A note on ‘A best proximity point theorem for Geraghty-contractions’

Abstract

In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

MSC:47H05, 47H09, 47H10.

1 Introduction and preliminaries

Let A and B be nonempty subsets of a metric space $(X,d)$. An operator $T:A\to B$ is said to be contractive if there exists $k\in [0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for any $x,y\in A$. The well-known Banach contraction principle says: Let $(X,d)$ be a complete metric space, and $T:X\to X$ be a contraction of X into itself. Then T has a unique fixed point in X.

In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2.

Definition 1.1 ([1])

Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is said to be a Geraghty-contraction if there exists such that for any $x,y\in X$

$$d(Tx,Ty)\le \beta (d(x,y))\cdot d(x,y),$$

where the class Γ denotes those functions $\beta :[0,\mathrm{\infty })\to [0,1)$ satisfying the following condition:

$$\beta (t_n)\to 1\Rightarrow t_n\to 0.$$

Theorem 1.2 ([1])

Let $(X,d)$ be a complete metric space and $T:X\to X$ be an operator. Suppose that there exists such that for any $x,y\in X$,

$$d(Tx,Ty)\le \beta (d(x,y))\cdot d(x,y).$$

Then T has a unique fixed point.

Obviously, Theorem 1.2 is an extensive version of Banach contraction principle. In 2012, Caballero et al. introduced generalized Geraghty-contraction as follows.

Definition 1.3 ([2])

Let A, B be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a Geraghty-contraction if there exists such that for any $x,y\in A$

$$d(Tx,Ty)\le \beta (d(x,y))\cdot d(x,y),$$

where the class denotes those functions $\beta :[0,\mathrm{\infty })\to [0,1)$ satisfying the following condition:

$$\beta (t_n)\to 1\Rightarrow t_n\to 0.$$

Now we need the following notations and basic facts.

Let A and B be two nonempty subsets of a metric space $(X,d)$. We denote by $A_0$ and $B_0$ the following sets:

where $d(A,B)=inf\{d(x,y):x\in A\text{ and }y\in B\}$.

In [3], the authors give sufficient conditions for when the sets $A_0$ and $B_0$ are nonempty. In [4], the author presents the following definition and proves that any pair $(A,B)$ of nonempty, closed and convex subsets of a real Hilbert space H satisfies the P-property.

Definition 1.4 ([2])

Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with $A_0\ne \mathrm{\varnothing }$. Then the pair $(A,B)$ is said to have the P-property if and only if for any $x_1,x_2\in A_0$ and $y_1,y_2\in B_0$,

$$\{\begin{array}{c}d(x_1,y_1)=d(A,B),\hfill \\ d(x_2,y_2)=d(A,B)\hfill \end{array}\Rightarrow d(x_1,x_2)=d(y_1,y_2).$$

Let A, B be two nonempty subsets of a complete metric space and consider a mapping $T:A\to B$. The best proximity point problem is whether we can find an element $x_0\in A$ such that $d(x_0,T{x}_0)=min\{d(x,Tx):x\in A\}$. Since $d(x,Tx)\ge d(A,B)$ for any $x\in A$, in fact, the optimal solution to this problem is the one for which the value $d(A,B)$ is attained.

In [2], the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.

Theorem 1.5 ([2])

Let $(A,B)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $A_0$ is nonempty. Let $T:A\to B$ be a Geraghty-contraction satisfying $T(A_0)\subseteq B_0$. Suppose that the pair $(A,B)$ has the P-property. Then there exists a unique $x^{\ast }$ in A such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$.

Remark In [2], the proof of Theorem 1.5 is unnecessarily complex. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

2 Main results

Before giving our main results, we first introduce the notion of weak P-property.

Weak P-property Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with $A_0\ne \mathrm{\varnothing }$. Then the pair $(A,B)$ is said to have the weak P-property if and only if for any $x_1,x_2\in A_0$ and $y_1,y_2\in B_0$,

$$\{\begin{array}{c}d(x_1,y_1)=d(A,B),\hfill \\ d(x_2,y_2)=d(A,B)\hfill \end{array}\Rightarrow d(x_1,x_2)\le d(y_1,y_2).$$

Now we are in a position to give our main results.

Theorem 2.1 Let $(A,B)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $A_0\ne \mathrm{\varnothing }$. Let $T:A\to B$ be a Geraghty-contraction satisfying $T(A_0)\subseteq B_0$. Suppose that the pair $(A,B)$ has the weak P-property. Then there exists a unique $x^{\ast }$ in A such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$.

Proof We first prove that $B_0$ is closed. Let $\{y_n\}\subseteq B_0$ be a sequence such that $y_n\to q\in B$. It follows from the weak P-property that

$$d(y_n,y_m)\to 0\Rightarrow d(x_n,x_m)\to 0,$$

as $n,m\to \mathrm{\infty }$, where $x_n,x_m\in A_0$ and $d(x_n,y_n)=d(A,B)$, $d(x_m,y_m)=d(A,B)$. Then $\{x_n\}$ is a Cauchy sequence so that $\{x_n\}$ converges strongly to a point $p\in A$. By the continuity of metric d we have $d(p,q)=d(A,B)$, that is, $q\in B_0$, and hence $B_0$ is closed.

Let ${\overline{A}}_0$ be the closure of $A_0$, we claim that $T({\overline{A}}_0)\subseteq B_0$. In fact, if $x\in {\overline{A}}_0\setminus A_0$, then there exists a sequence $\{x_n\}\subseteq A_0$ such that $x_n\to x$. By the continuity of T and the closeness of $B_0$, we have $Tx={lim}_{n\to \mathrm{\infty }}T{x}_n\in B_0$. That is $T({\overline{A}}_0)\subseteq B_0$.

Define an operator $P_{A_0}:T({\overline{A}}_0)\to A_0$, by $P_{A_0}y=\{x\in A_0:d(x,y)=d(A,B)\}$. Since the pair $(A,B)$ has weak P-property and T is a Geraghty-contraction, we have

$$d(P_{A_0}T{x}_1,P_{A_0}T{x}_2)\le d(T{x}_1,T{x}_2)\le \beta (d(x_1,x_2))\cdot d(x_1,x_2)$$

for any $x_1,x_2\in {\overline{A}}_0$. This shows that $P_{A_0}T:{\overline{A}}_0\to {\overline{A}}_0$ is a Geraghty-contraction from complete metric subspace ${\overline{A}}_0$ into itself. Using Theorem 1.2, we can get $P_{A_0}T$ has a unique fixed point $x^{\ast }$. That is $P_{A_0}T{x}^{\ast }=x^{\ast }\in A_0$. It implies that

$$d(x^{\ast },T{x}^{\ast })=d(A,B).$$

Therefore, $x^{\ast }$ is the unique one in $A_0$ such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$. It is easy to see that $x^{\ast }$ is also the unique one in A such that $d(x^{\ast },T{x}^{\ast })=d(A,B)$. □

Remark In Theorem 2.1, P-property is weakened to weak P-property. Therefore, Theorem 2.1 is an improved result of Theorem 1.5. In addition, our proof is shorter and simpler than that in [2]. In fact, our proof process is less than one page. However, the proof process in [2] is three pages.

3 Example

Now we present an example which satisfies weak P-property but not P-property.

Consider $(R^2,d)$, where d is the Euclidean distance and the subsets $A=\{(0,0)\}$ and $B=\{y=1+\sqrt{1-x^2}\}$.

Obviously, $A_0=\{(0,0)\}$, $B_0=\{(-1,1),(1,1)\}$ and $d(A,B)=\sqrt{2}$. Furthermore,

$$d((0,0),(-1,1))=d((0,0),(1,1))=\sqrt{2},$$

however,

$$0=d((0,0),(0,0))<d((-1,1),(1,1))=2.$$

We can see that the pair $(A,B)$ satisfies the weak P-property but not the P-property.