# 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, [16]) 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 [15] 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) [311] 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., [1119] and the references therein. Very recently, Du, Karapinar and Shahzad [11] 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 [[11], 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 [11], 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 [[11], 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 [7]

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 [[22], 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 [6] are quite useful for proving our main results.

Theorem 2.1 [[6], 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 [4], but we give the proof for the sake of completeness and the readers convenience.

Lemma 2.2 [[4], 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 [[4], 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. □

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## 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.

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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