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

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.

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

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

(c₂) there exist constants $b,c\in (0,1)$ with $c<b$ such that for any $x\in X$, there is $y\in T(x)$ satisfying

$$bd(x,y)\le f(x)\mathit{\text{and}}f(y)\le cd(x,y).$$

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

(c₃) there exist $b\in (0,1)$ and $\phi :{\mathbb{R}}^{+}\to [0,b)$ satisfying

$$\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\phi (r)<b,\mathrm{\forall }t\in {\mathbb{R}}^{+},$$

and for any $x\in X$, there is $y\in T(x)$ satisfying

$$bd(x,y)\le d(x,T(x))\mathit{\text{and}}f(y)\le \phi (d(x,y))d(x,y).$$

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

(c₄) there exists a function $\phi :{\mathbb{R}}^{+}\to [a,1)$, $0<a<1$, satisfying

$$\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\phi (r)<1,\mathrm{\forall }t\in {\mathbb{R}}^{+},$$

and for any $x\in X$, there is $y\in T(x)$ satisfying

$$\sqrt{\phi (f(x))}d(x,y)\le f(x)\mathit{\text{and}}f(y)\le \phi (f(x))d(x,y).$$

Then T has a fixed point in X.

Theorem 1.4 ([6])

Let T be a multi-valued mapping from a complete metric space $(X,d)$ into $CL(X)$ such that

$$\begin{array}{c}\mathit{\text{for each }}x\in X,\mathit{\text{there exists }}y\in T(x)\mathit{\text{ satisfying}}\hfill \\ \alpha (f(x))d(x,y)\le f(x)\mathit{\text{and}}f(y)\le \beta (f(x))d(x,y),\hfill \end{array}$$

where

$$B=\{\begin{array}{cc}[0,supf(X)]\hfill & \mathit{\text{if }}supf(X)<\mathrm{\infty },\hfill \\ [0,\mathrm{\infty })\hfill & \mathit{\text{if }}supf(X)=\mathrm{\infty },\hfill \end{array}$$

$\alpha :B\to (0,1]$ and $\beta :B\to [0,1)$ satisfy that

$$\underset{r\to 0^{+}}{lim\hspace{0.17em}inf}\alpha (r)>0\mathit{\text{and}}\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}<1,\mathrm{\forall }t\in [0,supf(X)).$$

Then

(a1) for each $x_0\in X$, there exist an orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ 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 $f(x)=d(x,T(x))$, $x\in X$, is T-orbitally lower semi-continuous at z.

Theorem 1.5 ([6])

Let T be a multi-valued mapping from a complete metric space $(X,d)$ into $CL(X)$ such that

$$\begin{array}{r}\mathit{\text{for each }}x\in X,\mathit{\text{there exists }}y\in T(x)\mathit{\text{ satisfying}}\\ \alpha (d(x,y))d(x,y)\le f(x)\mathit{\text{and}}f(y)\le \beta (d(x,y))d(x,y),\end{array}$$

where

$$A=\{\begin{array}{cc}[0,diam(X)]\hfill & \mathit{\text{if }}diam(X)<\mathrm{\infty },\hfill \\ [0,\mathrm{\infty })\hfill & \mathit{\text{if }}diam(X)=\mathrm{\infty },\hfill \end{array}$$

$\alpha :A\to (0,1]$ and $\beta :A\to [0,1)$ satisfy that

$$\underset{r\to t^{+}}{lim\hspace{0.17em}inf}\alpha (r)>0\mathit{\text{and}}\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}<1,\mathrm{\forall }t\in [0,diam(X)),$$

and one of α and β is nondecreasing. Then

(a1) for each $x_0\in X$, there exist an orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ 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 $f(x)=d(x,T(x))$, $x\in X$, 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 $(X,d)$ be a complete metric space with a w-distance w, and let T be a multi-valued mapping from X into $CL(X)$. Assume that

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

(c₆) there exists a function $\phi :{\mathbb{R}}^{+}\to [b,1)$, $0<b<1$, satisfying

$$\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\phi (r)<1,\mathrm{\forall }t\in {\mathbb{R}}^{+}$$

and for any $x\in X$, there is $y\in T(x)$ satisfying

$$\sqrt{\phi (f_w(x))}w(x,y)\le f_w(x)\mathit{\text{and}}{f}_w(y)\le \phi (f_w(x))w(x,y).$$

Then there exists $v_0\in X$ such that $f_w(v_0)=0$. Further, if $w(v_0,v_0)=0$, then $v_0\in T(v_0)$.

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 ${\mathbb{R}}^{+}=[0,\mathrm{\infty })$, ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$, where ℕ denotes the set of all positive integers.

Definition 1.7 ([1])

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

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

(w₂) for each $x\in X$, a mapping $w(x,\cdot ):X\to {\mathbb{R}}^{+}$ is lower semi-continuous, that is, if ${\{y_n\}}_{n\in \mathbb{N}}$ is a sequence in X with ${lim}_{n\to \mathrm{\infty }}{y}_n=y\in X$, then $w(x,y)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}w(x,y_n)$;

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

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

$$\begin{array}{c}d(u,D)=\underset{y\in D}{inf}d(u,y),w(u,D)=\underset{y\in D}{inf}w(u,y),\hfill \\ f(u)=d(u,T(u)),f_w(u)=w(u,T(u)),\hfill \\ diam(X)=sup\{d(x,y):x,y\in X\},diam(X_w)=sup\{w(x,y):x,y\in X\},\hfill \\ A_w=\{\begin{array}{cc}[0,diam(X_w)]\hfill & \text{if }diam(X_w)<\mathrm{\infty },\hfill \\ [0,\mathrm{\infty })\hfill & \text{if }diam(X_w)=\mathrm{\infty }\hfill \end{array}\hfill \end{array}$$

and

$$B_w=\{\begin{array}{cc}[0,sup{f}_w(X)]\hfill & \text{if }sup{f}_w(X)<\mathrm{\infty },\hfill \\ [0,\mathrm{\infty })\hfill & \text{if }sup{f}_w(X)=\mathrm{\infty }.\hfill \end{array}$$

A sequence ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ in X is called an orbit of T at $x_0\in X$ if $x_n\in T(x_{n-1})$ 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 $g(z)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}g(x_n)$ for each orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}\subset X$ 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 $\phi (s+t)\le \phi (s)+\phi (t)$ for all $s,t\in A_w$. A function $\phi :A_w\to {\mathbb{R}}^{+}$ is called strictly inverse in $A_w$ if $\phi (t)<\phi (s)$ implies that $t<s$.

Lemma 1.8 ([11])

Let $(X,d)$ be a metric space with a w-distance w and $D\in CL(X)$. Suppose that there exists $u\in X$ such that $w(u,u)=0$. Then $w(u,D)=0$ 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 $(X,d)$ be a complete metric space, w be a w-distance in X and T be a multi-valued mapping from X into $CL(X)$ such that

$$\begin{array}{r}\mathit{\text{for each }}x\in X,\mathit{\text{there exists }}y\in T(x)\mathit{\text{ satisfying}}\\ \alpha (f_w(x))\phi (w(x,y))\le f_w(x)\mathit{\text{and}}{f}_w(y)\le \beta (f_w(x))\psi (w(x,y)),\end{array}$$

(2.1)

where

$$\begin{array}{r}\alpha \mathit{\text{ and }}\beta \mathit{\text{ are functions from }}{B}_w\mathit{\text{ into }}(0,1]\mathit{\text{ and }}[0,1),\mathit{\text{respectively, with }}\\ \beta (0)<\alpha (0),\underset{r\to 0^{+}}{lim\hspace{0.17em}inf}\alpha (r)>0\mathit{\text{and}}\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}<1,\mathrm{\forall }t\in B_w,\end{array}$$

(2.2)

$$\begin{array}{r}\phi \mathit{\text{ and }}\psi \mathit{\text{ are functions from }}{A}_w\mathit{\text{ into }}{\mathbb{R}}^{+}\mathit{\text{ with }}\psi (t)\le \phi (t),\mathrm{\forall }t\in A_w\mathit{\text{and}}\\ \phi \mathit{\text{ is subadditive in }}{A}_w\mathit{\text{ and satisfies that either }}\end{array}$$

(2.3)

$$\phi \mathit{\text{ is strictly inverse in }}{A}_w,\phi (0)=0,\phi (t)>0,\mathrm{\forall }t\in A_w\setminus \{0\}$$

(2.4)

or

$$\begin{array}{r}\phi \mathit{\text{ is strictly increasing in }}{A}_w\mathit{\text{ and }}\underset{t\to 0^{+}}{lim}{\phi }^{-1}(t)=0,\mathit{\text{where }}{\phi }^{-1}\\ \mathit{\text{stands for the inverse function of }}\phi .\end{array}$$

(2.5)

Then

(a1) for each $x_0\in X$, there exists an orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ of T such that ${lim}_{n\to \mathrm{\infty }}{x}_n=u_0$ for some $u_0\in X$;

(a2) $f_w(u_0)=0$ if and only if the function $f_w$ is T-orbitally lower semi-continuous at $u_0$;

(a3) $u_0\in T(u_0)$ provided that $w(u_0,u_0)=0=f_w(u_0)$;

(a4) T has a fixed point in X if for each orbit ${\{z_n\}}_{n\in {\mathbb{N}}_0}$ of T in X and $v\in X$ with $v\notin T(v)$, one of the following conditions is satisfied:

$$inf\{w(z_n,v)+\phi (w(z_n,z_{n+1})):n\in {\mathbb{N}}_0\}>0;$$

(2.6)

$$inf\{w(z_n,v)+w(z_n,T(z_n)):n\in {\mathbb{N}}_0\}>0.$$

(2.7)

Proof Firstly, we prove (a1). Let

$$\gamma (t)=\frac{\beta (t)}{\alpha (t)},\mathrm{\forall }t\in B_w.$$

(2.8)

It follows from (2.1) that for each $x_0\in X$, there exists $x_1\in T(x_0)$ satisfying

$$\alpha (f_w(x_0))\phi (w(x_0,x_1))\le f_w(x_0)\text{and}{f}_w(x_1)\le \beta (f_w(x_0))\psi (w(x_0,x_1)),$$

which together with (2.3) and (2.8) yields that

$$\begin{array}{rl}{f}_w(x_1)& \le \beta (f_w(x_0))\psi (w(x_0,x_1))\le \beta (f_w(x_0))\phi (w(x_0,x_1))\\ \le \beta (f_w(x_0))\frac{f_w(x_0)}{\alpha (f_w(x_0))}=\gamma (f_w(x_0))f_w(x_0).\end{array}$$

Continuing this process, we choose easily an orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ of T satisfying

$$\begin{array}{r}{x}_{n+1}\in T(x_n),\alpha (f_w(x_n))\phi (w(x_n,x_{n+1}))\le f_w(x_n)\text{and}\\ f_w(x_{n+1})\le \beta (f_w(x_n))\psi (w(x_n,x_{n+1})),\mathrm{\forall }n\in {\mathbb{N}}_0.\end{array}$$

(2.9)

It follows from (2.3), (2.8) and (2.9) that

$$\begin{array}{rl}{f}_w(x_{n+1})& \le \beta (f_w(x_n))\psi (w(x_n,x_{n+1}))\le \beta (f_w(x_n))\phi (w(x_n,x_{n+1}))\\ \le \beta (f_w(x_n))\frac{f_w(x_n)}{\alpha (f_w(x_n))}=\gamma (f_w(x_n))f_w(x_n),\mathrm{\forall }n\in {\mathbb{N}}_0.\end{array}$$

(2.10)

Now we claim that

$$\underset{n\to \mathrm{\infty }}{lim}{f}_w(x_n)=0.$$

(2.11)

Notice that the ranges of α and β, (2.2) and (2.8) ensure that

$$0\le \gamma (t)<1,\mathrm{\forall }t\in B_w.$$

(2.12)

Using (2.10) and (2.12), we conclude that ${\{f_w(x_n)\}}_{n\in {\mathbb{N}}_0}$ is a nonnegative and nonincreasing sequence, which means that there is a constant $a\ge 0$ satisfying

$$\underset{n\to \mathrm{\infty }}{lim}{f}_w(x_n)=a.$$

(2.13)

Suppose that $a>0$. Using (2.2), (2.8), (2.10), (2.12) and (2.13), we obtain that

$$\begin{array}{rl}a& =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{f}_w(x_{n+1})\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}[\gamma (f_w(x_n))f_w(x_n)]\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\gamma (f_w(x_n))\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{f}_w(x_n)\\ \le a\underset{r\to a^{+}}{lim\hspace{0.17em}sup}\gamma (r)<a,\end{array}$$

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

Next we claim that ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence. Put

$$b=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\gamma (f_w(x_n))\text{and}c=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\alpha (f_w(x_n)).$$

(2.14)

It follows from (2.2), (2.8), (2.12) and (2.14) that

$$0\le b<1\text{and}c>0.$$

(2.15)

Let $p\in (0,c)$ and $q\in (b,1)$. Because of (2.14) and (2.15), we deduce that there exists some $n_0\in \mathbb{N}$ such that

$$\gamma (f_w(x_n))<q\text{and}\alpha (f_w(x_n))>p,\mathrm{\forall }n\ge n_0,$$

which together with (2.9) and (2.10) yields that

$$f_w(x_{n+1})\le q{f}_w(x_n)\text{and}\phi (w(x_n,x_{n+1}))\le \frac{f_w(x_n)}{p},\mathrm{\forall }n\ge n_0,$$

which implies that

$$f_w(x_{n+1})\le q^{n+1-n_0}{f}_w(x_{n_0})\text{and}\phi (w(x_n,x_{n+1}))\le \frac{f_w(x_{n_0})}{p}{q}^{n-n_0},\mathrm{\forall }n\ge n_0.$$

(2.16)

By means of (w₁), (2.3) and (2.16), we deduce that

$$\begin{array}{rl}\phi (w(x_n,x_m))& \le \sum _{k=n}^{m-1}\phi (w(x_k,x_{k+1}))\le \sum _{k=n}^{m-1}\frac{f_w(x_{n_0})}{p}{q}^{k-n_0}\\ \le \frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0},\mathrm{\forall }m>n\ge n_0.\end{array}$$

(2.17)

Given $\epsilon >0$, denote by δ the constant in (w₃) corresponding to ε. Assume that (2.4) holds. It follows from $\phi (\delta )>0$ and $q\in (b,1)$ that there exists a positive integer $N\ge n_0$ satisfying

$$\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}<\phi (\delta ),\mathrm{\forall }n\ge N.$$

(2.18)

Combining (2.17) and (2.18), we infer that

$$max\{\phi (w(x_N,x_m)),\phi (w(x_N,x_n))\}\le \frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}<\phi (\delta ),\mathrm{\forall }m>n\ge N,$$

which together with (2.4) guarantees that

$$max\{w(x_N,x_m),w(x_N,x_n)\}<\delta ,\mathrm{\forall }m>n>N.$$

(2.19)

It follows from (w₃) and (2.19) that

$$d(x_m,x_n)\le \epsilon ,\mathrm{\forall }m>n>N.$$

(2.20)

It is clear that (2.20) yields that ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence.

Assume that (2.5) holds. Since φ is strictly increasing, so does ${\phi }^{-1}$. It follows from (2.5) and $q\in (b,1)$ that there exists a positive integer $N\ge n_0$ satisfying

$${\phi }^{-1}\left(\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}\right)<\delta ,\mathrm{\forall }n\ge N,$$

which together with (2.5) and (2.17) means that

$$w(x_n,x_m)={\phi }^{-1}\left(\phi (w(x_n,x_m))\right)\le {\phi }^{-1}\left(\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}\right)<\delta ,\mathrm{\forall }m>n\ge N,$$

which ensures that (2.19) and (2.20) hold. Consequently, ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence.

It follows from completeness of $(X,d)$ 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 ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ be the orbit of T defined by (2.9) and satisfy (2.11). It follows from (2.11) that

$$0\le w(u_0,T(u_0))=f_w(u_0)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{f}_w(x_n)=0,$$

which means that $f_w(u_0)=0$. Conversely, suppose that $f_w(u_0)=0$ for some $u_0\in X$. Let ${\{y_n\}}_{n\in {\mathbb{N}}_0}$ be an arbitrary orbit of T in X with ${lim}_{n\to \mathrm{\infty }}{y}_n=u_0$. It follows that

$$f_w(u_0)=0\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{f}_w(y_n),$$

that is, $f_w$ is T-orbitally lower semi-continuous at $u_0$.

Thirdly, we prove (a3). Note that $T(u_0)$ is closed and

$$w(u_0,u_0)=0=f_w(u_0)=w(u_0,T(u_0)).$$

It follows from Lemma 1.8 that $u_0\in T(u_0)$.

Finally, we prove (a4). Assume that ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ 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 $q\in (b,1)$ mean that

$$\underset{n\to \mathrm{\infty }}{lim}\phi (w(x_n,x_{n+1}))=0.$$

(2.21)

Now we claim that

$$\underset{n\to \mathrm{\infty }}{lim}w(x_n,u_0)=0.$$

(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 $\phi (\epsilon )>0$ and $q\in (b,1)$. It follows that there exists a positive integer $N>n_0$ satisfying

$$\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}<\phi (\epsilon ),\mathrm{\forall }n\ge N,$$

which together with (2.17) yields that

$$\phi (w(x_n,x_m))\le \frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}<\phi (\epsilon ),\mathrm{\forall }m>n\ge N.$$

Since φ is strictly inverse, it follows that

$$w(x_n,x_m)<\epsilon ,\mathrm{\forall }m>n\ge N.$$

Letting $m\to \mathrm{\infty }$ in the above inequality and using (w₂), we get that

$$w(x_n,u_0)\le \underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}inf}w(x_n,x_m)\le \epsilon ,\mathrm{\forall }n\ge N,$$

that is, (2.22) holds.

Case 2. Assume that (2.5) holds. It follows from (2.5) and (2.17) that

$$w(x_n,x_m)={\phi }^{-1}\left(\phi (w(x_n,x_m))\right)\le {\phi }^{-1}\left(\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}\right),\mathrm{\forall }m>n\ge n_0,$$

which together with (w₂) and (2.5) ensures that

$$w(x_n,u_0)\le \underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}inf}w(x_n,x_m)\le {\phi }^{-1}\left(\frac{f_w(x_{n_0})}{p(1-q)}{q}^{n-n_0}\right)\to 0\text{as }n\to \mathrm{\infty },$$

that is, (2.22) holds.

Suppose that $u_0\notin T(u_0)$. 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

$$0<inf\{w(x_n,u_0)+\phi (w(x_n,x_{n+1})):n\in {\mathbb{N}}_0\}=0,$$

which is a contradiction. Assume that (2.7) holds. By virtue of (2.7), (2.11) and (2.22), we infer that

$$0<inf\{w(x_n,u_0)+w(x_n,x_{n+1}):n\in {\mathbb{N}}_0\}=0,$$

which is also a contradiction. Consequently, $u_0\in T(u_0)$. This completes the proof. □

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

$$\begin{array}{r}\mathit{\text{for each }}x\in X,\mathit{\text{there exists }}y\in T(x)\mathit{\text{ satisfying}}\\ \alpha (w(x,y))\phi (w(x,y))\le f_w(x)\mathit{\text{and}}{f}_w(y)\le \beta (w(x,y))\psi (w(x,y)),\end{array}$$

(2.23)

where

$$\begin{array}{r}\alpha \mathit{\text{ and }}\beta \mathit{\text{ are functions from }}{A}_w\mathit{\text{ into }}(0,1]\mathit{\text{ and }}[0,1),\mathit{\text{respectively, with}}\\ \beta (0)<\alpha (0),\underset{r\to 0^{+}}{lim\hspace{0.17em}inf}\alpha (r)>0\mathit{\text{and}}\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}<1,\mathrm{\forall }t\in A_w\end{array}$$

(2.24)

and

$$\mathit{\text{one of }}\alpha \mathit{\text{ and }}\beta \mathit{\text{ is nondecreasing in }}{A}_w.$$

(2.25)

Then (a1)-(a4) hold.

Proof Firstly, we prove (a1). Let

$$\gamma (t)=\frac{\beta (t)}{\alpha (t)},\mathrm{\forall }t\in A_w.$$

(2.26)

Notice that the ranges of α and β, (2.24) and (2.26) ensure that

$$0\le \gamma (t)<1,\mathrm{\forall }t\in A_w.$$

(2.27)

It follows from (2.23) that for each $x_0\in X$, there exists $x_1\in T(x_0)$ satisfying

$$\alpha (w(x_0,x_1))\phi (w(x_0,x_1))\le f_w(x_0)\text{and}{f}_w(x_1)\le \beta (w(x_0,x_1))\psi (w(x_0,x_1)),$$

which together with (2.3) and (2.26) means that

$$\begin{array}{rl}{f}_w(x_1)& \le \beta (w(x_0,x_1))\psi (w(x_0,x_1))\le \beta (w(x_0,x_1))\phi (w(x_0,x_1))\\ \le \beta (w(x_0,x_1))\frac{f_w(x_0)}{\alpha (w(x_0,x_1))}=\gamma (w(x_0,x_1))f_w(x_0).\end{array}$$

Continuing this process, we choose easily an orbit ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ of T satisfying

$$\begin{array}{r}{x}_{n+1}\in T(x_n),\alpha (w(x_n,x_{n+1}))\phi (w(x_n,x_{n+1}))\le f_w(x_n)\text{and}\\ f_w(x_{n+1})\le \beta (w(x_n,x_{n+1}))\psi (w(x_n,x_{n+1})),\mathrm{\forall }n\in {\mathbb{N}}_0,\end{array}$$

(2.28)

which together with (2.3) and (2.26) gives that

$$\begin{array}{rl}{f}_w(x_{n+1})& \le \beta (w(x_n,x_{n+1}))\psi (w(x_n,x_{n+1}))\le \beta (w(x_n,x_{n+1}))\phi (w(x_n,x_{n+1}))\\ \le \beta (w(x_n,x_{n+1}))\frac{f_w(x_n)}{\alpha (w(x_n,x_{n+1}))}=\gamma (w(x_n,x_{n+1}))f_w(x_n),\mathrm{\forall }n\in {\mathbb{N}}_0\end{array}$$

(2.29)

and

$$\begin{array}{rl}\phi (w(x_{n+1},x_{n+2}))& \le \frac{f_w(x_{n+1})}{\alpha (w(x_{n+1},x_{n+2}))}\\ \le \frac{\beta (w(x_n,x_{n+1}))}{\alpha (w(x_{n+1},x_{n+2}))}\psi (w(x_n,x_{n+1})),\mathrm{\forall }n\in {\mathbb{N}}_0.\end{array}$$

(2.30)

Now we claim that

$$w(x_{n+1},x_{n+2})\le w(x_n,x_{n+1}),\mathrm{\forall }n\in {\mathbb{N}}_0.$$

(2.31)

Suppose that there exists $n_0\in {\mathbb{N}}_0$ satisfying

$$w(x_{n_0+1},x_{n_0+2})>w(x_{n_0},x_{n_0+1}).$$

(2.32)

Let (2.4) hold. It follows from (2.3), (2.25), (2.26), (2.30) and (2.32) that

$$\begin{array}{rl}\phi (w(x_{n_0+1},x_{n_0+2}))& \le \frac{\beta (w(x_{n_0},x_{n_0+1}))}{\alpha (w(x_{n_0+1},x_{n_0+2}))}\psi (w(x_{n_0},x_{n_0+1}))\\ \le max\{\gamma (w(x_{n_0},x_{n_0+1})),\gamma (w(x_{n_0+1},x_{n_0+2}))\}\phi (w(x_{n_0},x_{n_0+1})).\end{array}$$

(2.33)

If $\phi (w(x_{n_0},x_{n_0+1}))=0$, it follows from (2.33) that $\phi (w(x_{n_0+1},x_{n_0+2}))=0$. Thus (2.4) and (2.32) guarantee that

$$0\le w(x_{n_0},x_{n_0+1})<w(x_{n_0+1},x_{n_0+2})=0,$$

which is a contradiction; if $\phi (w(x_{n_0},x_{n_0+1}))>0$, (2.4), (2.26), (2.27) and (2.33) yield that

$$\begin{array}{rl}\phi (w(x_{n_0+1},x_{n_0+2}))& \le max\{\gamma (w(x_{n_0},x_{n_0+1})),\gamma (w(x_{n_0+1},x_{n_0+2}))\}\phi (w(x_{n_0},x_{n_0+1}))\\ <\phi (w(x_{n_0},x_{n_0+1})).\end{array}$$

(2.34)

Since φ is strictly inverse, it follows from (2.32) and (2.34) that

$$w(x_{n_0+1},x_{n_0+2})<w(x_{n_0},x_{n_0+1})<w(x_{n_0+1},x_{n_0+2}),$$

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

$$\begin{array}{rl}\phi (w(x_{n_0+1},x_{n_0+2}))& \le \frac{\beta (w(x_{n_0},x_{n_0+1}))}{\alpha (w(x_{n_0+1},x_{n_0+2}))}\psi (w(x_{n_0},x_{n_0+1}))\\ \le max\{\gamma (w(x_{n_0},x_{n_0+1})),\gamma (w(x_{n_0+1},x_{n_0+2}))\}\phi (w(x_{n_0},x_{n_0+1}))\\ \le \phi (w(x_{n_0},x_{n_0+1}))\\ <\phi (w(x_{n_0+1},x_{n_0+2})),\end{array}$$

which is absurd. Hence (2.31) holds. That is, ${\{w(x_n,x_{n+1})\}}_{n\in {\mathbb{N}}_0}$ is a nonincreasing and nonnegative sequence. It follows that ${lim}_{n\to \mathrm{\infty }}w(x_n,x_{n+1})=d$ for some $d\ge 0$.

Now we claim that (2.11) holds. Using (2.27) and (2.29), we conclude that ${\{f_w(x_n)\}}_{n\in {\mathbb{N}}_0}$ 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

$$\begin{array}{rl}a& =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{f}_w(x_{n+1})\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}[\gamma (w(x_n,x_{n+1}))f_w(x_n)]\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\gamma (w(x_n,x_{n+1}))\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{f}_w(x_n)\le a\underset{t\to d^{+}}{lim\hspace{0.17em}sup}\gamma (t)\\ <a,\end{array}$$

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

Next we claim that ${\{x_n\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence. Put

$$b=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\gamma (w(x_n,x_{n+1}))\text{and}c=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\alpha (w(x_n,x_{n+1})).$$

(2.35)

It follows from (2.24), (2.27), (2.29) and (2.35) that (2.15) holds. Let $p\in (0,c)$ and $q\in (b,1)$. Because of (2.15) and (2.35), we deduce that there exists some $n_0\in \mathbb{N}$ such that

$$\gamma (w(x_n,x_{n+1}))<q\text{and}\alpha (w(x_n,x_{n+1}))>p,\mathrm{\forall }n\ge n_0,$$

which together with (2.28) and (2.29) yields that

$$f_w(x_{n+1})\le q{f}_w(x_n)\text{and}\phi (w(x_n,x_{n+1}))\le \frac{f_w(x_n)}{p},\mathrm{\forall }n\ge n_0.$$

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 $X=[0,1]\cup \{\frac{6}{5}\}$ be endowed with the Euclidean metric $d=|\cdot |$ and $u_0=0$. Define $w:X\times X\to {\mathbb{R}}^{+}$, $T:X\to CL(X)$, $\alpha :[0,\frac{1}{4}]\to (0,1]$, $\beta :[0,\frac{1}{4}]\to [0,1)$ and $\phi ,\psi :[0,\frac{6}{5}]\to {\mathbb{R}}^{+}$ by

$$\begin{array}{r}w(x,y)=y,\mathrm{\forall }x,y\in X,\\ T(x)=\{\begin{array}{cc}\{\frac{x}{4}\},\hfill & \mathrm{\forall }x\in [0,\frac{2}{5})\cup (\frac{2}{5},1],\hfill \\ \{\frac{1}{10},\frac{1}{3}\},\hfill & \mathrm{\forall }x\in \{\frac{2}{5},\frac{6}{5}\},\hfill \end{array}\\ \alpha (t)=\frac{8+t}{9},\beta (t)=\frac{2+t}{3},\mathrm{\forall }t\in [0,\frac{1}{4}]\end{array}$$

and

$$\phi (t)=t,\psi (t)=min\{t,|1-t|\},\mathrm{\forall }t\in [0,\frac{6}{5}].$$

It is easy to see that $A_w=[0,\frac{6}{5}]$, $B_w=[0,\frac{1}{4}]$, (2.3), (2.4) and (2.5) hold and

$$f_w(x)=w(x,T(x))=\{\begin{array}{cc}\frac{x}{4},\hfill & \mathrm{\forall }x\in [0,\frac{2}{5})\cup (\frac{2}{5},1],\hfill \\ \frac{1}{10},\hfill & \mathrm{\forall }x\in \{\frac{2}{5},\frac{6}{5}\},\hfill \end{array}$$

is T-orbitally lower semi-continuous at $u_0$,

$$\begin{array}{c}\beta (0)=\frac{2}{3}<\frac{8}{9}=\alpha (0),\underset{r\to 0^{+}}{lim\hspace{0.17em}inf}\alpha (r)=\frac{8}{9}>0,\hfill \\ \underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}=\underset{r\to t^{+}}{lim\hspace{0.17em}sup}(\frac{2+r}{3}\cdot \frac{9}{8+r})=\frac{6+3t}{8+t}<1,\mathrm{\forall }t\in [0,\frac{1}{4}].\hfill \end{array}$$

For $x\in [0,\frac{2}{5})\cup (\frac{2}{5},1]$, there exists $y=\frac{x}{4}\in T(x)=\{\frac{x}{4}\}$ satisfying

$$\alpha (f_w(x))\phi (w(x,y))=\frac{8+\frac{x}{4}}{9}\cdot \frac{x}{4}\le \frac{x}{4}=f_w(x)$$

and

$$f_w(y)=\frac{x}{16}\le \frac{2+\frac{x}{4}}{3}\cdot min\{\frac{x}{4},1-\frac{x}{4}\}=\beta (f_w(x))\psi (w(x,y)).$$

For $x\in \{\frac{2}{5},\frac{6}{5}\}$, there exists $y=\frac{1}{10}\in T(x)=\{\frac{1}{10},\frac{1}{3}\}$ satisfying

$$\alpha (f_w(x))\phi (w(x,y))=\frac{8+\frac{1}{10}}{9}\cdot \frac{1}{10}\le \frac{1}{10}=f_w(x)$$

and

$$f_w(y)=\frac{1}{40}\le \frac{2+\frac{1}{10}}{3}\cdot min\{\frac{1}{10},1-\frac{1}{10}\}=\beta (f_w(x))\psi (w(x,y)).$$

Put $v\in X\setminus \{0\}$ and ${\{z_n\}}_{n\in {\mathbb{N}}_0}$ is an orbit of T in X. It is easy to verify that ${lim}_{n\to \mathrm{\infty }}{z}_n=u_0=0$ and

$$\begin{array}{c}inf\{w(z_n,v)+\phi (w(z_n,z_{n+1})):n\in {\mathbb{N}}_0\}\hfill \\ =inf\{v+z_{n+1}:n\in {\mathbb{N}}_0\}\hfill \\ =v+u_0=v>0.\hfill \end{array}$$

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 $\phi :{\mathbb{R}}^{+}\to [a,1)$, $0<a<1$, such that

$$\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\phi (r)<1,\mathrm{\forall }t\in {\mathbb{R}}^{+},$$

(3.1)

and for any $x\in X$ there is $y\in T(x)$ satisfying

$$\sqrt{\phi (f(x))}d(x,y)\le f(x)$$

(3.2)

and

$$f(y)\le \phi (f(x))d(x,y).$$

(3.3)

Note that

$$f(x)=d(x,T(x))=\{\begin{array}{cc}\frac{3}{4}x,\hfill & \mathrm{\forall }x\in [0,\frac{2}{5})\cup (\frac{2}{5},1],\hfill \\ \frac{1}{15},\hfill & x=\frac{2}{5},\hfill \\ \frac{13}{15},\hfill & x=\frac{6}{5}.\hfill \end{array}$$

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

Case 1. $y=\frac{1}{10}$. It follows from (3.2) and (3.3) that

$$\frac{3}{10}\sqrt{\phi \left(\frac{1}{15}\right)}=\sqrt{\phi \left(f\left(\frac{2}{5}\right)\right)}d(\frac{2}{5},\frac{1}{10})=\sqrt{\phi (f(x))}d(x,y)\le f(x)=f\left(\frac{2}{5}\right)=\frac{1}{15}$$

and

$$\frac{3}{40}=f\left(\frac{1}{10}\right)=f(y)\le \phi (f(x))d(x,y)=\phi \left(f\left(\frac{2}{5}\right)\right)d(\frac{2}{5},\frac{1}{10})=\frac{3}{10}\phi \left(\frac{1}{15}\right),$$

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

$$\frac{1}{4}=f\left(\frac{1}{3}\right)=f(y)\le \phi (f(x))d(x,y)=\phi \left(f\left(\frac{2}{5}\right)\right)d(\frac{2}{5},\frac{1}{3})=\frac{1}{15}\phi \left(\frac{1}{15}\right),$$

which together with $\phi ({\mathbb{R}}^{+})\subseteq [a,1)$ 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 $\phi :{\mathbb{R}}^{+}\to [0,1)$ such that (3.1) holds, and for any $x\in X$ there is $y\in T(x)$ satisfying

$$d(x,y)\le (2-\phi (d(x,y)))f(x)$$

(3.4)

and

$$f(y)\le \phi (d(x,y))d(x,y).$$

(3.5)

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

Case 1. $y=\frac{1}{10}$. It follows from (3.4) that

$$\frac{3}{10}=d(\frac{2}{5},\frac{1}{10})=d(x,y)\le (2-\phi (d(x,y)))f(x)=(2-\phi \left(\frac{3}{10}\right))\frac{1}{15},$$

which together with $\phi ({\mathbb{R}}^{+})\subseteq [0,1)$ yields that

$$0\le \phi \left(\frac{3}{10}\right)\le -\frac{5}{2}<0,$$

which is a contradiction.

Case 2. $y=\frac{1}{3}$. It follows from (3.4) that

$$\frac{1}{4}=f\left(\frac{1}{3}\right)=f(y)\le \phi (d(x,y))d(x,y)=\phi \left(d(\frac{2}{5},\frac{1}{3})\right)d(\frac{2}{5},\frac{1}{3})=\frac{1}{15}\phi \left(\frac{1}{15}\right),$$

which together with $\phi ({\mathbb{R}}^{+})\subseteq [0,1)$ 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 $\phi :{\mathbb{R}}^{+}\to (0,1)$, $b:{\mathbb{R}}^{+}\to [b,1)$, $b>0$ such that

$$\phi (t)<b(t),\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\phi (r)<\underset{r\to t^{+}}{lim\hspace{0.17em}sup}b(r),\mathrm{\forall }t\in {\mathbb{R}}^{+},$$

(3.6)

and for any $x\in X$, there is $y\in T(x)$ satisfying (3.5) and

$$b(d(x,y))d(x,y)\le f(x).$$

(3.7)

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

Case 1. $y=\frac{1}{10}$. It follows from (3.7) and (3.5) that

$$\frac{3}{10}b\left(\frac{3}{10}\right)=b\left(d(\frac{2}{5},\frac{1}{10})\right)d(\frac{2}{5},\frac{1}{10})=b(d(x,y))d(x,y)\le f(x)=f\left(\frac{2}{5}\right)=\frac{1}{15}$$

and

$$\frac{3}{40}=f\left(\frac{1}{10}\right)=f(y)\le \phi (d(x,y))d(x,y)=\frac{3}{10}\phi \left(\frac{3}{10}\right),$$

which together with (3.6) means that

$$b\left(\frac{3}{10}\right)\le \frac{2}{9}<\frac{1}{4}\le \phi \left(\frac{3}{10}\right)<b\left(\frac{3}{10}\right),$$

which is absurd.

Case 2. $y=\frac{1}{3}$. It follows from (3.5) that

$$\frac{1}{4}=f\left(\frac{1}{3}\right)=f(y)\le \phi (d(x,y))d(x,y)=\phi \left(d(\frac{2}{5},\frac{1}{3})\right)d(\frac{2}{5},\frac{1}{3})=\frac{1}{15}\phi \left(\frac{1}{15}\right),$$

which together with $\phi ({\mathbb{R}}^{+})\subseteq [0,1)$ 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 $X=[0,\mathrm{\infty })$ be endowed with the Euclidean metric $d=|\cdot |$ and $p\ge 1$ be a constant. Put $u_0=0$. Define $w:X\times X\to {\mathbb{R}}^{+}$, $T:X\to CL(X)$, $\alpha :[0,\mathrm{\infty })\to (0,1]$ and $\phi ,\psi :[0,\mathrm{\infty })\to {\mathbb{R}}^{+}$ by $\beta :[0,\mathrm{\infty })\to [0,1)$ by

$$\begin{array}{c}w(x,y)=y^p,\mathrm{\forall }x,y\in X,\hfill \\ T(x)=\{\begin{array}{cc}[\frac{x^2}{2},\frac{x}{2}],\hfill & \mathrm{\forall }x\in [0,1),\hfill \\ [\frac{1}{9},\frac{1}{4}],\hfill & \mathrm{\forall }x\in [1,\mathrm{\infty }),\hfill \end{array}\hfill \\ \alpha (t)=\frac{5+t^{\frac{1}{p}}}{10},\beta (t)=\frac{3+t^{\frac{1}{p}}}{10},\mathrm{\forall }t\in [0,\mathrm{\infty })\hfill \end{array}$$

and

$$\phi (t)=t,\mathrm{\forall }t\in [0,\mathrm{\infty }),\psi (t)=\{\begin{array}{cc}t,\hfill & \mathrm{\forall }t\in [0,1),\hfill \\ \frac{1}{2},\hfill & \mathrm{\forall }t\in [1,\mathrm{\infty }).\hfill \end{array}$$

It is easy to see that $A_w=[0,\mathrm{\infty })$, (2.3), (2.4) and (2.5) hold, w is a w-distance in X and

$$f_w(x)=w(x,T(x))=\{\begin{array}{cc}{(\frac{x^2}{2})}^p,\hfill & \mathrm{\forall }x\in [0,1),\hfill \\ \frac{1}{9^p},\hfill & \mathrm{\forall }x\in [1,\mathrm{\infty })\hfill \end{array}$$

is T-orbitally lower semi-continuous in X, α and β are nondecreasing,

$$\beta (0)=\frac{3}{10}<\frac{1}{2}=\alpha (0),\underset{r\to 0^{+}}{lim\hspace{0.17em}inf}\alpha (r)=\frac{1}{2}>0$$

and

$$\underset{r\to t^{+}}{lim\hspace{0.17em}sup}\frac{\beta (r)}{\alpha (r)}=\frac{3+t^{\frac{1}{p}}}{5+t^{\frac{1}{p}}}<1,\mathrm{\forall }t\in A_w.$$

Put $x\in [0,1)$ and $y=\frac{x^2}{2}\in T(x)$. Note that

$$5+y\le 10\text{and}{\left(\frac{y}{2}\right)}^p\le \frac{1}{4^p}\le \frac{3+y}{10}$$

imply that

$$\alpha (w(x,y))\phi (w(x,y))=\frac{5+y}{10}\cdot y^p\le y^p=f_w(x)$$

and

$$f_w(y)={\left(\frac{y^2}{2}\right)}^p\le \frac{3+y}{10}\cdot y^p=\beta (w(x,y))\psi (w(x,y)).$$

Put $x\in [1,\mathrm{\infty })$ and $y=\frac{1}{9}\in T(x)=[\frac{1}{9},\frac{1}{4}]$. It follows that

$$\alpha (w(x,y))\phi (w(x,y))=\frac{5+\frac{1}{9}}{10}\cdot \frac{1}{9^p}\le \frac{1}{9^p}=f_w(x)$$

and

$$f_w(y)=\frac{1}{{182}^p}\le \frac{3+\frac{1}{9}}{10}\cdot \frac{1}{9^p}=\beta (w(x,y))\psi (w(x,y)).$$

Let $v\in X\setminus \{0\}$ and ${\{z_n\}}_{n\in {\mathbb{N}}_0}$ be an orbit of T. It is easy to verify that ${lim}_{n\to \mathrm{\infty }}{z}_n=0$ and

$$\begin{array}{c}inf\{w(z_n,v)+\phi (w(z_n,z_{n+1})):n\in {\mathbb{N}}_0\}\hfill \\ =inf\{v^p+z_{n+1}^p:n\in {\mathbb{N}}_0\}=v^p>0.\hfill \end{array}$$

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

$$f(x)=d(x,T(x))=\{\begin{array}{cc}\frac{x}{2},\hfill & \mathrm{\forall }x\in [0,1),\hfill \\ x-\frac{1}{4},\hfill & \mathrm{\forall }x\in [1,\mathrm{\infty })\hfill \end{array}$$

and

$$\underset{x\to 1}{lim\hspace{0.17em}inf}f(x)=\frac{1}{2}<\frac{3}{4}=f(1),$$

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.