We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions

which satisfy the following conditions:

- (a)

- (b)
*ψ* is subadditive, that is,
;

- (c)

- (d)
.

We introduce the following contraction.

**Definition 2.1** Let

*A*,

*B* be two nonempty subsets of a metric space

. A mapping

is said to be a

*ψ*-Geraghty contraction if there exists

such that

**Remark 2.1** Notice that since

, we have

We are now ready to state and prove our main theorem.

**Theorem 2.1**
*Let*
*be a pair of nonempty closed subsets of a complete metric space*
*such that*
*is nonempty*. *Let*
*be a*
*ψ*-*Geraghty contraction satisfying*
. *Suppose that the pair*
*has the*
*P*-*property*. *Then there exists a unique*
*in*
*A*
*such that*
.

*Proof* Regarding that

is nonempty, we take

. Since

, we can find

such that

. Analogously, regarding the assumption

, we determine

such that

. Recursively, we obtain a sequence

in

satisfying

Since

has the

*P*-property, we derive that

If there exists

such that

, then the proof is completed. Indeed,

and consequently,

. On the other hand, due to (8) we have

Therefore, we conclude that

For the rest of the proof, we suppose that

for any

. Since

*T* is a

*ψ*-Geraghty contraction, for any ℕ, we have that

Consequently,

is a nonincreasing sequence and bounded below, and so

exists. Let

. Assume that

. Then, from (6), we have

for each

, which implies that

On the other hand, since

, we conclude

, that is,

Notice that since

for any

, for fixed

, we have

, and since

satisfies the

*P*-property,

. In what follows, we prove that

is a Cauchy sequence. On the contrary, assume that we have

By using the triangular inequality,

By (12) and since

, by the comment mentioned above, regarding the discussion on the

*P*-property above together with (12), (15) and the property of the function

*ψ*, we derive that

By a simple manipulation, (16) yields that

By taking the properties of the function

*ψ* into account, together with (13) and

and

, the last inequality yields

Therefore

. By taking the fact

into account, we get

Regarding the properties of the function *ψ*, the limit above contradicts the assumption (14). Therefore,
is a Cauchy sequence.

Since
and *A* is a closed subset of the complete metric space
, we can find
such that
.

We claim that

. Suppose, on the contrary, that

. This means that we can find

such that for each

, there exists

with

Due to the properties of

*ψ*, we get

Using the fact that

*T* is a

*ψ*-Geraghty contraction, we have

for any

. Since

and

, we can find

such that for

Consequently, for

we have

a contradiction. Therefore,
.

Regarding the fact that the sequence

is a constant sequence with value

, we derive

which is equivalent to saying that
is the best proximity point of *T*. This completes the proof of the existence of a best proximity point.

We shall show the uniqueness of the best proximity point of

*T*. Suppose that

and

are two distinct best proximity points of

*T*, that is,

. This implies that

Using the

*P*-property, we have

Using the fact that

*T* is a

*ψ*-Geraghty contraction, we have

a contradiction. This completes the proof. □

Notice that the pair
satisfies the *P*-property for any nonempty subset *A* of *X*. Consequently, we have the following corollary.

**Corollary 2.1**
*Let*
*be a complete metric space and*
*A*
*be a nonempty closed subset of X*. *Let*
*be a*
*ψ*-*Geraghty*-*contraction*. *Then*
*T*
*has a unique fixed point*.

*Proof* Apply Theorem 2.1 with
. □

If we take
we obtain Theorem 1.2 as a corollary of Theorem 2.1.

**Corollary 2.2**
*Let*
*be a complete metric space and*
*A*
*be a nonempty closed subset of X*. *Let*
*be a Geraghty*-*contraction*. *Then*
*T*
*has a unique fixed point*.

*Proof* Apply Theorem 2.1 with
and
. □

In order to illustrate our results, we present the following example.

**Example 2.1** Suppose that

with the metric

and consider the closed subsets

and
and
.

Set

to be the mapping defined by

Since
, the pair
has the *P*-property.

Notice that
and
and
.

Without loss of generality, we assume that

. Moreover,

where
is defined as
.

Therefore,

*T* is a

*ψ*-Geraghty-contraction. Notice that the pair

satisfies the

*P*-property. Indeed, if

then

and

and hence

Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique

such that

More precisely, the point
is the best proximity point of *T*.

**Example 2.2** Let

and

be a metric on

*X*. Suppose

and

are two closed subsets of ℝ. Define

by

. Define

by

and

by

. Clearly,

. Now we have

Also,

. Further, clearly, the pair

has the

*P*-property. Let

. Note that, if

, then condition (6) holds. Hence, we assume that

. We shall show that (6) holds. Suppose, on the contrary, there exist

such that

which yields that
, a contradiction. Therefore condition (6) holds for all
. Hence, the conditions of Theorem 2.1 hold and *T* has a unique best proximity point. Here,
is the best proximity point of *T*.