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\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{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\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{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 \phantom{\rule{1em}{0ex}}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.