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 $(X,d)$ be a metric space. Denote by $\mathcal{N}(X)$ the family of all nonempty subsets of X and by $\mathcal{CB}(X)$ the family of all nonempty closed and bounded subsets of X. For each $x\in X$ and $A\subseteq X$, let $d(x,A)={inf}_{y\in A}d(x,y)$. A function $\mathcal{H}:\mathcal{CB}(X)\times \mathcal{CB}(X)\to [0,\mathrm{\infty })$ defined by

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

is said to be the Hausdorff metric on $\mathcal{CB}(X)$ induced by the metric d on X. The symbols ℕ and ℝ 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 $T:K\to \mathcal{N}(X)$ be a multivalued non-self-map. A point v in X is a coincidence point (see, for instance, [1–6]) 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 ${\mathcal{F}}_K(T)$ and ${\mathcal{COP}}_K(g,T)$, respectively. In particular, if $K\equiv X$, we use $\mathcal{F}(T)$ and $\mathcal{COP}(g,T)$ instead of ${\mathcal{F}}_K(T)$ and ${\mathcal{COP}}_K(g,T)$, respectively. The map T is said to have approximate fixed point property [1–5] on K provided ${inf}_{x\in K}d(x,Tx)=0$. It is obvious that ${\mathcal{F}}_K(T)\ne \mathrm{\varnothing }$ implies that T has approximate fixed point property.

A function $\phi :[0,\mathrm{\infty })\to [0,1)$ is said to be an $\mathcal{MT}$-function (or ℛ-function) [3–11] if ${lim\hspace{0.17em}sup}_{s\to t^{+}}\phi (s)<1$ for all $t\in [0,\mathrm{\infty })$. Clearly, if $\phi :[0,\mathrm{\infty })\to [0,1)$ 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., [11–19] 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 $(X,d)$ be a complete metric space, K be a nonempty closed subset of X, $T:K\to \mathcal{CB}(X)$ 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., $g(Tx\cap K)\subseteq Tx\cap K$) for each $x\in K$,

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

$$\begin{array}{rl}\mathcal{H}(Tx,Ty\cap K)\le & \gamma [d(x,Tx\cap K)+d(y,Tx\cap K)+d(y,Ty\cap K)]\\ +h(y)d(gy,Tx\cap K)\mathit{\text{for all}}x,y\in K.\end{array}$$

(1.1)

Then ${\mathcal{COP}}_K(g,T)\cap {\mathcal{F}}_K(T)\ne \mathrm{\varnothing }$.

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 $(X,d)$ be a complete metric space, K be a nonempty closed subset of X, $T:K\to \mathcal{CB}(X)$ 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 $\phi :[0,\mathrm{\infty })\to [0,1)$ and a function $h:K\to [0,\mathrm{\infty })$ such that

$$\mathcal{H}(Tx,Ty\cap K)\le \phi (d(x,y))d(x,y)+h(y)d(gy,Tx\cap K)\mathit{\text{for all}}x,y\in K,$$

(1.2)

then ${\mathcal{COP}}_K(g,T)\cap {\mathcal{F}}_K(T)\ne \mathrm{\varnothing }$.

In this work, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems of ${\mathcal{COP}}_K(g,T)$ and ${\mathcal{F}}_K(T)$ 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 $(X,d)$ be a metric space. Recall that a function $p:X\times X\to [0,\mathrm{\infty })$ is said to be a τ-function [3–5, 7, 8, 20–22], first introduced and studied by Lin and Du, if the following conditions hold:

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

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

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

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

Note that with the additional condition

(τ 5) $p(x,x)=0$ for all $x\in X$,

a τ-function becomes a ${\tau }^0$-function [3–5, 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), $p(x,y)=0$ if and only if $x=y$.

Example A [7]

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

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

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

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

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

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

• (a) φ is an $\mathcal{MT}$-function.

• (b) For each $t\in [0,\mathrm{\infty })$, there exist $r_t^{(1)}\in [0,1)$ and ${\epsilon }_t^{(1)}>0$ such that $\phi (s)\le r_t^{(1)}$ for all $s\in (t,t+{\epsilon }_t^{(1)})$.

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

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

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

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

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

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

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

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

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

$$p(y,Ty)\le \kappa \left(\tau (p(x,y))\right)p(x,y)\mathit{\text{for all}}y\in Tx.$$

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 $\phi :[0,\mathrm{\infty })\to [0,1)$ by $\phi (t)=\frac{1+\kappa (t)}{2}$. Then φ is also an $\mathcal{MT}$-function. Indeed, it is obvious that $0\le \kappa (t)<\phi (t)<1$ for all $t\in [0,\mathrm{\infty })$. Let ${\{x_n\}}_{n\in \mathbb{N}}$ be a strictly decreasing sequence in $[0,\mathrm{\infty })$. Since κ is an $\mathcal{MT}$-function, by (g) of Theorem 2.1, we get

$$0\le \underset{n\in \mathbb{N}}{sup}\kappa (x_n)<1$$

and hence

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

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 $p(x,y)>0$. By (Q2), we have

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

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

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

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

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

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

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

$$p(y,Ty)\le \kappa \left(\tau (p(x,y))\right)p(x,y)\mathit{\text{for all}}y\in Tx.$$

Then the following statements hold.

• (a) There exists a Cauchy sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in X such that

  • (i) $x_{n+1}\in T{x}_n$ for all $n\in \mathbb{N}$,

  • (ii) ${inf}_{n\in X}p(x_n,x_{n+1})={lim}_{n\to \mathrm{\infty }}p(x_n,x_{n+1})={lim}_{n\to \mathrm{\infty }}d(x_n,x_{n+1})={inf}_{n\in \mathbb{N}}d(x_n,x_{n+1})=0$.

• (b) ${inf}_{x\in X}p(x,Tx)={inf}_{x\in X}d(x,Tx)=0$; 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

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

and

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

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

$$\begin{array}{c}{w}_{n+1}=u\in Tu=T{w}_n\text{for all }n\in \mathbb{N},\hfill \\ \underset{n\to \mathrm{\infty }}{lim}p(w_n,w_{n+1})=\underset{n\in \mathbb{N}}{inf}p(w_n,w_{n+1})=p(u,u)=0,\hfill \end{array}$$

and

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

Clearly,

$$p(w_{n+1},w_{n+2})=0=\phi \left(\xi (p(w_n,w_{n+1}))\right)p(w_n,w_{n+1})\text{for all }n\in \mathbb{N}.$$

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 $p(x_1,x_2)>0$. Assume that condition (L1) is satisfied. Then there exists $x_3\in T{x}_2$ such that

$$p(x_2,x_3)\le \phi \left(\xi (p(x_1,x_2))\right)p(x_1,x_2).$$

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

$$p(x_3,x_4)\le \phi \left(\xi (p(x_2,x_3))\right)p(x_2,x_3).$$

By induction, we can obtain a sequence $\{x_n\}$ in X satisfying $x_{n+1}\in T{x}_n$ and

$$p(x_{n+1},x_{n+2})\le \phi \left(\xi (p(x_n,x_{n+1}))\right)p(x_n,x_{n+1})\text{for all }n\in \mathbb{N}.$$

(2.1)

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

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_{n+1})=\underset{n\in \mathbb{N}}{inf}p(x_n,x_{n+1})\ge 0\text{ exists}.$$

(2.2)

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

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

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

$$p(x_{n+1},x_{n+2})\le \lambda p(x_n,x_{n+1})\le \cdots \le {\lambda }^{n}p(x_1,x_2)\text{for each }n\in \mathbb{N}.$$

(2.3)

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

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

(2.4)

By (2.2) and (2.4), we obtain

$$\underset{n\in \mathbb{N}}{inf}p(x_n,x_{n+1})=\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_{n+1})=0.$$

(2.5)

Now, we claim that $\{x_n\}$ is a Cauchy sequence in X. Let ${\alpha }_n=\frac{{\lambda }^{n-1}}{1-\lambda }p(x_1,x_2)$, $n\in \mathbb{N}$. For $m,n\in \mathbb{N}$ with $m>n$, by (2.3), we have

$$p(x_n,x_m)\le \sum _{j=n}^{m-1}p(x_j,x_{j+1})<{\alpha }_n.$$

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

$$\underset{n\to \mathrm{\infty }}{lim}sup\{p(x_n,x_m):m>n\}=0.$$

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

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,x_{n+1})=\underset{n\in \mathbb{N}}{inf}d(x_n,x_{n+1})=0.$$

(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

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

(2.7)

and

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

(2.8)

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

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

The proof is completed. □

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

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

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

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

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

$$p(y,Ty)\le \beta (p(x,y))p(x,y)\mathit{\text{for all}}y\in Tx.$$

Then the following statements hold.

• (a) There exists a Cauchy sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in X such that

  • (i) $x_{n+1}\in T{x}_n$ for all $n\in \mathbb{N}$,

  • (ii) ${inf}_{n\in X}p(x_n,x_{n+1})={lim}_{n\to \mathrm{\infty }}p(x_n,x_{n+1})={lim}_{n\to \mathrm{\infty }}d(x_n,x_{n+1})={inf}_{n\in \mathbb{N}}d(x_n,x_{n+1})=0$.

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

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

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

$$0\le \underset{n\in \mathbb{N}}{sup}\kappa (\tau (x_n))<1,$$

or, equivalently,

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

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, $(K,d)$ 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, $d(y,Tx\cap K)=0$. By (D2), we have $d(gy,Tx\cap K)=0$. Hence inequality (1.1) implies

$$\mathcal{H}(Tx,Ty\cap K)\le \gamma [d(x,Tx\cap K)+\mathcal{H}(Tx,Ty\cap K)]\text{for all }y\in Tx\cap K.$$

(3.1)

Inequality (3.1) shows that

$$d(y,Ty\cap K)\le \mathcal{H}(Tx,Ty\cap K)\le kd(x,Tx\cap K)<\lambda d(x,y)\text{for all }y\in Tx\cap K.$$

(3.2)

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

$$Gx=Tx\cap K\text{for all }x\in K,$$

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

$$\eta (t)=\lambda \text{for all }t\in [0,\mathrm{\infty }).$$

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

$$d(y,Gy)\le \mu (d(x,y))d(x,y)\text{for all }y\in Gx.$$

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

$$x_{n+1}\in G{x}_n=T{x}_n\cap K\text{for all }n\in \mathbb{N}$$

(3.3)

and

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,x_{n+1})=\underset{n\in \mathbb{N}}{inf}d(x_n,x_{n+1})=0.$$

(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

$$g{x}_{n+1}\in T{x}_n\cap K\text{for each }n\in \mathbb{N}.$$

(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 $x\mapsto d(x,Tv)$ is continuous, by (1.1), (3.3), (3.4), (3.5) and (3.6), we get

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

which implies $d(v,Tv\cap K)=0$. By the closedness of Tv, we have $v\in Tv\cap K$. From (D2), $gv\in Tv\cap K\subseteq Tv$. Hence we verify $v\in {\mathcal{COP}}_K(g,T)\cap {\mathcal{F}}_K(T)$. 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, $(K,d)$ is also a complete metric space. Note first that for each $x\in K$, by (D2), we have $d(gy,Tx\cap K)=0$ for all $y\in Tx\cap K$. So, for each $x\in K$, by (1.2), we obtain

$$d(y,Ty\cap K)\le \phi (d(x,y))d(x,y)\text{for all }y\in Tx\cap K.$$

(3.7)

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

$$Gx=Tx\cap K\text{for all }x\in K.$$

From (3.7), we obtain

$$d(y,Gy)\le \phi (d(x,y))d(x,y)\text{for all }y\in Gx.$$

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

$$x_{n+1}\in G{x}_n=T{x}_n\cap K\text{for all }n\in \mathbb{N}$$

(3.8)

and

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,x_{n+1})=\underset{n\in \mathbb{N}}{inf}d(x_n,x_{n+1})=0.$$

(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

$$g{x}_{n+1}\in T{x}_n\cap K\text{for each }n\in \mathbb{N}.$$

(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 $x\mapsto d(x,Tv)$ is continuous, by (1.2), (3.8), (3.10) and (3.11), we get

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

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