Nonlinear conditions for the existence of best proximity points

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Research

Abstract

In this paper, we first introduce the new notion of MT Open image in new window-cyclic contraction and establish some new existence and convergence theorems of iterates of best proximity points for MT Open image in new window-cyclic contractions. Some nontrivial examples illustrating our results are also given.

MSC:41A17, 47H09.

Keywords

cyclic map best proximity point  MT Open image in new window-function ( R Open image in new window-function)  MT Open image in new window-cyclic contraction 

1 Introduction and preliminaries

Throughout this paper, we denote by N Open image in new window and R Open image in new window the sets of positive integers and real numbers, respectively. Let A and B be nonempty subsets of a nonempty set E. A map S : A B A B Open image in new window is called a cyclic map if S ( A ) B Open image in new window and S ( B ) A Open image in new window. Let ( X , d ) Open image in new window be a metric space and T : A B A B Open image in new window be a cyclic map. For any nonempty subsets A and B of X, let
dist ( A , B ) = inf { d ( x , y ) : x A , y B } . Open image in new window

A point x A B Open image in new window is called to be a best proximity point for T if d ( x , T x ) = dist ( A , B ) Open image in new window.

Definition 1.1 ([1])

Let A and B be nonempty subsets of a metric space ( X , d ) Open image in new window. A map T : A B A B Open image in new window is called a cyclic contraction if the following conditions hold:
  1. (1)

    T ( A ) B Open image in new window and T ( B ) A Open image in new window;

     
  2. (2)

    there exists k ( 0 , 1 ) Open image in new window such that d ( T x , T y ) k d ( x , y ) + ( 1 k ) dist ( A , B ) Open image in new window for all x A Open image in new window, y B Open image in new window.

     

Remark 1.1 Let A and B be nonempty closed subsets of a complete metric space ( X , d ) Open image in new window and T : A B A B Open image in new window be a cyclic contraction. If A B Open image in new window, then dist ( A , B ) = 0 Open image in new window and T is a contraction on the complete metric space ( A B , d ) Open image in new window. Hence, applying the Banach contraction principle, we know that T has a unique fixed point in A B Open image in new window.

Recently, under some weaker assumptions over a map T, the existence, uniqueness and convergence of iterates to the best proximity point were investigated by several authors; see [1, 2, 3, 4, 5, 6] and references therein. In [1], Eldred and Veeramani first proved the following interesting best proximity point theorem.

Theorem EV ([[1], Proposition 3.2])

LetAandBbe nonempty closed subsets of a complete metric spaceX. Let T : A B A B Open image in new windowbe a cyclic contraction map, x 1 A Open image in new windowand define x n + 1 = T x n Open image in new window, n N Open image in new window. Suppose { x 2 n 1 } Open image in new windowhas a convergent subsequence inA. Then there exists x A Open image in new windowsuch that d ( x , T x ) = dist ( A , B ) Open image in new window.

Let f be a real-valued function defined on R Open image in new window. For c R Open image in new window, we recall that
lim sup x c f ( x ) = inf ε > 0 sup 0 < | x c | < ε f ( x ) Open image in new window
and
lim sup x c + f ( x ) = inf ε > 0 sup 0 < x c < ε f ( x ) . Open image in new window

Definition 1.2 ([7, 8, 9, 10, 11, 12, 13])

A function φ : [ 0 , ) [ 0 , 1 ) Open image in new window is said to be an MT Open image in new window-function (or R Open image in new window-function) if lim sup s t + φ ( s ) < 1 Open image in new window for all t [ 0 , ) Open image in new window.

It is obvious that if φ : [ 0 , ) [ 0 , 1 ) Open image in new window is a nondecreasing function or a nonincreasing function, then φ is an MT Open image in new window-function. So the set of MT Open image in new window-functions is a rich class.

Very recently, Du [10] first proved some characterizations of MT Open image in new window-functions.

Theorem D ([10])

Let φ : [ 0 , ) [ 0 , 1 ) Open image in new windowbe a function. Then the following statements are equivalent.
  1. (a)

    φis an MT Open image in new window-function.

     
  2. (b)

    For each t [ 0 , ) Open image in new window, there exist r t ( 1 ) [ 0 , 1 ) Open image in new windowand ε t ( 1 ) > 0 Open image in new windowsuch that φ ( s ) r t ( 1 ) Open image in new windowfor all s ( t , t + ε t ( 1 ) ) Open image in new window.

     
  3. (c)

    For each t [ 0 , ) Open image in new window, there exist r t ( 2 ) [ 0 , 1 ) Open image in new windowand ε t ( 2 ) > 0 Open image in new windowsuch that φ ( s ) r t ( 2 ) Open image in new windowfor all s [ t , t + ε t ( 2 ) ] Open image in new window.

     
  4. (d)

    For each t [ 0 , ) Open image in new window, there exist r t ( 3 ) [ 0 , 1 ) Open image in new windowand ε t ( 3 ) > 0 Open image in new windowsuch that φ ( s ) r t ( 3 ) Open image in new windowfor all s ( t , t + ε t ( 3 ) ] Open image in new window.

     
  5. (e)

    For each t [ 0 , ) Open image in new window, there exist r t ( 4 ) [ 0 , 1 ) Open image in new windowand ε t ( 4 ) > 0 Open image in new windowsuch that φ ( s ) r t ( 4 ) Open image in new windowfor all s [ t , t + ε t ( 4 ) ) Open image in new window.

     
  6. (f)

    For any nonincreasing sequence { x n } n N Open image in new windowin [ 0 , ) Open image in new window, we have 0 sup n N φ ( x n ) < 1 Open image in new window.

     
  7. (g)

    φis a function of contractive factor [8]; that is, for any strictly decreasing sequence { x n } n N Open image in new windowin [ 0 , ) Open image in new window, we have 0 sup n N φ ( x n ) < 1 Open image in new window.

     

Motivated by the concepts of cyclic contractions and MT Open image in new window-functions, we first introduce the concept of MT Open image in new window-cyclic contractions.

Definition 1.3 Let A and B be nonempty subsets of a metric space ( X , d ) Open image in new window. If a map T : A B A B Open image in new window satisfies

(MT1) T ( A ) B Open image in new window and T ( B ) A Open image in new window;

(MT2) there exists an MT Open image in new window-function φ : [ 0 , ) [ 0 , 1 ) Open image in new window such that
d ( T x , T y ) φ ( d ( x , y ) ) d ( x , y ) + ( 1 φ ( d ( x , y ) ) ) dist ( A , B ) for any  x A  and  y B , Open image in new window

then T is called an MT Open image in new window-cyclic contraction with respect toφ on A B Open image in new window.

Remark 1.2 It is obvious that (MT2) implies that T satisfies d ( T x , T y ) d ( x , y ) Open image in new window for any x A Open image in new window and y B Open image in new window.

The following example gives a map T which is an MT Open image in new window-cyclic contraction but not a cyclic contraction.

Example A Let X = { v 1 , v 2 , v 3 , } Open image in new window be a countable set and { τ n } Open image in new window be a strictly increasing convergent sequence of positive real numbers. Denote by τ : = lim n τ n Open image in new window. Then τ 2 < τ Open image in new window. Let d : X × X [ 0 , ) Open image in new window be defined by d ( v n , v n ) = 0 Open image in new window for all n N Open image in new window and d ( v n , v m ) = d ( v m , v n ) = τ m Open image in new window if m > n Open image in new window. Then d is a metric on X. Set A = { v 1 , v 3 , v 5 , } Open image in new window, B = { v 2 , v 4 , v 6 , } Open image in new window. So A B = X Open image in new window and dist ( A , B ) = τ 2 Open image in new window. Now we define a map T : A B A B Open image in new window by
T v n = def { v 2 , if  n = 1 , v n 1 , if  n > 1 Open image in new window
for n N Open image in new window. It is easy to see that T ( A ) = B Open image in new window and T ( B ) = A Open image in new window, and so (MT1) holds. Define φ : [ 0 , ) [ 0 , 1 ) Open image in new window as
φ ( t ) = def { τ n 1 τ n , if  t = τ n  for some  n N  with  n > 2 , 0 , otherwise . Open image in new window
Since { τ n } Open image in new window is strictly increasing, lim sup s t + φ ( s ) = 0 < 1 Open image in new window for all t [ 0 , ) Open image in new window. Hence φ is an MT Open image in new window-function. Clearly, lim n d ( v n , v n + 1 ) = lim n τ n + 1 = τ Open image in new window and d ( T v 1 , T v 2 ) = τ 2 Open image in new window. For m , n N Open image in new window with m > n Open image in new window and m > 2 Open image in new window, d ( T v n , T v m ) = τ m 1 Open image in new window. So lim n d ( T v n , T v n + 1 ) = lim n τ n = τ Open image in new window. We claim that T is not a cyclic contraction on A B Open image in new window. Indeed, suppose that T is a cyclic contraction on A B Open image in new window. Thus there exists k [ 0 , 1 ) Open image in new window such that

From (∗), we get τ τ 2 Open image in new window, which is a contradiction. Therefore, T is not a cyclic contraction on A B Open image in new window.

Next, we show that T is an MT Open image in new window-cyclic contraction with respect to φ. To verify (MT2), we need to observe the following cases:
  1. (i)
    Since φ ( d ( v 2 , v 1 ) ) = φ ( τ 2 ) = 0 Open image in new window, we obtain
    φ ( d ( v 2 , v 1 ) ) d ( v 2 , v 1 ) + ( 1 φ ( d ( v 2 , v 1 ) ) ) dist ( A , B ) = τ 2 = d ( T v 1 , T v 2 ) ; Open image in new window
     
  2. (ii)
    For m , n N Open image in new window with m > n Open image in new window and m > 2 Open image in new window, we have
    φ ( d ( v n , v m ) ) d ( v n , v m ) + ( 1 φ ( d ( v n , v m ) ) ) dist ( A , B ) = φ ( τ m ) τ m + ( 1 φ ( τ m ) ) τ 2 > τ m 1 = d ( T v n , T v m ) . Open image in new window
     

From above, we can prove that (MT2) holds. Hence T is an MT Open image in new window-cyclic contraction with respect to φ. Moreover, since d ( v 1 , T v 1 ) = dist ( A , B ) Open image in new window, v 1 A Open image in new window is a best proximity point for T.

In this paper, we establish some new existence and convergence theorems of iterates of best proximity points for MT Open image in new window-cyclic contractions. Our results include some known results in the literature as special cases.

2 Existence and convergence theorems for best proximity points

First, we establish the following convergence theorem for MT Open image in new window-cyclic contractions, which is one of the main results in this paper.

Theorem 2.1LetAandBbe nonempty subsets of a metric space ( X , d ) Open image in new windowand T : A B A B Open image in new windowbe an MT Open image in new window-cyclic contraction with respect toφ. Then there exists a sequence { x n } n N Open image in new windowsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) Open image in new window.

Proof Let x 1 A B Open image in new window be given. Define an iterative sequence { x n } n N Open image in new window by x n + 1 = T x n Open image in new window for n N Open image in new window. Clearly, dist ( A , B ) d ( x n , x n + 1 ) Open image in new window for all n N Open image in new window. If there exists j N Open image in new window such that x j = x j + 1 A B Open image in new window, then lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = dist ( A , B ) = 0 Open image in new window. So it suffices to consider the case x n + 1 x n Open image in new window for all n N Open image in new window. By Remark 1.2, it is easy to see that the sequence { d ( x n , x n + 1 ) } Open image in new window is nonincreasing in ( 0 , ) Open image in new window. Then
t 0 : = lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) 0 . Open image in new window
(2.1)
Since φ is an MT Open image in new window-function, applying Theorem D, we get
0 sup n N φ ( d ( x n , x n + 1 ) ) < 1 . Open image in new window
Let λ : = sup n N φ ( d ( x n , x n + 1 ) ) Open image in new window. Then 0 φ ( d ( x n , x n + 1 ) ) λ < 1 Open image in new window for all n N Open image in new window. If x 1 A Open image in new window, then, by (MT1), we have x 2 n 1 A Open image in new window and x 2 n B Open image in new window for all n N Open image in new window. Notice first that (MT2) implies that
d ( x 2 , x 3 ) = d ( T x 1 , T x 2 ) φ ( d ( x 1 , x 2 ) ) d ( x 1 , x 2 ) + ( 1 φ ( d ( x 1 , x 2 ) ) ) dist ( A , B ) λ d ( x 1 , x 2 ) + dist ( A , B ) Open image in new window
and
d ( x 3 , x 4 ) = d ( T x 2 , T x 3 ) φ ( d ( x 2 , x 3 ) ) d ( x 2 , x 3 ) + ( 1 φ ( d ( x 2 , x 3 ) ) ) dist ( A , B ) φ ( d ( x 2 , x 3 ) ) [ λ d ( x 1 , x 2 ) + dist ( A , B ) ] + ( 1 φ ( d ( x 2 , x 3 ) ) ) dist ( A , B ) = φ ( d ( x 2 , x 3 ) ) λ d ( x 1 , x 2 ) + dist ( A , B ) λ 2 d ( x 1 , x 2 ) + dist ( A , B ) . Open image in new window
On the other hand, if x 1 B Open image in new window, then x 2 n A Open image in new window and x 2 n 1 B Open image in new window for all n N Open image in new window. Applying (MT2) again, we also have
d ( x 2 , x 3 ) λ d ( x 1 , x 2 ) + dist ( A , B ) Open image in new window
and
d ( x 3 , x 4 ) λ 2 d ( x 1 , x 2 ) + dist ( A , B ) . Open image in new window
Hence, by induction, one can obtain
dist ( A , B ) d ( x n + 1 , x n + 2 ) λ n d ( x 1 , x 2 ) + dist ( A , B ) for all  n N . Open image in new window
(2.2)

Since λ [ 0 , 1 ) Open image in new window, lim n λ n = 0 Open image in new window. Using (2.1) and (2.2), we obtain lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = dist ( A , B ) Open image in new window. The proof is completed. □

Here we give a nontrivial example illustrating Theorem 2.1.

Example B Let X = { v 1 , v 2 , v 3 , } Open image in new window be a countable set. Define a strictly decreasing sequence { τ n } Open image in new window of positive real numbers by τ n = 1 2 + 1 n Open image in new window for all n N Open image in new window. Then lim n τ n = 1 2 Open image in new window. Let d : X × X [ 0 , ) Open image in new window be defined by d ( v n , v n ) = 0 Open image in new window for all n N Open image in new window and d ( v n , v m ) = d ( v m , v n ) = τ n Open image in new window if m > n Open image in new window. Then d is a metric on X. Set A = { v 1 , v 3 , v 5 , } Open image in new window and B = { v 2 , v 4 , v 6 , } Open image in new window. So
lim n d ( v n , v n + 1 ) = inf n N d ( v n , v n + 1 ) = lim n τ n = dist ( A , B ) = 1 2 . Open image in new window
Let T : A B A B Open image in new window be defined by
T v n = def v n + 1 for  n N . Open image in new window
It is easy to see that T ( A ) = B Open image in new window and T ( B ) A Open image in new window and so (MT1) holds. Let x 1 = v 1 A Open image in new window be given. Define an iterative sequence { x n } n N Open image in new window by x n + 1 = T x n Open image in new window for n N Open image in new window. So { x n } Open image in new window and { v n } Open image in new window are identical, and hence lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = dist ( A , B ) = 1 2 Open image in new window. Define φ : [ 0 , ) [ 0 , 1 ) Open image in new window as
φ ( t ) = def { τ n + 1 τ n , if  t = τ n  for some  n N , 0 , otherwise . Open image in new window
Since lim sup s t + φ ( s ) = 0 < 1 Open image in new window for all t [ 0 , ) Open image in new window, φ is an MT Open image in new window-function. Now, we verify (MT2). For m , n N Open image in new window with m > n Open image in new window, since { τ n } Open image in new window is strictly decreasing and τ n + 1 τ n < 1 Open image in new window,
φ ( d ( x n , x m ) ) d ( x n , x m ) + ( 1 φ ( d ( x n , x m ) ) ) dist ( A , B ) = τ n + 1 + 1 2 ( 1 τ n + 1 τ n ) > τ n + 1 = d ( T x n , T x m ) , Open image in new window

which prove that (MT2) holds. Hence T is an MT Open image in new window-cyclic contraction with respect to φ. Therefore, all the assumptions of Theorem 2.1 are satisfied and the conclusion can follow from Theorem 2.1.

The following best proximity point theorem can be given immediately from Theorem 2.1.

Theorem 2.2LetAandBbe nonempty subsets of a metric space ( X , d ) Open image in new windowand T : A B A B Open image in new windowbe a cyclic map. Suppose that there exists a nondecreasing (or nonincreasing) function τ : [ 0 , ) [ 0 , 1 ) Open image in new windowsuch that
d ( T x , T y ) τ ( d ( x , y ) ) d ( x , y ) + ( 1 τ ( d ( x , y ) ) ) dist ( A , B ) for any  x A  and  y B . Open image in new window

Then there exists a sequence { x n } n N Open image in new windowsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) Open image in new window.

Corollary 2.1 ([1])

LetAandBbe nonempty subsets of a metric space ( X , d ) Open image in new windowand T : A B A B Open image in new windowbe a cyclic contraction. Then there exists a sequence { x n } n N Open image in new windowsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) Open image in new window.

Here, we give an existence theorem for best proximity points.

Theorem 2.3LetAandBbe nonempty subsets of a metric space ( X , d ) Open image in new windowand T : A B A B Open image in new windowbe a cyclic map. Let x 1 A Open image in new windowbe given. Define an iterative sequence { x n } n N Open image in new windowby x n + 1 = T x n Open image in new windowfor n N Open image in new window. Suppose that
  1. (i)

    d ( T x , T y ) d ( x , y ) Open image in new windowfor any x A Open image in new windowand y B Open image in new window;

     
  2. (ii)

    { x 2 n 1 } Open image in new windowhas a convergent subsequence inA;

     
  3. (iii)

    lim n d ( x n , x n + 1 ) = dist ( A , B ) Open image in new window.

     

Then there exists v A Open image in new windowsuch that d ( v , T v ) = dist ( A , B ) Open image in new window.

Proof Since T is a cyclic map and x 1 A Open image in new window, x 2 n 1 A Open image in new window and x 2 n B Open image in new window for all n N Open image in new window. By (ii), { x 2 n 1 } Open image in new window has a convergent subsequence { x 2 n k 1 } Open image in new window and x 2 n k 1 v Open image in new window as k Open image in new window for some v A Open image in new window. Since
dist ( A , B ) d ( v , x 2 n k ) d ( v , x 2 n k 1 ) + d ( x 2 n k 1 , x 2 n k ) for all  k N , Open image in new window
it follows from lim n d ( v , x 2 n k 1 ) = 0 Open image in new window and the condition (iii) that lim n d ( v , x 2 n k ) = dist ( A , B ) Open image in new window. By (i), we have
dist ( A , B ) d ( T v , x 2 n k + 1 ) d ( v , x 2 n k ) for all  k N , Open image in new window

which implies d ( v , T v ) = dist ( A , B ) Open image in new window. □

Applying Theorems 2.1 and 2.3, we establish the following new best proximity point theorem for MT Open image in new window-cyclic contractions.

Theorem 2.4LetAandBbe nonempty subsets of a metric space ( X , d ) Open image in new windowand T : A B A B Open image in new windowbe an MT Open image in new window-cyclic contraction with respect toφ. Let x 1 A Open image in new windowbe given. Define an iterative sequence { x n } n N Open image in new windowby x n + 1 = T x n Open image in new windowfor n N Open image in new window. Suppose that { x 2 n 1 } Open image in new windowhas a convergent subsequence inA, then there exists v A Open image in new windowsuch that d ( v , T v ) = dist ( A , B ) Open image in new window.

Remark 2.1 ([[1], Proposition 3.2])

(i.e. Theorem EV) is a special case of Theorem 2.4.

Notes

Acknowledgements

The first author was supported partially by the grant No. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.

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Copyright information

© Du and Lakzian; licensee Springer 2012

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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNational Kaohsiung Normal UniversityKaohsiungTaiwan
  2. 2.Department of MathematicsPayame Noor UniversityTehranIran

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