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

Authors: B. Saadaoui; S. Lazaiz; M. Aamri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## 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 nonselfG-contraction and G-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.


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## Body

## 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 nonselfG-contraction mappings in locally convex spaces endowed with a digraph G. 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 P-property condition. In addition, we discuss some existence results for nonself G-nonexpansive mappings.


## 2. Preliminaries

To start this work, we discuss some of the basic notations and terminologies which we will be using later. LetE be a Hausdorff locally convex space and P be a family of continuous seminorms which generates the topology of E. Let A,B be two subsets of E and p∈P.(1)δpA=suppx−x′:x,x′∈A,distpA,B=infpx−y:x∈A;y∈B.Recall that a mappingT:A⟶A is said to be contraction if there is α∈0,1 for any x,y∈A:(2)pTx−Ty≤αpx−y,and T:A⟶A is said to be nonexpansive if for any x,y∈A,(3)pTx−Ty≤px−y.Recall that a pairA,B satisfies a property if both A and B has that property. For example, A,B⊆C,D⇔A⊆C and B⊆D.LetA,B be a pair of sets in a Hausdorff locally convex space E,P. Define(4)A0p=x∈A:px−y′=distpA,Bfor somey′∈B,B0p=y∈B:px′−y=distpA,Bfor somex′∈A.Clearly, we have(5)A0p,B0p⊆A,B.A directed graph or digraphG is determined by a nonempty set VG of its vertices and the set EG⊂VG×VG 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 digraphG=V,E.If wheneverx,y∈EG⇒y,x∉EG, then the digraph G is called an oriented graph.A digraphG is transitive whenever x,y∈EGandy,z∈EG⇒x,z∈EG, for any x,y,z∈VG.A dipath ofG is a sequence a0, a1,…, an, …with ai,ai+1∈EG for each i∈ℕ.A finite dipath of lengthn from x to y is a sequence of n+1 vertices a0,a1,…,an with ai,ai+1∈EG and x=a0, y=an.xG is the set of all vertices which are contained in some path beginning at x (i.e., y∈xG⇔there exista0,a1,…,anwithai,ai+1∈EGandx=a0,y=an).A digraph is weakly convex if and only if for anyx,y,z, and h in E,(6)x∈zG,y∈hG,andλ∈0,1.
We have(7)λx+1−λy∈λz+1−λhG.
The letterG˜ denotes the undirected graph obtained from G by ignoring the direction of edges.
LetA,B be a pair such that A0p is nonempty.Definition 2.
LetE,P be a locally convex space endowed with a digraph G and A,B⊂E. A mapping T:A⟶B is said to be G-contraction mapping if there exists k∈0,1, for any x,y∈A such that y∈xG:(a)
pTx−Ty≤kpx−y.(b)
For anyx1,x2∈A0p,(8)px1−Tx=distpA,Bpx2−Ty=distpA,B⇒x2∈x1G.Definition 3.
LetE,P be a locally convex space endowed with a digraph G and A,B⊂E. A mapping T:A⟶B is said to be G-nonexpansive mapping if for any x,y∈A such that y∈xG:(a)
pTx−Ty≤px−y.(b)
For anyx1,x2∈A0p,(9)px1−Tx=distpA,Bpx2−Ty=distpA,B⇒x2∈x1G.Definition 4.
LetA,B be a pair of sets in a Hausdorff locally convex space E,P. Define A0p≠∅. Then, the pair A,B is said to have the P−property if and only if(10)px1−y1=distpA,Bpx2−y2=distpA,B⇒px1−x2=py1−y2,for allx1,x2∈Aandy1,y2∈B.Definition 5.
We will say that a nonempty subsetA of E is(1)
.G−compact if and only if for every net xαα∈Ω in A such that xα∈xβG, there is a subnet xα′α′∈Ω which is converging to a point x∈A and x∈xα′Gfor anyα′∈Ω.(2)
Sequentially.G−compact if and only if for every sequence xnn≥1 in A such that xn+1∈xnG, there exists a subsequence xknk of xnn≥1 which is converging to a point in A and x∈xnG.


## 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.
LetE,P be a locally convex space endowed with a reflexive digraph G. Let A,B be a nonempty pair in E such that A,B has the P-property and A is convex complete. Let T:A⟶B be a G-contraction mapping such that TA0p⊆B0p. Assume that(i)
There existx0 and x1 in A0p such that x1∈x0G⊆A0p and px1−Tx0=distpA,B.(ii)
For any sequencesnn∈ℕ in A with sn⟶s and sn+1∈snGfor anyn∈ℕ,s∈snG.
Then, there existx∗∈A0p such that px∗−Tx∗=distpA,B.Proof.
Letp∈P. By (i), there exist two points x0,x1∈A0p such that x1∈x0G and(11)px1−Tx0=distpA,B,and a finite sequence y0ii=0i=N of A0p such that x0=y00,y01,…,y0N=x1 and y0i∈y0i−1G for all i∈1,2,…,N.
Asy01∈A0p and TA0p⊆B0, there exist y11∈A0p such that(12)py11−Ty01=distpA,B,and since px1−Tx0=distpA,B and y01∈x0G, by the condition (b) of Definition 2, we get(13)y11∈x1G.
Similarly, fori=2,3,…,N, there exist y1i∈A0p such that(14)py1i−Ty0i=distpA,B,y1i∈y1i−1G,for all 1≤i≤N+1.
Now, letx2=y1N. Thus, the finite sequence y1ii=0i=N is a path from x1=y0N=y10 to y1N=x2.
Again, sincey1i∈A0p and TA0p⊆B0p for each i=0,1,2,…,N, there exist y2i∈A0p such that(15)py2i−Ty1i=distpA,B,y2i∈y2i−1G,for all i∈1,…,N+1. Also, we have(16)px2−Tx1=distpA,B,x2∈x1G.
Continuing in this manner for alln∈ℕ, we obtain a sequence xnn∈ℕ where(17)xn+1∈xnG,pxn+1−Txn=distpA,B.
Produce a pathynii=0i=N from xn=yn0 to ynN=xn+1 (see Figure 1 for illustration) in such a way that(18)pyni+1−Tyni=distpA,B,∀i=0,1,…,N−1.
Using theP-property of A,B and equation (18) for any n∈ℕ, we have(19)pyni−1−yni=pTyn−1i−1−Tyn−1i,∀i=1,…,N+1.
Now, for eachn∈ℕ,(20)pxn−xn+1≤pyn0−ynN≤pyn0−yn1+pyn1−yn2+pyn2−yn3+…+pynN−1−ynN=∑i=1N+1pyni−1−yni=∑i=1N+1pTyn−1i−1−Tyn−1i.
Since for alln∈ℕ and 1≤i≤N+1, yni∈yni−1G and T is a G-contraction, it follows that for any n∈ℕ,(21)pxn−xn+1≤α∑i=1N+1pyn−1i−1−yn−1i,for some α∈0,1. Repeating the process, it follows that for all n∈ℕ,(22)pxn−xn+1≤Mαn,where M=∑i=1N+1py0i−y0i−1. Hence, xnn is a Cauchy sequence. So, xn⟶x∗∈A. By condition (ii), we get x∗∈xnG for any n∈ℕ. Hence,(23)pTxn−Tx∗≤αpxn−x∗,i.e.,(24)pTxn−Tx∗⟶0,where by continuity of p, we get(25)pxn+1−Txn⟶px∗−Tx∗.
Sincepxn+1−Txn=distpA,B, we obtain(26)px∗−Tx∗=distpA,B.Figure 1
xn sequence.Theorem 2.
LetE,P be a locally convex space endowed with a weakly convex and reflexive digraph G. Let A,B be a nonempty pair that has the P-property in E. Let T:A⟶B be a G-nonexpansive mapping such that A is convex complete and TA0p⊆B0p. Assume that(i)
There existx0 and x1 in A0p such that x1∈x0G⊆A0p and px1−Tx0=distpA,B.(ii)
For any sequencesnn in A with sn⟶s and sn+1∈snG, s∈snG.
Then,(27)infx∈Apx−Tx=distpA,B.Proof.
x0∈A0p. Let λ∈0,1 and define Tλ:A⟶B by(28)Tλx=Tλx0+1−λx.
Tλ is G-contraction. By Theorem 1, there exists a best proximity point wλ∈A0p of Tλ, i.e.,(29)pwλ−Tλwλ=distpA,B.
Hence,(30)pλx0+1−λwλ−Tλx0+1−λwλ=pλx0−λwλ+wλ−Tλx0+1−λwλ≤λpx0−wλ+distpA,B.
Then, we have(31)pλx0+1−λwλ−Tλx0+1−λwλ≤λδpA+distpA,B.
Setxn=1/nx0+1−1/nw1/n for each n≥1. Hence by equation (31), we get(32)pxn−Txn≤δpAn+distpA,B,for all n≥1. Thus,(33)infx∈Apx−Tx≤limn⟶∞pxn−Txn=distpA,B.Theorem 3.
LetE,P be a locally convex space endowed with a weakly convex and reflexive digraph G. Let A,B be a nonempty pair with the P-property in E. Let T:A⟶B be a G-nonexpansive mapping such that A is convex complete and sequentially .G-compact and TA0p⊆B0p. Assume that(i)
There existx00 and x10 in A0p such that x10∈x00G⊆A0p and px10−Tx00=distpA,B.(ii) For any sequencesnn in A with sn⟶s and sn+1∈snG, s∈snG.
Then,T has a best proximity point x in the sense that(34)px−Tx=distpA,B.Proof.
Letp∈P and λ1∈0,1. Set(35)T1x=Tλ1x00+1−λ1x.
SinceA0p is convex and TA0p⊆B0p, we get T1A0p⊆B0p, and since T1x00=Tx00, there exists x01∈A0p such that(36)x01∈x00G,px01−T1x00=distpA,B.
AsT1 is G-contraction, by Theorem 1, T1 has a best point proximity x1∗ such that(37)x1∗∈x00G.
Again, letλ2∈0,1 with λ1<λ2. Set T2x=Tλ2x1∗+1−λ2x. We have(38)T2x1∗=Tx1∗∈B0p.
So, there existsx11∈A0p such that(39)px11−T2x1∗=distpA,B,and by the above construction, we also have px11−T2x1∗=px1∗−T1x1∗. That is,(40)px11−Tλ1x1∗+1−λ1x1∗=distpA,B,px1∗−Tλ1x00+1−λ1x1∗=distpA,B.
SinceT is G-nonexpansive and G is weakly reflexive convex graph, we obtain(41)x11∈x1∗G,px11−T2x1∗=distpA,B,and as T2 is G-contraction, then T2 has a best proximity point x2∗∈A0p that satisfies(42)x2∗∈x11G.
Continuing in this manner we construct a sequence of mappingsTn satisfying for all n≥0:(43)Tn+1x=Tλn+1xn∗+1−λn+1x,xn+1∗∈xn∗G,x0∗=x00,where λn is a nondecreasing sequence of 0,1.
Note thatTn+1 is a G-contraction with xn+1∗ as its best proximity point. Since xn+1∗∈xn∗G for any n∈ℕ and A is .G-compact, there exists a subsequence xnk∗k witch converges to x∗ in A and satisfies(44)x∗∈xnk∗G,for any k. T is G-nonexpansive mapping; thus,(45)pTxnk∗−Tx∗≤pxnk∗−x∗⟶0,i.e.,(46)Txnk∗⟶Tx∗.
The above yields to(47)px∗−Tx∗≤px∗−xnk+1∗+pxnk+1∗−Tnk+1xnk+1∗+pTnk+1xnk+1∗−Tnk+1xnk∗+pTxnk∗−Tx∗≤px∗−xnk+1∗+pxnk+1∗−Tnk+1xnk+1∗+1−λnk+1pxnk+1∗−xnk∗+pTxnk∗−Tx∗,and since(48)pxnk+1∗−Tnk+1xnk+1∗=distpA,B,for any k∈ℕ, we get(49)px∗−Tx∗=distpA,B.Example 1.
LetX be the space of continuous real functions, i.e.,(50)X=Cℝ=f:ℝ⟶ℝ:continuous.
LetP=pKK⊂ℝ be the family of seminorms generated the topology of X such that for every f∈X and K, a compact of ℝ, we have(51)pKf=maxx∈Kfx.
Let(52)A=λ1−1n+1−λ1−1m:λ∈0,1andm,n∈ℕ∗∪1,B=λ2+1n+1−λ2+1m:λ∈0,1andm,n∈ℕ∗∪2.
Then,A is convex. Moreover, we have(53)pK1−2=1=distpKA,B.
Thus,(54)A0pK=1.
So,A,B has the P-property.
Letf1,f2∈A. We define a digraph G on A as follows:(55)f1,f2∈EG⇔∃c∈1,2such thatf1≤cf2.
Then,G is weakly convex and reflexive digraph. Obviously, A is sequentially .G-compact. Since pK1−T1=distpKA,B, assumption (i) of Theorem 3 holds.
Now, letT:A⟶B be the mapping defined by(56)Tλ1−1n+1−λ1−1m=λ2+1n+1−λ2+1m,T1=2.
T is G-nonexpansive since we have(57)pKTλ1−1n+1−λ1−1m−Tλ′1−1n′+1−λ′1−1m′=pKλ2+1n+1−λ2+1m−λ′2+1n′−1−λ′2+1m′=pKλ1−1n+1−λ1−1m−λ′1−1n′−1−λ′1−1m′.
So,T is G-nonexpansive and TA0pK⊆B0pK. Then, there exists f∈A such that(58)pKf−Tf=distpKA,B.
Indeed the only such point isfsuch thatfx=1for anyx∈ℝ.
If the digraphG is partial order, we get the following interesting consequence.Corollary 1.
LetE,P be a locally convex space endowed with a partial order ≺¯. Let A,B be a nonempty pair with the P-property in E. Let T:A⟶B be a ≺¯-nonexpansive mapping such that A is convex complete and sequentially ≺¯-compact and TA0p⊆B0p. Assume that(i)
There existx00 and x10 in A0p such that x10⪯¯x00 and px10−Tx00=distpA,B.(ii)
For any sequencesnn in A with sn⟶s and sn+1≺¯sn, we have s≺¯sn.
Then,T has a best proximity point x in the sense that(59)px−Tx=distpA,B.Example 3.6.
LetX be the space of real sequences, i.e.,(60)X=ℝℕ=x=xii∈ℕ:xi∈ℝ.
LetP=pnn∈ℕ be the family of seminorms generated the topology of X such that for every x=xii≥1∈X,(61)pnx=max1≤i≤nxi.
Suppose thatekk∈ℕ is the canonical basis of X and let(62)A=λe1+1−λe2:λ∈0,1,B=λe3+1−λe4:λ∈0,1.
Then,A is weakly compact subset of X since it is convex bounded and closed. Moreover, as for each λ∈0,1, we have 1/2≤maxλ;1−λ; thus, for all n∈ℕ, we have(63)pne1+e22−e3+e42=12=distpnA,B.
Thus,(64)A0pn=e1+e22.
So,A,B has P-property.
Letx1,x2∈A. We define a partial order on A as follows:(65)x1≺¯x2⇔λ1≤λ2,where xi=λie1+1−λie2 for i=1,2.
A is obviously sequentially ≺¯-compact.
Now, letT:A⟶B be the mapping defined by(66)Tλe1+1−λe2=λe3+1−λe4,for each λ∈0,1. T is ≺¯-nonexpansive since for every i∈ℕ, we have(67)pnTλie1+1−λie2−Tλi+1e1+1−λi+1e2=pnλie3+1−λie4−λi+1e3+1−λi+1e4=pnλie1+1−λie2−λi+1e1+1−λi+1e2.
So,T is ≺¯-nonexpansive and TA0p⊆B0p=e3+e4/2. Then, there exists x∈A such that(68)pnx−Tx=distpnA,B.
Indeed the only such point isx=e1+e2/2.
In order to get a common best proximity point with respect to every seminormpα, we have the next result.Theorem 4.
LetE,P=pαα∈Ω be a locally convex space endowed with a digraph G which is weakly convex and reflexive. Let A,B be a nonempty pair with the P-property in E. Let T:A⟶B be a G-nonexpansive mapping such that A is convex complete and .G-compact and TA0pα⊆B0pα. Assume that(i)
There existx0 and x1 in A0pα such that x1∈x0G⊆A0pα and pαx1−Tx0=distpαA,B for all α∈Ω.(ii)
For any two elementsx,y in A, there is a finite path between them.(iii)
For any netsαα∈Ω in A with sα⟶s and sα∈sβG∀β<α,we haves∈sαG∀α∈Ω.
Then,T has a best proximity point in the sense that there is x∗∈A such that(69)px∗−Tx∗=distpA,Bfor anyp∈P.Proof.
Letα∈Ω. By Theorem 3, there exist xα∈A0pα such that(70)pαxα−Txα=distpαA,B.
Applying (ii) and the fact thatA is .G-compact, the net xαα∈Ω has a converging subnet xαα′∈Ω′ which is converging to x∗ with x∗∈xα′G for any α′∈Ω′. Moreover, we have for all α′∈Ω′,(71)pα′x∗−Tx∗≤pα′x∗−xα′+pα′xα′−Txα′+pα′Txα′−Tx∗≤pα′x∗−xα′+pα′xα′−Txα′+pxα′−x∗≤2px∗−xα′+pxα′−Txα′=2px∗−xα′+distα′A,B≤ε+distα′A,B,for all ε>0. The proof is completed.



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