Best proximity points of generalized almost ψ-Geraghty contractive non-self-mappings

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R E S E A R C H

Open Access

Best proximity points of generalized almost

ψ

-Geraghty contractive non-self-mappings

Hassen Aydi

1

_{, Erdal Karapınar}

2,3

_{, ˙Inci M Erhan}

2*

_{and Peyman Salimi}

4

*_{Correspondence:}

[email protected]; [email protected] 2_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara 06836, Turkey

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the new notion of almost

ψ-Geraghty contractive

mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weakP-property. We provide an example to validate our best proximity point theorem. The obtained results extend, generalize, and complement some known ﬁxed and best proximity point results from the literature.

MSC: 47H10; 54H25; 46J10; 46J15

Keywords: ﬁxed point; metric space; best proximity point; generalized almost

ψ-Geraghty contractions

1 Introduction and preliminaries

Non-self-mappings are among the intriguing research directions in ﬁxed point theory. This is evident from the increase of the number of publications related with such maps. A great deal of articles on the subject investigate the non-self-contraction mappings on metric spaces. Let (X,d) be a metric space and A and B be nonempty subsets of X. A mappingT :A→B is said to be ak-contraction if there exists k∈[, ) such that

d(Tx,Ty)≤kd(x,y) for anyx,y∈A. It is clear that ak-contraction coincides with the cel-ebrated Banach ﬁxed point theorem (Banach contraction principle) [] if one takesA=B

where the induced metric space (A,d|A) is complete.

In nonlinear analysis, the theory of fixed points is an essential instrument to solve the equationTx=xfor a self-mappingT defined on a subset of an abstract space such as a metric space, a normed linear space or a topological vector space. Following the Banach contraction principle, most of the fixed point results have been proved for a self-mapping defined on an abstract space. It is quite natural to investigate the existence and unique-ness of a non-self-mappingT:A→Bwhich does not possess a fixed point. If a non-self-mappingT:A→Bhas no fixed point, then the answer of the following question makes sense: Is there a pointx∈Xsuch that the distance betweenxandTxis closest in some sense? Roughly speaking, best proximity theory investigates the existence and uniqueness of such a closest pointx. We refer the reader to [–] and [–] for further discussion of best proximity.

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Deﬁnition . Let (X,d) be a metric space andA,B⊂X. We say thatx∗∈Ais a best proximity point of the non-self-mappingT:A→Bif the following equality holds:

dx∗,Tx∗=d(A,B), ()

whered(A,B) =inf{d(x,y) :x∈A,y∈B}.

It is clear that the notion of a ﬁxed point coincided with the notion of a best proximity point when the underlying mapping is a self-mapping.

Let (X,d) be a metric space. Suppose thatAandBare nonempty subsets of a metric space (X,d). We deﬁne the following sets:

A=

x∈A:d(x,y) =d(A,B) for somey∈B,

B=

y∈B:d(x,y) =d(A,B) for somex∈A.

()

In [], the authors presented suﬃcient conditions for the setsAandBto be nonempty.

In  Geraghty [] introduced the classSof functionsβ: [,∞)→[, ) satisfying the following condition:

β(tn)→ implies tn→. ()

The author deﬁned contraction mappings via functions from this class and proved the following result.

Theorem .(Geraghty []) Let(X,d)be a complete metric space and T:X→X be an operator.If T satisﬁes the following inequality:

d(Tx,Ty)≤βd(x,y)d(x,y) for any x,y∈X, ()

whereβ∈S,then T has a unique ﬁxed point.

Recently, Caballeroet al.[] introduced the following contraction.

Deﬁnition .([]) LetA,Bbe two nonempty subsets of a metric space (X,d). A mapping

T:A→Bis said to be a Geraghty-contraction if there existsβ∈Ssuch that

d(Tx,Ty)≤βd(x,y)d(x,y) for any x,y∈A. ()

Based on Deﬁnition ., the authors [] obtained the following result.

Theorem .(See []) Let(A,B)be a pair of nonempty closet subsets of a complete metric space(X,d)such that Ais nonempty.Let T:A→B be a continuous,Geraghty-contraction

satisfying T(A)⊆B.Suppose that the pair(A,B)has the P-property,then there exists a

unique x∗in A such that d(x∗,Tx∗) =d(A,B).

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Deﬁnition . Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with

A=∅. Then the pair (A,B) is said to have theP-property if and only if for anyx,x∈A

andy,y∈B,

d(x,y) =d(A,B) and d(x,y) =d(A,B) ⇒ d(x,x) =d(y,y). ()

It is easily seen that for any nonempty subset A of (X,d), the pair (A,A) has the

P-property. In [], the author proved that any pair (A,B) of nonempty closed convex subsets of a real Hilbert spaceHsatisﬁes theP-property.

Recently, Zhang et al. [] deﬁned the following notion, which is weaker than the

P-property.

Deﬁnition . Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with

A=∅. Then the pair (A,B) is said to have the weak P-property if and only if for any

x,x∈Aandy,y∈B,

d(x,y) =d(A,B) and d(x,y) =d(A,B) ⇒ d(x,x)≤d(y,y). ()

Letdenote the class of functionsψ: [,∞)→[,∞) satisfying the following condi-tions:

(a) ψis nondecreasing;

(b) ψis subadditive, that is,ψ(s+t)≤ψ(s) +ψ(t); (c) ψis continuous;

(d) ψ(t) = ⇔t= .

The notion ofψ-Geraghty contraction has been introduced very recently in [], as an extension of Deﬁnition ..

Deﬁnition . LetA, Bbe two nonempty subsets of a metric space (X,d). A mapping

T:A→Bis said to be aψ-Geraghty contraction if there existβ∈Sandψ∈such that

ψd(Tx,Ty)≤βψd(x,y)ψd(x,y) for any x,y∈A. ()

Remark . Notice that sinceβ: [,∞)→[, ), we have

ψd(Tx,Ty)≤βψd(x,y)ψd(x,y)<ψd(x,y)

for anyx,y∈Awithx=y. () In [], the author also proved the following best proximity point theorem.

Theorem .(See []) Let(A,B)be a pair of nonempty closed subsets of a complete metric space(X,d)such that Ais nonempty.Let T:A→B be aψ-Geraghty contraction satisfying

T(A)⊆B.Suppose that the pair(A,B)has the P-property.Then there exists a unique x∗

in A such that d(x∗,Tx∗) =d(A,B).

2 Main results

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Deﬁnition . LetA,Bbe two nonempty subsets of a metric space (X,d). A mapping

T:A→Bis said to be a generalized almostψ-Geraghty contraction if there existβ∈S

andψ∈such that

ψd(Tx,Ty)≤βψM(x,y)ψM(x,y) –d(A,B)+LψN(x,y) –d(A,B) ()

for allx,y∈AwhereL≥,

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),

N(x,y) =mind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Now, we state and prove our main theorem about existence and uniqueness of a best proximity point for a non-self-mapping satisfying a generalized almostψ-Geraghty con-traction.

Theorem . Let(A,B)be a pair of nonempty closed subsets of a complete metric space

(X,d)such that Ais nonempty.Let T:A→B be a generalized almostψ-Geraghty

con-traction satisfying T(A)⊆B.Assume that the pair(A,B)has the weak P-property.Then

T has a unique best proximity point in A.

Proof Since the subsetAis not empty, we can takexinA. Taking into account that

Tx∈ T(A)⊆ B, we can ﬁnd x ∈A such that d(x,Tx) =d(A,B). Further, since

Tx∈T(A)⊆B, it follows that there is an elementxinAsuch thatd(x,Tx) =d(A,B).

Recursively, we obtain a sequence{xn}inAsatisfying

d(xn+,Txn) =d(A,B) for anyn∈N. ()

Since the pair (A,B) has the weakP-property, we deduce

d(xn,xn+)≤d(Txn–,Txn) for anyn∈N. ()

Due to the triangle inequality together with the equality () we have

d(xn–,Txn–)≤d(xn–,xn) +d(xn,Txn–) =d(xn–,xn) +d(A,B).

Analogously, combining the equalities () and () with the triangle inequality we obtain

d(xn,Txn)≤d(xn,xn+) +d(xn+,Txn) =d(xn,xn+) +d(A,B). ()

Consequently, we have

M(xn–,xn) =max

d(xn–,xn),d(xn–,Txn–),d(xn,Txn)

≤maxd(xn–,xn),d(xn,xn+)

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Also note that

N(xn–,xn) –d(A,B)

=mind(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(xn,Txn–)

–d(A,B)

≤mind(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(A,B)

–d(A,B)

=d(A,B) –d(A,B) = . () If there existsn∈Nsuch thatd(xn,xn+) = , then the proof is completed. Indeed,

 =d(xn,xn+) =d(Txn–,Txn), ()

and consequently,Txn–=Txn. Therefore, we conclude that

d(A,B) =d(xn,Txn–) =d(xn,Txn). ()

For the rest of the proof, we suppose thatd(xn,xn+) >  for alln∈N. In view of the fact

thatTis a generalized almostψ-Geraghty contraction, we have

ψd(xn,xn+)

≤ψd(Txn–,Txn)

≤βψM(xn–,xn)

ψM(xn–,xn) –d(A,B)

+LψN(xn–,xn) –d(A,B)

=βψM(xn–,xn)

+Lψ() =βψM(xn–,xn)

<ψM(xn–,xn) –d(A,B)

. ()

Taking into account the inequalities () and (), we deduce that

ψd(xn,xn+)

<ψM(xn–,xn) –d(A,B)

≤ψmaxd(xn–,xn),d(xn,xn+)

. If for somen,max{d(xn–,xn),d(xn,xn+)}=d(xn,xn+), then we get

ψd(xn,xn+)

<ψd(xn,xn+)

,

which is a contradiction. Therefore, we must have

M(xn–,xn)≤max

d(xn–,xn),d(xn,xn+)

+d(A,B) =d(xn–,xn) +d(A,B) ()

for alln∈N. Regarding the inequality (), we see that

ψd(xn,xn+)

=ψd(Txn–,Txn)

≤βψM(xn–,xn)

ψd(xn–,xn)

<ψd(xn–,xn)

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holds for all n∈ N. Since ψ is nondecreasing, then d(xn,xn+) <d(xn–,xn) for all n.

Consequently, the sequence{d(xn,xn+)}is decreasing and is bounded below and hence

limn→∞d(xn,xn+) =s≥ exists. Assume thats> . Rewrite () as

ψ(d(xn+,xn+))

ψ(d(xn,xn+)) ≤

βψM(xn,xn+)

≤

for eachn≥. Taking the limit of both sides asn→ ∞, we ﬁnd

lim

n→∞β

ψM(xn,xn+)

= .

On the other hand, sinceβ∈S, we concludelimn→∞ψ(M(xn,xn+)) = , that is,

s= lim

n→∞d(xn,xn+) = . ()

Sinced(xn,Txn–) =d(A,B) holds for alln∈Nand (A,B) satisﬁes the weakP-property,

then for allm,n∈N, we can write

d(xm,xn)≤d(Txm–,Txn–). ()

From (), we deduce

M(xm,xn) =max

d(xm,xn),d(xm,Txm),d(xn,Txn)

≤maxd(xm,xn),d(xm,xm+),d(xn,xn+)

+d(A,B). By usinglimn→∞d(xn,xn+) = , we get

lim

m,n→∞

M(xm,xn) –d(A,B)

≤ lim

m,n→∞d(xm,xn). ()

On the other hand,

≤N(xm,xn) –d(A,B)

=mind(xm,Txm),d(xn,Txn),d(xm,Txn),d(xn,Txm)

–d(A,B)

≤mind(xm,xm+) +d(A,B),d(xn,Txn),d(xm,Txn),d(xn,Txm)

–d(A,B). () Due to the fact thatlimn→∞d(xn,xn+) = , we obtain

lim

m,n→∞

N(xm,xn) –d(A,B)

= . ()

We shall show next that{xn}is a Cauchy sequence. Assume on the contrary that

ε=lim sup

m,n→∞d(xn,xm) > . ()

Employing the triangular inequality and (), we get

d(xn,xm)≤d(xn,xn+) +d(xn+,xm+) +d(xm+,xm)

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Combining () and (), and regarding the properties ofψ, we obtain

ψd(xn,xm)

≤ψd(xn,xn+) +d(Txn,Txm) +d(xm+,xm)

≤ψd(xn,xn+)

+ψd(Txn,Txm)

+ψd(xm+,xm)

≤ψd(xn,xn+)

+βψM(xn,xm)

ψM(xn,xm) –d(A,B)

+LψN(xn,xm) –d(A,B)

+ψd(xm+,xm)

. () From (), (), (), and by usinglim_n_→∞d(xn,xn+) = , we have

lim

m,n→∞ψ

d(xn,xm)

≤ lim

m,n→∞β

ψM(xn,xm)

lim

m,n→∞ψ

M(xm,xn) –d(A,B)

≤ lim

m,n→∞β

ψM(xn,xm)

lim

m,n→∞ψ

d(xm,xn)

.

So by (), we get

≤ lim

m,n→∞β

ψM(xn,xm)

,

that is, limm,n→∞β(ψ(M(xn,xm))) = . Therefore, limm,n→∞M(xn,xm) = . This implies

thatlimm,n→∞d(xn,xm) = , which is a contradiction. Therefore,{xn}is a Cauchy sequence.

Since{xn} ⊂AandAis a closed subset of the complete metric space (X,d), we can ﬁnd

x∗∈Asuch thatxn→x∗asn→ ∞. We shall show thatd(x∗,Tx∗) =d(A,B). Ifx∗=Tx∗,

thenA∩B=∅, andd(x∗,Tx∗) =d(A,B) = ,i.e.,x∗is a best proximity point ofT. Hence, we assume thatd(x∗,Tx∗) > . Suppose on the contrary thatx∗is not a best proximity point ofT, that is,d(x∗,Tx∗) >d(A,B). First note that

dx∗,Tx∗≤dx∗,Txn

+dTxn,Tx∗

≤dx∗,xn+

+d(xn+,Txn) +d

Txn,Tx∗

≤dx∗,xn+

+d(A,B) +dTxn,Tx∗

.

Taking the limit asn→ ∞in the above inequality, we obtain

dx∗,Tx∗–d(A,B)≤ lim

n→∞d

Txn,Tx∗

. Sinceψis nondecreasing and continuous, then

ψdx∗,Tx∗–d(A,B)≤ψ

lim

n→∞d

Txn,Tx∗ = lim n→∞ψ

dTxn,Tx∗

. () Also, lettingn→ ∞in () results in

lim

n→∞d(xn,Txn)≤d(A,B),

that is,limn→∞d(xn,Txn) =d(A,B). Then we get

lim

n→∞M

xn,x∗

=max

lim

n→∞d

x∗,xn

, lim

n→∞d(xn,Txn),d

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and therefore

lim

n→∞ψ

Mxn,x∗

–d(A,B)=ψdx∗,Tx∗–d(A,B). ()

Further,

lim

n→∞N

xn,x∗

–d(A,B) =min

lim

n→∞d(xn,Txn),d

x∗,Tx∗, lim

n→∞d

xn,Tx∗

, lim

n→∞d

x∗,Txn

–d(A,B), which implies

lim

n→∞N

xn,x∗

–d(A,B) = . () Therefore, combining (), (), (), and () we deduce

ψdx∗,Tx∗–d(A,B)≤ lim

n→∞ψ

dTxn,Tx∗

≤ lim

n→∞β

ψMxn,x∗

lim

n→∞ψ

Mxn,x∗

–d(A,B) +L lim

n→∞ψ

Nxn,x∗

–d(A,B) = lim

n→∞β

ψMxn,x∗

ψdx∗,Tx∗–d(A,B). () Now, sinceψ(d(x∗,Tx∗) –d(A,B)) > , and making use of (), we get

≤ lim

n→∞β

ψMxn,x∗

, that is,

lim

n→∞β

ψMxn,x∗

= ,

which implies

lim

n→∞M

xn,x∗

=dx∗,Tx∗= ,

and sod(x∗,Tx∗) =  >d(A,B), which is a contradiction. Therefore,d(x∗,Tx∗)≤d(A,B), that is,d(x∗,Tx∗) =d(A,B). In other words,x∗is a best proximity point ofT. This com-pletes the proof of the existence of a best proximity point.

We shall show next the uniqueness of the best proximity point ofT. Suppose thatx∗and

y∗are two best proximity points ofT, such thatx∗=y∗. This implies that

dx∗,Tx∗=d(A,B) =dy∗,Ty∗, ()

whered(x∗,y∗) > . Due to the weakP-property of the pair (A,B), we have

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Observe that in this case

Mx∗,y∗=maxdx∗,y∗,dx∗,Tx∗,dy∗,Ty∗

=maxdx∗,y∗,d(A,B),d(A,B).

Also, note that

Nx∗,y∗–d(A,B) =mindx∗,Tx∗,dy∗,Ty∗,dx∗,Ty∗,dy∗,Tx∗–d(A,B) =mind(A,B),d(A,B),dx∗,Ty∗,dy∗,Tx∗–d(A,B) =d(A,B) –d(A,B) = .

Using the fact thatTis a generalized almostψ-Geraghty contraction, we derive

ψdx∗,y∗≤ψdTx∗,Ty∗

≤βψMx∗,y∗ψMx∗,y∗–d(A,B)+LψNx∗,y∗–d(A,B) =βψMx∗,y∗ψMx∗,y∗–d(A,B)

<ψMx∗,y∗–d(A,B).

IfM(x∗,y∗) =d(A,B), due to the fact thatd(x∗,y∗) > , the inequality above becomes

 <ψdx∗,y∗<ψ(), ()

which impliesd(x∗,y∗) =  and contradicts the assumptiond(x∗,y∗) > . Else, ifM(x∗,y∗) =

d(x∗,y∗), we deduce

 <ψdx∗,y∗<ψdx∗,y∗–d(A,B), ()

which is not possible, sinceψ is nondecreasing. Therefore, we must haved(x∗,y∗) = . This completes the proof. To illustrate our result given in Theorem ., we present the following example, which shows that Theorem . is a proper generalization of Theorem ..

Example . Consider the spaceX=Rwith Euclidean metric. Take the sets

A= (–∞, –] and B= [, +∞).

Obviously,d(A,B) = . LetT:A→Bbe deﬁned byTx= –x. Notice thatA={–},B={}

andT(A)⊆B. Also, it is clear that the pair (A,B) has the weakP-property.

Consider

β(t) =



+t, if ≤t< , t

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andψ(t) =αt(withα≥_) for allt≥. Note thatβ∈Sandψ∈. For allx,y∈A, we have

d(Tx,Ty) =|x–y| and M(x,y) =max|x–y|, –x, –y.

We shall show thatT is a generalized almostψ-Geraghty contraction. Without loss of generality, consider the case wherex≥y. Then we haveM(x,y) = –yandd(Tx,Ty) =x–y.

In this case, we see that

ψd(Tx,Ty)=α(x–y)≤α(–x–y– )

≤α(–x–y– )≤[–αy]α(–x–y– ) = [–αy]α(–y– ) –α(x–y)

=ψM(x,y)ψM(x,y) –d(A,B)–ψd(Tx,Ty).

Therefore

ψd(Tx,Ty)≤ ψ(M(x,y))  +ψ(M(x,y))ψ

M(x,y) –d(A,B). ()

On the other hand, we know thatψ(M(x,y)) = –αy≥ for allx,y∈Awithx≥y. Hence,

βψM(x,y)= ψ(M(x,y))  +ψ(M(x,y)), and from () we deduce

ψd(Tx,Ty)≤βψM(x,y)ψM(x,y) –d(A,B).

Thus, all hypotheses of Theorem . are satisﬁed, andx∗= – is the unique best proximity point of the mapT.

On the other hand,T is not a Geraghty contraction. Indeed, takingx= – andy= –, we get

d(Tx,Ty) =  >  =β

d(x,y)d(x,y).

Then Theorem . (the main result of Caballeroet al.[]) is not applicable.

Similarly, we cannot apply Theorem . becauseTis not aψ-Geraghty contraction. Let

x= –,y= – andψ(t) =αtwithα< . ThenT does not satisfy ().

If in Theorem . we takeψ(t) =tfor allt≥, then we deduce the following corollary.

Corollary . Let(A,B)be a pair of nonempty closed subsets of a complete metric space

(X,d)such that Ais nonempty.Let T:A→B be a non-self-mapping satisfying T(A)⊆B

and

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for all x,y∈A whereβ∈S,L≥,

Assume that the pair(A,B)has the weak P-property.Then T has a unique best proximity point in A.

If further in the above corollary we takeβ(t) =rwhere ≤r< , then we deduce another particular result.

Corollary . Let(A,B)be a pair of nonempty closed subsets of a complete metric space

(X,d)such that Ais nonempty.Let T:A→B be a non-self-mapping satisfying T(A)⊆B

and

d(Tx,Ty)≤rM(x,y) –d(A,B)+LN(x,y) –d(A,B)

for all x,y∈A where≤r< ,L≥,

Assume that the pair(A,B)has the weak P-property.Then T has a unique best proximity point in A.

3 Application to ﬁxed point theory

The caseA=Bin Theorem . corresponds to a self-mapping and results in an existence and uniqueness theorem for a ﬁxed point of the mapT. We state this case in the next theorem.

Theorem . Let(X,d)be a complete metric space.Suppose that A is a nonempty closed subset of X.Let T:A→A be a mapping such that

ψd(Tx,Ty)≤βψM(x,y)ψM(x,y)+LψN(x,y) for any x,y∈A, ()

whereψ∈,β∈S,L≥,

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty) and N(x,y) =mind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Then T has a unique ﬁxed point.

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Corollary . Let(X,d)be a complete metric space.Suppose that A is a nonempty closed subset of X.Let T:A→A be a mapping such that

d(Tx,Ty)≤βM(x,y)M(x,y) +LN(x,y) for any x,y∈A, ()

whereβ∈S,L≥,

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty) and N(x,y) =mind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Then T has a unique ﬁxed point.

Remark . The best proximity theorem given in this work, more precisely Theorem ., is a quite general result. It is a generalization of Theorem . in [], Theorem  in [], and also Theorem . given in Section . In addition, Corollary . improves Theorem ..

Remark . Very recently, Karapınar and Samet [] proved that the functiondϕ=ϕ◦d

on the setX, whereϕ∈ is also a metric onX. Therefore, some of the ﬁxed theorems regarding contraction mappings deﬁned via auxiliary functions from the set can be in fact deduced from the existing ones in the literature. However, our main result given in Theorem . is not a consequence of any existing theorems due to the fact that the contraction condition contains the termd(A,B).

On the other hand, the deﬁnition ofdϕ=ϕ◦dcan be used to show that Theorem . follows from Corollary .. Nevertheless, Corollary . and hence Theorem . are still new results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the manuscript.

Author details

1_{Department of Mathematics, Jubail College of Education, Dammam University, P.O. Box 12020, Industrial Jubail, 31961,}

Saudi Arabia.2_{Department of Mathematics, Atilim University, ˙Incek, Ankara 06836, Turkey.}3_{Nonlinear Analysis and} Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia.4_{Young Researchers and} Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran.

Acknowledgements

The authors thank to the referees for their careful reading and valuable comments and remarks which contributed to the improvement of the article.

Received: 11 November 2013 Accepted: 24 January 2014 Published:11 Feb 2014 References

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