International Journal of Mathematics and Mathematical Sciences
Volume 2020 (2020), Article ID 7481060, 7 pages
https://doi.org/10.1155/2020/7481060

Research Article

On Best Proximity Point Theorems in Locally Convex Spaces Endowed with a Graph

B. Saadaoui,¹ S. Lazaiz,² and M. Aamri¹

¹Laboratory of Algebra, Analysis and Applications, Department of Mathematics, Ben M’sik Faculty of Sciences, University of Hassan II, Casablanca, Morocco
²LASMA Laboratory, Department of Mathematics, Dhar El Mahraz Faculty of Sciences, University of Sidi Mohamed Ben Abdellah, Fes, Morocco

Correspondence should be addressed to B. Saadaoui; saadaoui.brahim2015@gmail.com

Received 20 April 2020; Accepted 30 June 2020; Published 1 August 2020

Academic Editor: Jean Michel Rakotoson

Copyright © 2020 B. Saadaoui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 -contraction and -nonexpansive mappings, and we show that for each seminorm, we have a best proximity point. In addition, we conclude our work with a result showing the existence of best proximity point for every seminorm.

1. Introduction

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 [1–6]).

Fan in [7] gave a common generalization of both theorems of Kakutani and Tychonoff. He proved under compactness of the domain in a locally convex space that every upper semicontinuous set valued mapping has a fixed point in the sense that there is an element which belongs to his image.

In [8] Ling established some properties relating the concepts of normal structure and submeans in a Hausdorff locally convex space and obtained a fixed point theorem for left reversible semigroups of nonexpansive mapping with a compactness of the domain. This extends a result obtained by Lau and Takahashi in Banach space [9] (generalization of Lim’s fixed point) and shows that Lau and Takahashi’s result remains valid in the more general setting of a locally convex space.

Vuong in [10] established a fixed point theorem for nonexpansive mappings in a locally convex space with normal structure and the compactness of the domain.

In this paper, we define the concept of nonself -contraction mappings in locally convex spaces endowed with a digraph . Then, we obtain sufficient conditions for the existence of a best proximity point for such mappings when both the domain and the range satisfy the -property condition. In addition, we discuss some existence results for nonself -nonexpansive mappings.

2. Preliminaries

To start this work, we discuss some of the basic notations and terminologies which we will be using later. Let be a Hausdorff locally convex space and be a family of continuous seminorms which generates the topology of . Let be two subsets of and .

Recall that a mapping is said to be contraction if there is for any :

and is said to be nonexpansive if for any ,

Recall that a pair satisfies a property if both and has that property. For example, and .

Let be a pair of sets in a Hausdorff locally convex space . Define

Clearly, we have

A directed graph or digraph is determined by a nonempty set of its vertices and the set of its directed edges. A digraph is reflexive if each vertex has a loop. For more details, one can consult the book [11].

Definition 1. Given a digraph .

 If whenever , then the digraph is called an oriented graph. A digraph G is transitive whenever , for any . A dipath of is a sequence , ,…, , …with for each . A finite dipath of length from x to y is a sequence of vertices with and , .  is the set of all vertices which are contained in some path beginning at (i.e., ). A digraph is weakly convex if and only if for any , and in ,

We have

The letter denotes the undirected graph obtained from by ignoring the direction of edges.
Let be a pair such that is nonempty.

Definition 2. Let be a locally convex space endowed with a digraph and . A mapping is said to be -contraction mapping if there exists for any such that :

(a).(b),

Definition 3. Let be a locally convex space endowed with a digraph and . A mapping is said to be -nonexpansive mapping if for any such that :

(a).(b),

Definition 4. Let be a pair of sets in a Hausdorff locally convex space Define . Then, the pair is said to have the property if and only if

.

Definition 5. We will say that a nonempty subset of is

(1)compact if and only if for every net in such that , there is a subnet which is converging to a point and (2)Sequentially compact if and only if for every sequence in such that , there exists a subsequence of which is converging to a point in and .

3. Main Results

The proof of the next result follows the same pattern of ([12], Theorem 3.2). For the sake of completeness, we give the proof.

Theorem 1. Let be a locally convex space endowed with a reflexive digraph . Let be a nonempty pair in such that has the -property and is convex complete. Let be a -contraction mapping such that . Assume that

(i)There exist and in such that and .(ii)For any sequence in with and .

Then, there exist such that .

Proof. Let . By (i), there exist two points such that and

and a finite sequence of such that and for all .
As and , there exist such that

and since and , by the condition (b) of Definition 2, we get

Similarly, for , there exist such that

for all .
Now, let . Thus, the finite sequence is a path from to .
Again, since and for each , there exist such that

for all . Also, we have

Continuing in this manner for all , we obtain a sequence where

Produce a path from to (see Figure 1 for illustration) in such a way that

Using the -property of and equation (18) for any , we have

Now, for each ,

Since for all and , and is a -contraction, it follows that for any ,

for some . Repeating the process, it follows that for all ,

where . Hence, is a Cauchy sequence. So, . By condition (ii), we get for any . Hence,

i.e.,

where by continuity of , we get

Since , we obtain

Figure 1: sequence.

Theorem 2. Let be a locally convex space endowed with a weakly convex and reflexive digraph . Let be a nonempty pair that has the -property in . Let be a -nonexpansive mapping such that is convex complete and . Assume that

(i)There exist and in such that and .(ii)For any sequence in with and , .

Then,

Proof. . Let and define by

is -contraction. By Theorem 1, there exists a best proximity point of , i.e.,

Hence,

Then, we have

Set for each . Hence by equation (31), we get

for all . Thus,

Theorem 3. Let be a locally convex space endowed with a weakly convex and reflexive digraph . Let be a nonempty pair with the -property in . Let be a -nonexpansive mapping such that is convex complete and sequentially -compact and . Assume that

(i)There exist and in such that and . (ii) For any sequence in with and , .

Then, has a best proximity point in the sense that

Proof. Let and . Set

Since is convex and , we get , and since , there exists such that

As is -contraction, by Theorem 1, has a best point proximity such that

Again, let with . Set . We have

So, there exists such that

and by the above construction, we also have . That is,

Since is -nonexpansive and is weakly reflexive convex graph, we obtain

and as is -contraction, then has a best proximity point that satisfies

Continuing in this manner we construct a sequence of mappings satisfying for all

where is a nondecreasing sequence of .
Note that is a -contraction with as its best proximity point. Since for any and is -compact, there exists a subsequence witch converges to in and satisfies

for any . is -nonexpansive mapping; thus,

i.e.,

The above yields to

and since

for any , we get

Example 1. Let be the space of continuous real functions, i.e.,

Let be the family of seminorms generated the topology of such that for every and , a compact of , we have

Let

Then, is convex. Moreover, we have

Thus,

So, has the -property.
Let . We define a digraph on as follows:

Then, is weakly convex and reflexive digraph. Obviously, is sequentially -compact. Since , assumption (i) of Theorem 3 holds.
Now, let be the mapping defined by

is -nonexpansive since we have

So, is -nonexpansive and . Then, there exists such that

Indeed the only such point is .
If the digraph is partial order, we get the following interesting consequence.

Corollary 1. Let be a locally convex space endowed with a partial order . Let be a nonempty pair with the -property in . Let be a -nonexpansive mapping such that is convex complete and sequentially -compact and . Assume that

(i)There exist and in such that and .(ii)For any sequence in with and , we have .

Then, has a best proximity point in the sense that

Example 3.6. Let be the space of real sequences, i.e.,

Let be the family of seminorms generated the topology of such that for every ,

Suppose that is the canonical basis of and let

Then, is weakly compact subset of since it is convex bounded and closed. Moreover, as for each , we have ; thus, for all , we have

Thus,

So, has -property.
Let . We define a partial order on as follows:

where for .
is obviously sequentially -compact.
Now, let be the mapping defined by

for each . is -nonexpansive since for every , we have

So, is -nonexpansive and . Then, there exists such that

Indeed the only such point is .
In order to get a common best proximity point with respect to every seminorm , we have the next result.

Theorem 4. Let be a locally convex space endowed with a digraph which is weakly convex and reflexive. Let be a nonempty pair with the -property in . Let be a -nonexpansive mapping such that is convex complete and -compact and . Assume that

(i)There exist and in such that and for all .(ii)For any two elements in , there is a finite path between them.(iii)For any net in A with and .

Then, has a best proximity point in the sense that there is such that

Proof. Let . By Theorem 3, there exist such that

Applying (ii) and the fact that is -compact, the net has a converging subnet which is converging to with for any . Moreover, we have for all ,

for all . The proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

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

References

H. Aydi, H. Lakzian, Z. D. Mitrović, and S. Radenović, “Best proximity points of mt-cyclic contractions with property UC,” Numerical Functional Analysis and Optimization, vol. 41, no. 7, pp. 871–882, 2020.

K. Chaira and S. Lazaiz, “Best proximity point theorems for cyclic relatively-nonexpansive mappings in modular spaces,” Abstract and Applied Analysis, vol. 2018, Article ID 8084712, 8 pages, 2018.

A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, pp. 283–293, 2005.

M. Gabeleh, “Cyclic relatively nonexpansive mappings in convex metric spaces,” Miskolc Mathematical Notes, vol. 16, no. 1, pp. 133–144, 2015.

S. Sahmim, A. Felhi, and H. Aydi, “Convergence and best proximity points for generalized contraction pairs,” Mathematics, vol. 7, no. 2, p. 176, 2019.

V. Sankar R. and P. Veeramani, “Best proximity pair theorems for relatively nonexpansive mappings,” Applied General Topology, vol. 10, no. 1, pp. 21–28, 2009.

K. Fan, “Fixed-point and minimax theorems in locally convex topological linear spaces,” Proceedings of the National Academy of Sciences, vol. 38, no. 2, pp. 121–126, 1952.

J. M. Ling, “Fixed points of nonexpansive maps on locally convex spaces,” Bulletin of the Korean Mathematical Society, vol. 37, no. 1, pp. 21–36, 2000.

A. T.-M. Lau and W. Takahashi, “Invariant submeans and semigroups of nonexpansive mappings on banach spaces with normal structure,” Journal of Functional Analysis, vol. 142, no. 1, pp. 79–88, 1996.

H. D. Vuong, “A fixed point theorem for nonexpansive mappings in locally convex spaces,” Vietnam Journal of Mathematics, vol. 34, 2006.

M. Rigo, Advanced Graph Theory and Combinatorics, John Wiley & Sons, Hoboken, NJ, USA, 2016.

A. Sultana and V. Vetrivel, “Best proximity points of contractive mappings on a metric space with a graph and applications,” Applied General Topology, vol. 18, no. 1, pp. 13–21, 2017.