Aydi et al. Fixed Point Theory and Applications 2014, 2014:32http://www.fixedpointtheoryandapplications.com/content/2014/1/32

RESEARCH Open Access

Best proximity points of generalized almostψ-Geraghty contractive non-self-mappingsHassen Aydi1, Erdal Karapınar2,3, Inci M Erhan2* and Peyman Salimi4

*Correspondence:ierhan@atilim.edu.tr;incierhan@gmail.com2 Department of Mathematics,Atilim University, Incek, Ankara06836, TurkeyFull list of author information isavailable at the end of the article

AbstractIn this paper, we introduce the new notion of almost ψ -Geraghty contractivemappings and investigate the existence of a best proximity point for such mappingsin complete metric spaces via the weak P-property. We provide an example tovalidate our best proximity point theorem. The obtained results extend, generalize,and complement some known fixed and best proximity point results from theliterature.MSC: 47H10; 54H25; 46J10; 46J15

Keywords: fixed point; metric space; best proximity point; generalized almostψ -Geraghty contractions

1 Introduction and preliminariesNon-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 onmetric spaces. Let (X,d) be a metric space and A and B be nonempty subsets of X.A mapping T : A → B is said to be a k-contraction if there exists k ∈ [, ) such thatd(Tx,Ty)≤ kd(x, y) for any x, y ∈ A. It is clear that a k-contraction coincides with the cel-ebrated Banach fixed point theorem (Banach contraction principle) [] if one takes A = Bwhere the induced metric space (A,d|A) is complete.In nonlinear analysis, the theory of fixed points is an essential instrument to solve the

equation Tx = x for a self-mapping T defined on a subset of an abstract space such as ametric space, a normed linear space or a topological vector space. Following the Banachcontraction principle, most of the fixed point results have been proved for a self-mappingdefined on an abstract space. It is quite natural to investigate the existence and unique-ness of a non-self-mapping T : A → B which does not possess a fixed point. If a non-self-mapping T : A → B has no fixed point, then the answer of the following question makessense: Is there a point x ∈ X such that the distance between x and Tx is closest in somesense? Roughly speaking, best proximity theory investigates the existence and uniquenessof such a closest point x. We refer the reader to [–] and [–] for further discussionof best proximity.

©2014 Aydi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 2 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Definition . Let (X,d) be a metric space and A,B ⊂ X. We say that x∗ ∈ A is a bestproximity point of the non-self-mapping T : A→ B if the following equality holds:

d(x∗,Tx∗) = d(A,B), ()

where d(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 proximitypoint when the underlying mapping is a self-mapping.Let (X,d) be a metric space. Suppose that A and B are nonempty subsets of a metric

space (X,d). We define the following sets:

A ={x ∈ A : d(x, y) = d(A,B) for some y ∈ B

},

B ={y ∈ B : d(x, y) = d(A,B) for some x ∈ A

}.

()

In [], the authors presented sufficient conditions for the setsA andB to be nonempty.In Geraghty [] introduced the class S of functions β : [,∞) → [, ) satisfying

the following condition:

β(tn)→ implies tn → . ()

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

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

d(Tx,Ty)≤ β(d(x, y)

)d(x, y) for any x, y ∈ X, ()

where β ∈ S, then T has a unique fixed point.

Recently, Caballero et al. [] introduced the following contraction.

Definition . ([]) LetA, B be two nonempty subsets of ametric space (X,d). AmappingT : A→ B is said to be a Geraghty-contraction if there exists β ∈ S such that

d(Tx,Ty)≤ β(d(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 metricspace (X,d) such that A is nonempty. Let T : A → B be a continuous,Geraghty-contractionsatisfying T(A) ⊆ B. Suppose that the pair (A,B) has the P-property, then there exists aunique x∗ in A such that d(x∗,Tx∗) = d(A,B).

The P-property mentioned in the theorem above has been introduced in [].

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 3 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Definition . Let (A,B) be a pair of nonempty subsets of a metric space (X,d) withA = ∅. Then the pair (A,B) is said to have the P-property if and only if for any x,x ∈ A

and y, 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 theP-property. In [], the author proved that any pair (A,B) of nonempty closed convexsubsets of a real Hilbert space H satisfies the P-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) withA = ∅. Then the pair (A,B) is said to have the weak P-property if and only if for anyx,x ∈ A and y, y ∈ B,

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

Let � denote the class of functions ψ : [,∞) → [,∞) satisfying the following condi-tions:(a) ψ is nondecreasing;(b) ψ is subadditive, that is, ψ(s + t)≤ ψ(s) +ψ(t);(c) ψ is continuous;(d) ψ(t) = ⇔ t = .The notion of ψ-Geraghty contraction has been introduced very recently in [], as an

extension of Definition ..

Definition . Let A, B be two nonempty subsets of a metric space (X,d). A mappingT : A→ B is said to be a ψ-Geraghty contraction if there exist β ∈ S and ψ ∈ � such that

ψ(d(Tx,Ty)

) ≤ β(ψ

(d(x, y)

))ψ

(d(x, y)

)for any x, y ∈ A. ()

Remark . Notice that since β : [,∞)→ [, ), we have

ψ(d(Tx,Ty)

) ≤ β(ψ

(d(x, y)

))ψ

(d(x, y)

)<ψ

(d(x, y)

)for any x, y ∈ A with x = 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 completemetricspace (X,d) such that A is nonempty.Let T : A → B be aψ-Geraghty contraction satisfyingT(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 resultsOur main results are based on the following definition which is a generalization of Defi-nition ..

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 4 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Definition . Let A, B be two nonempty subsets of a metric space (X,d). A mappingT : A → B is said to be a generalized almost ψ-Geraghty contraction if there exist β ∈ Sand ψ ∈ � such that

ψ(d(Tx,Ty)

) ≤ β(ψ

(M(x, y)

))ψ

(M(x, y) – d(A,B)

)+ Lψ

(N(x, y) – d(A,B)

)()

for all x, y ∈ A where L≥ ,

M(x, y) = max{d(x, y),d(x,Tx),d(y,Ty)

},

N(x, y) = min{d(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 bestproximity point for a non-self-mapping satisfying a generalized almost ψ-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 almost ψ-Geraghty con-traction satisfying T(A)⊆ B. Assume that the pair (A,B) has the weak P-property. ThenT has a unique best proximity point in A.

Proof Since the subset A is not empty, we can take x in A. Taking into account thatTx ∈ T(A) ⊆ B, we can find x ∈ A such that d(x,Tx) = d(A,B). Further, sinceTx ∈ T(A)⊆ B, it follows that there is an element x in A such that d(x,Tx) = d(A,B).Recursively, we obtain a sequence {xn} in A satisfying

d(xn+,Txn) = d(A,B) for any n ∈N. ()

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

d(xn,xn+)≤ d(Txn–,Txn) for any n ∈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)

}≤ max

{d(xn–,xn),d(xn,xn+)

}+ d(A,B). ()

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 5 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Also note that

N(xn–,xn) – d(A,B)

= min{d(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(xn,Txn–)

}– d(A,B)

≤ min{d(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(A,B)

}– d(A,B)

= d(A,B) – d(A,B) = . ()

If there exists n ∈N such that d(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 that d(xn,xn+) > for all n ∈N. In view of the factthat T is a generalized almost ψ-Geraghty contraction, we have

ψ(d(xn,xn+)

) ≤ ψ(d(Txn–,Txn)

)≤ β

(ψ

(M(xn–,xn)

))ψ

(M(xn–,xn) – d(A,B)

)+ Lψ

(N(xn–,xn) – d(A,B)

)= β

(ψ

(M(xn–,xn)

))ψ

(M(xn–,xn) – d(A,B)

)+ Lψ()

= β(ψ

(M(xn–,xn)

))ψ

(M(xn–,xn) – d(A,B)

)<ψ

(M(xn–,xn) – d(A,B)

). ()

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

ψ(d(xn,xn+)

)<ψ

(M(xn–,xn) – d(A,B)

) ≤ ψ(max

{d(xn–,xn),d(xn,xn+)

}).

If for some n, max{d(xn–,xn),d(xn,xn+)} = d(xn,xn+), then we get

ψ(d(xn,xn+)

)<ψ

(d(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 all n ∈N. Regarding the inequality (), we see that

ψ(d(xn,xn+)

)=ψ

(d(Txn–,Txn)

)≤ β

(ψ

(M(xn–,xn)

))ψ

(d(xn–,xn)

)<ψ

(d(xn–,xn)

)()

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 6 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

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 hencelimn→∞ d(xn,xn+) = s≥ exists. Assume that s > . Rewrite () as

ψ(d(xn+,xn+))ψ(d(xn,xn+))

≤ β(ψ

(M(xn,xn+)

)) ≤

for each n≥ . Taking the limit of both sides as n→ ∞, we find

limn→∞β

(ψ

(M(xn,xn+)

))= .

On the other hand, since β ∈ S, we conclude limn→∞ ψ(M(xn,xn+)) = , that is,

s = limn→∞d(xn,xn+) = . ()

Since d(xn,Txn–) = d(A,B) holds for all n ∈ N and (A,B) satisfies the weak P-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)

}≤ max

{d(xm,xn),d(xm,xm+),d(xn,xn+)

}+ d(A,B).

By using limn→∞ d(xn,xn+) = , we get

limm,n→∞

(M(xm,xn) – d(A,B)

) ≤ limm,n→∞d(xm,xn). ()

On the other hand,

≤N(xm,xn) – d(A,B)

= min{d(xm,Txm),d(xn,Txn),d(xm,Txn),d(xn,Txm)

}– d(A,B)

≤ min{d(xm,xm+) + d(A,B),d(xn,Txn),d(xm,Txn),d(xn,Txm)

}– d(A,B). ()

Due to the fact that limn→∞ d(xn,xn+) = , we obtain

limm,n→∞

[N(xm,xn) – d(A,B)

]= . ()

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

ε = lim supm,n→∞

d(xn,xm) > . ()

Employing the triangular inequality and (), we get

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

≤ d(xn,xn+) + d(Txn,Txm) + d(xm+,xm). ()

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 7 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Combining () and (), and regarding the properties of ψ , we obtain

ψ(d(xn,xm)

) ≤ ψ(d(xn,xn+) + d(Txn,Txm) + d(xm+,xm)

)≤ ψ

(d(xn,xn+)

)+ψ

(d(Txn,Txm)

)+ψ

(d(xm+,xm)

)≤ ψ

(d(xn,xn+)

)+ β

(ψ

(M(xn,xm)

))ψ

(M(xn,xm) – d(A,B)

)+ Lψ

(N(xn,xm) – d(A,B)

)+ψ

(d(xm+,xm)

). ()

From (), (), (), and by using limn→∞ d(xn,xn+) = , we have

limm,n→∞ψ

(d(xn,xm)

) ≤ limm,n→∞β

(ψ

(M(xn,xm)

))lim

m,n→∞ψ(M(xm,xn) – d(A,B)

)≤ lim

m,n→∞β(ψ

(M(xn,xm)

))lim

m,n→∞ψ(d(xm,xn)

).

So by (), we get

≤ limm,n→∞β

(ψ

(M(xn,xm)

)),

that is, limm,n→∞ β(ψ(M(xn,xm))) = . Therefore, limm,n→∞ M(xn,xm) = . This impliesthat limm,n→∞ d(xn,xm) = ,which is a contradiction. Therefore, {xn} is aCauchy sequence.Since {xn} ⊂ A and A is a closed subset of the complete metric space (X,d), we can find

x∗ ∈ A such that xn → x∗ as n → ∞. We shall show that d(x∗,Tx∗) = d(A,B). If x∗ = Tx∗,thenA∩B = ∅, and d(x∗,Tx∗) = d(A,B) = , i.e., x∗ is a best proximity point ofT . Hence, weassume that d(x∗,Tx∗) > . Suppose on the contrary that x∗ is not a best proximity pointof T , that is, d(x∗,Tx∗) > d(A,B). First note that

d(x∗,Tx∗) ≤ d

(x∗,Txn

)+ d

(Txn,Tx∗)

≤ d(x∗,xn+

)+ d(xn+,Txn) + d

(Txn,Tx∗)

≤ d(x∗,xn+

)+ d(A,B) + d

(Txn,Tx∗).

Taking the limit as n→ ∞ in the above inequality, we obtain

d(x∗,Tx∗) – d(A,B)≤ lim

n→∞d(Txn,Tx∗).

Since ψ is nondecreasing and continuous, then

ψ(d(x∗,Tx∗) – d(A,B)

) ≤ ψ(

limn→∞d

(Txn,Tx∗)) = lim

n→∞ψ(d(Txn,Tx∗)). ()

Also, letting n → ∞ in () results in

limn→∞d(xn,Txn)≤ d(A,B),

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

limn→∞M

(xn,x∗) = max

{limn→∞d

(x∗,xn

), limn→∞d(xn,Txn),d

(x∗,Tx∗)} = d

(x∗,Tx∗),

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 8 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

and therefore

limn→∞ψ

(M

(xn,x∗) – d(A,B)

)=ψ

(d(x∗,Tx∗) – d(A,B)

). ()

Further,

limn→∞N

(xn,x∗) – d(A,B)

= min{

limn→∞d(xn,Txn),d

(x∗,Tx∗), lim

n→∞d(xn,Tx∗), lim

n→∞d(x∗,Txn

)}– d(A,B),

which implies

limn→∞N

(xn,x∗) – d(A,B) = . ()

Therefore, combining (), (), (), and () we deduce

ψ(d(x∗,Tx∗) – d(A,B)

) ≤ limn→∞ψ

(d(Txn,Tx∗))

≤ limn→∞β

(ψ

(M

(xn,x∗))) lim

n→∞ψ(M

(xn,x∗) – d(A,B)

)+ L lim

n→∞ψ(N

(xn,x∗) – d(A,B)

)= lim

n→∞β(ψ

(M

(xn,x∗)))ψ(

d(x∗,Tx∗) – d(A,B)

). ()

Now, since ψ(d(x∗,Tx∗) – d(A,B)) > , and making use of (), we get

≤ limn→∞β

(ψ

(M

(xn,x∗))),

that is,

limn→∞β

(ψ

(M

(xn,x∗))) = ,

which implies

limn→∞M

(xn,x∗) = d

(x∗,Tx∗) = ,

and so d(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 of T . This com-pletes the proof of the existence of a best proximity point.We shall show next the uniqueness of the best proximity point of T . Suppose that x∗ and

y∗ are two best proximity points of T , such that x∗ = y∗. This implies that

d(x∗,Tx∗) = d(A,B) = d

(y∗,Ty∗), ()

where d(x∗, y∗) > . Due to the weak P-property of the pair (A,B), we have

d(x∗, y∗) ≤ d

(Tx∗,Ty∗). ()

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 9 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

Observe that in this case

M(x∗, y∗) = max

{d(x∗, y∗),d(

x∗,Tx∗),d(y∗,Ty∗)}

= max{d(x∗, y∗),d(A,B),d(A,B)}.

Also, note that

N(x∗, y∗) – d(A,B) = min

{d(x∗,Tx∗),d(

y∗,Ty∗),d(x∗,Ty∗),d(

y∗,Tx∗)} – d(A,B)

= min{d(A,B),d(A,B),d

(x∗,Ty∗),d(

y∗,Tx∗)} – d(A,B)

= d(A,B) – d(A,B) = .

Using the fact that T is a generalized almost ψ-Geraghty contraction, we derive

ψ(d(x∗, y∗)) ≤ ψ

(d(Tx∗,Ty∗))

≤ β(ψ

(M

(x∗, y∗)))ψ(

M(x∗, y∗) – d(A,B)

)+ Lψ

(N

(x∗, y∗) – d(A,B)

)= β

(ψ

(M

(x∗, y∗)))ψ(

M(x∗, y∗) – d(A,B)

)< ψ

(M

(x∗, y∗) – d(A,B)

).

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

<ψ(d(x∗, y∗)) <ψ(), ()

which implies d(x∗, y∗) = and contradicts the assumption d(x∗, y∗) > . Else, ifM(x∗, y∗) =d(x∗, y∗), we deduce

<ψ(d(x∗, y∗)) <ψ

(d(x∗, y∗) – d(A,B)

), ()

which is not possible, since ψ is nondecreasing. Therefore, we must have d(x∗, y∗) = .This completes the proof. �

To illustrate our result given in Theorem ., we present the following example, whichshows that Theorem . is a proper generalization of Theorem ..

Example . Consider the space X =R with Euclidean metric. Take the sets

A = (–∞, –] and B = [, +∞).

Obviously, d(A,B) = . LetT : A→ B be defined byTx = –x. Notice thatA = {–}, B = {}and T(A)⊆ B. Also, it is clear that the pair (A,B) has the weak P-property.Consider

β(t) ={

+t , if ≤ t < ,t+t , if t ≥ ,

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 10 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

and ψ(t) = αt (with α ≥ ) for all t ≥ . Note that β ∈ S and ψ ∈ � . For all x, y ∈ A, we

have

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

We shall show that T is a generalized almost ψ-Geraghty contraction. Without loss ofgenerality, consider the case where x ≥ y. Thenwe haveM(x, y) = –y and d(Tx,Ty) = x–y.In this case, we see that

ψ(d(Tx,Ty)

)= α(x – y)≤ α(–x – y – )

≤ α(–x – y – )≤ [–αy][α(–x – y – )

]= [–αy]

[α(–y – ) – α(x – y)

]=ψ

(M(x, y)

)[ψ

(M(x, y) – d(A,B)

)–ψ

(d(Tx,Ty)

)].

Therefore

ψ(d(Tx,Ty)

) ≤ ψ(M(x, y)) +ψ(M(x, y))

ψ(M(x, y) – d(A,B)

). ()

On the other hand, we know that ψ(M(x, y)) = –αy≥ for all x, y ∈ A with x ≥ y. Hence,

β(ψ

(M(x, y)

))=

ψ(M(x, y)) +ψ(M(x, y))

,

and from () we deduce

ψ(d(Tx,Ty)

) ≤ β(ψ

(M(x, y)

))ψ

(M(x, y) – d(A,B)

).

Thus, all hypotheses of Theorem . are satisfied, and x∗ = – is the unique best proximitypoint of the map T .On the other hand, T is not a Geraghty contraction. Indeed, taking x = – and y = –,

we get

d(Tx,Ty) = > = β

(d(x, y)

)d(x, y).

Then Theorem . (the main result of Caballero et al. []) is not applicable.Similarly, we cannot apply Theorem . because T is not a ψ-Geraghty contraction. Let

x = –, y = – and ψ(t) = αt with α < . Then T does not satisfy ().

If in Theorem . we take ψ(t) = t for all t ≥ , 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)⊆ Band

d(Tx,Ty)≤ β(M(x, y)

)[M(x, y) – d(A,B)

]+ L

[N(x, y) – d(A,B)

]

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 11 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

for all x, y ∈ A where β ∈ S, L ≥ ,

M(x, y) = max{d(x, y),d(x,Tx),d(y,Ty)

},

N(x, y) = min{d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)

}.

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

If further in the above corollary we take β(t) = r where ≤ r < , thenwe deduce anotherparticular 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)⊆ Band

d(Tx,Ty)≤ r[M(x, y) – d(A,B)

]+ L

[N(x, y) – d(A,B)

]

for all x, y ∈ A where ≤ r < , L ≥ ,

M(x, y) = max{d(x, y),d(x,Tx),d(y,Ty)

},

N(x, y) = min{d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)

}.

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

3 Application to fixed point theoryThe case A = B in Theorem . corresponds to a self-mapping and results in an existenceand uniqueness theorem for a fixed point of the map T . We state this case in the nexttheorem.

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

ψ(d(Tx,Ty)

) ≤ β(ψ

(M(x, y)

))ψ

(M(x, y)

)+ Lψ

(N(x, y)

)for any x, y ∈ A, ()

where ψ ∈ � , β ∈ S, L ≥ ,

M(x, y) = max{d(x, y),d(x,Tx),d(y,Ty)

}and

N(x, y) = min{d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)

}.

Then T has a unique fixed point.

Finally, taking ψ(t) = t in Theorem ., we get another fixed point result.

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 12 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

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

d(Tx,Ty)≤ β(M(x, y)

)M(x, y) + LN(x, y) for any x, y ∈ A, ()

where β ∈ S, L ≥ ,

M(x, y) = max{d(x, y),d(x,Tx),d(y,Ty)

}and

N(x, y) = min{d(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 [], andalso Theorem . given in Section . In addition, Corollary . improves Theorem ..

Remark . Very recently, Karapınar and Samet [] proved that the function dϕ = ϕ ◦ don the set X, where ϕ ∈ � is also a metric on X. Therefore, some of the fixed theoremsregarding contraction mappings defined via auxiliary functions from the set � can bein fact deduced from the existing ones in the literature. However, our main result givenin Theorem . is not a consequence of any existing theorems due to the fact that thecontraction condition contains the term d(A,B).On the other hand, the definition of dϕ = ϕ ◦ d can be used to show that Theorem .

follows from Corollary .. Nevertheless, Corollary . and hence Theorem . are stillnew results.

Competing interestsThe authors declare that they have no competing interests.

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

Author details1 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 andApplied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia. 4 Young Researchers andElite Club, Rasht Branch, Islamic Azad University, Rasht, Iran.

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

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

References1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam.

Math. 3, 133-181 (1922)2. Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal. 70,

3665-3671 (2009)3. Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl. 156, 2942-2948 (2009)4. Basha, SS, Veeramani, P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103,

119-129 (2000)5. Bilgili, N, Karapınar, E, Sadarangani, K: A generalization for the best proximity point of Geraghty-contractions.

J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-2866. Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory

Appl. (2012). doi:10.1186/1687-1812-2012-231

Aydi et al. Fixed Point Theory and Applications 2014, 2014:32 Page 13 of 13http://www.fixedpointtheoryandapplications.com/content/2014/1/32

7. Jleli, M, Samet, B: Best proximity points for α-ψ -proximal contractive type mappings and applications. Bull. Sci. Math.137, 977-995 (2013). doi:10.1016/j.bulsci.2013.02.003

8. Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001-1006(2006)

9. De la Sen, M: Fixed point and best proximity theorems under two classes of integral-type contractive conditions inuniform metric spaces. Fixed Point Theory Appl. (2010). doi:10.1155/2010/510974

10. Karapınar, E: Best proximity points of cyclic mappings. Appl. Math. Lett. 25, 1761-1766 (2012)11. Karapınar, E: On best proximity point of ψ -Geraghty contractions. Preprint12. Karapınar, E, Erhan, IM: Best proximity point on different type contractions. Appl. Math. Inform. Sci. 3, 342-353 (2011)13. Karapınar, E: Best proximity points of Kannan type cylic weak φ-contractions in ordered metric spaces. An. Univ.

“Ovidius” Constanta, Ser. Mat. 20, 51-64 (2012)14. Karapınar, E: On best proximity point of ψ -Geraghty contractions. Fixed Point Theory Appl. (2013).

doi:10.1186/1687-1812-2013-20015. Karapınar, E, Samet, B: A note on ‘ψ -Geraghty type contractions’. Fixed Point Theory Appl. (2014).

doi:10.1186/1687-1812-2014-2616. Karapınar, E, Pragadeeswarar, V, Marudai, M: Best proximity point for generalized proximal weak contractions in

complete metric space. J. Appl. Math. 2013, Article ID 150941 (2013)17. Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24,

851-862 (2003)18. Kumam, P, Aydi, H, Karapınar, E, Sintunavarat, W: Best proximity points and extension of Mizoguchi-Takahashi’s fixed

point theorem. Fixed Point Theory Appl. (2013). doi:10.1186/1687-1812-2013-24219. Markin, J, Shahzad, N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex

spaces. Nonlinear Anal. 70, 2435-2441 (2009)20. Mongholkeha, C, Cho, YJ, Kumam, P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point

Theory Appl. (2013). doi:10.1186/1687-1812-2013-18021. Mongholkeha, C, Cho, YJ, Kumam, P: Best proximity points for generalized proximal C-contraction mappings in

metric spaces with partial orders. J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-9422. Mongholkeha, C, Kumam, P: Best proximity points for asymptotic proximal pointwise weaker Meir-Keeler type

ψ -contraction mappings. J. Egypt. Math. Soc. 21(2), 87-90 (2013)23. Mongholkeha, C, Kumam, P: Some common best proximity points for proximity commuting mappings. Optim. Lett.

7(8), 1825-1836 (2013)24. Mongholkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric

spaces. J. Optim. Theory Appl. 155, 215-226 (2012)25. Nashine, HK, Vetro, C, Kumam, P: Best proximity point theorems for rationasl proximal contractions, for proximity

commuting mappings. Fixed Point Theory Appl. (2013). doi:10.1186/1687-1812-2013-9526. Pragadeeswarar, V, Marudai, M: Best proximity points: approximation and optimization in partially ordered metric

spaces. Optim. Lett. (2012). doi:10.1007/s11590-012-0529-x27. Raj, VS, Veeramani, P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 10, 21-28

(2009)28. Raj, VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 74, 4804-4808 (2011)29. Raj, VS: Banach’s contraction principle for non-self mappings. Preprint30. Samet, B: Some results on best proximity points. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0269-931. Shahzad, N, Basha, SS, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148,

69-78 (2011)32. Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math. 67, 421-429 (2001)33. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973)34. Zhang, J, Su, Y, Cheng, Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory

Appl. (2013). doi:10.1186/1687-1812-2013-99

10.1186/1687-1812-2014-32Cite this article as: Aydi et al.: Best proximity points of generalized almost ψ -Geraghty contractivenon-self-mappings. Fixed Point Theory and Applications 2014, 2014:32