# A note on approximate fixed point property and Du-Karapinar-Shahzad’s intersection theorems

## Abstract

In this note, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems for multivalued non-self-maps in complete metric spaces.

MSC:47H10, 54H25.

## 1 Introduction and preliminaries

Let us begin with some basic definitions and notations that will be needed in this paper. Let $\left(X,d\right)$ be a metric space. Denote by $\mathcal{N}\left(X\right)$ the family of all nonempty subsets of X and by $\mathcal{CB}\left(X\right)$ the family of all nonempty closed and bounded subsets of X. For each $x\in X$ and $A\subseteq X$, let $d\left(x,A\right)={inf}_{y\in A}d\left(x,y\right)$. A function $\mathcal{H}:\mathcal{CB}\left(X\right)×\mathcal{CB}\left(X\right)\to \left[0,\mathrm{\infty }\right)$ defined by

$\mathcal{H}\left(A,B\right)=max\left\{\underset{x\in B}{sup}d\left(x,A\right),\underset{x\in A}{sup}d\left(x,B\right)\right\}$

is said to be the Hausdorff metric on $\mathcal{CB}\left(X\right)$ induced by the metric d on X. The symbols and are used to denote the sets of positive integers and real numbers, respectively.

Let K be a nonempty subset of X, $g:K\to X$ be a single-valued non-self-map and $T:K\to \mathcal{N}\left(X\right)$ be a multivalued non-self-map. A point v in X is a coincidence point (see, for instance, ) of g and T if $gv\in Tx$. If $g=id$ is the identity map, then $v=gv\in Tv$ and call v a fixed point of T. The set of fixed points of T and the set of coincidence points of g and T are denoted by ${\mathcal{F}}_{K}\left(T\right)$ and ${\mathcal{COP}}_{K}\left(g,T\right)$, respectively. In particular, if $K\equiv X$, we use $\mathcal{F}\left(T\right)$ and $\mathcal{COP}\left(g,T\right)$ instead of ${\mathcal{F}}_{K}\left(T\right)$ and ${\mathcal{COP}}_{K}\left(g,T\right)$, respectively. The map T is said to have approximate fixed point property  on K provided ${inf}_{x\in K}d\left(x,Tx\right)=0$. It is obvious that ${\mathcal{F}}_{K}\left(T\right)\ne \mathrm{\varnothing }$ implies that T has approximate fixed point property.

A function $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ is said to be an $\mathcal{MT}$-function (or -function)  if ${lim sup}_{s\to {t}^{+}}\phi \left(s\right)<1$ for all $t\in \left[0,\mathrm{\infty }\right)$. Clearly, if $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ is a nondecreasing function or a nonincreasing function, then φ is an $\mathcal{MT}$-function. So, the set of $\mathcal{MT}$-functions is a rich class and has the questions many of which are worth studying.

The study of fixed points for single-valued non-self-maps or multivalued non-self-maps satisfying certain contractive conditions is an interesting and important direction of research in metric fixed point theory. A great deal of such research has been investigated by several authors, see, e.g.,  and the references therein. Very recently, Du, Karapinar and Shahzad  established the following intersection existence theorem of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type.

Theorem 1.1 [, Theorem 8]

Let $\left(X,d\right)$ be a complete metric space, K be a nonempty closed subset of X, $T:K\to \mathcal{CB}\left(X\right)$ be a multivalued map and $g:K\to X$ be a continuous map. Suppose that

(D1) $Tx\cap K\ne \mathrm{\varnothing }$ for all $x\in K$,

(D2) $Tx\cap K$ is g-invariant (i.e., $g\left(Tx\cap K\right)\subseteq Tx\cap K$) for each $x\in K$,

(D3) there exist a function $h:K\to \left[0,\mathrm{\infty }\right)$ and $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$\begin{array}{rl}\mathcal{H}\left(Tx,Ty\cap K\right)\le & \gamma \left[d\left(x,Tx\cap K\right)+d\left(y,Tx\cap K\right)+d\left(y,Ty\cap K\right)\right]\\ +h\left(y\right)d\left(gy,Tx\cap K\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}x,y\in K.\end{array}$
(1.1)

Then ${\mathcal{COP}}_{K}\left(g,T\right)\cap {\mathcal{F}}_{K}\left(T\right)\ne \mathrm{\varnothing }$.

In , they also gave some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type and Du type.

Theorem 1.2 [, Theorem 19]

Let $\left(X,d\right)$ be a complete metric space, K be a nonempty closed subset of X, $T:K\to \mathcal{CB}\left(X\right)$ be a multivalued map and $g:K\to X$ be a continuous map. Suppose that conditions (D1) and (D2) as in Theorem  1.1 hold. If there exist an $\mathcal{MT}$-function $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ and a function $h:K\to \left[0,\mathrm{\infty }\right)$ such that

$\mathcal{H}\left(Tx,Ty\cap K\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)+h\left(y\right)d\left(gy,Tx\cap K\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}x,y\in K,$
(1.2)

then ${\mathcal{COP}}_{K}\left(g,T\right)\cap {\mathcal{F}}_{K}\left(T\right)\ne \mathrm{\varnothing }$.

In this work, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems of ${\mathcal{COP}}_{K}\left(g,T\right)$ and ${\mathcal{F}}_{K}\left(T\right)$ for multivalued non-self-maps (i.e., Theorems 1.1 and 1.2) by applying an existence theorem for approximate fixed point property.

## 2 Some auxiliary key results

Let $\left(X,d\right)$ be a metric space. Recall that a function $p:X×X\to \left[0,\mathrm{\infty }\right)$ is said to be a τ-function [35, 7, 8, 2022], first introduced and studied by Lin and Du, if the following conditions hold:

(τ 1) $p\left(x,z\right)\le p\left(x,y\right)+p\left(y,z\right)$ for all $x,y,z\in X$;

(τ 2) if $x\in X$ and $\left\{{y}_{n}\right\}$ in X with ${lim}_{n\to \mathrm{\infty }}{y}_{n}=y$ such that $p\left(x,{y}_{n}\right)\le M$ for some $M=M\left(x\right)>0$, then $p\left(x,y\right)\le M$;

(τ 3) for any sequence $\left\{{x}_{n}\right\}$ in X with ${lim}_{n\to \mathrm{\infty }}sup\left\{p\left({x}_{n},{x}_{m}\right):m>n\right\}=0$, if there exists a sequence $\left\{{y}_{n}\right\}$ in X such that ${lim}_{n\to \mathrm{\infty }}p\left({x}_{n},{y}_{n}\right)=0$, then ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{y}_{n}\right)=0$;

(τ 4) for $x,y,z\in X$, $p\left(x,y\right)=0$ and $p\left(x,z\right)=0$ imply $y=z$.

Note that with the additional condition

(τ 5) $p\left(x,x\right)=0$ for all $x\in X$,

a τ-function becomes a ${\tau }^{0}$-function [35, 7, 8] introduced by Du.

Clearly, any metric d is a ${\tau }^{0}$-function. Observe further that if p is a ${\tau }^{0}$-function, then, from (τ 4) and (τ 5), $p\left(x,y\right)=0$ if and only if $x=y$.

Example A 

Let $X=\mathbb{R}$ with the metric $d\left(x,y\right)=|x-y|$ and $0. Define the function $p:X×X\to \left[0,\mathrm{\infty }\right)$ by

$p\left(x,y\right)=max\left\{a\left(y-x\right),b\left(x-y\right)\right\}.$

Then p is nonsymmetric and hence p is not a metric. It is easy to see that p is a ${\tau }^{0}$-function.

Lemma 2.1 [, Lemma 2.1]

Let $\left(X,d\right)$ be a metric space and $p:X×X\to \left[0,\mathrm{\infty }\right)$ be a function. Assume that p satisfies the condition (τ 3). If a sequence $\left\{{x}_{n}\right\}$ in X with ${lim}_{n\to \mathrm{\infty }}sup\left\{p\left({x}_{n},{x}_{m}\right):m>n\right\}=0$, then $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X.

Let $\left(X,d\right)$ be a metric space and p be a τ-function. A multivalued map $T:X\to \mathcal{N}\left(X\right)$ is said to have p-approximate fixed point property on X provided

$\underset{x\in X}{inf}p\left(x,Tx\right)=0.$

The following characterizations of $\mathcal{MT}$-functions proved first by Du  are quite useful for proving our main results.

Theorem 2.1 [, Theorem 2.1]

Let $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ be a function. Then the following statements are equivalent.

1. (a)

φ is an $\mathcal{MT}$-function.

2. (b)

For each $t\in \left[0,\mathrm{\infty }\right)$, there exist ${r}_{t}^{\left(1\right)}\in \left[0,1\right)$ and ${\epsilon }_{t}^{\left(1\right)}>0$ such that $\phi \left(s\right)\le {r}_{t}^{\left(1\right)}$ for all $s\in \left(t,t+{\epsilon }_{t}^{\left(1\right)}\right)$.

3. (c)

For each $t\in \left[0,\mathrm{\infty }\right)$, there exist ${r}_{t}^{\left(2\right)}\in \left[0,1\right)$ and ${\epsilon }_{t}^{\left(2\right)}>0$ such that $\phi \left(s\right)\le {r}_{t}^{\left(2\right)}$ for all $s\in \left[t,t+{\epsilon }_{t}^{\left(2\right)}\right]$.

4. (d)

For each $t\in \left[0,\mathrm{\infty }\right)$, there exist ${r}_{t}^{\left(3\right)}\in \left[0,1\right)$ and ${\epsilon }_{t}^{\left(3\right)}>0$ such that $\phi \left(s\right)\le {r}_{t}^{\left(3\right)}$ for all $s\in \left(t,t+{\epsilon }_{t}^{\left(3\right)}\right]$.

5. (e)

For each $t\in \left[0,\mathrm{\infty }\right)$, there exist ${r}_{t}^{\left(4\right)}\in \left[0,1\right)$ and ${\epsilon }_{t}^{\left(4\right)}>0$ such that $\phi \left(s\right)\le {r}_{t}^{\left(4\right)}$ for all $s\in \left[t,t+{\epsilon }_{t}^{\left(4\right)}\right)$.

6. (f)

For any nonincreasing sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in $\left[0,\mathrm{\infty }\right)$, we have $0\le {sup}_{n\in \mathbb{N}}\phi \left({x}_{n}\right)<1$.

7. (g)

φ is a function of contractive factor; that is, for any strictly decreasing sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in $\left[0,\mathrm{\infty }\right)$, we have $0\le {sup}_{n\in \mathbb{N}}\phi \left({x}_{n}\right)<1$.

The following result was essentially proved by Du et al. in , but we give the proof for the sake of completeness and the readers convenience.

Lemma 2.2 [, Lemma 3.1]

Let $\left(X,d\right)$ be a metric space, p be a ${\tau }^{0}$-function and $T:X\to \mathcal{N}\left(X\right)$ be a multivalued map. Then the following statements are equivalent.

(Q1) There exist a function $\xi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ and an $\mathcal{MT}$-function $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$, if $y\in Tx$ with $y\ne x$, then there exists $z\in Ty$ such that

$p\left(y,z\right)\le \phi \left(\xi \left(p\left(x,y\right)\right)\right)p\left(x,y\right).$

(Q2) There exist a function $\tau :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ and an $\mathcal{MT}$-function $\kappa :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$,

$p\left(y,Ty\right)\le \kappa \left(\tau \left(p\left(x,y\right)\right)\right)p\left(x,y\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}y\in Tx.$

Proof If (Q1) holds, then it is easy to verify that (Q2) also holds with $\kappa \equiv \phi$ and $\tau \equiv \xi$. So it suffices to prove that ‘(Q2) (Q1)’. Suppose that (Q2) holds. Define $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ by $\phi \left(t\right)=\frac{1+\kappa \left(t\right)}{2}$. Then φ is also an $\mathcal{MT}$-function. Indeed, it is obvious that $0\le \kappa \left(t\right)<\phi \left(t\right)<1$ for all $t\in \left[0,\mathrm{\infty }\right)$. Let ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ be a strictly decreasing sequence in $\left[0,\mathrm{\infty }\right)$. Since κ is an $\mathcal{MT}$-function, by (g) of Theorem 2.1, we get

$0\le \underset{n\in \mathbb{N}}{sup}\kappa \left({x}_{n}\right)<1$

and hence

$0<\underset{n\in \mathbb{N}}{sup}\phi \left({x}_{n}\right)=\frac{1}{2}\left[1+\underset{n\in \mathbb{N}}{sup}\kappa \left({x}_{n}\right)\right]<1.$

So, by Theorem 2.1 again, we prove that φ is an $\mathcal{MT}$-function.

For each $x\in X$, let $y\in Tx$ with $y\ne x$. Then $p\left(x,y\right)>0$. By (Q2), we have

$p\left(y,Ty\right)<\phi \left(\tau \left(p\left(x,y\right)\right)\right)p\left(x,y\right).$

Since $\phi \left(t\right)>0$ for all $t\in \left[0,\mathrm{\infty }\right)$, there exists $z\in Ty$ such that

$p\left(y,z\right)<\phi \left(\tau \left(p\left(x,y\right)\right)\right)p\left(x,y\right),$

which shows that (Q1) holds with $\xi \equiv \tau$. So, by above, we prove ‘(Q1) (Q2)’. □

Now, we present an existence theorem for p-approximate fixed point property and approximate fixed point property, which is indeed a somewhat generalized form of [, Theorem 3.3] and is one of the key technical devices in the new short proofs of Theorems 1.1 and 1.2.

Theorem 2.2 Let $\left(X,d\right)$ be a metric space, p be a ${\tau }^{0}$-function and $T:X\to \mathcal{N}\left(X\right)$ be a multivalued map. Assume that one of (L1) and (L2) is satisfied, where

(L1) there exist a nondecreasing function $\xi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ and an $\mathcal{MT}$-function $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$, if $y\in Tx$ with $y\ne x$, then there exists $z\in Ty$ such that

$p\left(y,z\right)\le \phi \left(\xi \left(p\left(x,y\right)\right)\right)p\left(x,y\right);$

(L2) there exist a nondecreasing function $\tau :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ and an $\mathcal{MT}$-function $\kappa :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$,

$p\left(y,Ty\right)\le \kappa \left(\tau \left(p\left(x,y\right)\right)\right)p\left(x,y\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}y\in Tx.$

Then the following statements hold.

1. (a)

There exists a Cauchy sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in X such that

1. (i)

${x}_{n+1}\in T{x}_{n}$ for all $n\in \mathbb{N}$,

2. (ii)

${inf}_{n\in X}p\left({x}_{n},{x}_{n+1}\right)={lim}_{n\to \mathrm{\infty }}p\left({x}_{n},{x}_{n+1}\right)={lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{x}_{n+1}\right)={inf}_{n\in \mathbb{N}}d\left({x}_{n},{x}_{n+1}\right)=0$.

2. (b)

${inf}_{x\in X}p\left(x,Tx\right)={inf}_{x\in X}d\left(x,Tx\right)=0$; that is, T has p-approximate fixed point property and approximate fixed point property on X.

Proof By Lemma 2.2, it suffices to prove that the conclusions hold under assumption (L1). Let $u\in X$ be given. If $u\in Tu$, then

$\underset{x\in X}{inf}p\left(x,Tx\right)\le p\left(u,Tu\right)\le p\left(u,u\right)=0,$

and

$\underset{x\in X}{inf}d\left(x,Tx\right)\le d\left(u,u\right)=0,$

which implies that ${inf}_{x\in X}p\left(x,Tx\right)={inf}_{x\in X}d\left(x,Tx\right)=0$. Let ${w}_{n}=u$ for all $n\in \mathbb{N}$. Thus we have

and

$\underset{n\to \mathrm{\infty }}{lim}d\left({w}_{n},{w}_{n+1}\right)=\underset{n\in \mathbb{N}}{inf}d\left({w}_{n},{w}_{n+1}\right)=d\left(u,u\right)=0.$

Clearly,

So, conclusions (a) and (b) hold in this case $u\in Tu$, no matter what condition one begins with. Suppose that $u\notin Tu$. Put ${x}_{1}=u$ and ${x}_{2}\in T{x}_{1}$. Then ${x}_{2}\ne {x}_{1}$ and hence $p\left({x}_{1},{x}_{2}\right)>0$. Assume that condition (L1) is satisfied. Then there exists ${x}_{3}\in T{x}_{2}$ such that

$p\left({x}_{2},{x}_{3}\right)\le \phi \left(\xi \left(p\left({x}_{1},{x}_{2}\right)\right)\right)p\left({x}_{1},{x}_{2}\right).$

If ${x}_{2}={x}_{3}\in T{x}_{2}$, then, following a similar argument as above, the conclusions are also proved. If ${x}_{3}\ne {x}_{2}$, then there exists ${x}_{4}\in T{x}_{3}$ such that

$p\left({x}_{3},{x}_{4}\right)\le \phi \left(\xi \left(p\left({x}_{2},{x}_{3}\right)\right)\right)p\left({x}_{2},{x}_{3}\right).$

By induction, we can obtain a sequence $\left\{{x}_{n}\right\}$ in X satisfying ${x}_{n+1}\in T{x}_{n}$ and

(2.1)

Since $\phi \left(t\right)<1$ for all $t\in \left[0,\mathrm{\infty }\right)$, inequality (2.1) implies that the sequence ${\left\{p\left({x}_{n},{x}_{n+1}\right)\right\}}_{n\in \mathbb{N}}$ is strictly decreasing in $\left[0,\mathrm{\infty }\right)$. Hence

(2.2)

Since ξ is nondecreasing, ${\left\{\xi \left(p\left({x}_{n},{x}_{n+1}\right)\right)\right\}}_{n\in \mathbb{N}}$ is a nonincreasing sequence in $\left[0,\mathrm{\infty }\right)$. Since φ is an $\mathcal{MT}$-function, by (g) of Theorem 2.1, we have

$0\le \underset{n\in \mathbb{N}}{sup}\phi \left(\xi \left(p\left({x}_{n},{x}_{n+1}\right)\right)\right)<1.$

Let $\lambda :={sup}_{n\in \mathbb{N}}\phi \left(\xi \left(p\left({x}_{n},{x}_{n+1}\right)\right)\right)$. So $\lambda \in \left[0,1\right)$ and we get from (2.1) that

(2.3)

Since $\lambda \in \left[0,1\right)$, ${lim}_{n\to \mathrm{\infty }}{\lambda }^{n}=0$ and hence the last inequality implies

$\underset{n\to \mathrm{\infty }}{lim}p\left({x}_{n},{x}_{n+1}\right)=0.$
(2.4)

By (2.2) and (2.4), we obtain

$\underset{n\in \mathbb{N}}{inf}p\left({x}_{n},{x}_{n+1}\right)=\underset{n\to \mathrm{\infty }}{lim}p\left({x}_{n},{x}_{n+1}\right)=0.$
(2.5)

Now, we claim that $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X. Let ${\alpha }_{n}=\frac{{\lambda }^{n-1}}{1-\lambda }p\left({x}_{1},{x}_{2}\right)$, $n\in \mathbb{N}$. For $m,n\in \mathbb{N}$ with $m>n$, by (2.3), we have

$p\left({x}_{n},{x}_{m}\right)\le \sum _{j=n}^{m-1}p\left({x}_{j},{x}_{j+1}\right)<{\alpha }_{n}.$

Since $\lambda \in \left[0,1\right)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and hence

$\underset{n\to \mathrm{\infty }}{lim}sup\left\{p\left({x}_{n},{x}_{m}\right):m>n\right\}=0.$

Applying Lemma 2.1, we show that $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X. Hence ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{x}_{n+1}\right)=0$. Since ${inf}_{n\in \mathbb{N}}d\left({x}_{n},{x}_{n+1}\right)\le d\left({x}_{m},{x}_{m+1}\right)$ for all $m\in \mathbb{N}$ and ${lim}_{m\to \mathrm{\infty }}d\left({x}_{m},{x}_{m+1}\right)=0$, one also obtains

$\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{x}_{n+1}\right)=\underset{n\in \mathbb{N}}{inf}d\left({x}_{n},{x}_{n+1}\right)=0.$
(2.6)

So conclusion (a) is proved. To see (b), since ${x}_{n+1}\in T{x}_{n}$ for each $n\in \mathbb{N}$, we have

$\underset{x\in X}{inf}p\left(x,Tx\right)\le p\left({x}_{n},T{x}_{n}\right)\le p\left({x}_{n},f{x}_{n+1}\right)$
(2.7)

and

$\underset{x\in X}{inf}d\left(x,Tx\right)\le d\left({x}_{n},T{x}_{n}\right)\le d\left({x}_{n},f{x}_{n+1}\right)$
(2.8)

for all $n\in \mathbb{N}$. Combining (2.6), (2.7) and (2.8), we get

$\underset{x\in X}{inf}p\left(x,Tx\right)=\underset{x\in X}{inf}d\left(x,Tx\right)=0.$

The proof is completed. □

The following existence theorem is obviously an immediate result from Theorem 2.2.

Theorem 2.3 Let $\left(X,d\right)$ be a metric space, p be a ${\tau }^{0}$-function and $T:X\to \mathcal{N}\left(X\right)$ be a multivalued map. Assume that one of (H1) and (H2) is satisfied, where

(H1) there exists an $\mathcal{MT}$-function $\alpha :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$, if $y\in Tx$ with $y\ne x$, then there exists $z\in Ty$ such that

$p\left(y,z\right)\le \alpha \left(p\left(x,y\right)\right)p\left(x,y\right);$

(H2) there exists an $\mathcal{MT}$-function $\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ such that for each $x\in X$,

$p\left(y,Ty\right)\le \beta \left(p\left(x,y\right)\right)p\left(x,y\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}y\in Tx.$

Then the following statements hold.

1. (a)

There exists a Cauchy sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in X such that

1. (i)

${x}_{n+1}\in T{x}_{n}$ for all $n\in \mathbb{N}$,

2. (ii)

${inf}_{n\in X}p\left({x}_{n},{x}_{n+1}\right)={lim}_{n\to \mathrm{\infty }}p\left({x}_{n},{x}_{n+1}\right)={lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{x}_{n+1}\right)={inf}_{n\in \mathbb{N}}d\left({x}_{n},{x}_{n+1}\right)=0$.

2. (b)

${inf}_{x\in X}p\left(x,Tx\right)={inf}_{x\in X}d\left(x,Tx\right)=0$; that is, T has p-approximate fixed point property and approximate fixed point property on X.

Lemma 2.3 Let $\tau :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be a nondecreasing function and $\kappa :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ be an $\mathcal{MT}$-function. Then $\kappa \circ \tau$ is an $\mathcal{MT}$-function.

Proof Let ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ be a strictly decreasing sequence in $\left[0,\mathrm{\infty }\right)$. Since τ is a nondecreasing function, ${\left\{\tau \left({x}_{n}\right)\right\}}_{n\in \mathbb{N}}$ is a nonincreasing sequence in $\left[0,\mathrm{\infty }\right)$. Since κ is an $\mathcal{MT}$-function, by (f) of Theorem 2.1, we get

$0\le \underset{n\in \mathbb{N}}{sup}\kappa \left(\tau \left({x}_{n}\right)\right)<1,$

or, equivalently,

$0\le \underset{n\in \mathbb{N}}{sup}\left(\kappa \circ \tau \right)\left({x}_{n}\right)<1.$

So, by Theorem 2.1 again, we prove that $\kappa \circ \tau$ is an $\mathcal{MT}$-function. □

Applying Lemma 2.3, we conclude that Theorem 2.2 is also a special case of Theorem 2.3. Therefore we obtain the following important fact.

Theorem 2.4 Theorem  2.2 and Theorem  2.3 are equivalent.

## 3 Short proofs of Theorems 1.1 and 1.2

Let us see how we can utilize Theorem 2.3 to prove Theorem 1.1.

Short proof of Theorem 1.1 Since K is a nonempty closed subset of X and X is complete, $\left(K,d\right)$ is also a complete metric space. Let $x\in K$. Put $k=\frac{\gamma }{1-\gamma }$ and $\lambda =\frac{1+k}{2}$. So, $0\le k<\lambda <1$. Let $y\in Tx\cap K$ be arbitrary. So, $d\left(y,Tx\cap K\right)=0$. By (D2), we have $d\left(gy,Tx\cap K\right)=0$. Hence inequality (1.1) implies

(3.1)

Inequality (3.1) shows that

(3.2)

Define $G:K\to \mathcal{CB}\left(K\right)$ by

and let $\mu :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ be defined by

Then μ is an $\mathcal{MT}$-function. By (3.2), we obtain

Applying Theorem 2.3 with $p\equiv d$, there exists a Cauchy sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in K such that

(3.3)

and

$\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{x}_{n+1}\right)=\underset{n\in \mathbb{N}}{inf}d\left({x}_{n},{x}_{n+1}\right)=0.$
(3.4)

By the completeness of K, there exists $v\in K$ such that ${x}_{n}\to v$ as $n\to \mathrm{\infty }$. By (3.3) and (D2), we have

(3.5)

Since g is continuous and ${lim}_{n\to \mathrm{\infty }}{x}_{n}=v$, we have

$\underset{n\to \mathrm{\infty }}{lim}g{x}_{n}=gv.$
(3.6)

Since the function $x↦d\left(x,Tv\right)$ is continuous, by (1.1), (3.3), (3.4), (3.5) and (3.6), we get

$\begin{array}{rcl}d\left(v,Tv\cap K\right)& =& \underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n+1},Tv\cap K\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim}\mathcal{H}\left(T{x}_{n},Tv\cap K\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim}\left\{\gamma \left[d\left({x}_{n},T{x}_{n}\cap K\right)+d\left(v,T{x}_{n}\cap K\right)+d\left(v,Tv\cap K\right)\right]\\ +h\left(v\right)d\left(gv,T{x}_{n}\cap K\right)\right\}\\ \le & \underset{n\to \mathrm{\infty }}{lim}\left\{\gamma \left[d\left({x}_{n},{x}_{n+1}\right)+d\left(v,{x}_{n+1}\right)+d\left(v,Tv\cap K\right)\right]+h\left(v\right)d\left(gv,g{x}_{n+1}\right)\right\}\\ =& \gamma d\left(v,Tv\cap K\right),\end{array}$

which implies $d\left(v,Tv\cap K\right)=0$. By the closedness of Tv, we have $v\in Tv\cap K$. From (D2), $gv\in Tv\cap K\subseteq Tv$. Hence we verify $v\in {\mathcal{COP}}_{K}\left(g,T\right)\cap {\mathcal{F}}_{K}\left(T\right)$. The proof is complete. □

In order to finish off our work, let us prove Theorem 1.2 by applying Theorem 2.3.

Short proof of Theorem 1.2 Since K is a nonempty closed subset of X and X is complete, $\left(K,d\right)$ is also a complete metric space. Note first that for each $x\in K$, by (D2), we have $d\left(gy,Tx\cap K\right)=0$ for all $y\in Tx\cap K$. So, for each $x\in K$, by (1.2), we obtain

(3.7)

Define $G:K\to \mathcal{CB}\left(K\right)$ by

From (3.7), we obtain

By using Theorem 2.3, there exists a Cauchy sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in K such that

(3.8)

and

$\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{x}_{n+1}\right)=\underset{n\in \mathbb{N}}{inf}d\left({x}_{n},{x}_{n+1}\right)=0.$
(3.9)

By the completeness of K, there exists $v\in K$ such that ${x}_{n}\to v$ as $n\to \mathrm{\infty }$. Thanks to (3.8) and (D2), we have

(3.10)

Since g is continuous and ${lim}_{n\to \mathrm{\infty }}{x}_{n}=v$, we have

$\underset{n\to \mathrm{\infty }}{lim}g{x}_{n}=gv.$
(3.11)

Since the function $x↦d\left(x,Tv\right)$ is continuous, by (1.2), (3.8), (3.10) and (3.11), we get

$\begin{array}{rcl}d\left(v,Tv\cap K\right)& =& \underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n+1},Tv\cap K\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim}\mathcal{H}\left(T{x}_{n},Tv\cap K\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim}\left\{\phi \left(d\left({x}_{n},v\right)\right)d\left({x}_{n},v\right)+h\left(v\right)d\left(gv,T{x}_{n}\cap K\right)\right\}\\ \le & \underset{n\to \mathrm{\infty }}{lim}\left\{\phi \left(d\left({x}_{n},v\right)\right)d\left({x}_{n},v\right)+h\left(v\right)d\left(gv,g{x}_{n+1}\right)\right\}=0,\end{array}$

which implies $d\left(v,Tv\cap K\right)=0$. By the closedness of Tv, we have $v\in Tv\cap K$. By (D2), $gv\in Tv\cap K\subseteq Tv$ and hence $v\in {\mathcal{COP}}_{K}\left(g,T\right)\cap {\mathcal{F}}_{K}\left(T\right)$. The proof is complete. □

## References

1. Hussain N, Amini-Harandi A, Cho YJ: Approximate endpoints for set-valued contractions in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 614867 10.1155/2010/614867

2. Khamsi MA: On asymptotically nonexpansive mappings in hyperconvex metric spaces. Proc. Am. Math. Soc. 2004, 132: 365–373. 10.1090/S0002-9939-03-07172-7

3. Du W-S: On approximate coincidence point properties and their applications to fixed point theory. J. Appl. Math. 2012., 2012: Article ID 302830 10.1155/2012/302830

4. Du W-S, He Z, Chen Y-L: New existence theorems for approximate coincidence point property and approximate fixed point property with applications to metric fixed point theory. J. Nonlinear Convex Anal. 2012, 13(3):459–474.

5. Du W-S: New existence results and generalizations for coincidence points and fixed points without global completeness. Abstr. Appl. Anal. 2013., 2013: Article ID 214230 10.1155/2013/214230

6. Du W-S: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol. Appl. 2012, 159: 49–56. 10.1016/j.topol.2011.07.021

7. Du W-S: Some new results and generalizations in metric fixed point theory. Nonlinear Anal. 2010, 73: 1439–1446. 10.1016/j.na.2010.05.007

8. Du W-S: On generalized weakly directional contractions and approximate fixed point property with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 6 10.1186/1687-1812-2012-6

9. Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25(1):52–57. 10.1016/j.aml.2011.05.047

10. Kumam, P, Aydi, H, Karapinar, E, Sintunavarat, W: Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems. Fixed Point Theory Appl. (in press)

11. Du W-S, Karapinar E, Shahzad N: The study of fixed point theory for various multivalued non-self-maps. Abstr. Appl. Anal. 2013., 2013: Article ID 938724 10.1155/2013/938724

12. Browder FE: Nonexpansive nonlinear operators in Banach space. Proc. Natl. Acad. Sci. USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041

13. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6

14. Kirk WA: Remarks on pseudo-contractive map. Proc. Am. Math. Soc. 1970, 25: 820–823. 10.1090/S0002-9939-1970-0264481-X

15. Kirk WA: Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings. Pac. J. Math. 1971, 38: 89–94. 10.2140/pjm.1971.38.89

16. Assad NA, Kirk WA: Fixed point theorems for set-valued mappings of contractive type. Pac. J. Math. 1972, 43: 553–562. 10.2140/pjm.1972.43.553

17. Reich S: Fixed points of condensing functions. J. Math. Anal. Appl. 1973, 41: 460–467. 10.1016/0022-247X(73)90220-5

18. Assad NA: A fixed point theorem for some non-self-mappings. Tamkang J. Math. 1990, 21(4):387–393.

19. Alghamdi MA, Berinde V, Shahzad N: Fixed points of multivalued nonself almost contractions. J. Appl. Math. 2013., 2013: Article ID 621614 10.1155/2013/621614

20. Lin L-J, Du W-S: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 2006, 323: 360–370. 10.1016/j.jmaa.2005.10.005

21. Lin L-J, Du W-S: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal. 2008, 68: 1246–1262. 10.1016/j.na.2006.12.018

22. Du W-S: Critical point theorems for nonlinear dynamical systems and their applications. Fixed Point Theory Appl. 2010., 2010: Article ID 246382 10.1155/2010/246382

## Acknowledgements

In this research, the author was supported by grant No. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China.

## Author information

Authors

### Corresponding author

Correspondence to Wei-Shih Du.

### Competing interests

The author declares that he has no competing interests.

## Rights and permissions

Reprints and Permissions

Du, WS. A note on approximate fixed point property and Du-Karapinar-Shahzad’s intersection theorems. J Inequal Appl 2013, 506 (2013). https://doi.org/10.1186/1029-242X-2013-506

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1029-242X-2013-506

### Keywords

• τ-function
• $\mathcal{MT}$-function (-function)
• coincidence point
• fixed point
• approximate fixed point property 