Best proximity points of implicit relation type modified α3 proximal contractions

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R E S E A R C H

Open Access

Best proximity points of implicit relation type

modiﬁed

α



-proximal contractions

Farzaneh Zabihi and Abdolrahman Razani

*

*_{Correspondence: [email protected]} Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

In this paper, we introduce the concept of an

α

3_{-proximal admissible mappings and} establish the existence of best proximity point theorems for implicit relation type modiﬁed

α

3_{-proximal contractions. As applications of our theorems, we derive some} new best proximity point results for implicit relation type contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature. Several interesting consequences of our theorems are also provided.

MSC: 46N40; 47H10; 54H25; 46T99

Keywords: ﬁxed point; best proximity point;

α

3_{-proximal admissible mapping;} implicit relation type

α

3_{-proximal contractions; metric space endowed with graph}

1 Introduction

In nonlinear functional analysis, one of the most significant research areas is fixed point theory. On the other hand, fixed point theory has an application in distinct branches of mathematics and also in different sciences, such as engineering, computer science, eco-nomics,etc. In , Banach proved that every contraction in a complete metric space has a unique fixed point. Following this celebrated result, many authors have generalized, improved, and extended this result in the context of different abstract spaces for various operators.

On the other hand, several classical fixed point theorems and common fixed point the-orems have been recently unified by considering general contractive conditions expressed by an implicit relation (see Popa [, ]). Following Popa’s approach, many results on fixed point, common fixed points, and coincidence points have been obtained, in various am-bient spaces (see [–], and references therein). On the other hand, Sametet al.[] in-troduced and studiedα-ψ-contractive mappings in complete metric spaces and provided applications of the results to ordinary differential equations. More recently, Salimiet al. [] modified the notions ofα-ψ-contractive andα-admissible mappings and established fixed point theorems to modify the results in []. For more details and applications of this line of research, we refer the reader to some related papers [–] and references therein. In this paper, we introduce the concept of anα-proximal admissible mappings and es-tablish the existence of best proximity point theorems for implicit relation type modi-fiedα-proximal contractions. As applications of our theorems, we derive some new best proximity point results for implicit relation type contractions whenever the range space

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is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature.

2 Main results

LetAandBbe two nonempty subsets of metric space (X,d) andT:A→Bbe a nonself mapping. We say thatx∗is a best proximity ofTif

dx∗,Tx∗=d(A,B),

where

d(A,B) =infd(x,y) :x∈A,y∈B.

We deﬁneAandBas follows:

A=

x∈A:d(x,y) =d(A,B) for somey∈B

and

B=

y∈B:d(x,y) =d(A,B) for somex∈A.

We denote byΨ the set of all nondecreasing functionsψ: [, +∞)→[, +∞) such that

∞

n=ψn(t) < +∞for allt> , whereψnis thenth iterate ofψ.

LetF be the set of all continuous functionsF:R

+→Rsatisfying the following

asser-tions:

(F) ifF(u,v,v,u,u+v, )≤, whereu,v> , thenu≤ψ(v); (F) F(t, . . . ,t)is decreasing int;

(F) ifF(u,v, ,u+v,u,v)≤, whereu,v≥, thenu≤ψ(v); (F) F(u,u, , ,u,u) > for allu> .

Example  Let

F(t,t,t,t,t,t) =t–ψ

max

t,t,t,

t+t

 –Lmin{t,t,t,t},

whereL≥ andψ∈Ψ. ThenF∈F.

Example  Let

F(t,t,t,t,t,t) =t–at–

b[ +t]t

 +t

–c[t+t] –d[t+t],

wherea+b+ c+ d< . ThenF∈F.

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if, for allx,x,u,u∈A, ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

α(x,x)≥,

α(x,x)≥,

α(x,x)≥,

d(u,Tx) =d(A,B),

d(u,Tx) =d(A,B)

⇒

⎧ ⎪ ⎨ ⎪ ⎩

α(u,u)≥,

α(u,u)≥,

α(u,u)≥.

Deﬁnition  LetAandBbe nonempty subsets of a metric space (X,d) andα:A×A→ [,∞) be a function. ThenT :A→Bis said to be an implicit relation type modiﬁed α-proximal contraction, if for allx,y,u,v∈A,

⎧ ⎪ ⎨ ⎪ ⎩

α(x,y)≥, d(u,Tx) =d(A,B), d(v,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(x,v),d(y,u)≤L –α(x,x)α(y,y), (.)

whereL≥ andF∈F.

Deﬁnition  Let (X,d) be a metric space andAandBbe two nonempty subsets ofX. ThenBis said to be approximatively compact with respect toAif every sequence{yn}inB, satisfying the conditiond(x,yn)→d(x,B) for somexinA, has a convergent subsequence.

Theorem  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete and Ais nonempty.Assume that T:A→B is a continuous implicit relation type

modiﬁedα-proximal contraction such that the following conditions hold:

(i) Tis anα-proximal admissible mapping and

T(A)⊆B,

(ii) there existx,x∈Asuch that

d(x,Tx) =d(A,B), α(x,x)≥, α(x,x)≥ and α(x,x)≥.

Then T has a best proximity point.Further,the best proximity point is unique if

(iii) for everyx,y∈Awithd(x,Tx) =d(A,B) =d(y,Ty),we haveα(x,y)≥,α(x,x)≥, andα(y,y)≥.

Proof By (ii) there existx,x∈Asuch that

d(x,Tx) =d(A,B), α(x,x)≥, α(x,x)≥ and α(x,x)≥.

On the other hand,T(A)⊆B, then there existsx∈Asuch that

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Now, sinceTisα_{-proximal admissible, we have}

α(x,x)≥, α(x,x)≥ and α(x,x)≥.

Hence,

d(x,Tx) =d(A,B), α(x,x)≥, α(x,x)≥ and α(x,x)≥.

SinceT(A)⊆B, there existsx∈Asuch that

d(x,Tx) =d(A,B).

Then we have

d(x,Tx) =d(A,B), d(x,Tx) =d(A,B), α(x,x)≥,

α(x,x)≥ and α(x,x)≥.

Again, sinceTisα_{-proximal admissible, we obtain}

α(x,x)≥, α(x,x)≥ and α(x,x)≥.

Also, there existsx∈Asuch that

d(x,Tx) =d(A,B),

and hence

d(x,Tx) =d(A,B), d(x,Tx) =d(A,B), α(x,x)≥,

α(x,x)≥ and α(x,x)≥.

By continuing this process, we construct a sequence{xn}such that

α(xn,xn)≥, α(xn–,xn–)≥ and ⎧ ⎪ ⎨ ⎪ ⎩

α(xn–,xn)≥,

d(xn,Txn–) =d(A,B),

d(xn+,Txn) =d(A,B)

(.)

for alln∈N. Now, from (.) withu=xn,v=xn+,x=xn–, andy=xn, we get

Fd(xn,xn+),d(xn–,xn),d(xn–,xn),d(xn,xn+),d(xn–,xn+),d(xn,xn)

≤L –α(xn–,xn–)α(xn,xn)

.

On the other hand from (.) we obtain

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That is,  –α(xn–,xn–)α(xn,xn)≤ for alln∈N. Therefore,

Fd(xn,xn+),d(xn–,xn),d(xn–,xn),d(xn,xn+),d(xn–,xn+),d(xn,xn)

≤L –α(xn–,xn–)α(xn,xn)

≤.

Now, sinceFis decreasing int

Fd(xn,xn+),d(xn–,xn),d(xn–,xn),d(xn,xn+),d(xn,xn+) +d(xn–,xn), 

≤,

and so from (F) we get

d(xn,xn+)≤ψ

d(xn–,xn)

.

By induction, we have

d(xn,xn+)≤ψn

d(x,x)

.

Fix> , there existsN∈Nsuch that

n≥N

ψnd(x,x)

< for alln∈N.

Letm,n∈Nwithm>n≥N. Then by the triangular inequality, we get

d(xn,xm)≤ m–

k=n

d(xk,xk+)≤

n≥N

ψnd(x,x)

<.

Consequentlylim_m_,_n_,→+∞d(xn,xm) = . Hence{xn}is a Cauchy sequence. SinceAis com-plete, there isz∈Asuch thatxn→z. SinceTis continuous,Txn→Tzasn→ ∞. Hence,

d(A,B) = lim

n→∞d(xn+,Txn) =d(z,Tz).

Thuszis the desired best proximity point ofT.

Letx,y∈Abe two best proximity point ofTsuch thatx=y. That is,d(x,Tx) =d(A,B) = d(y,Ty). From (iii), we getα(x,y)≥,α(x,x)≥, andα(y,y)≥. So by (.) we derive

Fd(x,y),d(x,y),d(x,x),d(y,y),d(y,x),d(x,y)≤L –α(x,x)α(y,y)≤,

which implies

Fd(x,y),d(x,y), , ,d(y,x),d(x,y)≤,

which is a contradiction to (F). Hence,Thas a unique best proximity point.

Theorem  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete,B is approximatively compact with respect to A,and Ais nonempty.Assume that

T:A→B is an implicit relation type modiﬁedα_{-proximal contraction such that the}

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(i) Tis anα_{-proximal admissible mapping and}_T₍_A )⊆B, (ii) there existx,x∈Asuch that

d(x,Tx) =d(A,B), α(x,x)≥, α(x,x)≥ and α(x,x)≥,

(iii) if{xn}is a sequence inXsuch thatα(xn,xn+)≥for alln∈N∪ {}withxn→xas

n→ ∞,thenα(xn,x)≥andα(x,x)≥.

(iv) for everyx,y∈A,whered(x,Tx) =d(A,B) =d(y,Ty),we haveα(x,y)≥,α(x,x)≥, andα(y,y)≥.

Proof Following the proof of Theorem , there exist a Cauchy sequence{xn} ⊆Aandz∈A such that (.) holds andxn→zasn→+∞. On the other hand, for alln∈N, we can write

d(z,B)≤d(z,Txn)

≤d(z,xn+) +d(xn+,Txn)

=d(z,xn+) +d(A,B).

Taking the limit asn→+∞in the above inequality, we get

lim

n→+∞d(z,Txn) =d(z,B) =d(A,B). (.)

SinceBis approximatively compact with respect toA, the sequence{Txn}has a subse-quence{Txnk}that converges to somey∗∈B. Hence,

dz,y∗= lim

n→∞d(xnk+,Txnk) =d(A,B)

and soz∈A. Now, sinceT(A)⊆B, we haved(w,Tz) =d(A,B) for somew∈A. By (iii)

and (.), we haveα(xn,z)≥,α(z,z)≥, andd(xn+,Txn) =d(A,B) for alln∈N∪ {}.

Also, sinceT is an implicit relation typeα-proximal contraction, we get

Fd(xn+,w),d(xn,z),d(xn,xn+),d(z,w),d(xn,w),d(z,xn+)

≤.

Taking the limit asn→+∞in the above inequality and applying continuity ofF, we have

Fd(z,w), , ,d(z,w),d(z,w), ≤.

Now, if we takeu=d(z,w) andv= , then we have

F(u,v, ,u+v,u,v)≤

and so from (F) we getu≤ψ(v). That is,d(z,w)≤ψ() = . Thus,z=w. Hencezis a best proximity point ofT. Uniqueness follows similarly to the proof of Theorem .

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Corollary  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete,B is approximatively compact with respect to A,and Ais nonempty.Assume that

T:A→B is a nonself mapping satisfying the following conditions:

(i) Tis anα-proximal admissible mapping andT(A)⊆B, (ii) there existx,x∈Asuch that

n→ ∞,thenα(xn,x)≥andα(x,x)≥,

(iv) there exist nonnegative real numbersa,b,c,dwitha+b+ c+ d< ,such that for allx,x,u,u∈A,

⎧ ⎪ ⎨ ⎪ ⎩

α(x,x)≥,

d(u,Tx) =d(A,B),

⇒ d(u,u) +Lα(x,x)α(x,x)≤ad(x,x) +b

[ +d(x,u)]d(x,u)

 +d(x,x)

+cd(x,u) +d(x,u)

+dd(x,u) +d(x,u)

+L,

whereL≥.

(v) for everyx,y∈A,whered(x,Tx) =d(A,B) =d(y,Ty),we haveα(x,y)≥,α(x,x)≥, andα(y,y)≥.

If in Corollary  we takeb=c=d= , then we have the following corollary.

Corollary  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete,B is approximatively compact with respect to A,and Ais nonempty.Assume that

T:A→B is a nonself mapping satisfying the following conditions:

(i) Tis anα_{-proximal admissible mapping and}_T₍_A )⊆B, (ii) there existx,x∈Asuch that

n→ ∞,thenα(xn,x)≥andα(x,x)≥,

(iv) there exists a nonnegative real numberawitha< ,such that for allx,x,u,u∈A, ⎧

⎪ ⎨ ⎪ ⎩

α(x,x)≥,

d(u,Tx) =d(A,B),

⇒ d(u,u) +Lα(x,x)α(x,x)≤ad(x,x) +L,

whereL≥.

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(v) for everyx,y∈A,whered(x,Tx) =d(A,B) =d(y,Ty),we haveα(x,y)≥,α(x,x)≥, andα(y,y)≥.

Example  LetX=Rbe endowed with the usual metricd(x,y) =|x–y|, for allx,y∈X. ConsiderA= (–∞, –],B= [, +∞) and deﬁneT:A→Bby

Tx= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, ifx∈(–∞, –), , ifx∈[–, –), , ifx∈[–, –), , ifx∈[–, –), , ifx∈[–, –), , ifx∈[–, –), , ifx∈[–, –), , ifx∈[–, –].

Deﬁneα:X×X→[, +∞) by

α(x,y) =

, ifx,y∈[–, –],



, otherwise.

Clearly,Bis approximatively compact with respect toAandd(A,B) = . ThenA={–}

andB={}. Clearly,T(A)⊆B,d(–,T(–)) =d(A,B) = , andα(–, –)≥.

Assume ⎧ ⎪ ⎨ ⎪ ⎩

α(x,x)≥,

d(u,Tx) =d(A,B) = ,

d(u,Tx) =d(A,B) = ,

then ⎧ ⎪ ⎨ ⎪ ⎩

x,x∈[–, –],

d(u,Tx) = ,

d(u,Tx) = .

Therefore,u=u= –, that is,α(u,u)≥,α(u,u)≥, andα(u,u)≥. Further,

d(u,u)≤ad(x,x) +b

[ +d(x,u)]d(x,u)

 +d(x,x)

+cd(x,u) +d(x,u)

+dd(x,u) +d(x,u)

+L –α(x,x)α(x,x)

,

that is,T is anα-proximal admissible mapping and condition (iv) of Corollary  holds true. Moreover, if{xn}is a sequence such thatα(xn,xn+)≥ for alln∈N∪ {}andxn→x

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If in Corollary  we takeα(x,y) = , then we have the following corollary.

Corollary (Theorem . of []) Let A and B be nonempty closed subsets of a complete metric space(X,d)such that B is approximatively compact with respect to A.Assume that a+b+c+d< .Let Aand Bbe nonempty and T:A→B be a nonself mapping satisfying

the following assertions:

(i) T(A)⊆B, (ii)

d(u,Tx) =d(A,B),

⇒ d(u,u)≤ad(x,x) +b

[ +d(x,u)]d(x,u)

 +d(x,x)

+cd(x,u) +d(x,u)

+dd(x,u) +d(x,u)

.

Then there exists z∈A such that

d(z,Tz) =d(A,B).

By takingα(x,y) =  in Theorem , we deduce the following corollary.

Corollary  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete,B is approximatively compact with respect to A,and Ais nonempty.Assume that

T:A→B is a nonself mapping such that TA⊆Band for all x,y,u,v∈A,

d(u,Tx) =d(A,B), d(v,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤,

where F∈F.Then T has a unique best proximity point.

Using Example  and Corollary , we deduce the following result.

Corollary  Let A,B be two nonempty subsets of a metric space(X,d)such that A is com-plete,B is approximatively compact with respect to A,and Ais nonempty.Assume that

T:A→B is a nonself mapping such that TA⊆Band,for all x,y,u,v∈A,

d(u,Tx) =d(A,B), d(v,Ty) =d(A,B)

⇒ d(u,v)≤ψ

max

d(x,y),d(x,u),d(y,v),d(y,u) +d(x,v) 

+Lmind(x,u),d(y,v),d(y,u),d(x,v),

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3 Some results in metric spaces endowed with a graph

Consistent with Jachymski [], let (X,d) be a metric space andΔdenotes the diagonal of the Cartesian productX×X. Consider a directed graphGsuch that the setV(G) of its vertices coincides withX, and the setE(G) of its edges contains all loops,i.e.,E(G)⊇Δ. We assumeGhas no parallel edges, so we can identifyGwith the pair (V(G),E(G)). More-over, we may treatGas a weighted graph (see []) by assigning to each edge the distance between its vertices. Ifxandyare vertices in a graphG, then a path inGfromxtoy of lengthN (N∈N) is a sequence{xi}N

i=ofN+  vertices such thatx=x,xN =yand

(xn–,xn)∈E(G) fori= , . . . ,N. A graphGis connected if there is a path between any two

vertices.Gis weakly connected ifG˜ is connected (see for details [, , ]).

In , Espínola and Kirk [] established an important combination of ﬁxed point theory and graph theory.

Deﬁnition  LetA,Bbe two nonempty closed subsets of a metric space (X,d) endowed with a graphG. ThenT:A→Bis said to be an implicit relation typeG-proximal con-traction, if, for allx,y,u,v∈A,

⎧ ⎪ ⎨ ⎪ ⎩

(x,y)∈E(G), d(u,Tx) =d(A,B), d(v,Ty) =d(A,B)

⇒ (u,v)∈E(G)

and

⎧ ⎪ ⎨ ⎪ ⎩

(x,y)∈E(G), d(u,Tx) =d(A,B), d(v,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤,

whereF∈F.

Theorem  Let A,B be two nonempty closed subsets of a metric space(X,d)endowed with a graph G.Assume that A is complete,Ais nonempty,and T:A→B is a continuous

implicit relation type G-proximal contraction such that the following conditions hold:

(i) T(A)⊆B,

(ii) there exist elementsx,x∈Asuch that

d(x,Tx) =d(A,B) and (x,x)∈E(G).

Then T has a best proximity point.Further,the best proximity point is unique if,for every x,y∈A such that d(x,Tx) =d(A,B) =d(y,Ty),we have(x,y)∈E(G).

Proof Deﬁneα:X×X→[, +∞) by

α(x,y) =

, if (x,y)∈E(G),



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Firstly, we prove thatTis anα_{-proximal admissible mapping. To this aim, assume} ⎧

⎪ ⎨ ⎪ ⎩

α(x,y)≥, d(u,Tx) =d(A,B), d(v,Ty) =d(A,B).

Therefore, we have

⎧ ⎪ ⎨ ⎪ ⎩

(x,y)∈E(G), d(u,Tx) =d(A,B), d(v,Ty) =d(A,B).

SinceT is an implicit relation typeG-proximal contraction, we get (u,v)∈E(G). Also, sinceΔ⊆E(G), (u,u), (v,v)∈E(G). That is,α(u,v)≥,α(u,u)≥,α(v,v)≥, and

Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤ =L –α(x,x)α(y,y)

when L= . ThusT is an α-proximal admissible mapping withT(A)⊆Band

con-tinuous implicit relation type G-proximal contraction. From (ii) there exist x,x∈A

such thatd(x,Tx) =d(A,B) and (x,x)∈E(G), that is,d(x,Tx) =d(A,B),α(x,x)≥,

α(x,x)≥, andα(x,x)≥. Hence, all the conditions of Theorem  are satisﬁed andT

has a best proximity point.

Similarly, by using Theorem , we can prove the following theorem.

Theorem  Let A,B be two nonempty closed subsets of a metric space (X,d)endowed with a graph G.Assume that A is complete,B is approximatively compact with respect to A,and Ais nonempty.Also suppose that T:A→B is an implicit relation type G-proximal

contraction mapping such that the following conditions hold:

(i) T(A)⊆B,

d(x,Tx) =d(A,B) and (x,x)∈E(G),

(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)for alln∈N∪ {}andxn→x asn→+∞,then(xn,x)∈E(G)for alln∈N∪ {}.

Corollary  Let A, B be two nonempty closed subsets of a metric space(X,d)endowed with a graph G.Assume that A is complete,B is approximatively compact with respect to A,and Ais nonempty.Assume a+b+ c+ d< .Also,suppose that T:A→B satisﬁes

the following conditions:

(i) T(A)⊆B,

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(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)for alln∈N∪ {}andxn→x asn→+∞,then(xn,x)∈E(G)for alln∈N∪ {},

(iv) forx,x,u,u∈A, ⎧

⎪ ⎨ ⎪ ⎩

(x,x)∈E(G),

d(u,Tx) =d(A,B),

⇒ d(u,u)≤ad(x,x) +b

[ +d(x,u)]d(x,u)

 +d(x,x)

+cd(x,u) +d(x,u)

+dd(x,u) +d(x,u)

.

Corollary  Let A,B be two nonempty closed subsets of a metric space(X,d)endowed with a graph G.Assume that A is complete,B is approximatively compact with respect to A,and Ais nonempty.Also,suppose that T:A→B satisﬁes the following conditions:

(i) T(A)⊆B,

d(x,Tx) =d(A,B) and (x,x)∈E(G),

(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)for alln∈N∪ {}andxn→x asn→+∞,then(xn,x)∈E(G)for alln∈N∪ {},

(iv) forx,x,u,u∈A, ⎧

⎪ ⎨ ⎪ ⎩

(x,x)∈E(G),

d(u,Tx) =d(A,B),

⇒ d(u,u)≤ψ

max

d(x,x),d(x,u),d(x,u),

d(x,u) +d(x,u)



+Lmind(x,u),d(x,u),d(x,u),d(x,u)

,

whereψ∈Ψ.

4 Some results in metric spaces endowed with a partially ordered

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section, as an application of our results we derive some new best proximity point results whenever the range space is endowed with a partial order.

Deﬁnition [] Let (X,d,) be a partially ordered metric space. We say that a nonself mappingT:A→Bis proximally ordered-preserving if and only if, for allx,x,u,u∈A,

⎧ ⎪ ⎨ ⎪ ⎩

xx,

d(u,Tx) =d(A,B),

⇒ uu.

Theorem  Let A,B be two nonempty closed subsets of a partially ordered metric space (X,d,)such that A is complete,B is approximatively compact with respect to A,and A

is nonempty.Assume that T:A→B satisﬁes the following conditions:

(i) Tis continuous and proximally ordered-preserving such thatT(A)⊆B, (ii) there exist elementsx,x∈Asuch that

d(x,Tx) =d(A,B) and xx,

(iii) for allx,y,u,v∈A, ⎧

⎪ ⎨ ⎪ ⎩

xy,

d(u,Tx) =d(A,B), d(y,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤. (.)

Then T has a best proximity point.

Proof Deﬁneα:A×A→[, +∞) by

α(x,y) =

, ifxy,



, otherwise.

Firstly, we prove thatTis anα-proximal admissible mapping. To this aim, assume

⎧ ⎪ ⎨ ⎪ ⎩

α(x,y)≥, d(u,Tx) =d(A,B), d(v,Ty) =d(A,B).

Therefore, we have

⎧ ⎪ ⎨ ⎪ ⎩

xy,

d(u,Tx) =d(A,B), d(v,Ty) =d(A,B).

Now, sinceTis proximally ordered-preserving, thenuv, that is,α(u,v)≥. Further, by (ii) we have

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Moreover, from (iii) we get

⎧ ⎪ ⎨ ⎪ ⎩

α(x,y)≥, d(u,Tx) =d(A,B), d(y,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤.

Thus all the conditions of Theorem  hold (when L= ) andT has a best proximity

point.

Theorem  Let A,B be two nonempty closed subsets of a partially ordered metric space (X,d,)such that A is complete,B is approximatively compact with respect to A,and A

is nonempty.Assume that T:A→B satisﬁes the following conditions:

(i) Tis proximally ordered-preserving such thatT(A)⊆B, (ii) there exist elementsx,x∈Asuch that

(iii) for allx,y,u,v∈A, ⎧

⎪ ⎨ ⎪ ⎩

xy,

d(u,Tx) =d(A,B), d(y,Ty) =d(A,B)

⇒ Fd(u,v),d(x,y),d(x,u),d(y,v),d(y,u),d(x,v)≤, (.)

(iv) if{xn}is an increasing sequence inAconverging tox∈A,thenxnxfor alln∈N. Then T has a best proximity point.

Corollary  Let A,B be two nonempty closed subsets of a partially ordered metric space (X,d,)such that A is complete,B is approximatively compact with respect to A,and A

is nonempty.Assume a+b+ c+ d< .Also,suppose that T:A→B satisﬁes the following conditions:

(i) T(A)⊆B,

(iii) if{xn}is a sequence inXsuch thatxnxn+for alln∈N∪ {}andxn→xas

n→+∞,thenxnxfor alln∈N∪ {}, (iv) forx,x,u,u∈A,

⎧ ⎪ ⎨ ⎪ ⎩

xx,

d(u,Tx) =d(A,B),

⇒ d(u,u)≤ad(x,x) +b

[ +d(x,u)]d(x,u)

 +d(x,x)

+cd(x,u) +d(x,u)

+dd(x,u) +d(x,u)

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Then T has a best proximity point.Further,the best proximity point is unique if,for every x,y∈A such that d(x,Tx) =d(A,B) =d(y,Ty),we have xy.

Corollary  Let A,B be two nonempty closed subsets of a partially ordered metric space (X,d,)such that A is complete,B is approximatively compact with respect to A,and A

is nonempty.Also,suppose that T:A→B satisﬁes the following conditions:

(i) T(A)⊆B,

(iii) if{xn}is a sequence inXsuch thatxnxn+for alln∈N∪ {}andxn→xas n→+∞,thenxnxfor alln∈N∪ {},

(iv) forx,x,u,u∈A, ⎧

⎪ ⎨ ⎪ ⎩

xx,

d(u,Tx) =d(A,B),

⇒ d(u,u)≤ψ

max

d(x,x),d(x,u),d(x,u),

d(x,u) +d(x,u)



+Lmind(x,u),d(x,u),d(x,u),d(x,u)

,

whereψ∈Ψ.

Then T has a best proximity point.Further,the best proximity point is unique if,for every x,y∈A such that d(x,Tx) =d(A,B) =d(y,Ty),we have xy.

5 Application to ﬁxed point theory

5.1 Implicit relation type modiﬁed

α

-contraction

Deﬁnition [] LetT be a self-mapping onXandα:X×X→[, +∞) be a function. We say thatT is anα-admissible mapping if

x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.

Remark  Note that everyα-admissible mappings areα-proximal admissible mappings whenA=B=X.

Deﬁnition  Let (X,d) be a metric space andα:A×A→[,∞) be a function. Then T :X→Xis said to be an implicit relation type α-contraction, if for allx,y∈X with α(x,y)≥, we have

Fd(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty)

≤L –α(x,x)α(y,y), (.)

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Theorem  Let(X,d)be a complete metric space.Assume that T:X→X is a continuous self-mapping satisfying the following conditions:

(i) Tisα-admissible,

(ii) there existsxinXsuch thatα(x,x)≥andα(x,Tx)≥, (iii) Tis an implicit relation type modiﬁedα-contraction.

Then T has a ﬁxed point.

Theorem  Let (X,d) be a complete metric space. Assume that T :X→X is a self-mapping and the following conditions hold:

(ii) there existsxinXsuch thatα(x,x)≥andα(x,Tx)≥, (iii) Tis an implicit relation type modiﬁedα-contraction,

(iv) if{xn}is a sequence inXsuch thatα(xn,xn+)≥andxn→xasn→+∞,then

α(x,x)≥andα(xn,x)≥for alln∈N.

Using Example  and Theorem , we deduce the following result.

Corollary  Let (X,d) be a complete metric space. Assume that T:X→X is a self-mapping and the following conditions hold:

(ii) there existsxinXsuch thatα(x,x)≥andα(x,Tx)≥, (iii) for allx,y∈Xwithα(x,y)≥we have

d(Tx,Ty) +Lα(x,x)α(y,y)≤ad(x,y) +b[ +d(x,Tx)]d(y,Ty)  +d(x,y) +cd(x,Tx) +d(y,Ty)

+dd(y,Tx) +d(x,Ty)+L,

wherea+b+ c+ d< andL≥,

Corollary  Let(X,d)be a complete metric space. Assume that T :X→X is a self-mapping and the following conditions hold:

(ii) there existsxinXsuch thatα(x,x)≥andα(x,Tx)≥, (iii) for allx,y∈Xwithα(x,y)≥we have

d(Tx,Ty) +Lα(x,x)α(y,y)≤ad(x,y) +L,

where≤a< andL≥,

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5.2 Implicit relation typeG-contraction

Deﬁnition [] We say that a mappingT:X→Xis a BanachG-contraction or simply G-contraction ifTpreserves edges ofG,i.e.,

∀x,y∈X (x,y)∈E(G)⇒T(x),T(y)∈E(G)

andTdecreases weights of edges ofGin the following way:

∃α∈(, ),∀x,y∈X (x,y)∈E(G)⇒dT(x),T(y)≤αd(x,y).

Deﬁnition [] A mappingT:X→Xis calledG-continuous, if for givenx∈Xand sequence{xn}

xn→x asn→ ∞ and (xn,xn+)∈E(G) for alln∈N imply Txn→Tx.

Deﬁnition  Let (X,d) be a metric space endowed with a graphG. ThenT:X→Xis said to be an implicit relation typeG-contraction, if, for allx,y∈X,

(x,y)∈E(G) ⇒ (Tx,Ty)∈E(G)

and

(x,y)∈E(G) ⇒ Fd(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty)≤,

whereF∈F.

Theorem  Let(X,d)be a complete metric space endowed with a graph G.Assume that T:X→X is a continuous self-mapping satisfying the following conditions:

(i) there existsxinXsuch that(x,Tx)∈E(G), (ii) T is an implicit relation typeG-contraction.

Theorem  Let(X,d)be a complete metric space endowed with a graph G.Assume that T:X→X is a self-mapping satisfying the following conditions:

(i) there existsxinXsuch that(x,Tx)∈E(G), (ii) Tis an implicit relation typeG-contraction,

(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈E(G)andxn→xasn→+∞,then (xn,x)∈E(G)for alln∈N.

5.3 Implicit relation type ordered contraction

Theorem ([], Theorem .) Let(X,d,)be a partially ordered complete metric space. Assume that T:X→X is a self-mapping that satisﬁes the following conditions:

(i) there existsxinXsuch thatxTx, (ii) for allx,y∈Xwithxywe have

Fd(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty)≤,

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(iii) eitherTis continuous or if{xn}is an increasing sequence inXsuch thatxn→xas n→+∞,thenxnxfor alln∈N.

Corollary  Let(X,d,)be complete metric space. Assume a+b+ c+ d< .Also, suppose that T:X→X is a self-mapping that satisﬁes the following conditions:

(i) there exists an elementx∈Xsuch thatxTx,

(ii) if{xn}is an increasing sequence inXsuch thatxn→xasn→+∞,thenxnxfor alln∈N∪ {},

(iii) forx,y∈Xwithxy,

d(Tx,Ty)≤ad(x,y) +b[ +d(x,Tx)]d(y,Ty)  +d(x,y)

+cd(x,Tx) +d(y,Ty)+dd(x,Ty) +d(y,Tx).

Corollary  Let(X,d,)be complete metric space.Assume that T :X→X is a self-mapping that satisﬁes the following conditions:

(i) there exist elementx∈Xsuch thatxTx,

(ii) if{xn}is an increasing sequence inXsuch thatxn→xasn→+∞,thenxnxfor alln∈N∪ {},

(iii) forx,y∈Xwithxy,

d(Tx,Ty)≤ψ

max

d(x,y),d(x,Tx),d(y,Ty),d(y,Tx) +d(x,Ty) 

+Lmind(x,Tx),d(y,Ty),d(y,Tx),d(x,Ty),

whereψ∈Ψ.Then T has a ﬁxed point.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Received: 29 June 2014 Accepted: 9 September 2014 Published:24 Sep 2014 References

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10.1186/1029-242X-2014-365

Cite this article as:Zabihi and Razani:Best proximity points of implicit relation type modiﬁedα3_-proximal