# Approximate Best Proximity Pairs in Metric Space

Authors: S. A. M. Mohsenalhosseini; H. Mazaheri; M. A. Dehghan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

LetA and B be nonempty subsets of a metric space X and also T:A∪B→A∪B and T(A)⊆B, T(B)⊆A. We are going to consider element x∈A such that d(x,Tx)≤d(A,B)+ϵ for some ϵ>0. We call pair (A,B) an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, T-minimizing a sequence are given in a metric space.


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## Body

## 1. Introduction

LetX be a metric space and A and B nonempty subsets of X, and d(A,B) is distance of A and B. If d(x0,y0)=d(A,B), then the pair (x0,y0) is called a best proximity pair for A and B and put (1.1)prox(A,B):={(x,y)∈A×B:d(x,y)=d(A,B)}
as the set of all best proximity pair (A,B). Best proximity pair evolves as a generalization of the concept of best approximation. That reader can find some important result of it in [1–4].Now, as in [5] (see also [4, 6–11]), we can find the best proximity points of the sets A and B, by considering a map T:A∪B→A∪B such that T(A)⊆B and T(B)⊆A. Best proximity pair also evolves as a generalization of the concept of fixed point of mappings. Because if A∩B≠∅, every best proximity point is a fixed point of  T.We say that the pointx∈A∪B is an approximate best proximity point of the pair (A,B), if d(x,Tx)≤d(A,B)+ϵ, for some ϵ>0.In the following, we introduce a concept of approximate proximity pair that is stronger than proximity pair.Definition 1.1.
LetA and B be nonempty subsets of a metric space X and T:A∪B→A∪B a map such that T(A)⊆B, T(B)⊆A. put
(1.2)PTa(A,B)={x∈A∪B:d(x,Tx)≤d(A,B)+ϵforsomeϵ>0}.
We say that the pair (A,B) is an approximate best proximity pair if PTa(A,B)≠∅.Example 1.2.
Suppose thatX=R2, A={(x,y)∈X:(x-y)2+y2≤1}, and B={(x,y)∈X:(x+y)2+y2≤1} with T(x,y)=(-x,y) for (x,y)∈X. Then d((x,y),T(x,y))≤d(A,B)+ϵ for some ϵ>0. Hence PTa(A,B)≠∅.


## 2. Approximate Best Proximity

In this section, we will consider the existence of approximate best proximity points for the mapT:A∪B→A∪B, such that T(A)⊆B, T(B)⊆A, and its diameter.Theorem 2.1.
LetA and  B be nonempty subsets of a metric space  X. Suppose that the mapping  T:A∪B→A∪B is satisfying  T(A)⊆B,  T(B)⊆A, and
(2.1)lim⁡n→∞d(Tnx,Tn+1x)=d(A,B)forsomex∈A∪B.
Then the pair (A,B) is an approximate best proximity pair.Proof.
Letϵ>0 be given and x∈A∪B such that limn→∞d(Tnx,Tn+1x)=d(A,B); then there exists N0>0 such that
(2.2)∀n≥N0:d(Tnx,Tn+1x)<d(A,B)+ϵ.
If n=N0, then d(TN0(x),T(TN0(x)))<d(A,B)+ϵ, and TN0(x)∈PTa(A,B) and PTa(A,B)≠∅.Theorem 2.2.
LetA and  B be nonempty subsets of a metric space  X. Suppose that the mapping  T:A∪B→A∪B is satisfying  T(A)⊆B,  T(B)⊆A and
(2.3)d(Tx,Ty)≤αd(x,y)+β[d(x,Tx)+d(y,Ty)]+γd(A,B)
for all x,y∈A∪B, where α,β,γ≥0 and α+2β+γ<1. Then the pair (A,B) is an approximate best proximity pair.Proof.
Ifx∈A∪B, then
(2.4)d(Tx,T2x)≤αd(x,Tx)+β[d(x,Tx)+d(Tx,T2x)]+γd(A,B).
Therefore,
(2.5)d(Tx,T2x)≤α+β1-βd(x,Tx)+γ1-βd(A,B).
Now if k=(α+β)/(1-β), then
(2.6)d(Tx,T2x)≤kd(x,Tx)+(1-k)d(A,B)
also
(2.7)d(T2x,T3x)≤k2d(x,Tx)+(1-k2)d(A,B).
Therefore,
(2.8)d(Tnx,Tn+1x)≤knd(x,Tx)+(1-kn)d(A,B),
and so
(2.9)d(Tnx,Tn+1x)⟶d(A,B),asn⟶∞.
Therefore, by Theorem 2.1, PTa(A,B)≠∅; then pair (A,B) is an approximate best proximity pair.Definition 2.3.
LetA and B be nonempty subsets of a metric space X. Suppose that the mapping T:A∪B→A∪B is satisfying T(A)⊆B, T(B)⊆A. We say that the sequence {zn}⊆A∪B is T-minimizing if
(2.10)lim⁡n→∞d(zn,Tzn)=d(A,B).Theorem 2.4.
LetA and  B be nonempty subsets of a metric space  X, suppose that the mapping  T:A∪B→A∪B is satisfying  T(A)⊆B,  T(B)⊆A. If  {Tnx} is a T-minimizing for some  x∈A∪B, then  (A,B) is an approximate best pair proximity.Proof.
Since(2.11)lim⁡n→∞d(Tnx,Tn+1x)=d(A,B)forsomex∈A∪B,
therefore, by Theorem 2.1, PTa(A,B)≠∅; then pair (A,B) is an approximate best proximity pair.Theorem 2.5.
LetA and  B be nonempty subsets of a normed space  X such that  A∪B is compact. Suppose that the mapping  T:A∪B→A∪B is satisfying  T(A)⊆B,  T(B)⊆A,  T is continuous and
(2.12)‖Tx-Ty‖≤‖x-y‖,
where (x,y)∈A×B. Then PTa(A,B) is nonempty and compact.Proof.
SinceA∪B compact, there exists a z0∈A∪B such that
*‖z0-Tz0‖=inf⁡z∈A∪B‖z-Tz‖⋅
If ∥z0-Tz0∥>d(A,B), then ∥Tz0-T2z0∥<∥z0-Tz0∥ which contradict to the definition of z0, (Tz0∈A∪B and by (*) ∥Tz0-T(Tz0)∥≥∥z0-Tz0∥). Therefore, ∥z0-Tz0∥=d(A,B)≤d(A,B)+ϵ for some ϵ>0 and z0∈PTa(A,B). Therefore, PTa(A,B) is nonempty.
Also, if{zn}⊆PTϵ(A,B), then ∥zn-Tzn∥<d(A,B)+ϵ, for some ϵ>0, and by compactness of A∪B, there exists a subsequence znk and a z0∈A∪B such that znk→z0 and so
(2.13)‖z0-Tz0‖=lim⁡k→∞‖znk-Tznk‖<d(A,B)+ϵ
for some ϵ>0, hence PTa(A,B) is compact.Example 2.6.
IfA=[-3,-1],B=[1,3], and T:A∪B→A∪B such that
(2.14)T(x)={1-x2,x∈A,-1-x2,x∈B,
then PTa(A,B) is compact, and we have
(2.15)PTa(A,B)={x∈A∪B:d(x,Tx)<d(A,B)+ϵforsomeϵ>0}={x∈A∪B:d(x,Tx)<2+ϵforsomeϵ>0}={1,-1}.
That is compact.In the following, bydiam(PTa(A,B)) for a set PTa(A,B)≠∅, we will understand the diameter of the set PTa(A,B).Definition 2.7.
LetT:A∪B→A∪B be a continuous map such that T(A)⊆B, T(B)⊆A and ϵ>0. We define diameter PTa(A,B) by
(2.16)diam⁡(PTa(A,B))=sup⁡⁡{d(x,y):x,y∈PTa(A,B)}.Theorem 2.8.
LetT:A∪B→A∪B, such that  T(A)⊆B,  T(B)⊆A and  ϵ>0. If there exists an  α∈[0,1] such that for all  (x,y)∈A×B(2.17)d(Tx,Ty)≤αd(x,y),
then
(2.18)diam⁡(PTa(A,B))≤2ϵ1-α+2d(A,B)1-α.Proof.
Ifx,y∈PTa(A,B), then
(2.19)d(x,y)≤d(x,Tx)+d(Tx,Ty)+d(Ty,y)≤ϵ1+αd(x,y)+2d(A,B)+ϵ2.
Put ϵ=Max{ϵ1,ϵ2}, therefore, d(x,y)≤2ϵ/(1-α)+(2d(A,B))/(1-α). Hence diam(PTa(A,B))≤2ϵ/(1-α)+(2d(A,B))/(1-α).


## 3. Approximate Best Proximity for Two Maps

In this section, we will consider the existence of approximate best proximity points for two mapsT:A∪B→A∪B and S:A∪B→A∪B, and its diameter.Definition 3.1.
LetA and B be nonempty subsets of a metric space (X,d) and let T:A∪B→A∪BS:A∪B→A∪B two maps such that T(A)⊆B, S(B)⊆A. A point (x,y) in A×B is said to be an approximate-pair fixed point for (T,S) in X if there exists ϵ>0(3.1)d(Tx,Sy)≤d(A,B)+ϵ.
We say that the pair (T,S) has the approximate-pair fixed property in X if P(T,S)a(A,B)≠∅, where
(3.2)P(T,S)a(A,B)={(x,y)∈A×B:d(Tx,Sy)≤d(A,B)+ϵforsomeϵ>0}.Theorem 3.2.
LetA and  B be nonempty subsets of a metric space  (X,d) and let  T:A∪B→A∪B and  S:A∪B→A∪B be two maps such that  T(A)⊆B,  S(B)⊆A. If, for every  (x,y)∈A×B,
(3.3)d(Tn(x),Sn(y))⟶d(A,B),
then (T,S) has the approximate-pair fixed property.Proof.
Forϵ>0, Suppose (x,y)∈A×B. Since
(3.4)d(Tn(x),Sn(y))⟶d(A,B),∃n0>0s.t.∀n≥n0:d(Tn(x),Sn(y))<d(A,B)+ϵ,
then d(T(Tn-1(x),S(Sn-1(y))<d(A,B)+ϵ for every n≥n0. Put x0=Tn0-1(x) and y0=Sn0-1(y)). Hence d(T(x0),S(y0))≤d(A,B)+ϵ and P(T,S)a(A,B)≠∅.Theorem 3.3.
LetA and  B be nonempty subsets of a metric space  (X,d) and let  T:A∪B→A∪B and  S:A∪B→A∪B be two maps such that  T(A)⊆B,  S(B)⊆A and, for every  (x,y)∈A×B,
(3.5)d(Tx,Sy)≤αd(x,y)+β[d(x,Tx)+d(y,Sy)]+γd(A,B),
where α,β,γ≥0 and α+2β+γ<1. Then if x is an approximate fixed point for T, or y is an approximate fixed point for S, then P(T,S)a(A,B)≠∅.Proof.
If(x,y)∈A×B, then
(3.6)d(Tx,S(Tx))≤αd(x,Tx)+β[d(x,Tx)+d(Tx,S(Tx))]+γd(A,B).
Therefore,
(3.7)d(Tx,S(Tx))≤α+β1-βd(x,Tx)+γ1-βd(A,B).
Now if k=(α+β)/(1-β), then
*d(Tx,S(Tx))≤kd(x,Tx)+(1-k)d(A,B)
also
**d(Sy,T(Sy))≤kd(y,Sy)+(1-k)d(A,B).
If x is an approximate fixed point for T, then there exists a ϵ>0 and by (*)
(3.8)d(Tx,S(Tx))≤kd(x,Tx)+(1-k)d(A,B)≤k(d(A,B)+ϵ)+(1-k)d(A,B)=d(A,B)+kϵ<d(A,B)+ϵ.
And (x,Tx)∈P(T,S)a(A,B); also if y is an approximate fixed point for S, then there exists a ϵ>0 and by (**)
(3.9)d(Sy,T(Sy))≤kd(y,Sy)+(1-k)d(A,B)≤k(d(A,B)+ϵ)+(1-k)d(A,B)=d(A,B)+kϵ<d(A,B)+ϵ.
And (y,Sy)∈P(T,S)a(A,B). Therefore, P(T,S)a(A,B)≠∅.Theorem 3.4.
LetA and  B be nonempty subsets of a metric space  (X,d) and let  T:A∪B→A∪B and  S:A∪B→A∪B be two continuous maps such that  T(A)⊆B,  S(B)⊆A. If, for every  (x,y)∈A×B,
(3.10)d(Tx,Sy)≤αd(x,y)+γd(A,B),
where α,γ≥0 and α+γ=1, also let {xn} and {yn} be as follows:
(3.11)xn+1=Syn,yn+1=Txnforsome(x1,y1)∈A×B,n∈N.
If {xn} has a convergent subsequence in A, then there exists a x0∈A such that d(x0,Tx0)=d(A,B).Proof.
We have(3.12)d(xn+1,yn+1)=d(Txn,Syn)≤αd(xn,yn)+γ(d(A,B)≤⋯≤αn+1d(x0,y0)+(1+α+⋯+αn)γd(A,B).
If {xnk}k≥1 converges to x1∈A, that is, xnk→x1, then
(3.13)d(xnK+1,ynk+1)≤αnk+1d(x0,y0)+(1+α+⋯+αkn)γd(A,B).
Since T is continuous, then
(3.14)d(xnk+1,Txnk)⟶γ1-αd(A,B)=d(A,B).
Therefore, d(x1,Tx1)=d(A,B).Definition 3.5.
LetT:A∪B→A∪BandS:A∪B→A∪Bbe continues maps such thatT(A)⊆BandS(B)⊆A. We define diameterP(T,S)a(A,B)by(3.15)diam⁡(P(T,S)a(A,B))=sup⁡⁡{d(x,y):d(Tx,Ty)≤ϵ+d(A,B)forsomeϵ>0}.Example 3.6.
SupposeA={(x,0):0≤x≤1}, B={(x,1):0≤x≤1}, T(x,0)=T(x,1)=(1/2,1), and S(x,1)=S(x,0)=(1/2,0). Then d(T(x,0),S(y,1))=1 and diam(P(T,S)a(A,B))=diam(A×B)=2.Theorem 3.7.
LetT:A∪B→A∪B and  S:A∪B→A∪B be continuous maps such that  T(A)⊆B,  S(B)⊆A. If there exists  a  k∈[0,1],
(3.16)d(x,Tx)+d(Sy,y)≤kd(x,y),
then
(3.17)diam⁡(P(T,S)a(A,B))≤ϵ1-k+d(A,B)1-kforsomeϵ>0.Proof.
If(x,y)∈P(T,S)a(A,B), then
(3.18)d(x,y)≤d(x,Tx)+d(Tx,Sy)+d(Sy,y)≤ϵ+kd(x,y)+d(A,B).
Therefore, d(x,y)≤ϵ/(1-k)+(d(A,B))/(1-k). Then diam(P(T,S)a(A,B))≤ϵ/(1-k)+(d(A,B))/(1-k).



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