R E S E A R C H
Open Access
Common best proximity point theorem
for multivalued mappings in partially ordered
metric spaces
V Pragadeeswarar
1, G Poonguzali
2, M Marudai
2and Stojan Radenovi´c
3,4**Correspondence:
3Nonlinear Analysis Research Group,
Ton Duc Thang University, Ho Chi Minh City, Vietnam
4Faculty of Mathematics and
Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article
Abstract
In this paper, we prove the existence of a common best proximity point for a pair of multivalued non-self mappings in partially ordered metric spaces. Also, we provide some interesting examples to illustrate our main results.
MSC: 41A65; 90C30; 47H10
Keywords: partially ordered set; proximal relation;P-property; altering distance function; best proximity point; common best proximity point
1 Introduction
The study of multivalued mappings plays a vital role in pure and applied mathematics because of its many applications, for instance, in real and complex analysis. In the litera-ture, there are many researchers focusing on the study of abstract and practical problems which involve multivalued mappings. As a matter of fact, amongst the various approaches to develop this theory, one of the most interesting is based on best proximity point theory. One can find the existence and convergence of best proximity points in [–]. For the existence of a best proximity point in the setting of partially ordered metric spaces, see [– ]. Also, for more results on a best proximity point of multivalued non-self mappings, we suggest [–].
2 Preliminaries
In this section, we give some basic definitions and notions that will be used frequently. LetXbe a nonempty set such that (X,d,) is a partially ordered metric space. Consider AandBto be nonempty subsets of the metric space (X,d). We denote byCB(X) the class of all nonempty closed and bounded subsets ofX.
δ(A,B) :=supd(a,b) :a∈Aandb∈B, D(A,B) :=infd(a,b) :a∈Aandb∈B, A=
a∈A:d(a,b) =D(A,B) for someb∈B, B=
b∈B:d(a,b) =D(A,B) for somea∈A.
Definition . LetT :A→B be any multivalued mapping. Then an elementx∈Ais
said to be a best proximity point ifD(x,Tx) =D(A,B).
Definition . Given multivalued non-self mappingsS:A→BandT:A→B, an
el-ementa∈Ais called acommon best proximity pointof the mappings if they satisfy the condition thatD(a,Sa) =D(a,Ta) =D(A,B).
Definition . ([]) A functionψ : [,∞)→[,∞) is said to be an altering distance function if it satisfies the following conditions:
(i) ψis continuous and nondecreasing. (ii) ψ(t) = if and only ift= .
Example . Defineψ: [,∞)→[,∞) byψ(t) =kt, wherek< . Thenψis an altering distance function.
Definition .([]) Let (A,B) be a pair of nonempty subsets of a metric spaceX with A=∅. Then the pair (A,B) is said to have theP-property if and only if
d(a,b) =D(A,B)
d(a,b) =D(A,B) ⎫ ⎬
⎭ ⇒ d(a,a) =d(b,b),
wherea,a∈Aandb,b∈B.
The existence of fixed points in partially ordered metric spaces was first established by Nieto and Rodriguez-Lopez []. In this direction, Choudhury and Metiya [] proved the existence of a fixed point for multivalued self mappings in partially ordered metric spaces. In this paper, our main objective is to establish the existence of best proximity points and common best proximity points of multivalued mappings in partially ordered metric spaces. Also, our results generalize the corresponding results of []. In particular, the aim of this paper is to initiate the study of common best proximity points of multivalued mappings in partially ordered metric spaces.
Here we define the notion of proximal relation between two subsets ofX.
Definition .([]) LetAandBbe two nonempty subsets of a partially ordered met-ric space (X,d,) such thatA=∅. LetB andB be two nonempty subsets ofB. The proximal relations betweenBandBare denoted and defined as follows:
(i) B≺()Bif, for everyb∈Bwithd(a,b) =D(A,B), there existsb∈Bwith d(a,b) =D(A,B)such thataa.
(ii) B≺()Bif, for everyb∈Bwithd(a,b) =D(A,B), there existsb∈Bwith d(a,b) =D(A,B)such thataa.
(iii) B≺()BifB≺()BandB≺()B. 3 Main results
Now, we state our first main result in this section.
P-property.Let T:A→CB(B)be a multivalued mapping such that the following conditions are satisfied:
(i) There exist elementsa,ainAandb∈Tasuch that d(a,b) =D(A,B) and aa.
(ii) Tais included inBfor alla∈Aand
δ(Ta,Tb)≤ψM(a,b)+LN(a,b) for all comparablea,b∈A, () whereM(a,b) =max{d(a,b),D(a,Ta) –D(A,B),D(b,Tb) –D(A,B),
D(a,Tb)+D(b,Ta)
–D(A,B)},L≥,N(a,b) =min{D(a,Ta) –D(A,B),D(b,Tb) –D(A,B), D(a,Tb) –D(A,B),D(b,Ta) –D(A,B)}andψ: [,∞)→[,∞)is a nondecreasing and upper-semicontinuous function withψ(t) <tfor eacht> .
(iii) Fora,b∈A,abimpliesTa≺()Tb.
(iv) If{an}is a nondecreasing sequence inAsuch thatan→a,thenanafor alln.
Then there exists an element a in A such that
D(a,Ta) =D(A,B).
Proof By assumption (i), there exist two elements a,a in A andb∈Ta such that d(a,b) =D(A,B) andaa. By assumption (iii),Ta≺()Ta, there existsb∈Tawith d(a,b) =D(A,B) such thataa. In general, for eachn∈N, there existan+∈Aand bn∈Tansuch thatd(an+,bn) =D(A,B). Hence, we obtain
d(an+,bn) =D(an+,Tan) =D(A,B)
for alln∈Nwithaaa · · · anan+ · · ·. () If there existsnsuch thatan=an+, thenD(an+,Tan) =D(an,Tan) =D(A,B). This
means thatanis a best proximity point ofT and hence the proof. Thus, we can suppose
thatan=an+for alln. Sinced(an+,bn) =D(A,B) andd(an,bn–) =D(A,B) and (A,B) has theP-property,
d(an,an+) =d(bn–,bn) for alln∈N. ()
Sincean–≺an,
d(an,an+) =d(bn–,bn)≤δ(Tan–,Tan)≤ψ
M(an–,an)
+LN(an–,an). ()
By the triangle inequality ofd, we have
M(an–,an)
=max d(an–,an),D(an–,Tan–) –D(A,B),D(an,Tan) –D(A,B),
D(an–,Tan) +D(an,Tan–)
–D(A,B)
≤max d(an–,an),d(an–,bn–) –D(A,B),d(an,bn) –D(A,B),
d(an–,bn) +d(an,bn–)
–D(A,B)
≤max d(an–,an),d(an–,bn–) +d(bn–,bn–) –D(A,B),d(an,bn–)
+d(bn–,bn) –D(A,B),
d(an–,bn–) +d(bn–,bn–) +d(bn–,bn) +d(an,bn–)
–D(A,B)
≤max d(an–,an),D(A,B) +d(an–,an) –D(A,B),D(A,B) +d(an,an+) –D(A,B), D(A,B) +d(an–,an) +d(an,an+) +D(A,B)
–D(A,B)
=maxd(an–,an),d(an,an+)
. ()
Also, we have
N(an–,an) = min
D(an–,Tan–) –D(A,B),D(an,Tan)
–D(A,B),D(an–,Tan) –D(A,B),
D(an,Tan–) –D(A,B)
≤mind(an–,bn–) –D(A,B),d(an,bn) –D(A,B),
d(an–,bn) –D(A,B),d(an,bn–) –D(A,B)
.
Sinced(an,bn–) =D(A,B), henceN(an–,an) = for alln∈N.
Using () and the inequality in (), we get
d(an,an+)≤ψ
maxd(an–,an),d(an,an+)
. ()
Ifd(an,an+) >d(an–,an). From () we obtain
d(an,an+)≤ψ
d(an,an+)
<d(an,an+),
which is a contradiction. So, we have
d(an,an+)≤d(an–,an). ()
Hence, the sequence{d(an,an+)}is monotone nonincreasing and bounded below. Thus, there existsk≥ such that
lim
n→∞d(an,an+) =k≥. ()
Suppose thatlimn→∞d(an,an+) =k> . Using (), inequality () becomes
d(an,an+)≤ψ
d(an–,an)
Takingn→ ∞in the above inequality and using the properties ofψ, we have k≤lim sup
n→∞ ψ
d(an–,an)
≤ψ(k),
which is a contradiction unlessk= . Hence,
lim
n→∞d(an,an+) = . ()
Now, we claim that the sequence{an}is a Cauchy sequence. Suppose that{an}is not a
Cauchy sequence. Then there exists > with subsequences{am(r)}and{an(r)} of{an}
such thatn(r) is the smallest index for whichn(r) >m(r) >r,d(am(r),an(r))≥. This means that
d(am(r),an(r)–) <. ()
Now, we have
≤d(am(r),an(r))
≤d(am(r),an(r)–) +d(an(r)–,an(r)) <+d(an(r)–,an(r)).
Lettingr→ ∞and using (), we can conclude that
lim
r→∞d(am(r),an(r)) =. ()
Again,
d(am(r),an(r)–)≤d(am(r),an(r)) +d(an(r),an(r)–) and
d(am(r),an(r))≤d(am(r),an(r)–) +d(an(r),an(r)–). Therefore,
d(am(r),an(r)–) –d(am(r),an(r))≤d(an(r),an(r)–).
Takingr→ ∞and using () and (), we get
lim
r→∞d(am(r),an(r)–) =. ()
Similarly, we can prove that
lim
r→∞d(am(r)–,an(r)) =r→∞limd(am(r)–,an(r)–) =r→∞limd(am(r)+,an(r))
= lim
Sincem(r) <n(r),am(r)–an(r)–, from () and (), we have
d(am(r),an(r))≤δ(Tam(r)–,Tan(r)–)≤ψ
M(am(r)–,an(r)–)
+LN(am(r)–,an(r)–), ()
where
M(am(r)–,an(r)–)
=max d(am(r)–,an(r)–),D(am(r)–,Tam(r)–) –D(A,B),
D(an(r)–,Tan(r)–) –D(A,B),
D(am(r)–,Tan(r)–) +D(an(r)–,Tam(r)–)
–D(A,B)
≤max d(am(r)–,an(r)–),d(am(r)–,bm(r)–) –D(A,B),d(an(r)–,bn(r)–)
–D(A,B),d(am(r)–,bn(r)–) +d(an(r)–,bm(r)–) – D(A,B)
≤max d(am(r)–,an(r)–),d(am(r)–,am(r)) +d(am(r),bm(r)–) –D(A,B),
d(an(r)–,an(r)) +d(an(r),bn(r)–) –D(A,B),
d(am(r)–,an(r))
+d(an(r),bn(r)–) +d(an(r)–,am(r)) +d(am(r),bm(r)–) – D(A,B)
.
Usingd(an+,bn) =D(A,B) in the above inequality, we get
M(am(r)–,an(r)–)≤max d(am(r)–,an(r)–),d(am(r)–,am(r)),d(an(r)–,an(r)), d(am(r)–,an(r)) +d(an(r)–,am(r))
()
and
N(am(r)–,an(r)–)
=minD(am(r)–,Tam(r)–) –D(A,B),D(an(r)–,Tan(r)–) –D(A,B),
D(am(r)–,Tan(r)–) –D(A,B),D(an(r)–,Tam(r)–) –D(A,B)
≤mind(am(r)–,bm(r)–) –D(A,B),d(an(r)–,bn(r)–) –D(A,B), d(am(r)–,bn(r)–) –D(A,B),d(an(r)–,bm(r)–) –D(A,B)
≤mind(am(r)–,am(r)) +d(am(r),bm(r)–) –D(A,B),d(an(r)–,an(r)) +d(an(r),bn(r)–) –D(A,B),d(am(r)–,an(r)) +d(an(r),bn(r)–) –D(A,B),
d(an(r)–,am(r)) +d(am(r),bm(r)–) –D(A,B)
Usingd(an+,bn) =D(A,B) in the above inequality, we get
N(am(r)–,an(r)–)≤min
d(am(r)–,am(r)),d(an(r)–,an(r)),
d(am(r)–,an(r)),d(an(r)–,am(r))
. ()
Using () and () in () and takingr→ ∞, from (), (), () and (), we get
≤ψmax{, , ,}+Lmin{, ,,} ()
=ψ() <, ()
which is a contradiction to the property ofψ. Thus,{an}is a Cauchy sequence inAand
hence it converges to some elementainA. Sinced(an,an+) =d(bn–,bn), the sequence{bn}
inBis Cauchy and hence it is convergent. Suppose thatbn→b. By the relationd(an+,bn) =
D(A,B), for alln, we conclude thatd(a,b) =D(A,B). We now claim thatb∈Ta. Since{an}is an increasing sequence inAandan→a, by hypothesis (iv),ana,∀n.
D(bn,Ta)
≤δ(Tan,Ta)
≤ψ
max d(an,a),D(an,Tan) –D(A,B),D(a,Ta) –D(A,B),
D(an,Ta) +D(a,Tan)
–D(A,B)
+LminD(an,Tan) –D(A,B),
D(a,Ta) –D(A,B),D(an,Ta) –D(A,B),D(a,Tan) –D(A,B)
≤ψ(max d(an,a),d(an,bn) –D(A,B),D(a,Ta) –D(A,B),
D(an,Ta) +d(a,bn)
–D(A,B)
+Lmind(an,bn) –D(A,B),D(a,Ta) –D(A,B),D(an,Ta) –D(A,B),
d(a,bn) –D(A,B)
.
Asn→ ∞in the above inequality, usingan→a,bn→b,d(a,b) =D(A,B) and sinceψis
upper-semicontinuous, we get
D(b,Ta)≤ψ
max , ,D(a,Ta) –D(A,B),D(a,Ta) +D(A,B)
–D(A,B)
+Lmin,D(a,Ta) –D(A,B),D(a,Ta) –D(A,B),
=ψD(a,Ta) –D(A,B)
≤ψd(a,b) +D(b,Ta) –D(A,B) =ψD(b,Ta)<D(b,Ta),
which is a contradiction unlessD(b,Ta) = .
This implies that b∈Ta, and henceD(a,Ta) =D(A,B). That is,ais a best proximity
Example . Let X :=R with the order defined as follows: for (a
,b), (a,b)∈ X, (a,b)(a,b) if and only ifa≤a,b≤b, where≤is the usual order inR. Then (X,,d) becomes a complete partially ordered metric space with respect to a metric d((a,b), (a,b)) =|a–a|+|b–b|for each (a,b), (a,b)∈X.
LetA={(, ), (, ), (, ), (, )}andB={(, –), (, ), (, ), (, )}. Then (A,B) satisfies theP-property, andD(A,B) = . LetT:A→CB(B) be defined as follows:
Ta= ⎧ ⎨ ⎩
{(, ), (, )} ifa= (, ),
{(, )} otherwise,
and defineψ: [,∞)→[,∞) asψ(t) =t
.
It is easy to see that for all comparablea,b∈XandL≥,Tsatisfies the following:
δ(Ta,Tb)≤ψ
max d(a,b),D(a,Ta) –D(A,B),D(b,Tb) –D(A,B),
D(a,Tb) +D(b,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B),
D(b,Tb) –D(A,B),D(a,Tb) –D(A,B),D(b,Ta) –D(A,B).
Also, it is easy to verify that thisT satisfies all the conditions in Theorem .. It is clear that (, ) is a best proximity point ofT.
The following corollary is a particular case of Theorem . when A=B. Also, it is a partial generalization of Theorem . in [].
Corollary . Let (X,,d) be a partially ordered complete metric space. Let A be a nonempty closed subset of X and T:A→CB(A)be a multivalued mapping such that the following conditions are satisfied:
(i) There exist elementsa,ainAandb∈Tasuch thatd(a,b) = and aa=b.
(ii) Tsatisfies
δ(Ta,Tb)≤ψ
max D(a,b),D(a,Ta),D(b,Tb),D(a,Tb) +D(b,Ta)
+LminD(a,Ta),D(b,Tb),D(a,Tb),D(b,Ta)
for all comparablea,b∈A,whereL≥andψis an altering distance function. (iii) Fora,b∈A,abimpliesTa≺()Tb.
(iv) If{an}is a nondecreasing sequence inAsuch thatan→a,thenanafor alln.
Then there exists an element a in A such that D(a,Ta) = .That is,a is a fixed point of the mapping T.
Corollary . Let (X,,d) be a partially ordered complete metric space. Let A be a nonempty closed subset of X.Let T:A→A be a single-valued mapping such that the fol-lowing conditions are satisfied:
(i) There exist elementsaandainAsuch that
d(a,Ta) = and aa. (ii) Tsatisfies
δ(Ta,Tb)≤ψ
max d(a,b),d(a,Ta),d(b,Tb),d(a,Tb) +d(b,Ta)
+Lmind(a,Ta),d(b,Tb),d(a,Tb),d(b,Ta)
for all comparablea,b∈A,whereL≥,andψis an altering distance function. (iii) Fora,b∈A,abimplies{Ta} ≺(){Tb}.
(iv) If{an}is a nondecreasing sequence inAsuch thatan→a,thenanafor alln.
Then there exists an element a in A such that d(a,Ta) = .That is,a is a fixed point of the mapping T.
Now, we state our second main result in this section.
Theorem . Let(X,,d)be a partially ordered complete metric space.Let A and B be nonempty closed subsets of the metric space (X,d)such that A=∅and(A,B)satisfies the P-property.Let S,T :A→CB(B)be a multivalued mapping such that the following conditions are satisfied:
(i) Sa,Ta⊆Bfor alla∈A.
(ii) There existsa∈Awithd(a,b) =D(A,B)for someb∈Bwith{b} ≺()Ta. (iii) For anya,b∈AwithabimpliesSb≺()Ta.
(iv) If{an}is any sequence inAwhose consecutive terms are comparable withan→a,
thenanafor alln.
(v) TandSsatisfy
δ(Ta,Sb)≤αM(a,b) +LN(a,b) for all comparablea,b∈A, () whereM(a,b) =max{d(a,b),D(a,Ta) –D(A,B),D(b,Sb) –D(A,B),
D(a,Sb)+D(b,Ta)
–D(A,B)},L≥andN(a,b) =min{D(a,Ta) –D(A,B), D(b,Sb) –D(A,B),D(a,Sb) –D(A,B),D(b,Ta) –D(A,B)}, <α< . Then T and S have a common best proximity point.
Proof By assumption (ii), there existsa∈Awithd(a,b) =D(A,B) such that{b} ≺() Ta. For thisb∈B, there existsb∈Tawithd(a,b) =D(A,B) such thataa. By assumption (iii),Sa≺()Ta, which impliesSa≺()Ta. So, for thisb∈Ta, there exists b∈Sawithd(a,b) =D(A,B) such thataa. Again, by assumption (iii),Sa≺()Ta, which impliesSa≺()Ta. Therefore, there existsb∈Tawithd(a,b) =D(A,B) such thataa. Continuing in this way, we can construct a sequence{an}such that
() for eachn,an∈Aandbn+∈Tanandbn+∈San+withd(an,bn) =D(A,B);
First we claim that any best proximity point ofT is a best proximity point ofSand con-versely. Now suppose thatpis a best proximity point ofTbut not a best proximity point ofS. Consider
D(p,Sp)≤D(p,Tp) +δ(Tp,Sp) =D(A,B) +δ(Tp,Sp).
Hence,D(p,Sp) –D(A,B)≤δ(Tp,Sp). Then, by condition (v),
D(p,Sp) –D(A,B)≤δ(Tp,Sp)
≤αmax d(p,p),D(p,Tp) –D(A,B),D(p,Sp) –D(A,B), D(p,Sp) +D(p,Tp)
–D(A,B)
+LminD(p,Tp) –D(A,B),
D(p,Sp) –D(A,B),D(p,Sp) –D(A,B),D(p,Tp) –D(A,B)
=αmax , ,D(p,Sp) –D(A,B),D(p,Sp) –D(A,B)
+Lmin,D(p,Sp) –D(A,B)
=αD(p,Sp) –D(A,B),
which is a contradiction, unlessD(p,Sp) =D(A,B). Hencepis a best proximity point toS. Using a similar argument, we can prove that any best proximity ofSis a best proximity point ofT.
If there exists a positive integer N such thataN =aN+, thenaN becomes a
com-mon best proximity point. Similarly, the same conclusion holds if aN+ =aN+ for someN. Therefore, we may assume thatan=an+for alln≥. We know thatd(an,bn) =
d(an+,bn+) =D(A,B) and so by theP-property, we haved(an,an+) =d(bn,bn+). Now,
d(an+,an+) =d(bn+,bn+)≤δ(Tan,San+). ()
Consider
M(an,an+) = max d(an,an+),D(an,Tan) –D(A,B),D(an+,San+) –D(A,B),
D(an,San+) +D(an+,Tan)
–D(A,B)
≤max d(an,an+),d(an,bn+) –D(A,B),d(an+,bn+) –D(A,B), d(an,bn+) +d(an+,bn+)
–D(A,B)
≤max d(an,an+),d(an,bn) +d(bn,bn+) –D(A,B),
d(an,bn) +d(bn,bn+) +d(bn+,bn+) +D(A,B)
–D(A,B)
=max d(an,an+),d(an+,an+),
d(an,an+) +d(an+,an+)
=maxd(an,an+),d(an+,an+)
and
N(an,an+) =min
D(an,Tan) –D(A,B),D(an+,San+) –D(A,B),
D(an,San+) –D(A,B),D(an+,Tan) –D(A,B)
≤mind(an,bn+) –D(A,B),d(an+,bn+) –D(A,B), d(an,bn+) –D(A,B),d(an+,bn+) –D(A,B)
=mind(an,bn+) –D(A,B),d(an+,bn+)
–D(A,B),d(an,bn+) –D(A,B),
= .
Therefore, by inequality () and by inequality (), we get
d(an+,an+)≤αmax
d(an,an+),d(an+,an+)
. ()
Suppose thatd(an,an+)≤d(an+,an+) for some positive integern. Then from () we have
d(an+,an+)≤αd(an+,an+) <d(an+,an+),
which is a contradiction. Hence,d(an+,an+) <d(an,an+) for alln≥. In a similar way, we can prove thatd(an+,an+) <d(an+,an+) for alln> . Then{d(an,an+)}is a monotone decreasing sequence of nonnegative real numbers. Hence there existsk≥ such thatlimn→∞d(an,an+) =k. We will now claim thatk= .
From the above discussion, we have
d(an+,an+)≤αd(an,an+) for alln≥. Takingn→ ∞in the above inequality, we get
k≤αk,
which is a contradiction unlessk= . Therefore,
lim
n→∞d(an,an+) = .
Now we will prove that{an}is a Cauchy sequence. Letm>n. Then
d(am,an)≤d(am,am–) +d(am–,am–) +· · ·+d(an+,an)
≤αm–+αm–+· · ·+αnd(a,a)≤ αn
which implies that{an}is a Cauchy sequence inA. From the completeness ofX, there
existsa∈Xsuch thatan→aasn→ ∞. SinceAis closed,a∈A. Also, by assumption
(iv),anafor alln. Sinced(an,an+) =d(bn,bn+), the sequence{bn}becomes Cauchy.
Hence it converges to someb∈X. By the relationd(an,bn) =D(A,B) for alln, we conclude
thatd(a,b) =D(A,B). We now claim thatb∈Ta.
D(Ta,bn+)
≤δ(Ta,San+)
≤αmax d(a,an+),D(a,Ta) –D(A,B),D(an+,San+) –D(A,B), D(a,San+) +D(an+,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B),
D(an+,San+) –D(A,B),D(an+,Ta) –D(A,B),D(a,San+) –D(A,B)
≤αmax d(a,an+),D(a,Ta) –D(A,B),d(an+,bn+) –D(A,B), d(a,bn+) +D(an+,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B),
d(an+,bn+) –D(A,B),D(an+,Ta) –D(A,B),d(a,bn+) –D(A,B)
≤αmax d(a,an+),D(a,Ta) –D(A,B),d(bn+,bn+), d(a,bn+) +D(an+,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B),
d(bn+,bn+),D(an+,Ta) –D(A,B),d(a,bn+) –D(A,B)
.
Asn→ ∞in the above inequality, and using the properties of sequences{an}and{bn},
we get
D(Ta,b)≤αmax ,D(a,Ta) –D(A,B), ,d(a,b) +D(a,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B), ,D(a,Ta) –D(A,B),d(a,b) –D(A,B)
≤αmax d(a,b) +D(b,Ta) –D(A,B),d(a,b) +d(a,b) +D(b,Ta)
–D(A,B)
=αD(b,Ta),
which is true only ifD(b,Ta) = . Henceb∈Ta, that is,ais the best proximity point ofT. By what we have proved already,ais a common best proximity point ofTandS.
Example . Let X:=R with the order defined as follows: for (a
LetA={(, ), (, ), (, ), (, )}andB={(, –), (, ), (, ), (, )}. Then (A,B) satisfies theP-property, andD(A,B) = . LetT:A→CB(B) andS:A→CB(B) be defined as fol-lows:
Ta= ⎧ ⎨ ⎩
{(, ), (, )} ifa= (, ),
{(, )} otherwise,
andSa={(, )}for alla∈Arespectively. It is easy to see that for all comparablea,b∈X, α=
andL≥,TandSsatisfy the following:
δ(Ta,Sb)≤αmax d(a,b),D(a,Ta) –D(A,B),D(b,Sb) –D(A,B), D(a,Sb) +D(b,Ta)
–D(A,B)
+LminD(a,Ta) –D(A,B),
D(b,Sb) –D(A,B),D(a,Sb) –D(A,B),D(b,Ta) –D(A,B).
Also,TandSsatisfy all the other conditions in Theorem .. It is easy to check that (, ) is a common best proximity point toSandT.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of the paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Amrita University, Coimbatore, India.2Department of Mathematics, Bharathidasan
University, Tiruchirappalli, Tamil Nadu 620 024, India.3Nonlinear Analysis Research Group, Ton Duc Thang University, Ho
Chi Minh City, Vietnam. 4Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
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Received: 29 June 2017 Accepted: 14 September 2017 References
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