A remark on Jleli–Samet’s best proximity point theorems for α-ψ-contraction mappings

Abstract

Inspired by the work of Jachymski, we slightly extend some fixed point theorems with a graph and show that some best proximity point theorems for α-ψ-contraction mappings of Jleli and Samet can be deduced by our results.

1 Introduction

Let f be a mapping on a nonempty set X. We say that $z\in X$ is a fixed point of f if $z=fz$. In 1922, Banach established one of the most famous fixed point theorems, namely the Banach contraction principle (see [2]), which has been generalized in many directions (for examples, see [1, 3, 6, 7]).

Let A, B be nonempty subsets of a metric space $(X,d)$ and $T:A\to B$. We say that $z\in A$ is a best proximity point of T if $d(z,Tz)=d(A,B)$. Note that if $A\cap B\neq \emptyset$, then a best proximity point becomes a fixed point.

In this paper, we slightly extend some fixed point theorems with a graph, which were introduced by Jachymski [4], and show that some best proximity point theorems for α-ψ-contraction mappings of Jleli and Samet (see Theorems 3.1, 3.2 and 3.3 in [5]) can be deduced by our results.

The following two theorems were proved by Jachymski in 2008.

Theorem J1

Let $(X,d)$ be a complete metric space, $f:X\to X$ be a mapping and G be a directed graph. Suppose that

(A1):

for any sequence $\{x_{n}\}$ in X if $\lim_{n} x_{n}=x$ for some $x\in X$ and $(x_{n},x_{n+1})\in E(G)$ for all $n\geq 1$, then there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $(x_{n_{k}},x)\in E(G)$ for all $k\geq 1$.

Assume that f satisfies the followings:

• (1) For all $x,y\in X$ if $(x,y)\in E(G)$, then $(fx,fy)\in E(G)$;

• (2) There is $\lambda \in (0,1)$ such that for all $x,y\in X$ if $(x,y)\in E(G)$, then $d(fx,fy)\leq \lambda d(x,y)$.

Then $\operatorname{Fix}(f):=\{x:x=fx\}\neq \emptyset$ if and only if $X_{f}:=\{x:(x,fx)\in E(G)\}\neq \emptyset$.

Theorem J2

Let $(X,d)$ be a complete metric space, $f:X\to X$ be a mapping and G be a directed graph. Suppose that

(A2):

f is orbitally G-continuous.

Assume that f satisfies the followings:

• (1) For all $x,y\in X$ if $(x,y)\in E(G)$, then $(fx,fy)\in E(G)$;

• (2) There is $\lambda \in (0,1)$ such that for all $x,y\in X$ if $(x,y)\in E(G)$, then $d(fx,fy)\leq \lambda d(x,y)$.

Then $\operatorname{Fix}(f)\neq \emptyset$ if and only if $X_{f}\neq \emptyset$.

2 Basics concepts and notations

Let X be a nonempty set and $\Delta :=\{(x,x):x\in X\}$. In this paper, a directed graph G on X means the set of its vertices $V(G)$ is X and the set of its edges $E(G)$ is a subset of $X\times X$ and we assume that $\Delta \subset E(G)$ and G has no parallel edges.

Let G be a directed graph. The conversion of G, denoted by $G^{-1}$, is the graph such that $V(G^{-1})=V(G)$ and $E(G^{-1})=\{(x,y):(y,x)\in E(G)\}$. The undirected graph obtained from G, denoted by G̃, is the graph such that $V(\widetilde{G})=V(G)$ and $E(\widetilde{G})=E(G)\cup E(G^{-1})$.

For $x,y\in V(G)$, a path in a directed graph G from x to y of length N is a sequence $\{x_{i}\}_{i=0}^{N}$ such that $x_{0}=x$, $x_{N}=y$ and $(x_{i-1},x_{i})\in E(G)$ for $i=1,2,\ldots ,N$.

A directed graph G is connected if every pair of vertices has a path. A directed graph G is weakly connected if G̃ is connected.

The following definition was introduced by Jachymski.

Definition 2.1

([4])

Let $(X,d)$ be a metric space and G be a directed graph. A mapping $f:X\to X$ is called orbitally G-continuous if for all $x,y\in X$ and any sequence $\{k_{n}\}$ of positive integers

$$ \left . \begin{gathered} \lim_{n} f^{k_{n}}x=y \\ \bigl(f^{k_{n}}x,f^{k_{n+1}}x \bigr)\in E(G)\quad \text{for all } n>0 \end{gathered} \right \}\quad \Rightarrow \quad \lim _{n}f \bigl(f^{k_{n}}x \bigr)=fy. $$

By using the concept of orbitally G-continuity of f, we slightly extend Theorems J1 and J2 by weakening the continuity of f and the completeness of $(X,d)$.

Definition 2.2

Let $(X,d)$ be a metric space and G be a directed graph. A mapping $f:X\to X$ is called weakly orbitally G-continuous if for all $x,y\in X$,

$$ \left . \begin{gathered} \lim_{n} f^{n}x=y \\ \bigl(f^{n}x,f^{{n+1}}x \bigr)\in E(G) \quad \text{for all } n>0 \end{gathered} \right \}\quad \Rightarrow \quad \lim_{n}f \bigl(f^{n}x \bigr)=fy. $$

The following example shows that there is a weakly orbitally G-continuous mapping which is not orbitally G-continuous.

Example 2.1

Let $X=[0,\infty )$ with the usual metric $|\cdot |$. Let G be a directed graph on X with $E(G)=\Delta \cup \{(x,y): x,y\in (0,1)\}$. Suppose $f:X\to X$ is a mapping defined by

$$ f(x)=\textstyle\begin{cases} 1/2&\text{if } x=0 \text{ or } x=1, \\ 1/x&\text{if } 0< x< 1, \\ 1/x^{2}&\text{if } x>1. \end{cases} $$

Since there is no $x\in X$ such that $(f^{n}x,f^{n+1}x)\in E(G)$ for all $n>0$, we have that f is weakly orbitally G-continuous.

Note that $\lim_{n}f^{2n+1}(1)=0$ and $(f^{2n+1}(1),f^{2n+3}(1))\in E(G)$ for all $n>0$ but $\lim_{n}f(f^{2n+1}(1))$ does not exist. That is, f is not orbitally G-continuous.

Definition 2.3

Let $(X,d)$ be a metric space and $f:X\to X$ be a mapping. Let G be a directed graph. We say that $(X,d)$ is weakly $(f,G)$-orbitally complete if for all $x\in X$,

$$ \left . \begin{gathered} \bigl\{ f^{n}x \bigr\} \text{ is Cauchy } \\ \bigl(f^{n}x,f^{{n+1}}x \bigr)\in E(G)\quad \text{for all } n>0 \end{gathered} \right \}\quad \Rightarrow \quad \lim _{n}f^{n}x=y \quad \text{for some } y\in X. $$

3 Main results

We denote by Ψ the set of nondecreasing functions $\psi :[0,\infty )\to [0,\infty )$ such that $\sum_{n=1}^{\infty }\psi ^{n}(t)<\infty$ for all $t>0$.

Lemma 3.1

Let $\psi \in \Psi$. Then the followings hold:

• (1) $\psi (t)< t$ for all $t>0$;

• (2) $\psi (0)=0$.

Proof

(2) follows immediately from (1) and a proof of (1) can be found in [8]. □

Lemma 3.2

Let $(X,d)$ be a metric space with a directed graph G and f be a self-mapping on X. If $\operatorname{Fix}(f)\neq \emptyset$, then $X_{f}\neq \emptyset$.

Proof

Assume $\operatorname{Fix}(f)\neq \emptyset$. Let $z\in X$ such that $z=fz$. Since $E(G)$ contains all loops, we get $(z,fz)\in E(G)$, that is, $z\in X_{f}$. □

Theorem 3.1

Let $(X,d)$ be a metric space and G be a directed graph. Suppose that

(A1):

for any sequence $\{x_{n}\}$ in X if $\lim_{n} x_{n}=x$ for some $x\in X$ and $(x_{n},x_{n+1})\in E(G)$ for all $n\geq 1$, then there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $(x_{n_{k}},x)\in E(G)$ for all $k\geq 1$.

Suppose that $f:X\to X$ and $\psi \in \Psi$ satisfy the followings:

• (1) For all $x,y\in X$ if $(x,y)\in E(G)$, then $(fx,fy)\in E(G)$;

• (2) For all $x,y\in X$ if $(x,y)\in E(G)$, then $d(fx,fy)\leq \psi (d(x,y))$.

Suppose that X is weakly $(f,G)$-orbitally complete. Then $\operatorname{Fix}(f)\neq \emptyset$ if and only if $X_{f}\neq \emptyset$.

Proof

It follows immediately from Lemma 3.2 that $\operatorname{Fix}(f)\neq \emptyset$ implies $X_{f}\neq \emptyset$. On the other hand, we assume that there is $x_{0}\in X$ such that $(x_{0},fx_{0})\in E(G)$. For each $n\geq 1$, we have $(f^{n}x_{0},f^{n+1}x_{0})\in E(G)$ which implies that $d(f^{n+1}x_{0},f^{n+2}x_{0})\leq \psi (d(f^{n}x_{0},f^{n+1}x_{0}))$. Since ψ is nondecreasing, $d(f^{n}x_{0},f^{n+1}x_{0})\leq \psi ^{n}(d(x_{0},fx_{0}))$. Then

$$ \sum_{n=1}^{\infty }d \bigl(f^{n}x_{0},f^{n+1}x_{0} \bigr)\leq \sum_{n=1}^{ \infty}\psi ^{n} \bigl(d(x_{0},fx_{0}) \bigr)< \infty $$

which implies that $\{f^{n}x_{0}\}$ is a Cauchy sequence. Note that X is weakly $(f,G)$-orbitally complete. Therefore, there is $z\in X$ such that $\lim_{n} f^{n}x_{0}=z$. By Condition (A1), there exists a subsequence $\{f^{n_{k}}x_{0}\}$ of $\{f^{n}x_{0}\}$ such that $(f^{n_{k}}x_{0},z)\in E(G)$ for all $k\geq 1$. By Condition (2) and Lemma 3.1, it follows that

$$ d \bigl(f^{n_{k}+1}x_{0},fz \bigr)\leq \psi \bigl(d \bigl(f^{n_{k}}x_{0},z \bigr) \bigr)\leq d \bigl(f^{n_{k}}x_{0},z \bigr). $$

Therefore, $d(z,fz)=\lim_{k}d(f^{n_{k}+1}x_{0},fz)\leq \lim_{k}d(f^{n_{k}}x_{0},z)=0$, that is, $z\in \operatorname{Fix}(f)$. □

Theorem 3.2

Let $(X,d)$ be a metric space and G be a directed graph. Let $f:X\to X$ be a mapping such that

(A2^∗):

f is weakly orbitally G-continuous.

Let $\psi \in \Psi$. Suppose that f and ψ satisfy the followings:

• (1) For all $x,y\in X$ if $(x,y)\in E(G)$, then $(fx,fy)\in E(G)$;

• (2) For all $x,y\in X$ if $(x,y)\in E(G)$, then $d(fx,fy)\leq \psi (d(x,y))$.

Suppose that X is weakly $(f,G)$-orbitally complete. Then $\operatorname{Fix}(f)\neq \emptyset$ if and only if $X_{f}\neq \emptyset$.

Proof

Let $x_{0}\in X$ such that $(x_{0},fx_{0})\in E(G)$. By following the proof of Theorem 3.1, we have that $\lim_{n}f^{n}x_{0}=z$ for some $z\in X$. Since the condition (A2^∗) holds, we get $z=\lim_{n}f^{n+1}x_{0}=fz$ and this completes the proof. □

Remark 3.1

Theorems 3.1 and 3.2 extend Theorems J1 and J2 as follows:

• 1. The orbitally G-continuity of f in Theorem J2 implies that f is weakly orbitally G-continuous in Theorem 3.2;

• 2. The completeness of $(X,d)$ implies that X is weakly $(f,G)$-orbitally complete;

• 3. In Theorems J1 and J2, if we put $\psi (t)=\lambda t$ for all $t\in [0,\infty )$, then $\psi \in \Psi$.

Inspired by the work of Jachymski (see [4, Theorem 3.1]), the following theorem characterizes the uniqueness of a fixed point (if it exists) of a mapping in a metric space with a directed graph.

Theorem 3.3

Let $(X,d)$ be a metric space and G be a directed graph. The followings are equivalent:

• (1) G is weakly connected;

• (2) For any $f:X\to X$ with $\lim_{n} d(f^{n}x,f^{n}y)=0$ whenever $(x,y)\in E(G)$, $\operatorname{card}(\operatorname{Fix}(f))\leq 1$.

Proof

(1)⇒(2): We assume that G is weakly connected. Let $f:X\to X$ be a mapping such that $\lim_{n} d(f^{n}x,f^{n}y)=0$ whenever $(x,y)\in E(G)$. Let u and v be two fixed points of f, that is, $u=fu$ and $v=fv$. Since G is weakly connected, there is a path $\{x_{i}\}_{i=0} ^{N}$ of length N such that

$$ x_{0}=u, \qquad x_{N}=v \quad \text{and} \quad (x_{i-1},x_{i})\in E(\widetilde{G})\quad \text{for all } i=1,2,\ldots ,N. $$

Therefore,

$$ d(u,v)=\lim_{n}d \bigl(f^{n}u,f^{n}v \bigr)\leq \lim_{n}\sum_{i=1}^{N} d \bigl(f^{n}x_{i-1},f^{n}x_{i} \bigr)=0. $$

That is, $\operatorname{card}(\operatorname{Fix}(f))\leq 1$.

(2)⇒(1): Suppose that G is not weakly connected. Then there are $x,y\in X$ such that there is no path from x to y in G̃. Note that $x\neq y$. We define $f:X\to X$ by for all $w\in X$

$$ fw=\textstyle\begin{cases} x&\text{if there exists a path in } \widetilde{G} \text{ from } w \text{ to } x; \\ y & \text{otherwise}. \end{cases} $$

Note that x and y are two different fixed points of f. Let $(u,v)\in E(G)$. Then $fu=fv=x$ or $fu=fv=y$ which implies that $\lim_{n} d(f^{n}u,f^{n}v)=0$. This completes the proof. □

4 Discussion on best proximity fixed point results for α-ψ-proximal contrative type mappings

In this section, we discuss some best proximity fixed point results for α-ψ-proximal contrative type mappings of of Jleli and Samet in [5] that can be deduced by our results.

Let A and B be two nonempty subsets of a metric space $(X,d)$. We recall the following notations:

$$\begin{aligned}& d(A,B):= \inf \bigl\{ d(a,b):a\in A, b\in B \bigr\} ; \\& A_{0}:= \bigl\{ a\in A:d(a,b)=d(A,B) \text{ for some } b\in B \bigr\} ; \\& B_{0}:= \bigl\{ b\in B:d(a,b)=d(A,B) \text{ for some } a\in A \bigr\} . \end{aligned}$$

Definition 4.1

([5])

Let A and B be two nonempty subsets of a metric space $(X,d)$. Then the pair $(A,B)$ is said to have the P-property if

$$ d(x_{1},y_{1})=d(x_{2},y_{2})=d(A,B) \quad \Rightarrow\quad d(x_{1},x_{2})=d(y_{1},y_{2}), $$

where $x_{1},x_{2}\in A$ and $y_{1},y_{2}\in B$.

Definition 4.2

([5])

Let A and B be two nonempty subsets of a metric space $(X,d)$. Let $T:A\to B$ and $\alpha :A\times A\to [0,\infty )$. We say that T is α-proximal admissible if

$$\begin{aligned} \left . \begin{gathered} \alpha (x_{1},x_{2}) \geq 1 \\ d(u_{1},Tx_{1})=d(u_{2},Tx_{2})=d(A,B) \end{gathered} \right \} \quad \Rightarrow \quad \alpha (u_{1},u_{2}) \geq 1 \end{aligned}$$

for all $x_{1},x_{2},u_{1},u_{2}\in A$.

Definition 4.3

([5])

Let A and B be two nonempty subsets of a metric space $(X,d)$. Let $T:A\to B$ and $\alpha :A\times A\to [0,\infty )$ and $\psi \in \Psi$. We say that T is an α-ψ-proximal contraction if

$$ \alpha (x,y)d(Tx,Ty)\leq \psi \bigl(d(x,y) \bigr)\quad \text{for all } x,y\in A. $$

Definition 4.4

([5])

Let A and B be two nonempty subsets of a metric space $(X,d)$. Let $T:A\to B$ and $\alpha :A\times A\to [0,\infty )$. We say that T is $(\alpha ,d)$-regular if for all $(x,y)\in \alpha ^{-1}([0,1))$, there exists $z\in A_{0}$ such that

$$ \alpha (x,z)\geq 1 \quad \text{and} \quad \alpha (y,z)\geq 1. $$

The following three theorems were proved by Jleli and Samet in 2013.

Theorem JS1

Let A, B be nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha :A\times A\to [0,\infty )$ and $\psi \in \Psi$. Suppose that $T:A\to B$ satisfies the followings:

(B1):

$T(A_{0})\subset B_{0}$ and $(A,B)$ satisfies the P-property;

(B2):

T is α-proximal admissible;

(B3):

T is an α-ψ-proximal contraction;

(B4):

There are $u,v\in A_{0}$ such that $d(v,Tu)=d(A,B)$ and $\alpha (u,v)\geq 1$;

(B5):

If $\{x_{n}\}$ is a sequence in A such that $\alpha (x_{n},x_{n+1})\geq 1$ for all $n>0$ and $\lim_{n}x_{n}\to x\in A$, then there is a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\alpha (x_{n_{k}},x)\geq 1$ for all $k>0$.

Then there is $z\in A_{0}$ such that $d(z,Tz)=d(A,B)$.

Theorem JS2

Let A, B be nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha :A\times A\to [0,\infty )$ and $\psi \in \Psi$. Suppose that $T:A\to B$ satisfies the followings:

(B1):

$T(A_{0})\subset B_{0}$ and $(A,B)$ satisfies the P-property;

(B2):

T is α-proximal admissible;

(B3):

T is an α-ψ-proximal contraction;

(B4):

There are $u,v\in A_{0}$ such that $d(v,Tu)=d(A,B)$ and $\alpha (u,v)\geq 1$;

(B5′):

T is continuous.

Then there is $z\in A_{0}$ such that $d(z,Tz)=d(A,B)$.

Theorem JS3

Let A, B be nonempty subsets of a metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha :A\times A\to [0,\infty )$ and $\psi \in \Psi$. Suppose that $T:A\to B$ satisfies the followings:

(B1):

$T(A_{0})\subset B_{0}$ and $(A,B)$ satisfies the P-property;

(B2):

T is α-proximal admissible;

(B3):

T is an α-ψ-proximal contraction.

If T is $(\alpha ,d)$-regular, then T has at most one best proximity point.

To show that Theorems JS1, JS2 and JS3 are the consequences of our Theorems 3.1, 3.2 and 3.3, respectively, we need the following lemmas.

Lemma 4.1

Let $(X,d)$ be a metric space. Let $A,B\subset X$ such that $A_{0}$ is nonempty and $(A,B)$ has the P-property. Suppose that $T:A\to B$ is a mapping such that $T(A_{0})\subset B_{0}$. Then, for each $x\in A_{0}$, the set $\{u\in A_{0}:d(u,Tx)=d(A,B)\}$ is a singleton set.

Proof

Let $x\in A_{0}$. Put $P:=\{u\in A_{0}:d(u,Tx)=d(A,B)\}$. Since $T(A_{0})\subset B_{0}$, P is nonempty. Let $u_{1},u_{2}\in P$. Then $d(u_{1},Tx)=d(u_{2},Tx)=d(A,B)$. Since $(A,B)$ satisfy the P-property, we get $d(u_{1},u_{2})=d(Tx,Tx)=0$, that is, $u_{1}=u_{2}$. □

Lemma 4.2

Let A, B be nonempty subsets of a metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha :A\times A\to [0,\infty )$ and $\psi \in \Psi$. Suppose that $T:A\to B$ satisfies the followings:

(B1):

$T(A_{0})\subset B_{0}$ and $(A,B)$ satisfies the P-property;

(B2):

T is α-proximal admissible;

(B3):

T is an α-ψ-proximal contraction.

Let $f:A_{0}\to A_{0}$ be defined by for each $x\in A_{0}$

$$ fx=u\quad \textit{where } u\in A_{0} \textit{ with } d(u,Tx)=d(A,B) $$

and $G_{0}$ a directed graph defined by $V(G_{0})=A_{0}$ and

$$ E(G_{0})= \bigl\{ (x,y)\in A_{0}\times A_{0}: \alpha (x,y)\geq 1 \bigr\} \cup \bigl\{ (x,x):x \in A_{0} \bigr\} . $$

Then f and $G_{0}$ satisfy the followings:

• (1) f is well-defined;

• (2) For all $x,y\in X$ if $(x,y)\in E(G_{0})$, then $(fx,fy)\in E(G_{0})$;

• (3) For all $x,y\in X$ if $(x,y)\in E(G_{0})$, then $d(fx,fy)\leq \psi (d(x,y))$.

Proof

It follows from Lemma 4.1 that f is well-defined.

To see (2), let $(x,y)\in E(G_{0})$. If $x=y$, then $fx=fy$ which implies that $(fx,fy)\in E(G_{0})$. Otherwise, we assume that $\alpha (x,y)\geq 1$. Note that $d(fx,Tx)=d(fy,Ty)=d(A,B)$ and T is α-proximal admissible. We have $\alpha (fx,fy)\geq 1$, which implies that $(fx,fy)\in E(G_{0})$.

To see (3), let $(x,y)\in E(G_{0})$. If $x=y$, then $d(fx,fy)=0=\psi (d(x,y))$. Otherwise, we assume that $\alpha (x,y)\geq 1$. Note that $d(fx,Tx)=d(fy,Ty)=d(A,B)$. Since T is an α-ψ-proximal contraction and $(A,B)$ satisfies the P-property, we get $d(fx,fy)=d(Tx,Ty)\leq \alpha (x,y)d(Tx,Ty)\leq \psi (d(x,y))$, as desired. □

Lemma 4.3

Let A, B be nonempty closed subsets of a metric space $(X,d)$ such that $A_{0}$ is nonempty. Suppose that $T:A\to B$ satisfies $T(A_{0})\subset B_{0}$ and $(A,B)$ satisfies the P-property. Let $f:A_{0}\to A_{0}$ be defined by for each $x\in A_{0}$

$$ fx=u\quad \textit{where } u\in A_{0} \textit{ with } d(u,Tx)=d(A,B). $$

Then for each $z\in A_{0}$, $z=fz$ if and only if $d(z,Tz)=d(A,B)$.

Proof

It obtains immediately by the definition of f. □

The following is a proof of Theorem JS1 using our Theorem 3.1.

Proof of Theorem JS1

Let all assumptions in Theorem JS1 be satisfied. We define a mapping $f:A_{0}\to A_{0}$ and a graph $G_{0}$ as in Lemma 4.2. Then f and $G_{0}$ satisfy the conditions (1) and (2) in Theorem 3.1.

To see that the metric space $(A_{0},d)$ with the directed graph $G_{0}$ satisfies Condition (A1) in Theorem 3.1, let $\{x_{n}\}$ be a sequence in $A_{0}$ with $\lim_{n} x_{n}=x$ for some $x\in A_{0}$ and $(x_{n},x_{n+1})\in E(G_{0})$ for all $n\geq 1$. If $\{n:\alpha (x_{n},x_{n+1})\geq 1\}$ is finite, then there is $N\geq 1$ such that, for each $n\geq N$, $x_{n}=x$, that is, $(x_{n},x)\in E(G_{0})$. Otherwise, we assume that $\{n:\alpha (x_{n},x_{n+1})\geq 1\}$ is infinite. Note that $\alpha (x_{n},x_{n+1})\geq 1$ whenever $x_{n}\neq x_{n+1}$. Then there is a subsequence $\{\widetilde{x}_{n}\}$ of $\{x_{n}\}$ such that $\alpha (\widetilde{x}_{n},\widetilde{x}_{n+1})\geq 1$ for all $n\geq 1$. By the condition (B5), there is a subsequence $\{\widetilde{x}_{n_{k}}\}$ of $\{\widetilde{x}_{n}\}$ such that, for each $k\geq 1$, $\alpha (\widetilde{x}_{n_{k}},x)\geq 1$, that is, $(\widetilde{x}_{n_{k}},x)\in E(G_{0})$.

We now show that $(A_{0},d)$ is weakly $(f,G_{0})$-orbitally complete. Let $x\in A_{0}$. Assume that $\{f^{n}x\}$ is Cauchy and $(f^{n}x,f^{n+1}x)\in E(G_{0})$ for all $n\geq 1$. Since X is complete and A is closed, there is $z\in A$ such that $\lim_{n} f^{n}x=z$. If $\{n:\alpha (f^{n}x,f^{n+1}x)\geq 1\}$ is finite, then there is $N\geq 1$ such that $f^{n}x=z$ for all $n\geq N$, that is, $z\in A_{0}$. Otherwise,we assume that $\{n:\alpha (f^{n}x,f^{n+1}x)\geq 1\}$ is infinite. Using Condition (B5), there is a subsequence $\{f^{n_{k}}x\}$ of $\{f^{n}x\}$ such that $\alpha (f^{n_{k}}x,z)\geq 1$ for all $k\geq 1$. Then we have

$$ d \bigl(Tf^{n_{k}}x,Tz \bigr)\leq \alpha \bigl(f^{n_{k}}x,z \bigr)d \bigl(Tf^{n_{k}}x,Tz \bigr)\leq \psi \bigl(d \bigl(f^{n_{k}}x,z \bigr) \bigr)\leq d \bigl(f^{n_{k}}x,z \bigr). $$

We get $\lim_{k} Tf^{n_{k}}x=Tz$. By the definition of f, $d(f^{n_{k}+1}x,Tf^{n_{k}}x)=d(A,B)$ for all $k\geq 1$. Therefore,

$$ d(z,Tz)=\lim_{k}d \bigl(f^{n_{k}+1}x,Tf^{n_{k}}x \bigr)=d(A,B). $$

That is $z\in A_{0}$.

Finally, we show that $\{x:(x,fx)\in E(G_{0})\}\neq \emptyset$. Since $d(v,Tu)=d(A,B)$ and $\alpha (u,v)\geq 1$, we have $v=fu$ and hence $(u,fu)\in E(G_{0})$. By using our Theorem 3.1, there is $p\in A_{0}$ such that $p=fp$. By Lemma 4.3, we have $d(p,Tp)=d(A,B)$, as desired. □

The following is a proof of Theorem JS2 via Theorem 3.2.

Proof of Theorem JS2

Let all assumptions in Theorem JS2 be satisfied. We define a mapping $f:A_{0}\to A_{0}$ and a graph $G_{0}$ as in Lemma 4.2. Then we have f and $G_{0}$ satisfy (1) and (2) in Theorem 3.2.

To see that f satisfies Condition (A2^∗) in Theorem 3.2, let $x,y\in A_{0}$ such that $\lim_{n}f^{n}x=y$ and $(f^{n}x,f^{n+1}x)\in E(G_{0})$ for all $n\geq 1$. Note that $d(fw,Tw)=d(A,B)$ for all $w\in A_{0}$. Since $(A,B)$ has the P-property and T is continuous,

$$ \lim_{n}d \bigl(f \bigl(f^{n}x \bigr),fy \bigr)= \lim_{n}d \bigl(T \bigl(f^{n}x \bigr),Ty \bigr)=0. $$

Finally, we show that $(A_{0},d)$ is weakly $(f,G_{0})$-orbitally complete. Let $x\in A_{0}$. Assume that $\{f^{n}x\}$ is Cauchy and $(f^{n}x,f^{n+1}x)\in E(G_{0})$ for all $n\geq 1$. Since X is complete and A is closed, there is $z\in A$ such that $\lim_{n} f^{n}x=z$. By the continuity of T, we have $\lim_{n}T(f^{n}x)=Tz$. This implies that

$$ d(z,Tz)=\lim_{n}d \bigl(f \bigl(f^{n}x \bigr),T \bigl(f^{n}x \bigr) \bigr)=d(A,B). $$

That is $z\in A_{0}$. Note that Condition (B4) implies $(u,fu)\in E(G_{0})$. By using Theorem 3.2, there is $p\in A_{0}$ such that $p=fp$ which implies that $d(p,Tp)=d(A,B)$. □

Finally, we show a proof of Theorem JS3 by using our Theorem 3.3.

Proof of Theorem JS3

We assume that all the assumptions hold and suppose that T is $(\alpha ,d)$-regular. We define the mapping $f:A_{0}\to A_{0}$ and the directed graph $G_{0}$ as in Lemma 4.2. Since T is $(\alpha ,d)$-regular, we obtain immediately that $G_{0}$ is weakly connected. Note that for each $(x,y)\in E(G_{0})$,

$$ (fx,fy)\in E(G_{0}) \quad \text{and} \quad d(fx,fy)\leq \psi \bigl(d(x,y) \bigr). $$

Therefore,

$$ \lim_{n} d \bigl(f^{n}x,f^{n}y \bigr) \leq \lim_{n} \psi ^{n} \bigl(d(x,y) \bigr)=0. $$

By Theorem 3.3, we have $\operatorname{card}(\operatorname{Fix}(f))\leq 1$. Using Lemma 4.3, T has at most one best proximity point. The proof is complete. □