Journal of Inequalities and Applications

, 2012:206

Nonlinear conditions for the existence of best proximity points

Authors

    • Department of MathematicsNational Kaohsiung Normal University
  • Hossein Lakzian
    • Department of MathematicsPayame Noor University

DOI: 10.1186/1029-242X-2012-206

Abstract

In this paper, we first introduce the new notion of MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction and establish some new existence and convergence theorems of iterates of best proximity points for MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contractions. Some nontrivial examples illustrating our results are also given.

MSC:41A17, 47H09.

Keywords

cyclic map best proximity point MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function ( R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq2_HTML.gif-function) MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction

1 Introduction and preliminaries

Throughout this paper, we denote by N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq3_HTML.gif and R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq4_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq5_HTML.gif is called a cyclic map if S ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq6_HTML.gif and S ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq7_HTML.gif. Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gif be a metric space and T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif 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 } . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equa_HTML.gif

A point x A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq10_HTML.gif is called to be a best proximity point for T if d ( x , T x ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq11_HTML.gif.

Definition 1.1 ([1])

Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gif. A map T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif is called a cyclic contraction if the following conditions hold:
  1. (1)

    T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq12_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq13_HTML.gif;

     
  2. (2)

    there exists k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq14_HTML.gif such that d ( T x , T y ) k d ( x , y ) + ( 1 k ) dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq15_HTML.gif for all x A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq16_HTML.gif, y B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq17_HTML.gif.

     

Remark 1.1 Let A and B be nonempty closed subsets of a complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gif and T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif be a cyclic contraction. If A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq18_HTML.gif, then dist ( A , B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq19_HTML.gif and T is a contraction on the complete metric space ( A B , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq20_HTML.gif. Hence, applying the Banach contraction principle, we know that T has a unique fixed point in A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq21_HTML.gif.

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 [16] and references therein. In [1], Eldred and Veeramani first proved the following interesting best proximity point theorem.

Theorem EV ([[1], Proposition 3.2])

Let A and B be nonempty closed subsets of a complete metric space X. Let T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe a cyclic contraction map, x 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq22_HTML.gifand define x n + 1 = T x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq23_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Suppose { x 2 n 1 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq25_HTML.gifhas a convergent subsequence in A. Then there exists x A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq16_HTML.gifsuch that d ( x , T x ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq11_HTML.gif.

Let f be a real-valued function defined on R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq26_HTML.gif. For c R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq27_HTML.gif, we recall that
lim sup x c f ( x ) = inf ε > 0 sup 0 < | x c | < ε f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equb_HTML.gif
and
lim sup x c + f ( x ) = inf ε > 0 sup 0 < x c < ε f ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equc_HTML.gif

Definition 1.2 ([713])

A function φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq28_HTML.gif is said to be an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function (or R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq2_HTML.gif-function) if lim sup s t + φ ( s ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq29_HTML.gif for all t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif.

It is obvious that if φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq31_HTML.gif is a nondecreasing function or a nonincreasing function, then φ is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function. So the set of MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-functions is a rich class.

Very recently, Du [10] first proved some characterizations of MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq32_HTML.gif-functions.

Theorem D ([10])

Let φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq33_HTML.gifbe a function. Then the following statements are equivalent.
  1. (a)

    φ is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function.

     
  2. (b)

    For each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif, there exist r t ( 1 ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq34_HTML.gif and ε t ( 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq35_HTML.gif such that φ ( s ) r t ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq36_HTML.gif for all s ( t , t + ε t ( 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq37_HTML.gif.

     
  3. (c)

    For each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif, there exist r t ( 2 ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq38_HTML.gif and ε t ( 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq39_HTML.gif such that φ ( s ) r t ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq40_HTML.gif for all s [ t , t + ε t ( 2 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq41_HTML.gif.

     
  4. (d)

    For each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif, there exist r t ( 3 ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq42_HTML.gif and ε t ( 3 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq43_HTML.gif such that φ ( s ) r t ( 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq44_HTML.gif for all s ( t , t + ε t ( 3 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq45_HTML.gif.

     
  5. (e)

    For each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif, there exist r t ( 4 ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq46_HTML.gif and ε t ( 4 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq47_HTML.gif such that φ ( s ) r t ( 4 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq48_HTML.gif for all s [ t , t + ε t ( 4 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq49_HTML.gif.

     
  6. (f)

    For any nonincreasing sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gif in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq51_HTML.gif, we have 0 sup n N φ ( x n ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq52_HTML.gif.

     
  7. (g)

    φ is a function of contractive factor [8]; that is, for any strictly decreasing sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gif in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq51_HTML.gif, we have 0 sup n N φ ( x n ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq52_HTML.gif.

     

Motivated by the concepts of cyclic contractions and MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-functions, we first introduce the concept of MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contractions.

Definition 1.3 Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gif. If a map T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif satisfies

(MT1) T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq12_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq13_HTML.gif;

(MT2) there exists an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq53_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equd_HTML.gif

then T is called an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction with respect to φ on A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq54_HTML.gif.

Remark 1.2 It is obvious that (MT2) implies that T satisfies d ( T x , T y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq55_HTML.gif for any x A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq16_HTML.gif and y B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq17_HTML.gif.

The following example gives a map T which is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction but not a cyclic contraction.

Example A Let X = { v 1 , v 2 , v 3 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq56_HTML.gif be a countable set and { τ n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq57_HTML.gif be a strictly increasing convergent sequence of positive real numbers. Denote by τ : = lim n τ n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq58_HTML.gif. Then τ 2 < τ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq59_HTML.gif. Let d : X × X [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq60_HTML.gif be defined by d ( v n , v n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif and d ( v n , v m ) = d ( v m , v n ) = τ m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq62_HTML.gif if m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq63_HTML.gif. Then d is a metric on X. Set A = { v 1 , v 3 , v 5 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq64_HTML.gif, B = { v 2 , v 4 , v 6 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq65_HTML.gif. So A B = X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq66_HTML.gif and dist ( A , B ) = τ 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq67_HTML.gif. Now we define a map T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif by
T v n = def { v 2 , if  n = 1 , v n 1 , if  n > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Eque_HTML.gif
for n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. It is easy to see that T ( A ) = B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq68_HTML.gif and T ( B ) = A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq69_HTML.gif, and so (MT1) holds. Define φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq70_HTML.gif as
φ ( t ) = def { τ n 1 τ n , if  t = τ n  for some  n N  with  n > 2 , 0 , otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equf_HTML.gif
Since { τ n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq57_HTML.gif is strictly increasing, lim sup s t + φ ( s ) = 0 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq71_HTML.gif for all t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq30_HTML.gif. Hence φ is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function. Clearly, lim n d ( v n , v n + 1 ) = lim n τ n + 1 = τ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq72_HTML.gif and d ( T v 1 , T v 2 ) = τ 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq73_HTML.gif. For m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq74_HTML.gif with m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq63_HTML.gif and m > 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq75_HTML.gif, d ( T v n , T v m ) = τ m 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq76_HTML.gif. So lim n d ( T v n , T v n + 1 ) = lim n τ n = τ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq77_HTML.gif. We claim that T is not a cyclic contraction on A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq54_HTML.gif. Indeed, suppose that T is a cyclic contraction on A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq54_HTML.gif. Thus there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq78_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equg_HTML.gif

From (∗), we get τ τ 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq79_HTML.gif, which is a contradiction. Therefore, T is not a cyclic contraction on A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq54_HTML.gif.

Next, we show that T is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq80_HTML.gif, 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 ) ; https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equh_HTML.gif
     
  2. (ii)
    For m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq74_HTML.gif with m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq63_HTML.gif and m > 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq75_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equi_HTML.gif
     

From above, we can prove that (MT2) holds. Hence T is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction with respect to φ. Moreover, since d ( v 1 , T v 1 ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq81_HTML.gif, v 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq82_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq32_HTML.gif-cyclic contractions, which is one of the main results in this paper.

Theorem 2.1 Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gifand T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contraction with respect to φ. Then there exists a sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gifsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq83_HTML.gif.

Proof Let x 1 A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq84_HTML.gif be given. Define an iterative sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gif by x n + 1 = T x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq23_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Clearly, dist ( A , B ) d ( x n , x n + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq85_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. If there exists j N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq86_HTML.gif such that x j = x j + 1 A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq87_HTML.gif, then lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = dist ( A , B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq88_HTML.gif. So it suffices to consider the case x n + 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq89_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. By Remark 1.2, it is easy to see that the sequence { d ( x n , x n + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq90_HTML.gif is nonincreasing in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq91_HTML.gif. Then
t 0 : = lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equ1_HTML.gif
(2.1)
Since φ is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function, applying Theorem D, we get
0 sup n N φ ( d ( x n , x n + 1 ) ) < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equj_HTML.gif
Let λ : = sup n N φ ( d ( x n , x n + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq92_HTML.gif. Then 0 φ ( d ( x n , x n + 1 ) ) λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq93_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. If x 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq22_HTML.gif, then, by (MT1), we have x 2 n 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq94_HTML.gif and x 2 n B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq95_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equk_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equl_HTML.gif
On the other hand, if x 1 B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq96_HTML.gif, then x 2 n A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq97_HTML.gif and x 2 n 1 B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq98_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Applying (MT2) again, we also have
d ( x 2 , x 3 ) λ d ( x 1 , x 2 ) + dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equm_HTML.gif
and
d ( x 3 , x 4 ) λ 2 d ( x 1 , x 2 ) + dist ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equn_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equ2_HTML.gif
(2.2)

Since λ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq99_HTML.gif, lim n λ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq100_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq101_HTML.gif. The proof is completed. □

Here we give a nontrivial example illustrating Theorem 2.1.

Example B Let X = { v 1 , v 2 , v 3 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq56_HTML.gif be a countable set. Define a strictly decreasing sequence { τ n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq57_HTML.gif of positive real numbers by τ n = 1 2 + 1 n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq102_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Then lim n τ n = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq103_HTML.gif. Let d : X × X [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq60_HTML.gif be defined by d ( v n , v n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif and d ( v n , v m ) = d ( v m , v n ) = τ n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq104_HTML.gif if m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq63_HTML.gif. Then d is a metric on X. Set A = { v 1 , v 3 , v 5 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq64_HTML.gif and B = { v 2 , v 4 , v 6 , } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq105_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equo_HTML.gif
Let T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gif be defined by
T v n = def v n + 1 for  n N . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equp_HTML.gif
It is easy to see that T ( A ) = B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq68_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq13_HTML.gif and so (MT1) holds. Let x 1 = v 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq106_HTML.gif be given. Define an iterative sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gif by x n + 1 = T x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq23_HTML.gif for n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. So { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq107_HTML.gif and { v n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq108_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq109_HTML.gif. Define φ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq110_HTML.gif as
φ ( t ) = def { τ n + 1 τ n , if  t = τ n  for some  n N , 0 , otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equq_HTML.gif
Since lim sup s t + φ ( s ) = 0 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq71_HTML.gif for all t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq111_HTML.gif, φ is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-function. Now, we verify (MT2). For m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq74_HTML.gif with m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq63_HTML.gif, since { τ n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq57_HTML.gif is strictly decreasing and τ n + 1 τ n < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq112_HTML.gif,
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equr_HTML.gif

which prove that (MT2) holds. Hence T is an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-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 2.2 Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gifand T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe a cyclic map. Suppose that there exists a nondecreasing (or nonincreasing) function τ : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq113_HTML.gifsuch 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equs_HTML.gif

Then there exists a sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gifsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq83_HTML.gif.

Corollary 2.1 ([1])

Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gifand T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe a cyclic contraction. Then there exists a sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gifsuch that lim n d ( x n , T x n ) = inf n N d ( x n , T x n ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq114_HTML.gif.

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

Theorem 2.3 Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gifand T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe a cyclic map. Let x 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq22_HTML.gifbe given. Define an iterative sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gifby x n + 1 = T x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq23_HTML.giffor n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Suppose that
  1. (i)

    d ( T x , T y ) d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq55_HTML.gif for any x A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq16_HTML.gif and y B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq17_HTML.gif;

     
  2. (ii)

    { x 2 n 1 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq25_HTML.gif has a convergent subsequence in A;

     
  3. (iii)

    lim n d ( x n , x n + 1 ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq115_HTML.gif.

     

Then there exists v A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq116_HTML.gifsuch that d ( v , T v ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq117_HTML.gif.

Proof Since T is a cyclic map and x 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq22_HTML.gif, x 2 n 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq94_HTML.gif and x 2 n B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq118_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. By (ii), { x 2 n 1 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq25_HTML.gif has a convergent subsequence { x 2 n k 1 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq119_HTML.gif and x 2 n k 1 v https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq120_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq121_HTML.gif for some v A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq116_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equt_HTML.gif
it follows from lim n d ( v , x 2 n k 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq122_HTML.gif and the condition (iii) that lim n d ( v , x 2 n k ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq123_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_Equu_HTML.gif

which implies d ( v , T v ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq117_HTML.gif. □

Applying Theorems 2.1 and 2.3, we establish the following new best proximity point theorem for MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq1_HTML.gif-cyclic contractions.

Theorem 2.4 Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq8_HTML.gifand T : A B A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq9_HTML.gifbe an MT https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq124_HTML.gif-cyclic contraction with respect to φ. Let x 1 A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq22_HTML.gifbe given. Define an iterative sequence { x n } n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq50_HTML.gifby x n + 1 = T x n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq23_HTML.giffor n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq24_HTML.gif. Suppose that { x 2 n 1 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq25_HTML.gifhas a convergent subsequence in A, then there exists v A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq116_HTML.gifsuch that d ( v , T v ) = dist ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-206/MediaObjects/13660_2012_Article_307_IEq117_HTML.gif.

Remark 2.1 ([[1], Proposition 3.2])

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

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.

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.