^{1}

^{2}

^{2}

^{1}

^{2}

We introduce the concept of

The importance of fixed point theory emerges from the fact that it furnishes a unified approach and constitutes an important tool in solving equations which are not necessarily linear. A large number of problems can be formulated as nonlinear equations of the form

Best proximity point theorem analyzes the condition under which the optimisation problem, namely,

A best proximity point problem is a problem of achieving the minimum distance between two sets through a function defined on one of the sets to the other.

The very popular best approximation theorem is due to Fan [

Afterwards many authors such as Eldred and Veeramani [

In fixed point theory, other spaces of study other than metric spaces have been used by different authors. Pseudometric spaces interestingly generalise metric spaces. One of the spaces in literature that generalises the metric and pseudometric spaces is the uniform space.

Weil [

Also, Olatinwo [

In another development, Geraghty [

Another useful result is by Karapinar and Erhan [

Also, Basha [

Motivated by the results above, we develop the concept of

The following definitions are fundamental to our work.

A uniform space

If

If

If

If

If

Let

Another extension of a metric space is the

Let

Examples in literature to show that

Now, we introduce the concept of

Let

Let

Also, the following definition is required.

Let

If

A uniform space

A uniform structure

Observe that all the above maps are self-mappings.

A large number of articles investigate non-self-contractive mappings on metric spaces. Some of these are given below.

Let

Let

Among the generalisations of the Banach contraction is the proximal contraction given by Basha in [

Let

Let

Let

Given nonempty subsets

Let

An element

Now, we give the definition of

Let

For each

Let

It is easy to see that a self-mapping that is a

The following example shows that

Let

Clearly,

Now, taking

Clearly,

The following Lemma, which is true for self-mappings (see Lemma

Let

If

If

If

If

The major aim of this paper is to prove results similar to Theorem

We give the first theorem.

Let

Let

Since

Therefore, the sequence

Also, since

Let

Set

Let

Set

Let

Set

Now, we establish some results of best proximity point for

Let

the pair (

Let

Recall that

Suppose

Then by (

Hence,

Since

Next we prove the uniqueness of

Let

The proof follows from Theorem

Let

The proof follows from Theorem

We give the following example to show that (

Let

Let

(1)

Hence, (

But taking

(2)

We give the following examples to show that the

Consider the usual metric

We see that

But

The authors declare that they have no conflicts of interest.

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.