Open Access

The existence and convergence of best proximity points for generalized proximal contraction mappings

Fixed Point Theory and Applications20142014:228

https://doi.org/10.1186/1687-1812-2014-228

Received: 17 August 2014

Accepted: 15 October 2014

Published: 4 November 2014

Abstract

In 2011, Sadiq Basha (Nonlinear Anal. 74:5844-5850, 2011) studied and established best proximity point theorems for proximal contractions of the first and the second kinds which are more general than the fixed point theorems of self-contractions. The purpose of this paper is to extend the notion of proximal contraction mappings of the first and the second kinds. We also establish the existence and convergence of best proximity point theorems for these classes and give an example to validate our main results.

MSC:47H10, 47H09.

Keywords

fixed pointbest proximity pointproximal contraction mapping of the first kindproximal contraction mapping of the second kind

1 Introduction

Let X be an arbitrary nonempty set. A fixed point for a self-mapping T : X X is a point x X such that T x = x . The applications of fixed point theory are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering, and economics. One of the very popular tools of fixed point theory is the Banach contraction principle, which first appeared in 1922. It states that if ( X , d ) is a complete metric space and T : X X is a contraction mapping (i.e., d ( T x , T y ) α d ( x , y ) for all x , y X , where α [ 0 , 1 ) ), then T has a unique fixed point. It has been generalized in different ways by mathematicians over the years (see [14]). However, almost all such results relate to the existence of a fixed point for self-mappings.

One of the most interesting studies is the extension of Banach’s contraction principle to the case of non-self-mappings. In fact, given nonempty closed subsets A and B of a complete metric space ( X , d ) , a contraction non-self-mapping T : A B does not necessarily have a fixed point.

Eventually, it is quite natural to seek an element x such that d ( x , T x ) is minimum, which implies that x and Tx are in close proximity to each other. As a matter of fact, d ( x , T x ) is at least d ( A , B ) , and best proximity point theorems accentuate the preceding viewpoint further to guarantee the existence of an element x such that d ( x , T x ) assumes the least possible value d ( A , B ) , thereby accomplishing the highest possible closeness between x and Tx. A point x in A for which d ( x , T x ) = d ( A , B ) is called a best proximity point of T.

Whenever non-self-mapping T has no fixed point, a best proximity point represents an optimal approximate solution to the equation T x = x , for the error involved, d ( x , T x ) , attains the global minimal value d ( A , B ) for any best proximity point x. One finds that best proximity point theorems are natural generalizations of the contraction principle to the case of non-self-mappings because a best proximity point reduces to a fixed point if the underlying mapping is assumed to be self-mapping.

In 1969, a best approximation theorem was introduced by Fan [5]. Afterward, several authors have derived extensions of Fan’s theorem in many directions (see, e.g., [69]). Other works concerning the existence of a best proximity point theorems for single-valued and set-valued mappings have been established in [1029].

Recently, Sadiq Basha in [30] gave necessary and sufficient conditions to the claim of the existence of a best proximity point for proximal contractions of the first kind and the second kind which are non-self-mapping analogs of contraction self-mappings and also established some best proximity and convergence theorems. However, the main result of Sadiq Basha [30] is an essential tool to claim the existence of a best proximity point and a sequence that converges to a best proximity point for some non-self-mappings. It is most interesting to find another auxiliary tool for the claim of the existence of a best proximity point and a sequence that converges to this point.

In this work, we introduce a new class of non-self-mappings. Indeed, the classes of proximal contractions of the first kind and the second kind are proper subclasses of these classes. We prove the existence and convergence as regards best proximity point theorems for these classes and also give some illustrative examples of our main results. Our results generalize, extend, and unify several well-known comparable results in the literature and these results can be applied to a much wider class of problems.

2 Preliminaries

Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively. We also suppose that A and B are nonempty subsets of a metric space ( X , d ) and use the following notations:
d ( A , B ) : = inf { d ( x , y ) : x A  and  y B } , 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 } .

We observe that if A B , then A 0 and B 0 are nonempty. Also, if A 0 or B 0 is nonempty, then both A 0 and B 0 are nonempty. Further, it is interesting to notice that A 0 and B 0 are contained in the boundaries of A and B, respectively, provided A and B are closed subsets of a normed linear space such that d ( A , B ) > 0 .

Definition 1 ([30])

A mapping S : A B is said to be a proximal contraction of the first kind if there exists α [ 0 , 1 ) such that
d ( a , S x ) = d ( b , S y ) = d ( A , B ) d ( a , b ) α d ( x , y )

for all a , b , x , y A .

Clearly, a self-mapping that is a proximal contraction of the first kind is precisely a contraction. However, a non-self proximal contraction is not necessarily a contraction.

Definition 2 ([30])

A mapping S : A B is said to be a proximal contraction of the second kind if there exists α [ 0 , 1 ) such that
d ( a , S x ) = d ( b , S y ) = d ( A , B ) d ( S a , S b ) α d ( S x , S y )

for all a , b , x , y A .

The necessary condition for a self-mapping S to be a proximal contraction of the second kind is that
d ( S S x , S S y ) α d ( S x , S y )

for all x , y in the domain of S. Therefore, every contraction self-mapping is a proximal contraction of the second kind but the converse is not true (see the example below).

Example 1 Consider R endowed with the Euclidean metric. Let the self-mapping S : [ 0 , 1 ] [ 0 , 1 ] be defined as
S ( x ) = { 0 if  x  is rational , 1 otherwise .

Then S is a proximal contraction of the second kind but not a contraction.

The above example shows that a self-mapping that is a proximal contraction of the second kind is not necessarily continuous.

Definition 3 ([30])

Let S : A B and T : B A be two mappings. The pair ( S , T ) is said to be a proximal cyclic contraction pair if there exists α [ 0 , 1 ) such that
d ( a , S x ) = d ( b , T y ) = d ( A , B ) d ( a , b ) α d ( x , y ) + ( 1 α ) d ( A , B )

for all a , b , x , y A .

Definition 4 ([30])

Let S : A B be a mapping and g : A A be an isometry. The mapping S is said to preserve the isometric distance with respect to g if
d ( S g x , S g y ) = d ( S x , S y )

for all x , y A .

Definition 5 ([30])

A point x A is said to be a best proximity point of the mapping S : A B if it satisfies the condition that
d ( x , S x ) = d ( A , B ) .

It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

Definition 6 ([30])

A is said to be approximatively compact with respect to B if every sequence { x n } in A that satisfies the condition that d ( y , x n ) d ( y , A ) for some y B has a convergent subsequence.

We observe that every set is approximatively compact with respect to itself, and that every compact set is approximatively compact. Moreover, A 0 and B 0 are nonempty sets if A is compact and B is approximatively compact with respect to A.

3 Main results

For mappings S : A B and g : A A B , we let Ξ S ( g ) be a collection of mappings ξ S : A [ 0 , 1 ) which satisfies the following condition:
d ( g x , S y ) = d ( A , B ) ξ S ( x ) ξ S ( y )

for x , y A .

Definition 7 A mapping S : A B is said to be a generalized proximal contraction of the first kind with respect to g : A A B if there exists a mapping ξ S Ξ S ( g ) such that
d ( a , S x ) = d ( b , S y ) = d ( A , B ) d ( a , b ) ξ S ( x ) d ( x , y )

for all a , b , x , y A .

It is easy to see that a generalized proximal contraction of the first kind with respect to the mapping g reduces to proximal contraction of the first kind if we set ξ S ( x ) = α for all x A where α [ 0 , 1 ) . But the converse is not true (see the example below).

Example 2 Consider the metric space R n with Euclidean metric, where n N . Let
A = { ( y , 0 , π , π , , π ) R n : 1 < y < 1 }
and
B = { ( y , 1 , π , π , , π ) R n : 1 < y < 1 } .
Define two mappings S : A B and g : A A B as follows:
S ( y , 0 , π , π , , π ) = ( y 2 2 , 1 , π , π , , π )
and
g ( y , 0 , π , π , , π ) = ( y , 0 , π , π , , π ) .

Then it is easy to see that d ( A , B ) = 1 .

It is easy to show that there is no α [ 0 , 1 ) that satisfies
d ( a , S x ) = d ( b , S y ) = d ( A , B ) d ( a , b ) α d ( x , y )

for all a , b , x , y A . Therefore, S is not a proximal contraction of the first kind.

Next, we show that S is a generalized proximal contraction of the first kind with respect to g. Consider a function ξ S : A [ 0 , 1 ) defined by
ξ S ( y , 0 , π , π , , π ) = | y | + 1 2 .

It is easy to see that ξ S Ξ S ( g ) .

If ( y 1 , 0 , π , π , , π ) , ( y 2 , 0 , π , π , , π ) A such that
d ( a , S ( y 1 , 0 , π , π , , π ) ) = d ( A , B ) = 1
and
d ( b , S ( y 2 , 0 , π , π , , π ) ) = d ( A , B ) = 1
for all a , b A , then we have
a = ( y 1 2 2 , 0 , π , π , , π ) , b = ( y 2 2 2 , 0 , π , π , , π ) .
Therefore, it follows that
d ( a , b ) = d ( ( y 1 2 2 , 0 , π , π , , π ) , ( y 2 2 2 , 0 , π , π , , π ) ) = | y 1 2 2 y 2 2 2 | = ( | y 1 + y 2 | 2 ) | y 1 y 2 | ( | y 1 | + | y 2 | 2 ) | y 1 y 2 | ( | y 1 | + 1 2 ) | y 1 y 2 | = ξ S ( y 1 , 0 , π , π , , π ) d ( ( y 1 , 0 , π , π , , π ) , ( y 2 , 0 , π , π , , π ) ) .

This implies that the non-self-mapping S is a generalized proximal contraction of the first kind with respect to g with the function ξ S .

Definition 8 A mapping S : A B is said to be a generalized proximal contraction of the second kind with respect to g : A A B if there exists a mapping ξ S Ξ S ( g ) such that
d ( a , S x ) = d ( b , S y ) = d ( A , B ) d ( S a , S b ) ξ S ( x ) d ( S x , S y )

for all a , b , x , y A .

Remark 1 The class of generalized proximal contractions of the second kind with respect to g is more general than the class of proximal contractions of the second kind (Definition 2).

Next, we give the result for generalized proximal contractions of the first kind.

Theorem 1 Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Further, suppose that A 0 or B 0 is nonempty. Let S : A B , T : B A and g : A B A B satisfy the following conditions:
  1. (a)

    S is a generalized proximal contractions of the first kind with respect to g | A and T is a generalized proximal contractions of the first kind with respect to g | B .

     
  2. (b)

    g is an isometry.

     
  3. (c)

    The pair ( S , T ) is a proximal cyclic contraction.

     
  4. (d)

    S ( A 0 ) B 0 , T ( B 0 ) A 0 .

     
  5. (e)

    A 0 g ( A 0 ) and B 0 g ( B 0 ) .

     
Then there exists a unique point x A and there exists a unique point y B such that
d ( g x , S x ) = d ( g y , T y ) = d ( x , y ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( g x n , S x n 1 ) = d ( A , B ) ,
converges to the element x. For any fixed y 0 B 0 , the sequence { y n } , defined by
d ( g y n , T y n 1 ) = d ( A , B ) ,

converges to the element y.

Furthermore, a sequence { u n } in A converges to x if { ξ S ( x n ) : n N } is bounded with constant M < 1 and there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( g z n + 1 , S u n ) = d ( A , B ) .

Proof Let x 0 be a fixed element in A 0 . In view of the fact that S ( A 0 ) B 0 and A 0 g ( A 0 ) , it is ascertained that there exists an element x 1 A 0 such that
d ( g x 1 , S x 0 ) = d ( A , B ) .
(1)
Again, since S ( A 0 ) B 0 and A 0 g ( A 0 ) , there exists an element x 2 A 0 such that
d ( g x 2 , S x 1 ) = d ( A , B ) .
(2)
This process can be continued. Therefore, we can construct the sequence { x n } in A 0 such that
d ( g x n , S x n 1 ) = d ( A , B )
(3)

for all n N .

It follows from S being a generalized proximal contraction of the first kind with respect to g | A that
d ( g x n + 1 , g x n ) ξ S ( x n ) d ( x n , x n 1 )
(4)
for all n N . Since g is an isometry, we have
d ( x n + 1 , x n ) ξ S ( x n ) d ( x n , x n 1 )
(5)
for all n N . From (3) and the notion of a generalized proximal contraction of the first kind with respect to g | A , we get
d ( x n + 1 , x n ) ξ S ( x n ) d ( x n , x n 1 ) ξ S ( x n 1 ) d ( x n , x n 1 ) ξ S ( x n 2 ) d ( x n , x n 1 ) ξ S ( x 0 ) d ( x n , x n 1 )
(6)
for all n N . By repeating (6), we get
d ( x n + 1 , x n ) ζ n d ( x 1 , x 0 )
(7)
for all n N , where ζ = ξ S ( x 0 ) [ 0 , 1 ) . For positive integers m and n with n > m , it follows from (7) that
d ( x n , x m ) d ( x n , x n 1 ) + d ( x n 1 , x n 2 ) + + d ( x m + 1 , x m ) ζ n 1 d ( x 1 , x 0 ) + ζ n 2 d ( x 1 , x 0 ) + + ζ m d ( x 1 , x 0 ) ( ζ m 1 ζ ) d ( x 1 , x 0 ) ,
(8)

which implies that d ( x n , x m ) 0 as m , n and then { x n } is a Cauchy sequence in A. By the completeness of A, the sequence { x n } converges to some x A .

Similarly, in view of the fact that T ( B 0 ) A 0 and A 0 g ( A 0 ) , we can conclude that, for fixed y 0 B 0 , there is a sequence { y n } in B such that
d ( g y n , T y n 1 ) = d ( A , B )
(9)
for all n N . Since T is a generalized proximal contraction of the first kind with respect to g | B , we have
d ( g y n + 1 , g y n ) ξ T ( y n ) d ( y n , y n 1 ) .
(10)
From g being an isometry, we get
d ( y n + 1 , y n ) ξ T ( y n ) d ( y n , y n 1 )
(11)
for all n N . By virtue of (9) and T being a generalized proximal contraction of the first kind with respect to g | B , we get
d ( x n + 1 , y n ) ξ T ( y n ) d ( y n , y n 1 ) ξ T ( y n 1 ) d ( y n , y n 1 ) ξ T ( y n 2 ) d ( y n , y n 1 ) ξ T ( y 0 ) d ( y n , y n 1 )
(12)
for all n N . By repeating (12), we get
d ( y n + 1 , y n ) η n d ( y 1 , y 0 )
(13)
for all n N , where η = ξ T ( y 0 ) [ 0 , 1 ) . For positive integers m and n with n > m , it follows from (13) that
d ( y n , y m ) d ( y n , y n 1 ) + d ( y n 1 , y n 2 ) + + d ( y m + 1 , y m ) η n 1 d ( y 1 , y 0 ) + η n 2 d ( y 1 , y 0 ) + + η m d ( y 1 , y 0 ) ( η m 1 η ) d ( y 1 , y 0 ) ,
(14)

which implies that d ( y n , y m ) 0 as m , n and then { y n } is a Cauchy sequence in B. By the completeness of B, the sequence { y n } converges to some y B .

Since the pair ( S , T ) is a proximal cyclic contraction, we have
d ( x n + 1 , y n + 1 ) = d ( g x n + 1 , g y n + 1 ) α d ( x n , y n ) + ( 1 α ) d ( A , B ) .
(15)
We take the limit in (15) as n , and it follows that
d ( x , y ) = d ( A , B ) ,
(16)
which implies that x A 0 and y B 0 . It follows from S ( A 0 ) B 0 and T ( B 0 ) A 0 that there are p A and q B such that
d ( p , S x ) = d ( A , B )
(17)
and
d ( q , T y ) = d ( A , B ) .
(18)
From (3), (17), and the notion of a generalized proximal contraction of the first kind of S, we get
d ( p , g x n ) ξ S ( x ) d ( x , x n 1 )
(19)
for all n N . Letting n , we conclude that p = g x . Therefore
d ( g x , S x ) = d ( A , B ) .
(20)
Similarly, we can show that q = g y and then
d ( g y , T y ) = d ( A , B ) .
(21)
From (16), (20), and (21), we get
d ( x , y ) = d ( g x , S x ) = d ( g y , T y ) = d ( A , B ) .
For the uniqueness, let us suppose that there exist x A and y B such that
d ( x , y ) = d ( g x , S x ) = d ( g y , T y ) = d ( A , B ) .
Since g is an isometry and S and T are generalized proximal contractions of the first kind with respect to g | A and g | B , respectively, it follows that
d ( x , x ) = d ( g x , g x ) ξ S ( x ) d ( x , x )
and
d ( y , y ) = d ( g y , g y ) ξ T ( y ) d ( y , y ) .

It follows from ξ S ( x ) and ξ T ( y ) being contained in [ 0 , 1 ) that x = x and y = y .

On the other hand, let { u n } be a sequence in A and { ϵ n } be a sequence of positive real numbers such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,
where z n + 1 A satisfies the condition that d ( g z n + 1 , S u n ) = d ( A , B ) . Since g is an isometry and S is a generalized proximal contraction of the first kind with respect to g | A , we have
d ( x n + 1 , z n + 1 ) = d ( g x n + 1 , g z n + 1 ) ξ S ( x n ) d ( x n , u n )
(22)
and hence
d ( x n + 1 , z n + 1 ) M d ( x n , u n ) .
(23)
Given ϵ > 0 , we choose a positive integer N such that ϵ n ϵ for all n N . For each n N , we get
d ( x n + 1 , u n + 1 ) d ( x n + 1 , z n + 1 ) + d ( z n + 1 , u n + 1 ) M d ( x n , u n ) + ϵ n ,
(24)
which implies that d ( x n + 1 , u n + 1 ) M n d ( x 1 , x 0 ) + i = 1 n M n i ϵ i . Therefore, for each n N , we have
d ( u n + 1 , x ) d ( u n + 1 , x n + 1 ) + d ( x n + 1 , x ) M n d ( x 1 , x 0 ) + i = 1 n M n i ϵ i + d ( x n + 1 , x ) M n d ( x 1 , x 0 ) + M n N i = 1 N M N i ϵ i + ϵ i = N + 1 n M n i + d ( x n + 1 , x ) .

Letting n , we have lim n d ( u n + 1 , x ) ϵ 1 M . It follows from ϵ > 0 being arbitrary that { u n } is convergent and it converges to x. This completes the proof of the theorem. □

Now, we give an example to illustrate Theorem 1.

Example 3 Consider the complete metric space R 2 with Euclidean metric. Let
A = { ( 0 , y ) : 1 y 1 } and B = { ( 1 , y ) : 1 y 1 } .
Define three mappings S : A B , T : B A , and g : A B A B as follows:
S ( 0 , y ) = ( 1 , y 2 4 ) , T ( 1 , y ) = ( 0 , y 2 4 ) , g ( x , y ) = ( x , y ) .

Then it is easy to see that d ( A , B ) = 1 , A 0 = A , B 0 = B , and the mapping g is an isometry.

Next, we claim that S is a generalized proximal contraction of the first kind with respect to g | A and T is a generalized proximal contraction of the first kind with respect to g | B .

Consider a function ξ S : A [ 0 , 1 ) defined by
ξ S ( 0 , y ) = | y | + 1 4 .
Then ξ S Ξ S ( g | A ) . If ( 0 , y 1 ) , ( 0 , y 2 ) A such that
d ( a , S ( 0 , y 1 ) ) = d ( A , B ) = 1 and d ( b , S ( 0 , y 2 ) ) = d ( A , B ) = 1
for all a , b A , then we have
a = ( 0 , y 1 2 4 ) , b = ( 0 , y 2 2 4 ) .
Therefore, it follows that
d ( a , b ) = d ( ( 0 , y 1 2 4 ) , ( 0 , y 2 2 4 ) ) = | y 1 2 4 y 2 2 4 | = ( | y 1 + y 2 | 4 ) | y 1 y 2 | ( | y 1 | + | y 2 | 4 ) | y 1 y 2 | ( | y 1 | + 1 4 ) | y 1 y 2 | = ξ S ( 0 , y 1 ) d ( ( 0 , y 1 ) , ( 0 , y 2 ) ) .

Hence S is a generalized proximal contraction of the first kind with respect to g | A with the function ξ S .

Consider a function ξ T : B [ 0 , 1 ) defined by
ξ T ( 1 , y ) = | y | + 1 4 .
Then ξ T Ξ S ( g | B ) . If ( 1 , y 1 ) , ( 1 , y 2 ) B such that
d ( a , T ( 1 , y 1 ) ) = d ( A , B ) = 1 and d ( b , T ( 1 , y 2 ) ) = d ( A , B ) = 1
for all a , b B , then we get
a = ( 1 , y 1 2 4 ) , b = ( 1 , y 2 2 4 ) .
Since
d ( a , b ) = d ( ( 1 , y 1 2 4 ) , ( 1 , y 2 2 4 ) ) = | y 1 2 4 y 2 2 4 | = ( | y 1 + y 2 | 4 ) | y 1 y 2 | ( | y 1 | + | y 2 | 4 ) | y 1 y 2 | ( | y 1 | + 1 4 ) | y 1 y 2 | = ξ T ( 1 , y 1 ) d ( ( 1 , y 1 ) , ( 1 , y 2 ) ) ,

we can conclude that T is a generalized proximal contraction of the first kind with respect to g | B with the function ξ T .

Moreover, the pair ( S , T ) forms a proximal cyclic contraction and the other hypotheses of Theorem 1 are also satisfied. Further, it is easy to see that we have the unique elements ( 0 , 0 ) A and ( 1 , 0 ) B such that
d ( g ( 0 , 0 ) , S ( 0 , 0 ) ) = d ( g ( 1 , 0 ) , T ( 1 , 0 ) ) = d ( ( 0 , 0 ) , ( 1 , 0 ) ) = d ( A , B ) .

Corollary 1 (Theorem 3.1 in [30])

Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Further, suppose that A 0 or B 0 is nonempty. Let S : A B , T : B A , and g : A B A B satisfy the following conditions:
  1. (a)

    S and T are proximal contractions of the first kind.

     
  2. (b)

    g is an isometry.

     
  3. (c)

    The pair ( S , T ) is a proximal cyclic contraction.

     
  4. (d)

    S ( A 0 ) B 0 , T ( B 0 ) A 0 .

     
  5. (e)

    A 0 g ( A 0 ) and B 0 g ( B 0 ) .

     
Then there exists a unique point x A and there exists a unique point y B such that
d ( g x , S x ) = d ( g y , T y ) = d ( x , y ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( g x n + 1 , S x n ) = d ( A , B ) ,
converges to the element x. For any fixed y 0 B 0 , the sequence { y n } , defined by
d ( g y n + 1 , T y n ) = d ( A , B ) ,

converges to the element y.

Furthermore, a sequence { u n } in A converges to x if there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( g z n + 1 , S u n ) = d ( A , B ) .

Proof Since a proximal contractions of the first kind is a special case of a generalized proximal contraction of the first kind, we get this result from Theorem 1. □

If g is assumed to be the identity mapping, then Corollary 1 yields the following best proximity point theorem.

Corollary 2 (Corollary 3.3 in [30])

Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Further, suppose that A 0 or B 0 is nonempty. Let S : A B and T : B A satisfy the following conditions:
  1. (a)

    S and T are proximal contractions of the first kind.

     
  2. (b)

    The pair ( S , T ) is a proximal cyclic contraction.

     
  3. (c)

    S ( A 0 ) B 0 , T ( B 0 ) A 0 .

     
Then there exists a unique point x A and there exists a unique point y B such that
d ( x , S x ) = d ( y , T y ) = d ( x , y ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( x n + 1 , S x n ) = d ( A , B ) ,
converges to the element x. For any fixed y 0 B 0 , the sequence { y n } , defined by
d ( y n + 1 , T y n ) = d ( A , B ) ,

converges to the element y.

Furthermore, a sequence { u n } in A converges to x if there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( z n + 1 , S u n ) = d ( A , B ) .

Next, we establish a result for non-self-mappings which are generalized proximal contractions of the first kind and the second kind.

Theorem 2 Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Further, suppose that A 0 or B 0 is nonempty. Let S : A B and g : A A satisfy the following conditions:
  1. (a)

    S is generalized proximal contractions of first and second kinds with respect to g.

     
  2. (b)

    g is an isometry.

     
  3. (c)

    S preserves isometric distance with respect to g.

     
  4. (d)

    S ( A 0 ) B 0 .

     
  5. (e)

    A 0 g ( A 0 ) .

     
Then there exists a unique point x A such that
d ( g x , S x ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( g x n , S x n 1 ) = d ( A , B ) ,

converges to the element x.

Furthermore, a sequence { u n } in A converges to x if { ξ S ( x n ) : n N } bounded with constant M < 1 and there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( g z n + 1 , S u n ) = d ( A , B ) .

Proof For fixed x 0 A 0 , since S ( A 0 ) B 0 and A 0 g ( A 0 ) , we can construct the sequence { x n } in A 0 similarly to Theorem 1 such that
d ( g x n , S x n 1 ) = d ( A , B )
(25)
for all n N . It follows from g being an isometry and by virtue of the fact that we have a generalized proximal contraction of the first kind with respect to g of S that
d ( x n , x n + 1 ) = d ( g x n , g x n + 1 ) ξ S ( x n ) d ( x n , x n 1 )
for all n N . Similarly to the proof of Theorem 1, we can conclude that the sequence { x n } is a Cauchy sequence in A and so converges to some x A . As S is a generalized proximal contraction of the second kind with respect to g and preserves the isometric distance with respect to g,
d ( S x n , S x n + 1 ) = d ( S g x n , S g x n + 1 ) ξ S ( x n 1 ) d ( S x n 1 , S x n ) ξ S ( x n 2 ) d ( S x n 1 , S x n ) ξ S ( x 0 ) d ( S x n 1 , S x n ) ,
(26)
which implies that { S x n } is a Cauchy sequence in B and then it converges to some y B . Therefore, we can conclude that
d ( g x , y ) = lim n d ( g x n + 1 , S x n ) = d ( A , B ) ,
(27)
that is, g x A 0 . Since A 0 g ( A 0 ) , we have g x = g z for some z A 0 and then d ( g x , g z ) = 0 . By the fact that g is an isometry, we get d ( x , z ) = d ( g x , g z ) = 0 . Hence x and z must be identical and so x becomes a point in A 0 . As S ( A 0 ) B 0 ,
d ( u , S x ) = d ( A , B )
(28)
for some u A . It follows from (25), (28), and S being a generalized proximal contraction of the first kind with respect to g that
d ( u , g x n + 1 ) ξ S ( x ) d ( x , x n )
(29)

for all n N . Taking the limit as n , we see that the sequence { g x n } converges to a point u. Owing to the fact that g is continuous, { g x n } converge to a point gx. By the uniqueness of the limit of the sequence, we conclude that u = g x . Therefore, we have the result that d ( g x , S x ) = d ( u , S x ) = d ( A , B ) . The uniqueness and the remaining part of the proof follow as in Theorem 1. This completes the proof of the theorem. □

Corollary 3 (Theorem 3.4 in [30])

Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Further, suppose that A 0 or B 0 is nonempty. Let S : A B and g : A A satisfy the following conditions:
  1. (a)

    S is proximal contractions of first and second kinds.

     
  2. (b)

    g is an isometry.

     
  3. (c)

    S preserves isometric distance with respect to g.

     
  4. (d)

    S ( A 0 ) B 0 .

     
  5. (e)

    A 0 g ( A 0 ) .

     
Then there exists a unique point x A such that
d ( g x , S x ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( g x n + 1 , S x n ) = d ( A , B ) ,

converges to the element x.

Furthermore, a sequence { u n } in A converges to x if there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( g z n + 1 , S u n ) = d ( A , B ) .

Proof Since proximal contractions of the first kind and the second kind are special cases of generalized proximal contractions of the first and the second kinds, we get the result from Theorem 2. □

Corollary 4 (Corollary 3.5 in [30])

Let ( X , d ) be a complete metric space and A and B be nonempty, closed subsets of X. Suppose that A 0 or B 0 is nonempty and S : A B satisfy the following conditions:
  1. (a)

    S is proximal contractions of first and second kinds.

     
  2. (b)

    S ( A 0 ) B 0 .

     
Then there exists a unique point x A such that
d ( x , S x ) = d ( A , B ) .
Moreover, for any fixed x 0 A 0 , the sequence { x n } , defined by
d ( x n + 1 , S x n ) = d ( A , B ) ,

converges to the element x.

Furthermore, a sequence { u n } in A converges to x if there is a sequence of positive numbers { ϵ n } such that
lim n ϵ n = 0 and d ( u n + 1 , z n + 1 ) ϵ n ,

where z n + 1 A satisfies the condition that d ( z n + 1 , S u n ) = d ( A , B ) .

Declarations

Acknowledgements

The first author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript. The second author was supported by the Higher Education Research Promotion and National Research University Project of Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bang Mod, Thrung Khru

References

  1. An TV, Dung NV, Kadelburg Z, Radenović S: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2014. 10.1007/s13398-014-0173-7Google Scholar
  2. Roshan JR, Shobkolaei N, Sedhi S, Parvaneh V, Radenović S:Common fixed point theorems for three maps in discontinuous G b -metric spaces. Acta Math. Sci. Ser. B 2014, 34(5):1–12.Google Scholar
  3. Ćojbašić Rajić V, Radenović S, Chauhan S: Common fixed point of generalized weakly contractive maps in partial metric spaces. Acta Math. Sci. Ser. B 2014, 34(4):1345–1356. 10.1016/S0252-9602(14)60088-6View ArticleMathSciNetGoogle Scholar
  4. Alsulami HH, Roldan A, Karapinar E, Radenović S: Some inevitable remarks on ‘Tripled fixed point theorems for mixed monotone Kannan type contractive mappings’. J. Appl. Math. 2014., 2014: Article ID 392301Google Scholar
  5. Fan K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225View ArticleMathSciNetGoogle Scholar
  6. Prolla JB: Fixed point theorems for set-valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 1982/83, 5: 449–455.View ArticleMathSciNetGoogle Scholar
  7. Reich S: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 1978, 62: 104–113. 10.1016/0022-247X(78)90222-6View ArticleMathSciNetGoogle Scholar
  8. Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 1988, 102: 534–537.MathSciNetGoogle Scholar
  9. Sehgal VM, Singh SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 1989, 10: 181–184. 10.1080/01630568908816298View ArticleMathSciNetGoogle Scholar
  10. Al-Thagafi MA, Shahzad N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 2009, 70(3):1209–1216. 10.1016/j.na.2008.02.004View ArticleMathSciNetGoogle Scholar
  11. Al-Thagafi MA, Shahzad N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008., 2008: Article ID 457069Google Scholar
  12. Kim WK, Kum S, Lee KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 2008, 68(8):2216–2227. 10.1016/j.na.2007.01.057View ArticleMathSciNetGoogle Scholar
  13. Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380View ArticleMathSciNetGoogle Scholar
  14. Sadiq Basha S, Veeramani P: Best approximations and best proximity pairs. Acta Sci. Math. 1997, 63: 289–300.MathSciNetGoogle Scholar
  15. Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415View ArticleMathSciNetGoogle Scholar
  16. Sadiq Basha S, Veeramani P, Pai DV: Best proximity pair theorems. Indian J. Pure Appl. Math. 2001, 32: 1237–1246.MathSciNetGoogle Scholar
  17. Srinivasan PS: Best proximity pair theorems. Acta Sci. Math. 2001, 67: 421–429.Google Scholar
  18. Wlodarczyk K, Plebaniak R, Banach A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 2009, 70(9):3332–3341. 10.1016/j.na.2008.04.037View ArticleMathSciNetGoogle Scholar
  19. Wlodarczyk K, Plebaniak R, Banach A: Erratum to: ‘Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces’. Nonlinear Anal. 2009, 71: 3583–3586.MathSciNetGoogle Scholar
  20. Wlodarczyk K, Plebaniak R, Obczynski C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 2010, 72: 794–805. 10.1016/j.na.2009.07.024View ArticleMathSciNetGoogle Scholar
  21. Radenović S, Kadelburg Z: A note on some recent best proximity point results for non-self mappings. Gulf J. Math. 2013, 1: 36–41.Google Scholar
  22. Raj VS: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052View ArticleMathSciNetGoogle Scholar
  23. Kumam P, Roldán-López-de-Hierro AF: On existence and uniqueness of g -best proximity points under ( φ , θ , α , g ) -contractivity conditions and consequences. Abstr. Appl. Anal. 2014., 2014: Article ID 234027Google Scholar
  24. Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal C -contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013., 2013: Article ID 94Google Scholar
  25. Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 180Google Scholar
  26. Kumam P, Aydi H, Karapinar E, Sintunavarat W: Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 242Google Scholar
  27. Sintunavarat W, Kumam P: The existence theorems of an optimal approximate solution for generalized proximal contraction mappings. Abstr. Appl. Anal. 2013., 2013: Article ID 375604Google Scholar
  28. Karapinar E, Sintunavarat W: The existence of optimal approximate solution theorems for generalized α -proximal contraction non-self mappings and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 323Google Scholar
  29. Kutbi MA, Chandok S, Sintunavarat W:Optimal solutions for nonlinear proximal C N -contraction mapping in metric space. J. Inequal. Appl. 2014., 2014: Article ID 193Google Scholar
  30. Sadiq Basha S: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 2011, 74: 5844–5850. 10.1016/j.na.2011.04.017View ArticleMathSciNetGoogle Scholar

Copyright

© Sintunavarat and Kumam; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.