi \left(t\right)$ for all $s,t\in {A}_{w}$. A function $\phi :{A}_{w}\to {\mathbb{R}}^{+}$ is called strictly inverse in ${A}_{w}$ if $\phi \left(t\right)<\phi \left(s\right)$ implies that $t.

Lemma 1.8 ([11])

Let $\left(X,d\right)$ be a metric space with a w-distance w and $D\in CL\left(X\right)$. Suppose that there exists $u\in X$ such that $w\left(u,u\right)=0$. Then $w\left(u,D\right)=0$ if and only if $u\in D$.

## 2 Fixed point theorems

In this section we prove the existence of fixed points and iterative approximations for some nonlinear multi-valued contraction mappings in complete metric spaces with w-distance.

Theorem 2.1 Let $\left(X,d\right)$ be a complete metric space, w be a w-distance in X and T be a multi-valued mapping from X into $CL\left(X\right)$ such that
(2.1)
where
(2.2)
(2.3)
(2.4)
or
(2.5)

Then

(a1) for each ${x}_{0}\in X$, there exists an orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}={u}_{0}$ for some ${u}_{0}\in X$;

(a2) ${f}_{w}\left({u}_{0}\right)=0$ if and only if the function ${f}_{w}$ is T-orbitally lower semi-continuous at ${u}_{0}$;

(a3) ${u}_{0}\in T\left({u}_{0}\right)$ provided that $w\left({u}_{0},{u}_{0}\right)=0={f}_{w}\left({u}_{0}\right)$;

(a4) T has a fixed point in X if for each orbit ${\left\{{z}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T in X and $v\in X$ with $v\notin T\left(v\right)$, one of the following conditions is satisfied:
$inf\left\{w\left({z}_{n},v\right)+\phi \left(w\left({z}_{n},{z}_{n+1}\right)\right):n\in {\mathbb{N}}_{0}\right\}>0;$
(2.6)
$inf\left\{w\left({z}_{n},v\right)+w\left({z}_{n},T\left({z}_{n}\right)\right):n\in {\mathbb{N}}_{0}\right\}>0.$
(2.7)
Proof Firstly, we prove (a1). Let
$\gamma \left(t\right)=\frac{\beta \left(t\right)}{\alpha \left(t\right)},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {B}_{w}.$
(2.8)
It follows from (2.1) that for each ${x}_{0}\in X$, there exists ${x}_{1}\in T\left({x}_{0}\right)$ satisfying
$\alpha \left({f}_{w}\left({x}_{0}\right)\right)\phi \left(w\left({x}_{0},{x}_{1}\right)\right)\le {f}_{w}\left({x}_{0}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{w}\left({x}_{1}\right)\le \beta \left({f}_{w}\left({x}_{0}\right)\right)\psi \left(w\left({x}_{0},{x}_{1}\right)\right),$
which together with (2.3) and (2.8) yields that
$\begin{array}{rl}{f}_{w}\left({x}_{1}\right)& \le \beta \left({f}_{w}\left({x}_{0}\right)\right)\psi \left(w\left({x}_{0},{x}_{1}\right)\right)\le \beta \left({f}_{w}\left({x}_{0}\right)\right)\phi \left(w\left({x}_{0},{x}_{1}\right)\right)\\ \le \beta \left({f}_{w}\left({x}_{0}\right)\right)\frac{{f}_{w}\left({x}_{0}\right)}{\alpha \left({f}_{w}\left({x}_{0}\right)\right)}=\gamma \left({f}_{w}\left({x}_{0}\right)\right){f}_{w}\left({x}_{0}\right).\end{array}$
Continuing this process, we choose easily an orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T satisfying
$\begin{array}{r}{x}_{n+1}\in T\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}\alpha \left({f}_{w}\left({x}_{n}\right)\right)\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le {f}_{w}\left({x}_{n}\right)\phantom{\rule{1em}{0ex}}\text{and}\\ {f}_{w}\left({x}_{n+1}\right)\le \beta \left({f}_{w}\left({x}_{n}\right)\right)\psi \left(w\left({x}_{n},{x}_{n+1}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0}.\end{array}$
(2.9)
It follows from (2.3), (2.8) and (2.9) that
$\begin{array}{rl}{f}_{w}\left({x}_{n+1}\right)& \le \beta \left({f}_{w}\left({x}_{n}\right)\right)\psi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le \beta \left({f}_{w}\left({x}_{n}\right)\right)\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\\ \le \beta \left({f}_{w}\left({x}_{n}\right)\right)\frac{{f}_{w}\left({x}_{n}\right)}{\alpha \left({f}_{w}\left({x}_{n}\right)\right)}=\gamma \left({f}_{w}\left({x}_{n}\right)\right){f}_{w}\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0}.\end{array}$
(2.10)
Now we claim that
$\underset{n\to \mathrm{\infty }}{lim}{f}_{w}\left({x}_{n}\right)=0.$
(2.11)
Notice that the ranges of α and β, (2.2) and (2.8) ensure that
$0\le \gamma \left(t\right)<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {B}_{w}.$
(2.12)
Using (2.10) and (2.12), we conclude that ${\left\{{f}_{w}\left({x}_{n}\right)\right\}}_{n\in {\mathbb{N}}_{0}}$ is a nonnegative and nonincreasing sequence, which means that there is a constant $a\ge 0$ satisfying
$\underset{n\to \mathrm{\infty }}{lim}{f}_{w}\left({x}_{n}\right)=a.$
(2.13)
Suppose that $a>0$. Using (2.2), (2.8), (2.10), (2.12) and (2.13), we obtain that
$\begin{array}{rl}a& =\underset{n\to \mathrm{\infty }}{lim sup}{f}_{w}\left({x}_{n+1}\right)\le \underset{n\to \mathrm{\infty }}{lim sup}\left[\gamma \left({f}_{w}\left({x}_{n}\right)\right){f}_{w}\left({x}_{n}\right)\right]\\ \le \underset{n\to \mathrm{\infty }}{lim sup}\gamma \left({f}_{w}\left({x}_{n}\right)\right)\underset{n\to \mathrm{\infty }}{lim sup}{f}_{w}\left({x}_{n}\right)\\ \le a\underset{r\to {a}^{+}}{lim sup}\gamma \left(r\right)

which is a contradiction. Thus $a=0$, that is, (2.11) holds.

Next we claim that ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is a Cauchy sequence. Put
$b=\underset{n\to \mathrm{\infty }}{lim sup}\gamma \left({f}_{w}\left({x}_{n}\right)\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}c=\underset{n\to \mathrm{\infty }}{lim inf}\alpha \left({f}_{w}\left({x}_{n}\right)\right).$
(2.14)
It follows from (2.2), (2.8), (2.12) and (2.14) that
$0\le b<1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}c>0.$
(2.15)
Let $p\in \left(0,c\right)$ and $q\in \left(b,1\right)$. Because of (2.14) and (2.15), we deduce that there exists some ${n}_{0}\in \mathbb{N}$ such that
$\gamma \left({f}_{w}\left({x}_{n}\right)\right)p,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge {n}_{0},$
which together with (2.9) and (2.10) yields that
${f}_{w}\left({x}_{n+1}\right)\le q{f}_{w}\left({x}_{n}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le \frac{{f}_{w}\left({x}_{n}\right)}{p},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge {n}_{0},$
which implies that
${f}_{w}\left({x}_{n+1}\right)\le {q}^{n+1-{n}_{0}}{f}_{w}\left({x}_{{n}_{0}}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le \frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p}{q}^{n-{n}_{0}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge {n}_{0}.$
(2.16)
By means of (w1), (2.3) and (2.16), we deduce that
$\begin{array}{rl}\phi \left(w\left({x}_{n},{x}_{m}\right)\right)& \le \sum _{k=n}^{m-1}\phi \left(w\left({x}_{k},{x}_{k+1}\right)\right)\le \sum _{k=n}^{m-1}\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p}{q}^{k-{n}_{0}}\\ \le \frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge {n}_{0}.\end{array}$
(2.17)
Given $\epsilon >0$, denote by δ the constant in (w3) corresponding to ε. Assume that (2.4) holds. It follows from $\phi \left(\delta \right)>0$ and $q\in \left(b,1\right)$ that there exists a positive integer $N\ge {n}_{0}$ satisfying
$\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}<\phi \left(\delta \right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge N.$
(2.18)
Combining (2.17) and (2.18), we infer that
$max\left\{\phi \left(w\left({x}_{N},{x}_{m}\right)\right),\phi \left(w\left({x}_{N},{x}_{n}\right)\right)\right\}\le \frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}<\phi \left(\delta \right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge N,$
which together with (2.4) guarantees that
$max\left\{w\left({x}_{N},{x}_{m}\right),w\left({x}_{N},{x}_{n}\right)\right\}<\delta ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n>N.$
(2.19)
It follows from (w3) and (2.19) that
$d\left({x}_{m},{x}_{n}\right)\le \epsilon ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n>N.$
(2.20)

It is clear that (2.20) yields that ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is a Cauchy sequence.

Assume that (2.5) holds. Since φ is strictly increasing, so does ${\phi }^{-1}$. It follows from (2.5) and $q\in \left(b,1\right)$ that there exists a positive integer $N\ge {n}_{0}$ satisfying
${\phi }^{-1}\left(\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}\right)<\delta ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge N,$
which together with (2.5) and (2.17) means that
$w\left({x}_{n},{x}_{m}\right)={\phi }^{-1}\left(\phi \left(w\left({x}_{n},{x}_{m}\right)\right)\right)\le {\phi }^{-1}\left(\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}\right)<\delta ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge N,$

which ensures that (2.19) and (2.20) hold. Consequently, ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is a Cauchy sequence.

It follows from completeness of $\left(X,d\right)$ that there is some ${u}_{0}\in X$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}={u}_{0}$.

Secondly, we prove (a2). Suppose that ${f}_{w}$ is T-orbitally lower semi-continuous at ${u}_{0}$. Let ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ be the orbit of T defined by (2.9) and satisfy (2.11). It follows from (2.11) that
$0\le w\left({u}_{0},T\left({u}_{0}\right)\right)={f}_{w}\left({u}_{0}\right)\le \underset{n\to \mathrm{\infty }}{lim inf}{f}_{w}\left({x}_{n}\right)=0,$
which means that ${f}_{w}\left({u}_{0}\right)=0$. Conversely, suppose that ${f}_{w}\left({u}_{0}\right)=0$ for some ${u}_{0}\in X$. Let ${\left\{{y}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ be an arbitrary orbit of T in X with ${lim}_{n\to \mathrm{\infty }}{y}_{n}={u}_{0}$. It follows that
${f}_{w}\left({u}_{0}\right)=0\le \underset{n\to \mathrm{\infty }}{lim inf}{f}_{w}\left({y}_{n}\right),$

that is, ${f}_{w}$ is T-orbitally lower semi-continuous at ${u}_{0}$.

Thirdly, we prove (a3). Note that $T\left({u}_{0}\right)$ is closed and
$w\left({u}_{0},{u}_{0}\right)=0={f}_{w}\left({u}_{0}\right)=w\left({u}_{0},T\left({u}_{0}\right)\right).$

It follows from Lemma 1.8 that ${u}_{0}\in T\left({u}_{0}\right)$.

Finally, we prove (a4). Assume that ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is the orbit of T defined by (2.9) and that it satisfies (2.11), (2.16), (2.17) and ${lim}_{n\to \mathrm{\infty }}{x}_{n}={u}_{0}\in X$. Clearly, (2.16) and $q\in \left(b,1\right)$ mean that
$\underset{n\to \mathrm{\infty }}{lim}\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)=0.$
(2.21)
Now we claim that
$\underset{n\to \mathrm{\infty }}{lim}w\left({x}_{n},{u}_{0}\right)=0.$
(2.22)

In order to prove (2.22), we consider two possible cases as follows.

Case 1. Assume that (2.4) holds. Let $\epsilon >0$ be given. Notice that $\phi \left(\epsilon \right)>0$ and $q\in \left(b,1\right)$. It follows that there exists a positive integer $N>{n}_{0}$ satisfying
$\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}<\phi \left(\epsilon \right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge N,$
which together with (2.17) yields that
$\phi \left(w\left({x}_{n},{x}_{m}\right)\right)\le \frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}<\phi \left(\epsilon \right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge N.$
Since φ is strictly inverse, it follows that
$w\left({x}_{n},{x}_{m}\right)<\epsilon ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge N.$
Letting $m\to \mathrm{\infty }$ in the above inequality and using (w2), we get that
$w\left({x}_{n},{u}_{0}\right)\le \underset{m\to \mathrm{\infty }}{lim inf}w\left({x}_{n},{x}_{m}\right)\le \epsilon ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge N,$

that is, (2.22) holds.

Case 2. Assume that (2.5) holds. It follows from (2.5) and (2.17) that
$w\left({x}_{n},{x}_{m}\right)={\phi }^{-1}\left(\phi \left(w\left({x}_{n},{x}_{m}\right)\right)\right)\le {\phi }^{-1}\left(\frac{{f}_{w}\left({x}_{{n}_{0}}\right)}{p\left(1-q\right)}{q}^{n-{n}_{0}}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }m>n\ge {n}_{0},$
which together with (w2) and (2.5) ensures that

that is, (2.22) holds.

Suppose that ${u}_{0}\notin T\left({u}_{0}\right)$. Let $v={u}_{0}$ and ${z}_{n}={x}_{n}$ for each $n\in {\mathbb{N}}_{0}$. Assume that (2.6) holds. Making use of (2.6), (2.21) and (2.22), we conclude that
$0
which is a contradiction. Assume that (2.7) holds. By virtue of (2.7), (2.11) and (2.22), we infer that
$0

which is also a contradiction. Consequently, ${u}_{0}\in T\left({u}_{0}\right)$. This completes the proof. □

Theorem 2.2 Let $\left(X,d\right)$ be a complete metric space, w be a w-distance in X and T be a multi-valued mapping from X into $CL\left(X\right)$ such that (2.3) and one of (2.4) and (2.5) hold and
(2.23)
where
(2.24)
and
(2.25)

Then (a1)-(a4) hold.

Proof Firstly, we prove (a1). Let
$\gamma \left(t\right)=\frac{\beta \left(t\right)}{\alpha \left(t\right)},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {A}_{w}.$
(2.26)
Notice that the ranges of α and β, (2.24) and (2.26) ensure that
$0\le \gamma \left(t\right)<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {A}_{w}.$
(2.27)
It follows from (2.23) that for each ${x}_{0}\in X$, there exists ${x}_{1}\in T\left({x}_{0}\right)$ satisfying
$\alpha \left(w\left({x}_{0},{x}_{1}\right)\right)\phi \left(w\left({x}_{0},{x}_{1}\right)\right)\le {f}_{w}\left({x}_{0}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{w}\left({x}_{1}\right)\le \beta \left(w\left({x}_{0},{x}_{1}\right)\right)\psi \left(w\left({x}_{0},{x}_{1}\right)\right),$
which together with (2.3) and (2.26) means that
$\begin{array}{rl}{f}_{w}\left({x}_{1}\right)& \le \beta \left(w\left({x}_{0},{x}_{1}\right)\right)\psi \left(w\left({x}_{0},{x}_{1}\right)\right)\le \beta \left(w\left({x}_{0},{x}_{1}\right)\right)\phi \left(w\left({x}_{0},{x}_{1}\right)\right)\\ \le \beta \left(w\left({x}_{0},{x}_{1}\right)\right)\frac{{f}_{w}\left({x}_{0}\right)}{\alpha \left(w\left({x}_{0},{x}_{1}\right)\right)}=\gamma \left(w\left({x}_{0},{x}_{1}\right)\right){f}_{w}\left({x}_{0}\right).\end{array}$
Continuing this process, we choose easily an orbit ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ of T satisfying
$\begin{array}{r}{x}_{n+1}\in T\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}\alpha \left(w\left({x}_{n},{x}_{n+1}\right)\right)\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le {f}_{w}\left({x}_{n}\right)\phantom{\rule{1em}{0ex}}\text{and}\\ {f}_{w}\left({x}_{n+1}\right)\le \beta \left(w\left({x}_{n},{x}_{n+1}\right)\right)\psi \left(w\left({x}_{n},{x}_{n+1}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0},\end{array}$
(2.28)
which together with (2.3) and (2.26) gives that
$\begin{array}{rl}{f}_{w}\left({x}_{n+1}\right)& \le \beta \left(w\left({x}_{n},{x}_{n+1}\right)\right)\psi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le \beta \left(w\left({x}_{n},{x}_{n+1}\right)\right)\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\\ \le \beta \left(w\left({x}_{n},{x}_{n+1}\right)\right)\frac{{f}_{w}\left({x}_{n}\right)}{\alpha \left(w\left({x}_{n},{x}_{n+1}\right)\right)}=\gamma \left(w\left({x}_{n},{x}_{n+1}\right)\right){f}_{w}\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0}\end{array}$
(2.29)
and
$\begin{array}{rl}\phi \left(w\left({x}_{n+1},{x}_{n+2}\right)\right)& \le \frac{{f}_{w}\left({x}_{n+1}\right)}{\alpha \left(w\left({x}_{n+1},{x}_{n+2}\right)\right)}\\ \le \frac{\beta \left(w\left({x}_{n},{x}_{n+1}\right)\right)}{\alpha \left(w\left({x}_{n+1},{x}_{n+2}\right)\right)}\psi \left(w\left({x}_{n},{x}_{n+1}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0}.\end{array}$
(2.30)
Now we claim that
$w\left({x}_{n+1},{x}_{n+2}\right)\le w\left({x}_{n},{x}_{n+1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in {\mathbb{N}}_{0}.$
(2.31)
Suppose that there exists ${n}_{0}\in {\mathbb{N}}_{0}$ satisfying
$w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)>w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right).$
(2.32)
Let (2.4) hold. It follows from (2.3), (2.25), (2.26), (2.30) and (2.32) that
$\begin{array}{rl}\phi \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)& \le \frac{\beta \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)}{\alpha \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)}\psi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)\\ \le max\left\{\gamma \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right),\gamma \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)\right\}\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right).\end{array}$
(2.33)
If $\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)=0$, it follows from (2.33) that $\phi \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)=0$. Thus (2.4) and (2.32) guarantee that
$0\le w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)
which is a contradiction; if $\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)>0$, (2.4), (2.26), (2.27) and (2.33) yield that
$\begin{array}{rl}\phi \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)& \le max\left\{\gamma \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right),\gamma \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)\right\}\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)\\ <\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right).\end{array}$
(2.34)
Since φ is strictly inverse, it follows from (2.32) and (2.34) that
$w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)

which is impossible.

Let (2.5) hold. Notice that φ is strictly increasing. It follows from (2.3), (2.26), (2.27), (2.30) and (2.32) that
$\begin{array}{rl}\phi \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)& \le \frac{\beta \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)}{\alpha \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)}\psi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)\\ \le max\left\{\gamma \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right),\gamma \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)\right\}\phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)\\ \le \phi \left(w\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\right)\\ <\phi \left(w\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right),\end{array}$

which is absurd. Hence (2.31) holds. That is, ${\left\{w\left({x}_{n},{x}_{n+1}\right)\right\}}_{n\in {\mathbb{N}}_{0}}$ is a nonincreasing and nonnegative sequence. It follows that ${lim}_{n\to \mathrm{\infty }}w\left({x}_{n},{x}_{n+1}\right)=d$ for some $d\ge 0$.

Now we claim that (2.11) holds. Using (2.27) and (2.29), we conclude that ${\left\{{f}_{w}\left({x}_{n}\right)\right\}}_{n\in {\mathbb{N}}_{0}}$ is a nonnegative and nonincreasing sequence. Consequently, (2.13) is satisfied for some $a\ge 0$. Suppose that $a>0$. Using (2.13), (2.24), (2.27) and (2.29), we obtain that
$\begin{array}{rl}a& =\underset{n\to \mathrm{\infty }}{lim sup}{f}_{w}\left({x}_{n+1}\right)\le \underset{n\to \mathrm{\infty }}{lim sup}\left[\gamma \left(w\left({x}_{n},{x}_{n+1}\right)\right){f}_{w}\left({x}_{n}\right)\right]\\ \le \underset{n\to \mathrm{\infty }}{lim sup}\gamma \left(w\left({x}_{n},{x}_{n+1}\right)\right)\underset{n\to \mathrm{\infty }}{lim sup}{f}_{w}\left({x}_{n}\right)\le a\underset{t\to {d}^{+}}{lim sup}\gamma \left(t\right)\\

which is a contradiction. Thus $a=0$, that is, (2.11) holds.

Next we claim that ${\left\{{x}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is a Cauchy sequence. Put
$b=\underset{n\to \mathrm{\infty }}{lim sup}\gamma \left(w\left({x}_{n},{x}_{n+1}\right)\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}c=\underset{n\to \mathrm{\infty }}{lim inf}\alpha \left(w\left({x}_{n},{x}_{n+1}\right)\right).$
(2.35)
It follows from (2.24), (2.27), (2.29) and (2.35) that (2.15) holds. Let $p\in \left(0,c\right)$ and $q\in \left(b,1\right)$. Because of (2.15) and (2.35), we deduce that there exists some ${n}_{0}\in \mathbb{N}$ such that
$\gamma \left(w\left({x}_{n},{x}_{n+1}\right)\right)p,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge {n}_{0},$
which together with (2.28) and (2.29) yields that
${f}_{w}\left({x}_{n+1}\right)\le q{f}_{w}\left({x}_{n}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\phi \left(w\left({x}_{n},{x}_{n+1}\right)\right)\le \frac{{f}_{w}\left({x}_{n}\right)}{p},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge {n}_{0}.$

The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □

## 3 Remarks and illustrative examples

In this section we construct two nontrivial examples to illustrate the results in Section 2.

Remark 3.1 Theorem 2.1 extends Theorem 3.1 in [2], Theorem 2.1 in [5], Theorem 2.1 in [6], Theorems 2.1 and 2.2 in [7], Theorems 2.2 and 2.4 in [9], and Theorems 3.1 and 3.2 in [10]. Example 3.2 below shows that Theorem 2.1 extends substantially Theorem 3.1 in [2] and Theorem 2.1 in [5] and differs from Theorems 5 and 6 in [8] and Theorem 2.1 in [4].

Example 3.2 Let $X=\left[0,1\right]\cup \left\{\frac{6}{5}\right\}$ be endowed with the Euclidean metric $d=|\cdot |$ and ${u}_{0}=0$. Define $w:X×X\to {\mathbb{R}}^{+}$, $T:X\to CL\left(X\right)$, $\alpha :\left[0,\frac{1}{4}\right]\to \left(0,1\right]$, $\beta :\left[0,\frac{1}{4}\right]\to \left[0,1\right)$ and $\phi ,\psi :\left[0,\frac{6}{5}\right]\to {\mathbb{R}}^{+}$ by
$\begin{array}{r}w\left(x,y\right)=y,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in X,\\ T\left(x\right)=\left\{\begin{array}{cc}\left\{\frac{x}{4}\right\},\hfill & \mathrm{\forall }x\in \left[0,\frac{2}{5}\right)\cup \left(\frac{2}{5},1\right],\hfill \\ \left\{\frac{1}{10},\frac{1}{3}\right\},\hfill & \mathrm{\forall }x\in \left\{\frac{2}{5},\frac{6}{5}\right\},\hfill \end{array}\\ \alpha \left(t\right)=\frac{8+t}{9},\phantom{\rule{2em}{0ex}}\beta \left(t\right)=\frac{2+t}{3},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,\frac{1}{4}\right]\end{array}$
and
$\phi \left(t\right)=t,\phantom{\rule{2em}{0ex}}\psi \left(t\right)=min\left\{t,|1-t|\right\},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,\frac{6}{5}\right].$
It is easy to see that ${A}_{w}=\left[0,\frac{6}{5}\right]$, ${B}_{w}=\left[0,\frac{1}{4}\right]$, (2.3), (2.4) and (2.5) hold and
${f}_{w}\left(x\right)=w\left(x,T\left(x\right)\right)=\left\{\begin{array}{cc}\frac{x}{4},\hfill & \mathrm{\forall }x\in \left[0,\frac{2}{5}\right)\cup \left(\frac{2}{5},1\right],\hfill \\ \frac{1}{10},\hfill & \mathrm{\forall }x\in \left\{\frac{2}{5},\frac{6}{5}\right\},\hfill \end{array}$
is T-orbitally lower semi-continuous at ${u}_{0}$,
$\begin{array}{c}\beta \left(0\right)=\frac{2}{3}<\frac{8}{9}=\alpha \left(0\right),\phantom{\rule{2em}{0ex}}\underset{r\to {0}^{+}}{lim inf}\alpha \left(r\right)=\frac{8}{9}>0,\hfill \\ \underset{r\to {t}^{+}}{lim sup}\frac{\beta \left(r\right)}{\alpha \left(r\right)}=\underset{r\to {t}^{+}}{lim sup}\left(\frac{2+r}{3}\cdot \frac{9}{8+r}\right)=\frac{6+3t}{8+t}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,\frac{1}{4}\right].\hfill \end{array}$
For $x\in \left[0,\frac{2}{5}\right)\cup \left(\frac{2}{5},1\right]$, there exists $y=\frac{x}{4}\in T\left(x\right)=\left\{\frac{x}{4}\right\}$ satisfying
$\alpha \left({f}_{w}\left(x\right)\right)\phi \left(w\left(x,y\right)\right)=\frac{8+\frac{x}{4}}{9}\cdot \frac{x}{4}\le \frac{x}{4}={f}_{w}\left(x\right)$
and
${f}_{w}\left(y\right)=\frac{x}{16}\le \frac{2+\frac{x}{4}}{3}\cdot min\left\{\frac{x}{4},1-\frac{x}{4}\right\}=\beta \left({f}_{w}\left(x\right)\right)\psi \left(w\left(x,y\right)\right).$
For $x\in \left\{\frac{2}{5},\frac{6}{5}\right\}$, there exists $y=\frac{1}{10}\in T\left(x\right)=\left\{\frac{1}{10},\frac{1}{3}\right\}$ satisfying
$\alpha \left({f}_{w}\left(x\right)\right)\phi \left(w\left(x,y\right)\right)=\frac{8+\frac{1}{10}}{9}\cdot \frac{1}{10}\le \frac{1}{10}={f}_{w}\left(x\right)$
and
${f}_{w}\left(y\right)=\frac{1}{40}\le \frac{2+\frac{1}{10}}{3}\cdot min\left\{\frac{1}{10},1-\frac{1}{10}\right\}=\beta \left({f}_{w}\left(x\right)\right)\psi \left(w\left(x,y\right)\right).$
Put $v\in X\setminus \left\{0\right\}$ and ${\left\{{z}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ is an orbit of T in X. It is easy to verify that ${lim}_{n\to \mathrm{\infty }}{z}_{n}={u}_{0}=0$ and
$\begin{array}{c}inf\left\{w\left({z}_{n},v\right)+\phi \left(w\left({z}_{n},{z}_{n+1}\right)\right):n\in {\mathbb{N}}_{0}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=inf\left\{v+{z}_{n+1}:n\in {\mathbb{N}}_{0}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=v+{u}_{0}=v>0.\hfill \end{array}$

Hence (2.1), (2.2) and (2.6) hold, that is, the conditions of Theorem 2.1 are fulfilled. Thus Theorem 2.1 guarantees that (a1)-(a4) hold. Moreover, T has a fixed point ${u}_{0}=0\in X$.

Now we show that Theorem 2.1 in [5] is unapplicable in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exists a function $\phi :{\mathbb{R}}^{+}\to \left[a,1\right)$, $0, such that
$\underset{r\to {t}^{+}}{lim sup}\phi \left(r\right)<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {\mathbb{R}}^{+},$
(3.1)
and for any $x\in X$ there is $y\in T\left(x\right)$ satisfying
$\sqrt{\phi \left(f\left(x\right)\right)}d\left(x,y\right)\le f\left(x\right)$
(3.2)
and
$f\left(y\right)\le \phi \left(f\left(x\right)\right)d\left(x,y\right).$
(3.3)
Note that
$f\left(x\right)=d\left(x,T\left(x\right)\right)=\left\{\begin{array}{cc}\frac{3}{4}x,\hfill & \mathrm{\forall }x\in \left[0,\frac{2}{5}\right)\cup \left(\frac{2}{5},1\right],\hfill \\ \frac{1}{15},\hfill & x=\frac{2}{5},\hfill \\ \frac{13}{15},\hfill & x=\frac{6}{5}.\hfill \end{array}$

Put $x=\frac{2}{5}$. For $y\in T\left(x\right)=\left\{\frac{1}{10},\frac{1}{3}\right\}$, we discuss two cases as follows.

Case 1. $y=\frac{1}{10}$. It follows from (3.2) and (3.3) that
$\frac{3}{10}\sqrt{\phi \left(\frac{1}{15}\right)}=\sqrt{\phi \left(f\left(\frac{2}{5}\right)\right)}d\left(\frac{2}{5},\frac{1}{10}\right)=\sqrt{\phi \left(f\left(x\right)\right)}d\left(x,y\right)\le f\left(x\right)=f\left(\frac{2}{5}\right)=\frac{1}{15}$
and
$\frac{3}{40}=f\left(\frac{1}{10}\right)=f\left(y\right)\le \phi \left(f\left(x\right)\right)d\left(x,y\right)=\phi \left(f\left(\frac{2}{5}\right)\right)d\left(\frac{2}{5},\frac{1}{10}\right)=\frac{3}{10}\phi \left(\frac{1}{15}\right),$
which imply that
$0.25=\frac{1}{4}\le \phi \left(\frac{1}{15}\right)\le \frac{4}{81}=0.049,$

which is impossible.

Case 2. $y=\frac{1}{3}$. It follows from (3.3) that
$\frac{1}{4}=f\left(\frac{1}{3}\right)=f\left(y\right)\le \phi \left(f\left(x\right)\right)d\left(x,y\right)=\phi \left(f\left(\frac{2}{5}\right)\right)d\left(\frac{2}{5},\frac{1}{3}\right)=\frac{1}{15}\phi \left(\frac{1}{15}\right),$
which together with $\phi \left({\mathbb{R}}^{+}\right)\subseteq \left[a,1\right)$ yields that
$\frac{15}{4}\le \phi \left(\frac{1}{15}\right)<1,$

which is absurd.

Next we show that Theorem 5 in [8] is useless in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exists a function $\phi :{\mathbb{R}}^{+}\to \left[0,1\right)$ such that (3.1) holds, and for any $x\in X$ there is $y\in T\left(x\right)$ satisfying
$d\left(x,y\right)\le \left(2-\phi \left(d\left(x,y\right)\right)\right)f\left(x\right)$
(3.4)
and
$f\left(y\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right).$
(3.5)

Put $x=\frac{2}{5}$. For $y\in T\left(x\right)=\left\{\frac{1}{10},\frac{1}{3}\right\}$, we discuss two cases as follows.

Case 1. $y=\frac{1}{10}$. It follows from (3.4) that
$\frac{3}{10}=d\left(\frac{2}{5},\frac{1}{10}\right)=d\left(x,y\right)\le \left(2-\phi \left(d\left(x,y\right)\right)\right)f\left(x\right)=\left(2-\phi \left(\frac{3}{10}\right)\right)\frac{1}{15},$
which together with $\phi \left({\mathbb{R}}^{+}\right)\subseteq \left[0,1\right)$ yields that
$0\le \phi \left(\frac{3}{10}\right)\le -\frac{5}{2}<0,$

Case 2. $y=\frac{1}{3}$. It follows from (3.4) that
$\frac{1}{4}=f\left(\frac{1}{3}\right)=f\left(y\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)=\phi \left(d\left(\frac{2}{5},\frac{1}{3}\right)\right)d\left(\frac{2}{5},\frac{1}{3}\right)=\frac{1}{15}\phi \left(\frac{1}{15}\right),$
which together with $\phi \left({\mathbb{R}}^{+}\right)\subseteq \left[0,1\right)$ gives that
$\frac{15}{4}\le \phi \left(\frac{1}{15}\right)<1,$

which is impossible.

Finally we show that Theorem 6 in [8] is futile in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exist functions $\phi :{\mathbb{R}}^{+}\to \left(0,1\right)$, $b:{\mathbb{R}}^{+}\to \left[b,1\right)$, $b>0$ such that
$\phi \left(t\right)
(3.6)
and for any $x\in X$, there is $y\in T\left(x\right)$ satisfying (3.5) and
$b\left(d\left(x,y\right)\right)d\left(x,y\right)\le f\left(x\right).$
(3.7)

Put $x=\frac{2}{5}$. For $y\in T\left(x\right)=\left\{\frac{1}{10},\frac{1}{3}\right\}$, we discuss two cases as follows.

Case 1. $y=\frac{1}{10}$. It follows from (3.7) and (3.5) that
$\frac{3}{10}b\left(\frac{3}{10}\right)=b\left(d\left(\frac{2}{5},\frac{1}{10}\right)\right)d\left(\frac{2}{5},\frac{1}{10}\right)=b\left(d\left(x,y\right)\right)d\left(x,y\right)\le f\left(x\right)=f\left(\frac{2}{5}\right)=\frac{1}{15}$
and
$\frac{3}{40}=f\left(\frac{1}{10}\right)=f\left(y\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)=\frac{3}{10}\phi \left(\frac{3}{10}\right),$
which together with (3.6) means that
$b\left(\frac{3}{10}\right)\le \frac{2}{9}<\frac{1}{4}\le \phi \left(\frac{3}{10}\right)

which is absurd.

Case 2. $y=\frac{1}{3}$. It follows from (3.5) that
$\frac{1}{4}=f\left(\frac{1}{3}\right)=f\left(y\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)=\phi \left(d\left(\frac{2}{5},\frac{1}{3}\right)\right)d\left(\frac{2}{5},\frac{1}{3}\right)=\frac{1}{15}\phi \left(\frac{1}{15}\right),$
which together with $\phi \left({\mathbb{R}}^{+}\right)\subseteq \left[0,1\right)$ gives that
$\frac{15}{4}\le \phi \left(\frac{1}{15}\right)<1,$

which is impossible.

Observe that Theorem 6 in [8] extends Theorem 3.1 in [2], Theorem 2.1 in [4] and Theorem 2.2 in [5]. It follows that Theorem 3.1 in [2], Theorem 2.1 in [4] and Theorem 2.2 in [5] are not applicable in proving the existence of fixed points for the multi-valued mapping T.

Remark 3.3 Theorem 2.2 extends, improves and unifies Theorem 3.1 in [2], Theorem 2.1 in [4], Theorem 2.2 in [5], Theorem 2.3 in [6], Theorems 2.3 and 2.5 in [7], Theorem 6 in [8], and Theorems 3.3 and 3.4 in [10]. The following example reveals that Theorem 2.2 generalizes indeed the corresponding results in [2, 4, 5, 8].

Example 3.4 Let $X=\left[0,\mathrm{\infty }\right)$ be endowed with the Euclidean metric $d=|\cdot |$ and $p\ge 1$ be a constant. Put ${u}_{0}=0$. Define $w:X×X\to {\mathbb{R}}^{+}$, $T:X\to CL\left(X\right)$, $\alpha :\left[0,\mathrm{\infty }\right)\to \left(0,1\right]$ and $\phi ,\psi :\left[0,\mathrm{\infty }\right)\to {\mathbb{R}}^{+}$ by $\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ by
$\begin{array}{c}w\left(x,y\right)={y}^{p},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in X,\hfill \\ T\left(x\right)=\left\{\begin{array}{cc}\left[\frac{{x}^{2}}{2},\frac{x}{2}\right],\hfill & \mathrm{\forall }x\in \left[0,1\right),\hfill \\ \left[\frac{1}{9},\frac{1}{4}\right],\hfill & \mathrm{\forall }x\in \left[1,\mathrm{\infty }\right),\hfill \end{array}\hfill \\ \alpha \left(t\right)=\frac{5+{t}^{\frac{1}{p}}}{10},\phantom{\rule{2em}{0ex}}\beta \left(t\right)=\frac{3+{t}^{\frac{1}{p}}}{10},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,\mathrm{\infty }\right)\hfill \end{array}$
and
$\phi \left(t\right)=t,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left[0,\mathrm{\infty }\right),\phantom{\rule{2em}{0ex}}\psi \left(t\right)=\left\{\begin{array}{cc}t,\hfill & \mathrm{\forall }t\in \left[0,1\right),\hfill \\ \frac{1}{2},\hfill & \mathrm{\forall }t\in \left[1,\mathrm{\infty }\right).\hfill \end{array}$
It is easy to see that ${A}_{w}=\left[0,\mathrm{\infty }\right)$, (2.3), (2.4) and (2.5) hold, w is a w-distance in X and
${f}_{w}\left(x\right)=w\left(x,T\left(x\right)\right)=\left\{\begin{array}{cc}{\left(\frac{{x}^{2}}{2}\right)}^{p},\hfill & \mathrm{\forall }x\in \left[0,1\right),\hfill \\ \frac{1}{{9}^{p}},\hfill & \mathrm{\forall }x\in \left[1,\mathrm{\infty }\right)\hfill \end{array}$
is T-orbitally lower semi-continuous in X, α and β are nondecreasing,
$\beta \left(0\right)=\frac{3}{10}<\frac{1}{2}=\alpha \left(0\right),\phantom{\rule{2em}{0ex}}\underset{r\to {0}^{+}}{lim inf}\alpha \left(r\right)=\frac{1}{2}>0$
and
$\underset{r\to {t}^{+}}{lim sup}\frac{\beta \left(r\right)}{\alpha \left(r\right)}=\frac{3+{t}^{\frac{1}{p}}}{5+{t}^{\frac{1}{p}}}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in {A}_{w}.$
Put $x\in \left[0,1\right)$ and $y=\frac{{x}^{2}}{2}\in T\left(x\right)$. Note that
$5+y\le 10\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\left(\frac{y}{2}\right)}^{p}\le \frac{1}{{4}^{p}}\le \frac{3+y}{10}$
imply that
$\alpha \left(w\left(x,y\right)\right)\phi \left(w\left(x,y\right)\right)=\frac{5+y}{10}\cdot {y}^{p}\le {y}^{p}={f}_{w}\left(x\right)$
and
${f}_{w}\left(y\right)={\left(\frac{{y}^{2}}{2}\right)}^{p}\le \frac{3+y}{10}\cdot {y}^{p}=\beta \left(w\left(x,y\right)\right)\psi \left(w\left(x,y\right)\right).$
Put $x\in \left[1,\mathrm{\infty }\right)$ and $y=\frac{1}{9}\in T\left(x\right)=\left[\frac{1}{9},\frac{1}{4}\right]$. It follows that
$\alpha \left(w\left(x,y\right)\right)\phi \left(w\left(x,y\right)\right)=\frac{5+\frac{1}{9}}{10}\cdot \frac{1}{{9}^{p}}\le \frac{1}{{9}^{p}}={f}_{w}\left(x\right)$
and
${f}_{w}\left(y\right)=\frac{1}{{182}^{p}}\le \frac{3+\frac{1}{9}}{10}\cdot \frac{1}{{9}^{p}}=\beta \left(w\left(x,y\right)\right)\psi \left(w\left(x,y\right)\right).$
Let $v\in X\setminus \left\{0\right\}$ and ${\left\{{z}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$ be an orbit of T. It is easy to verify that ${lim}_{n\to \mathrm{\infty }}{z}_{n}=0$ and
$\begin{array}{c}inf\left\{w\left({z}_{n},v\right)+\phi \left(w\left({z}_{n},{z}_{n+1}\right)\right):n\in {\mathbb{N}}_{0}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=inf\left\{{v}^{p}+{z}_{n+1}^{p}:n\in {\mathbb{N}}_{0}\right\}={v}^{p}>0.\hfill \end{array}$

That is, (2.6) and (2.23)-(2.25) hold. Thus the conditions of Theorem 2.2 are satisfied. Consequently, Theorem 2.2 ensures that (a1)-(a4) hold and ${u}_{0}=0$ is a fixed point of the multi-valued mapping T in X.

Notice that
$f\left(x\right)=d\left(x,T\left(x\right)\right)=\left\{\begin{array}{cc}\frac{x}{2},\hfill & \mathrm{\forall }x\in \left[0,1\right),\hfill \\ x-\frac{1}{4},\hfill & \mathrm{\forall }x\in \left[1,\mathrm{\infty }\right)\hfill \end{array}$
and
$\underset{x\to 1}{lim inf}f\left(x\right)=\frac{1}{2}<\frac{3}{4}=f\left(1\right),$

which implies that f is not lower semi-continuous at 1. Thus Theorem 3.1 in [2], Theorem 2.1 in [4], Theorem 2.2 in [5] and Theorem 6 in [8] could not be used to judge the existence of fixed points of the multi-valued mapping T in X.

## Declarations

### Acknowledgements

The authors would like to thank the referees for useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

## Authors’ Affiliations

(1)
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning, 116029, People’s Republic of China
(2)
Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea
(3)
Department of Mathematics, Gyeongsang National University, Jinju, 660-701, Korea

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