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We consider the problem of best proximity point in locally convex spaces endowed with a weakly convex digraph. For that, we introduce the notions of nonself

Fixed point theorems deal with conditions under which maps (single or multivalued) have invariant points. The theory itself is a beautiful mixture of analysis (pure and applied), topology, and geometry. Over the last 50 years or so, the theory of fixed point has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics. If the fixed point equation of given mapping does not have a solution, then it is of interest to find an approximate solution for the fixed point equation. In other words, we are searching for an element in the domain of the mapping, whose image is as close to it as possible. This situation motivates the researchers to develop the notion of best proximity point theory. It is worth to note that the best proximity point theorems can be viewed as a generalization of fixed point theorems, since most fixed point theorems can be derived as corollaries of best proximity point results (for more details, see [

Fan in [

In [

Vuong in [

In this paper, we define the concept of nonself

To start this work, we discuss some of the basic notations and terminologies which we will be using later. Let

Recall that a mapping

Recall that a pair

Let

Clearly, we have

A directed graph or digraph

Given a digraph

If whenever

A digraph

A dipath of

A finite dipath of length

A digraph is weakly convex if and only if for any

We have

The letter

Let

Let

Let

Let

We will say that a nonempty subset

Sequentially

The proof of the next result follows the same pattern of ([

Let

There exist

For any sequence

Then, there exist

Let

As

Similarly, for

Now, let

Again, since

Continuing in this manner for all

Produce a path

Using the

Now, for each

Since for all

Since

Let

There exist

For any sequence

Then,

Hence,

Then, we have

Set

Let

There exist

(ii) For any sequence

Then,

Let

Since

As

Again, let

So, there exists

Since

Continuing in this manner we construct a sequence of mappings

Note that

The above yields to

Let

Let

Let

Then,

Thus,

So,

Let

Then,

Now, let

So,

Indeed the only such point is

If the digraph

Let

There exist

For any sequence

Then,

Let

Let

Suppose that

Then,

Thus,

So,

Let

Now, let

So,

Indeed the only such point is

In order to get a common best proximity point with respect to every seminorm

Let

There exist

For any two elements

For any net

Then,

Let

Applying (ii) and the fact that

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.