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

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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 weak P-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.

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 mapping T:AB is said to be a k-contraction if there exists k[0,1) such that d(Tx,Ty)kd(x,y) for any x,yA. It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem (Banach contraction principle) [1] if one takes A=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 equation Tx=x for a self-mapping T 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 uniqueness of a non-self-mapping T:AB which does not possess a fixed point. If a non-self-mapping T:AB has no fixed point, then the answer of the following question makes sense: Is there a point xX such that the distance between x and Tx is closest in some sense? Roughly speaking, best proximity theory investigates the existence and uniqueness of such a closest point x. We refer the reader to [29] and [1032] for further discussion of best proximity.

Definition 1.1 Let (X,d) be a metric space and A,BX. We say that x A is a best proximity point of the non-self-mapping T:AB if the following equality holds:

d ( x , T x ) =d(A,B),
(1)

where d(A,B)=inf{d(x,y):xA,yB}.

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 that A and B are nonempty subsets of a metric space (X,d). We define the following sets:

A 0 = { x A : d ( x , y ) = d ( A , B )  for some  y B } , B 0 = { y B : d ( x , y ) = d ( A , B )  for some  x A } .
(2)

In [17], the authors presented sufficient conditions for the sets A 0 and B 0 to be nonempty.

In 1973 Geraghty [33] introduced the class S of functions β:[0,)[0,1) satisfying the following condition:

β( t n )1implies t n 0.
(3)

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

Theorem 1.1 (Geraghty [33])

Let (X,d) be a complete metric space and T:XX be an operator. If T satisfies the following inequality:

d(Tx,Ty)β ( d ( x , y ) ) d(x,y)for any x,yX,
(4)

where βS, then T has a unique fixed point.

Recently, Caballero et al. [6] introduced the following contraction.

Definition 1.2 ([6])

Let A, B be two nonempty subsets of a metric space (X,d). A mapping T:AB is said to be a Geraghty-contraction if there exists βS such that

d(Tx,Ty)β ( d ( x , y ) ) d(x,y)for any x,yA.
(5)

Based on Definition 1.2, the authors [6] obtained the following result.

Theorem 1.2 (See [6])

Let (A,B) be a pair of nonempty closet subsets of a complete metric space (X,d) such that A 0 is nonempty. Let T:AB be a continuous, Geraghty-contraction satisfying T( A 0 ) B 0 . Suppose that the pair (A,B) has the P-property, then there exists a unique x in A such that d( x ,T x )=d(A,B).

The P-property mentioned in the theorem above has been introduced in [29].

Definition 1.3 Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with A 0 . Then the pair (A,B) is said to have the P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 ,

d( x 1 , y 1 )=d(A,B)andd( x 2 , y 2 )=d(A,B)d( x 1 , x 2 )=d( y 1 , y 2 ).
(6)

It is easily seen that for any nonempty subset A of (X,d), the pair (A,A) has the P-property. In [29], the author proved that any pair (A,B) of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.

Recently, Zhang et al. [34] defined the following notion, which is weaker than the P-property.

Definition 1.4 Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with A 0 . Then the pair (A,B) is said to have the weak P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 ,

d( x 1 , y 1 )=d(A,B)andd( x 2 , y 2 )=d(A,B)d( x 1 , x 2 )d( y 1 , y 2 ).
(7)

Let Ψ denote the class of functions ψ:[0,)[0,) satisfying the following conditions:

  1. (a)

    ψ is nondecreasing;

  2. (b)

    ψ is subadditive, that is, ψ(s+t)ψ(s)+ψ(t);

  3. (c)

    ψ is continuous;

  4. (d)

    ψ(t)=0t=0.

The notion of ψ-Geraghty contraction has been introduced very recently in [11], as an extension of Definition 1.2.

Definition 1.5 Let A, B be two nonempty subsets of a metric space (X,d). A mapping T:AB is said to be a ψ-Geraghty contraction if there exist βS and ψΨ such that

ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) for any x,yA.
(8)

Remark 1.1 Notice that since β:[0,)[0,1), we have

ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) < ψ ( d ( x , y ) ) for any  x , y A  with  x y .
(9)

In [11], the author also proved the following best proximity point theorem.

Theorem 1.3 (See [11])

Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A 0 is nonempty. Let T:AB be a ψ-Geraghty contraction satisfying T( A 0 ) B 0 . Suppose that the pair (A,B) has the P-property. Then there exists a unique x in A such that d( x ,T x )=d(A,B).

2 Main results

Our main results are based on the following definition which is a generalization of Definition 1.5.

Definition 2.1 Let A, B be two nonempty subsets of a metric space (X,d). A mapping T:AB is said to be a generalized almost ψ-Geraghty contraction if there exist βS and ψΨ such that

ψ ( d ( T x , T y ) ) β ( ψ ( M ( x , y ) ) ) ψ ( M ( x , y ) d ( A , B ) ) +Lψ ( N ( x , y ) d ( A , B ) )
(10)

for all x,yA where L0,

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } , N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

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 almost ψ-Geraghty contraction.

Theorem 2.1 Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A 0 is nonempty. Let T:AB be a generalized almost ψ-Geraghty contraction satisfying T( A 0 ) B 0 . Assume that the pair (A,B) has the weak P-property. Then T has a unique best proximity point in A.

Proof Since the subset A 0 is not empty, we can take x 0 in A 0 . Taking into account that T x 0 T( A 0 ) B 0 , we can find x 1 A 0 such that d( x 1 ,T x 0 )=d(A,B). Further, since T x 1 T( A 0 ) B 0 , it follows that there is an element x 2 in A 0 such that d( x 2 ,T x 1 )=d(A,B). Recursively, we obtain a sequence { x n } in A 0 satisfying

d( x n + 1 ,T x n )=d(A,B)for any nN.
(11)

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

d( x n , x n + 1 )d(T x n 1 ,T x n )for any nN.
(12)

Due to the triangle inequality together with the equality (11) we have

d( x n 1 ,T x n 1 )d( x n 1 , x n )+d( x n ,T x n 1 )=d( x n 1 , x n )+d(A,B).

Analogously, combining the equalities (11) and (12) with the triangle inequality we obtain

d( x n ,T x n )d( x n , x n + 1 )+d( x n + 1 ,T x n )=d( x n , x n + 1 )+d(A,B).
(13)

Consequently, we have

M ( x n 1 , x n ) = max { d ( x n 1 , x n ) , d ( x n 1 , T x n 1 ) , d ( x n , T x n ) } max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } + d ( A , B ) .
(14)

Also note that

N ( x n 1 , x n ) d ( A , B ) = min { d ( x n 1 , T x n 1 ) , d ( x n , T x n ) , d ( x n 1 , T x n ) , d ( x n , T x n 1 ) } d ( A , B ) min { d ( x n 1 , T x n 1 ) , d ( x n , T x n ) , d ( x n 1 , T x n ) , d ( A , B ) } d ( A , B ) = d ( A , B ) d ( A , B ) = 0 .
(15)

If there exists n 0 N such that d( x n 0 , x n 0 + 1 )=0, then the proof is completed. Indeed,

0=d( x n 0 , x n 0 + 1 )=d(T x n 0 1 ,T x n 0 ),
(16)

and consequently, T x n 0 1 =T x n 0 . Therefore, we conclude that

d(A,B)=d( x n 0 ,T x n 0 1 )=d( x n 0 ,T x n 0 ).
(17)

For the rest of the proof, we suppose that d( x n , x n + 1 )>0 for all nN. In view of the fact that T is a generalized almost ψ-Geraghty contraction, we have

ψ ( d ( x n , x n + 1 ) ) ψ ( d ( T x n 1 , T x n ) ) β ( ψ ( M ( x n 1 , x n ) ) ) ψ ( M ( x n 1 , x n ) d ( A , B ) ) + L ψ ( N ( x n 1 , x n ) d ( A , B ) ) = β ( ψ ( M ( x n 1 , x n ) ) ) ψ ( M ( x n 1 , x n ) d ( A , B ) ) + L ψ ( 0 ) = β ( ψ ( M ( x n 1 , x n ) ) ) ψ ( M ( x n 1 , x n ) d ( A , B ) ) < ψ ( M ( x n 1 , x n ) d ( A , B ) ) .
(18)

Taking into account the inequalities (14) and (18), we deduce that

ψ ( d ( x n , x n + 1 ) ) <ψ ( M ( x n 1 , x n ) d ( A , B ) ) ψ ( max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } ) .

If for some n, max{d( x n 1 , x n ),d( x n , x n + 1 )}=d( x n , x n + 1 ), then we get

ψ ( d ( x n , x n + 1 ) ) <ψ ( d ( x n , x n + 1 ) ) ,

which is a contradiction. Therefore, we must have

M( x n 1 , x n )max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } +d(A,B)=d( x n 1 , x n )+d(A,B)
(19)

for all nN. Regarding the inequality (18), we see that

ψ ( d ( x n , x n + 1 ) ) = ψ ( d ( T x n 1 , T x n ) ) β ( ψ ( M ( x n 1 , x n ) ) ) ψ ( d ( x n 1 , x n ) ) < ψ ( d ( x n 1 , x n ) )
(20)

holds for all nN. Since ψ is nondecreasing, then d( x n , x n + 1 )<d( x n 1 , x n ) for all n. Consequently, the sequence {d( x n , x n + 1 )} is decreasing and is bounded below and hence lim n d( x n , x n + 1 )=s0 exists. Assume that s>0. Rewrite (20) as

ψ ( d ( x n + 1 , x n + 2 ) ) ψ ( d ( x n , x n + 1 ) ) β ( ψ ( M ( x n , x n + 1 ) ) ) 1

for each n1. Taking the limit of both sides as n, we find

lim n β ( ψ ( M ( x n , x n + 1 ) ) ) =1.

On the other hand, since βS, we conclude lim n ψ(M( x n , x n + 1 ))=0, that is,

s= lim n d( x n , x n + 1 )=0.
(21)

Since d( x n ,T x n 1 )=d(A,B) holds for all nN and (A,B) satisfies the weak P-property, then for all m,nN, we can write

d( x m , x n )d(T x m 1 ,T x n 1 ).
(22)

From (13), we deduce

M ( x m , x n ) = max { d ( x m , x n ) , d ( x m , T x m ) , d ( x n , T x n ) } max { d ( x m , x n ) , d ( x m , x m + 1 ) , d ( x n , x n + 1 ) } + d ( A , B ) .

By using lim n d( x n , x n + 1 )=0, we get

lim m , n ( M ( x m , x n ) d ( A , B ) ) lim m , n d( x m , x n ).
(23)

On the other hand,

0 N ( x m , x n ) d ( A , B ) = min { d ( x m , T x m ) , d ( x n , T x n ) , d ( x m , T x n ) , d ( x n , T x m ) } d ( A , B ) min { d ( x m , x m + 1 ) + d ( A , B ) , d ( x n , T x n ) , d ( x m , T x n ) , d ( x n , T x m ) } d ( A , B ) .
(24)

Due to the fact that lim n d( x n , x n + 1 )=0, we obtain

lim m , n [ N ( x m , x n ) d ( A , B ) ] =0.
(25)

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

ε= lim sup m , n d( x n , x m )>0.
(26)

Employing the triangular inequality and (22), we get

d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x m + 1 ) + d ( x m + 1 , x m ) d ( x n , x n + 1 ) + d ( T x n , T x m ) + d ( x m + 1 , x m ) .
(27)

Combining (10) and (27), and regarding the properties of ψ, we obtain

ψ ( d ( x n , x m ) ) ψ ( d ( x n , x n + 1 ) + d ( T x n , T x m ) + d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + ψ ( d ( T x n , T x m ) ) + ψ ( d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + β ( ψ ( M ( x n , x m ) ) ) ψ ( M ( x n , x m ) d ( A , B ) ) + L ψ ( N ( x n , x m ) d ( A , B ) ) + ψ ( d ( x m + 1 , x m ) ) .
(28)

From (23), (25), (28), and by using lim n d( x n , x n + 1 )=0, we have

lim m , n ψ ( d ( x n , x m ) ) lim m , n β ( ψ ( M ( x n , x m ) ) ) lim m , n ψ ( M ( x m , x n ) d ( A , B ) ) lim m , n β ( ψ ( M ( x n , x m ) ) ) lim m , n ψ ( d ( x m , x n ) ) .

So by (26), we get

1 lim m , n β ( ψ ( M ( x n , x m ) ) ) ,

that is, lim m , n β(ψ(M( x n , x m )))=1. Therefore, lim m , n M( x n , x m )=0. This implies that lim m , n d( x n , x m )=0, which is a contradiction. Therefore, { x n } is a Cauchy sequence.

Since { x n }A and A is a closed subset of the complete metric space (X,d), we can find x A such that x n x as n. We shall show that d( x ,T x )=d(A,B). If x =T x , then AB, and d( x ,T x )=d(A,B)=0, i.e., x is a best proximity point of T. Hence, we assume that d( x ,T x )>0. Suppose on the contrary that x is not a best proximity point of T, that is, d( x ,T x )>d(A,B). First note that

d ( x , T x ) d ( x , T x n ) + d ( T x n , T x ) d ( x , x n + 1 ) + d ( x n + 1 , T x n ) + d ( T x n , T x ) d ( x , x n + 1 ) + d ( A , B ) + d ( T x n , T x ) .

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

d ( x , T x ) d(A,B) lim n d ( T x n , T x ) .

Since ψ is nondecreasing and continuous, then

ψ ( d ( x , T x ) d ( A , B ) ) ψ ( lim n d ( T x n , T x ) ) = lim n ψ ( d ( T x n , T x ) ) .
(29)

Also, letting n in (13) results in

lim n d( x n ,T x n )d(A,B),

that is, lim n d( x n ,T x n )=d(A,B). Then we get

lim n M ( x n , x ) =max { lim n d ( x , x n ) , lim n d ( x n , T x n ) , d ( x , T x ) } =d ( x , T x ) ,

and therefore

lim n ψ ( M ( x n , x ) d ( A , B ) ) =ψ ( d ( x , T x ) d ( A , B ) ) .
(30)

Further,

lim n N ( x n , x ) d ( A , B ) = min { lim n d ( x n , T x n ) , d ( x , T x ) , lim n d ( x n , T x ) , lim n d ( x , T x n ) } d ( A , B ) ,

which implies

lim n N ( x n , x ) d(A,B)=0.
(31)

Therefore, combining (10), (29), (30), and (31) we deduce

ψ ( d ( x , T x ) d ( A , B ) ) lim n ψ ( d ( T x n , T x ) ) lim n β ( ψ ( M ( x n , x ) ) ) lim n ψ ( M ( x n , x ) d ( A , B ) ) + L lim n ψ ( N ( x n , x ) d ( A , B ) ) = lim n β ( ψ ( M ( x n , x ) ) ) ψ ( d ( x , T x ) d ( A , B ) ) .
(32)

Now, since ψ(d( x ,T x )d(A,B))>0, and making use of (32), we get

1 lim n β ( ψ ( M ( x n , x ) ) ) ,

that is,

lim n β ( ψ ( M ( x n , x ) ) ) =1,

which implies

lim n M ( x n , x ) =d ( x , T x ) =0,

and so d( x ,T x )=0>d(A,B), which is a contradiction. Therefore, d( x ,T x )d(A,B), that is, d( x ,T x )=d(A,B). In other words, x is a best proximity point of T. This completes 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 , T x ) =d(A,B)=d ( y , T y ) ,
(33)

where d( x , y )>0. Due to the weak P-property of the pair (A,B), we have

d ( x , y ) d ( T x , T y ) .
(34)

Observe that in this case

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } = max { d ( x , y ) , d ( A , B ) , d ( A , B ) } .

Also, note that

N ( x , y ) d ( A , B ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } d ( A , B ) = min { d ( A , B ) , d ( A , B ) , d ( x , T y ) , d ( y , T x ) } d ( A , B ) = d ( A , B ) d ( A , B ) = 0 .

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

ψ ( d ( x , y ) ) ψ ( d ( T x , T y ) ) β ( ψ ( M ( x , y ) ) ) ψ ( M ( x , y ) d ( A , B ) ) + L ψ ( 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 ) ) .

If M( x , y )=d(A,B), due to the fact that d( x , y )>0, the inequality above becomes

0<ψ ( d ( x , y ) ) <ψ(0),
(35)

which implies d( x , y )=0 and contradicts the assumption d( x , y )>0. Else, if M( x , y )=d( x , y ), we deduce

0<ψ ( d ( x , y ) ) <ψ ( d ( x , y ) d ( A , B ) ) ,
(36)

which is not possible, since ψ is nondecreasing. Therefore, we must have d( x , y )=0. This completes the proof. □

To illustrate our result given in Theorem 2.1, we present the following example, which shows that Theorem 2.1 is a proper generalization of Theorem 1.2.

Example 2.1 Consider the space X=R with Euclidean metric. Take the sets

A=(,1]andB=[1,+).

Obviously, d(A,B)=2. Let T:AB be defined by Tx=x. Notice that A 0 ={1}, B 0 ={1} and T( A 0 ) B 0 . Also, it is clear that the pair (A,B) has the weak P-property.

Consider

β(t)= { 1 1 + t , if  0 t < 1 , t 1 + t , if  t 1 ,

and ψ(t)=αt (with α 1 2 ) for all t0. Note that βS and ψΨ. For all x,yA, we have

d(Tx,Ty)=|xy|andM(x,y)=max { | x y | , 2 x , 2 y } .

We shall show that T is a generalized almost ψ-Geraghty contraction. Without loss of generality, consider the case where xy. Then we have M(x,y)=2y and d(Tx,Ty)=xy.

In this case, we see that

ψ ( d ( T x , T y ) ) = α ( x y ) α ( x y 2 ) 2 α 2 ( x y 2 ) [ 2 α y ] [ α ( x y 2 ) ] = [ 2 α y ] [ α ( 2 y 2 ) α ( x y ) ] = ψ ( M ( x , y ) ) [ ψ ( M ( x , y ) d ( A , B ) ) ψ ( d ( T x , T y ) ) ] .

Therefore

ψ ( d ( T x , T y ) ) ψ ( M ( x , y ) ) 1 + ψ ( M ( x , y ) ) ψ ( M ( x , y ) d ( A , B ) ) .
(37)

On the other hand, we know that ψ(M(x,y))=2αy1 for all x,yA with xy. Hence,

β ( ψ ( M ( x , y ) ) ) = ψ ( M ( x , y ) ) 1 + ψ ( M ( x , y ) ) ,

and from (37) we deduce

ψ ( d ( T x , T y ) ) β ( ψ ( M ( x , y ) ) ) ψ ( M ( x , y ) d ( A , B ) ) .

Thus, all hypotheses of Theorem 2.1 are satisfied, and x =1 is the unique best proximity point of the map T.

On the other hand, T is not a Geraghty contraction. Indeed, taking x=1 and y=2, we get

d(Tx,Ty)=1> 1 2 =β ( d ( x , y ) ) d(x,y).

Then Theorem 1.2 (the main result of Caballero et al. [6]) is not applicable.

Similarly, we cannot apply Theorem 1.3 because T is not a ψ-Geraghty contraction. Let x=1, y=2 and ψ(t)=αt with α<2. Then T does not satisfy (8).

If in Theorem 2.1 we take ψ(t)=t for all t0, then we deduce the following corollary.

Corollary 2.1 Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A 0 is nonempty. Let T:AB be a non-self-mapping satisfying T( A 0 ) B 0 and

d(Tx,Ty)β ( M ( x , y ) ) [ M ( x , y ) d ( A , B ) ] +L [ N ( x , y ) d ( A , B ) ]

for all x,yA where βS, L0,

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } , N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

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 take β(t)=r where 0r<1, then we deduce another particular result.

Corollary 2.2 Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A 0 is nonempty. Let T:AB be a non-self-mapping satisfying T( A 0 ) B 0 and

d(Tx,Ty)r [ M ( x , y ) d ( A , B ) ] +L [ N ( x , y ) d ( A , B ) ]

for all x,yA where 0r<1, L0,

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } , N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

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 case A=B in Theorem 2.1 corresponds to a self-mapping and results in an existence and uniqueness theorem for a fixed point of the map T. We state this case in the next theorem.

Theorem 3.1 Let (X,d) be a complete metric space. Suppose that A is a nonempty closed subset of X. Let T:AA be a mapping such that

ψ ( d ( T x , T y ) ) β ( ψ ( M ( x , y ) ) ) ψ ( M ( x , y ) ) +Lψ ( N ( x , y ) ) for any x,yA,
(38)

where ψΨ, βS, L0,

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } and N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

Then T has a unique fixed point.

Finally, taking ψ(t)=t in Theorem 3.1, we get another fixed point result.

Corollary 3.1 Let (X,d) be a complete metric space. Suppose that A is a nonempty closed subset of X. Let T:AA be a mapping such that

d(Tx,Ty)β ( M ( x , y ) ) M(x,y)+LN(x,y)for any x,yA,
(39)

where βS, L0,

M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } and N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

Then T has a unique fixed point.

Remark 3.1 The best proximity theorem given in this work, more precisely Theorem 2.1, is a quite general result. It is a generalization of Theorem 2.1 in [14], Theorem 8 in [5], and also Theorem 1.2 given in Section 1. In addition, Corollary 3.1 improves Theorem 1.1.

Remark 3.2 Very recently, Karapınar and Samet [15] proved that the function d φ =φd on the set X, where φΨ is also a metric on X. Therefore, some of the fixed theorems regarding contraction mappings defined via auxiliary functions from the set Ψ can be in fact deduced from the existing ones in the literature. However, our main result given in Theorem 2.1 is not a consequence of any existing theorems due to the fact that the contraction condition contains the term d(A,B).

On the other hand, the definition of d φ =φd can be used to show that Theorem 3.1 follows from Corollary 3.1. Nevertheless, Corollary 3.1 and hence Theorem 3.1 are still new results.

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Acknowledgements

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

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Correspondence to İnci M Erhan.

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

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Keywords

  • fixed point
  • metric space
  • best proximity point
  • generalized almost ψ-Geraghty contractions