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Nonlinear conditions for the existence of best proximity points

Abstract

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

MSC:41A17, 47H09.

1 Introduction and preliminaries

Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Let A and B be nonempty subsets of a nonempty set E. A map S:ABAB is called a cyclic map if S(A)B and S(B)A. Let (X,d) be a metric space and T:ABAB 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 } .

A point xAB is called to be a best proximity point for T if d(x,Tx)=dist(A,B).

Definition 1.1 ([1])

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

  1. (1)

    T(A)B and T(B)A;

  2. (2)

    there exists k(0,1) such that d(Tx,Ty)kd(x,y)+(1k)dist(A,B) for all xA, yB.

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

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. LetT:ABABbe a cyclic contraction map, x 1 Aand define x n + 1 =T x n , nN. Suppose{ x 2 n 1 }has a convergent subsequence in A. Then there existsxAsuch thatd(x,Tx)=dist(A,B).

Let f be a real-valued function defined on R. For cR, we recall that

lim sup x c f(x)= inf ε > 0 sup 0 < | x c | < ε f(x)

and

lim sup x c + f(x)= inf ε > 0 sup 0 < x c < ε f(x).

Definition 1.2 ([713])

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

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

Very recently, Du [10] first proved some characterizations of MT-functions.

Theorem D ([10])

Letφ:[0,)[0,1)be a function. Then the following statements are equivalent.

  1. (a)

    φ is an MT-function.

  2. (b)

    For each t[0,), there exist r t ( 1 ) [0,1) and ε t ( 1 ) >0 such that φ(s) r t ( 1 ) for all s(t,t+ ε t ( 1 ) ).

  3. (c)

    For each t[0,), there exist r t ( 2 ) [0,1) and ε t ( 2 ) >0 such that φ(s) r t ( 2 ) for all s[t,t+ ε t ( 2 ) ].

  4. (d)

    For each t[0,), there exist r t ( 3 ) [0,1) and ε t ( 3 ) >0 such that φ(s) r t ( 3 ) for all s(t,t+ ε t ( 3 ) ].

  5. (e)

    For each t[0,), there exist r t ( 4 ) [0,1) and ε t ( 4 ) >0 such that φ(s) r t ( 4 ) for all s[t,t+ ε t ( 4 ) ).

  6. (f)

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

  7. (g)

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

Motivated by the concepts of cyclic contractions and MT-functions, we first introduce the concept of MT-cyclic contractions.

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

(MT1) T(A)B and T(B)A;

(MT2) there exists an MT-function φ:[0,)[0,1) such that

d(Tx,Ty)φ ( d ( x , y ) ) d(x,y)+ ( 1 φ ( d ( x , y ) ) ) dist(A,B)for any xA and yB,

then T is called an MT-cyclic contraction with respect to φ on AB.

Remark 1.2 It is obvious that (MT2) implies that T satisfies d(Tx,Ty)d(x,y) for any xA and yB.

The following example gives a map T which is an MT-cyclic contraction but not a cyclic contraction.

Example A Let X={ v 1 , v 2 , v 3 ,} be a countable set and { τ n } be a strictly increasing convergent sequence of positive real numbers. Denote by τ := lim n τ n . Then τ 2 < τ . Let d:X×X[0,) be defined by d( v n , v n )=0 for all nN and d( v n , v m )=d( v m , v n )= τ m if m>n. Then d is a metric on X. Set A={ v 1 , v 3 , v 5 ,}, B={ v 2 , v 4 , v 6 ,}. So AB=X and dist(A,B)= τ 2 . Now we define a map T:ABAB by

T v n = def { v 2 , if  n = 1 , v n 1 , if  n > 1

for nN. It is easy to see that T(A)=B and T(B)=A, and so (MT1) holds. Define φ:[0,)[0,1) as

φ(t) = def { τ n 1 τ n , if  t = τ n  for some  n N  with  n > 2 , 0 , otherwise .

Since { τ n } is strictly increasing, lim sup s t + φ(s)=0<1 for all t[0,). Hence φ is an MT-function. Clearly, lim n d( v n , v n + 1 )= lim n τ n + 1 = τ and d(T v 1 ,T v 2 )= τ 2 . For m,nN with m>n and m>2, d(T v n ,T v m )= τ m 1 . So lim n d(T v n ,T v n + 1 )= lim n τ n = τ . We claim that T is not a cyclic contraction on AB. Indeed, suppose that T is a cyclic contraction on AB. Thus there exists k[0,1) such that

From (), we get τ τ 2 , which is a contradiction. Therefore, T is not a cyclic contraction on AB.

Next, we show that T is an MT-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, 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 );
  2. (ii)

    For m,nN with m>n and m>2, 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 ) .

From above, we can prove that (MT2) holds. Hence T is an MT-cyclic contraction with respect to φ. Moreover, since d( v 1 ,T v 1 )=dist(A,B), v 1 A 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-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-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)andT:ABABbe anMT-cyclic contraction with respect to φ. Then there exists a sequence { x n } n N such that lim n d( x n ,T x n )= inf n N d( x n ,T x n )=dist(A,B).

Proof Let x 1 AB be given. Define an iterative sequence { x n } n N by x n + 1 =T x n for nN. Clearly, dist(A,B)d( x n , x n + 1 ) for all nN. If there exists jN such that x j = x j + 1 AB, then lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=dist(A,B)=0. So it suffices to consider the case x n + 1 x n for all nN. By Remark 1.2, it is easy to see that the sequence {d( x n , x n + 1 )} is nonincreasing in (0,). Then

t 0 := lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )0.
(2.1)

Since φ is an MT-function, applying Theorem D, we get

0 sup n N φ ( d ( x n , x n + 1 ) ) <1.

Let λ:= sup n N φ(d( x n , x n + 1 )). Then 0φ(d( x n , x n + 1 ))λ<1 for all nN. If x 1 A, then, by (MT1), we have x 2 n 1 A and x 2 n B for all nN. 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 )

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 ) .

On the other hand, if x 1 B, then x 2 n A and x 2 n 1 B for all nN. Applying (MT2) again, we also have

d( x 2 , x 3 )λd( x 1 , x 2 )+dist(A,B)

and

d( x 3 , x 4 ) λ 2 d( x 1 , x 2 )+dist(A,B).

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 nN.
(2.2)

Since λ[0,1), lim n λ n =0. 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). The proof is completed. □

Here we give a nontrivial example illustrating Theorem 2.1.

Example B Let X={ v 1 , v 2 , v 3 ,} be a countable set. Define a strictly decreasing sequence { τ n } of positive real numbers by τ n = 1 2 + 1 n for all nN. Then lim n τ n = 1 2 . Let d:X×X[0,) be defined by d( v n , v n )=0 for all nN and d( v n , v m )=d( v m , v n )= τ n if m>n. Then d is a metric on X. Set A={ v 1 , v 3 , v 5 ,} and B={ v 2 , v 4 , v 6 ,}. 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 .

Let T:ABAB be defined by

T v n = def v n + 1 for nN.

It is easy to see that T(A)=B and T(B)A and so (MT1) holds. Let x 1 = v 1 A be given. Define an iterative sequence { x n } n N by x n + 1 =T x n for nN. So { x n } and { v n } 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 . Define φ:[0,)[0,1) as

φ(t) = def { τ n + 1 τ n , if  t = τ n  for some  n N , 0 , otherwise .

Since lim sup s t + φ(s)=0<1 for all t[0,), φ is an MT-function. Now, we verify (MT2). For m,nN with m>n, since { τ n } is strictly decreasing and τ n + 1 τ n <1,

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 ) ,

which prove that (MT2) holds. Hence T is an MT-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)andT:ABABbe a cyclic map. Suppose that there exists a nondecreasing (or nonincreasing) functionτ:[0,)[0,1)such that

d(Tx,Ty)τ ( d ( x , y ) ) d(x,y)+ ( 1 τ ( d ( x , y ) ) ) dist(A,B)for any xA and yB.

Then there exists a sequence { x n } n N such that lim n d( x n ,T x n )= inf n N d( x n ,T x n )=dist(A,B).

Corollary 2.1 ([1])

Let A and B be nonempty subsets of a metric space(X,d)andT:ABABbe a cyclic contraction. Then there exists a sequence { x n } n N such that lim n d( x n ,T x n )= inf n N d( x n ,T x n )=dist(A,B).

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)andT:ABABbe a cyclic map. Let x 1 Abe given. Define an iterative sequence { x n } n N by x n + 1 =T x n fornN. Suppose that

  1. (i)

    d(Tx,Ty)d(x,y) for any xA and yB;

  2. (ii)

    { x 2 n 1 } has a convergent subsequence in A;

  3. (iii)

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

Then there existsvAsuch thatd(v,Tv)=dist(A,B).

Proof Since T is a cyclic map and x 1 A, x 2 n 1 A and x 2 n B for all nN. By (ii), { x 2 n 1 } has a convergent subsequence { x 2 n k 1 } and x 2 n k 1 v as k for some vA. 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 kN,

it follows from lim n d(v, x 2 n k 1 )=0 and the condition (iii) that lim n d(v, x 2 n k )=dist(A,B). By (i), we have

dist(A,B)d(Tv, x 2 n k + 1 )d(v, x 2 n k )for all kN,

which implies d(v,Tv)=dist(A,B). □

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

Theorem 2.4 Let A and B be nonempty subsets of a metric space(X,d)andT:ABABbe anMT-cyclic contraction with respect to φ. Let x 1 Abe given. Define an iterative sequence { x n } n N by x n + 1 =T x n fornN. Suppose that{ x 2 n 1 }has a convergent subsequence in A, then there existsvAsuch thatd(v,Tv)=dist(A,B).

Remark 2.1 ([[1], Proposition 3.2])

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

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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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Correspondence to Wei-Shih Du.

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Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

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Du, WS., Lakzian, H. Nonlinear conditions for the existence of best proximity points. J Inequal Appl 2012, 206 (2012). https://doi.org/10.1186/1029-242X-2012-206

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Keywords

  • cyclic map
  • best proximity point
  • MT-function (R-function)
  • MT-cyclic contraction