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Abstract and Applied Analysis
Volume 2011, Article ID 596971, 9 pages
doi:10.1155/2011/596971
Research Article
Approximate Best Proximity Pairs in Metric Space
S. A. M. Mohsenalhosseini,1 H. Mazaheri,2 and M. A. Dehghan1
1
2
Faculty of Mathematics, Valiasr Rafsanjan University, Rafsanjan, Iran
Faculty of Mathematics, Yazd University, Yazd, Iran
Correspondence should be addressed to S. A. M. Mohsenalhosseini, amah@vru.ac.ir2
Received 8 January 2011; Accepted 12 February 2011
Academic Editor: Norimichi Hirano
Copyright q 2011 S. A. M. Mohsenalhosseini et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let A 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 dx, T x ≤ dA, B for some > 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.
1. Introduction
Let X be a metric space and A and B nonempty subsets of X, and dA, B is distance of A and
B. If dx0 , y0 dA, B, then the pair x0 , y0 is called a best proximity pair for A and B and
put
proxA, B :
x, y ∈ A × B : d x, y dA, B
1.1
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 TA ⊆ B and TB ⊆ 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 point x ∈ A ∪ B is an approximate best proximity point of the pair
A, B, if dx, Tx ≤ dA, B , for some > 0.
In the following, we introduce a concept of approximate proximity pair that is stronger
than proximity pair.
2
Abstract and Applied Analysis
Definition 1.1. Let A and B be nonempty subsets of a metric space X and T : A ∪ B → A ∪ B
a map such that TA ⊆ B, TB ⊆ A. put
PTa A, B {x ∈ A ∪ B : dx, Tx ≤ dA, B for some > 0}.
1.2
∅.
We say that the pair A, B is an approximate best proximity pair if PTa A, B /
Example 1.2. Suppose that X R2 , A {x, y ∈ X : x − y2 y2 ≤ 1}, and B {x, y ∈ X :
x y2 y2 ≤ 1} with Tx, y −x, y for x, y ∈ X. Then dx, y, Tx, y ≤ dA, B for some > 0. Hence PTa A, B / ∅.
2. Approximate Best Proximity
In this section, we will consider the existence of approximate best proximity points for the
map T : A ∪ B → A ∪ B, such that TA ⊆ B, TB ⊆ A, and its diameter.
Theorem 2.1. Let A and B be nonempty subsets of a metric space X. Suppose that the mapping
T : A ∪ B → A ∪ B is satisfying TA ⊆ B, TB ⊆ A, and
lim d T n x, T n1 x dA, B for some x ∈ A ∪ B.
n→∞
2.1
Then the pair A, B is an approximate best proximity pair.
Proof. Let > 0 be given and x ∈ A ∪ B such that limn → ∞ dT n x, T n1 x dA, B; then there
exists N0 > 0 such that
∀n ≥ N0 : d T n x, T n1 x < dA, B .
2.2
If n N0 , then dT N0 x, TT N0 x < dA, B , and T N0 x ∈ PTa A, B and PTa A, B /
∅.
Theorem 2.2. Let A and B be nonempty subsets of a metric space X. Suppose that the mapping
T : A ∪ B → A ∪ B is satisfying TA ⊆ B, TB ⊆ A and
d Tx, Ty ≤ αd x, y β dx, Tx d y, Ty γ dA, B
2.3
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. If x ∈ A ∪ B, then
d Tx, T 2 x ≤ αdx, Tx β dx, Tx d Tx, T 2 x γ dA, B.
2.4
Abstract and Applied Analysis
3
Therefore,
αβ
γ
dx, Tx dA, B.
d Tx, T 2 x ≤
1−β
1−β
2.5
Now if k α β/1 − β, then
d Tx, T 2 x ≤ kdx, Tx 1 − kdA, B
2.6
d T 2 x, T 3 x ≤ k2 dx, Tx 1 − k2 dA, B.
2.7
d T n x, T n1 x ≤ kn dx, Tx 1 − kn dA, B,
2.8
also
Therefore,
and so
d T n x, T n1 x −→ dA, B,
as n −→ ∞.
2.9
Therefore, by Theorem 2.1, PTa A, B /
∅; then pair A, B is an approximate best proximity
pair.
Definition 2.3. Let A and B be nonempty subsets of a metric space X. Suppose that the
mapping T : A ∪ B → A ∪ B is satisfying TA ⊆ B, TB ⊆ A. We say that the sequence
{zn } ⊆ A ∪ B is T-minimizing if
lim dzn , Tzn dA, B.
n→∞
2.10
Theorem 2.4. Let A and B be nonempty subsets of a metric space X, suppose that the mapping T :
A ∪ B → A ∪ B is satisfying TA ⊆ B, TB ⊆ A. If {T n x} is a T-minimizing for some x ∈ A ∪ B,
then A, B is an approximate best pair proximity.
Proof. Since
lim d T n x, T n1 x dA, B
n→∞
for some x ∈ A ∪ B,
2.11
∅; then pair A, B is an approximate best proximity
therefore, by Theorem 2.1, PTa A, B /
pair.
4
Abstract and Applied Analysis
Theorem 2.5. Let A 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 TA ⊆ B, TB ⊆ A, T is continuous
and
Tx − Ty ≤ x − y,
2.12
where x, y ∈ A × B. Then PTa A, B is nonempty and compact.
Proof. Since A ∪ B compact, there exists a z0 ∈ A ∪ B such that
z0 − Tz0 inf z − Tz·
z∈A∪B
∗
If z0 − Tz0 > dA, B, then Tz0 − T 2 z0 < z0 − Tz0 which contradict to the definition of z0 ,
Tz0 ∈ A ∪ B and by ∗ Tz0 − TTz0 ≥ z0 − Tz0 . Therefore, z0 − Tz0 dA, B ≤
dA, B for some > 0 and z0 ∈ PTa A, B. Therefore, PTa A, B is nonempty.
Also, if {zn } ⊆ PT A, B, then zn − Tzn < dA, B , for some > 0, and by
compactness of A ∪ B, there exists a subsequence znk and a z0 ∈ A ∪ B such that znk → z0 and
so
z0 − Tz0 lim znk − Tznk < dA, B k→∞
2.13
for some > 0, hence PTa A, B is compact.
Example 2.6. If A −3, −1, B 1, 3, and T : A ∪ B → A ∪ B such that
Tx ⎧
1−x
⎪
⎪
⎨ 2 ,
x ∈ A,
⎪
⎪
⎩ −1 − x , x ∈ B,
2
2.14
then PTa A, B is compact, and we have
PTa A, B {x ∈ A ∪ B : dx, Tx < dA, B for some > 0}
{x ∈ A ∪ B : dx, Tx < 2 for some > 0}
2.15
{1, −1}.
That is compact.
In the following, by diamPTa A, B for a set PTa A, B /
∅, we will understand the
diameter of the set PTa A, B.
Abstract and Applied Analysis
5
Definition 2.7. Let T : A ∪ B → A ∪ B be a continuous map such that TA ⊆ B, TB ⊆ A and
> 0. We define diameter PTa A, B by
diam PTa A, B sup d x, y : x, y ∈ PTa A, B .
2.16
Theorem 2.8. Let T : A ∪ B → A ∪ B, such that TA ⊆ B, TB ⊆ A and > 0. If there exists an
α ∈ 0, 1 such that for all x, y ∈ A × B
d Tx, Ty ≤ αd x, y ,
2.17
2dA, B
2
.
diam PTa A, B ≤
1−α
1−α
2.18
then
Proof. If x, y ∈ PTa A, B, then
d x, y ≤ dx, Tx d Tx, Ty d Ty, y
≤ 1 αd x, y 2dA, B 2 .
2.19
Put Max{1 , 2 }, therefore, dx, y ≤ 2/1−α2dA, B/1−α. Hence diamPTa A, B
≤ 2/1 − α 2dA, 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
maps T : A ∪ B → A ∪ B and S : A ∪ B → A ∪ B, and its diameter.
Definition 3.1. Let A 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 TA ⊆ B, SB ⊆ 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 > 0
d Tx, Sy ≤ dA, B .
3.1
a
We say that the pair T, S has the approximate-pair fixed property in X if PT,S
A, B / ∅,
where
a
PT,S
A, B x, y ∈ A × B : d Tx, Sy ≤ dA, B for some > 0 .
3.2
Theorem 3.2. Let A 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 TA ⊆ B, SB ⊆ A. If, for every x, y ∈ A × B,
d T n x, Sn y −→ dA, B,
then T, S has the approximate-pair fixed property.
3.3
6
Abstract and Applied Analysis
Proof. For > 0, Suppose x, y ∈ A × B. Since
d T n x, Sn y −→ dA, B,
∃n0 > 0
s.t. ∀n ≥ n0 : d T n x, Sn y < dA, B ,
3.4
then dTT n−1 x, SSn−1 y < dA, B for every n ≥ n0 . Put x0 T n0 −1 x and y0 a
A, B /
∅.
Sn0 −1 y. Hence dTx0 , Sy0 ≤ dA, B and PT,S
Theorem 3.3. Let A 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 TA ⊆ B, SB ⊆ A and, for every x, y ∈ A × B,
d Tx, Sy ≤ αd x, y β dx, Tx d y, Sy γ dA, B,
3.5
where α, β, γ ≥ 0 and α2βγ < 1. Then if x is an approximate fixed point for T, or y is an approximate
a
A, B /
∅.
fixed point for S, then PT,S
Proof. If x, y ∈ A × B, then
dTx, STx ≤ αdx, Tx βdx, Tx dTx, STx γ dA, B.
3.6
Therefore,
dTx, STx ≤
αβ
γ
dx, Tx dA, B.
1−β
1−β
3.7
Now if k α β/1 − β, then
dTx, STx ≤ kdx, Tx 1 − kdA, B
∗
d Sy, T Sy ≤ kd y, Sy 1 − kdA, B.
∗∗
also
If x is an approximate fixed point for T, then there exists a > 0 and by ∗
dTx, STx ≤ kdx, Tx 1 − kdA, B
≤ kdA, B 1 − kdA, B
dA, B k
< dA, B .
3.8
Abstract and Applied Analysis
7
a
A, B; also if y is an approximate fixed point for S, then there exists
And x, Tx ∈ PT,S
a > 0 and by ∗∗
d Sy, T Sy ≤ kd y, Sy 1 − kdA, B
≤ kdA, B 1 − kdA, B
dA, B k
3.9
< dA, B .
a
a
And y, Sy ∈ PT,S
A, B. Therefore, PT,S
A, B /
∅.
Theorem 3.4. Let A 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 TA ⊆ B, SB ⊆ A. If, for every
x, y ∈ A × B,
d Tx, Sy ≤ αd x, y γ dA, B,
3.10
where α, γ ≥ 0 and α γ 1, also let {xn } and {yn } be as follows:
xn1 Syn ,
yn1 Txn
for some x1 , y1 ∈ A × B, n ∈ N.
3.11
If {xn } has a convergent subsequence in A, then there exists a x0 ∈ A such that dx0 , Tx0 dA, B.
Proof. We have
d xn1 , yn1 d Txn , Syn
≤ αd xn , yn γ dA, B
≤ ···
3.12
≤ αn1 d x0 , y0 1 α · · · αn γ dA, B.
If {xnk }k≥1 converges to x1 ∈ A, that is, xnk → x1 , then
d xnK1 , ynk1 ≤ αnk1 d x0 , y0 1 α · · · αnk γ dA, B.
3.13
Since T is continuous, then
dxnk1 , Txnk −→
Therefore, dx1 , Tx1 dA, B.
γ
dA, B dA, B.
1−α
3.14
8
Abstract and Applied Analysis
Definition 3.5. Let T : A ∪ B → A ∪ B and S : A ∪ B → A ∪ B be continues maps such that
a
A, B by
TA ⊆ B and SB ⊆ A. We define diameter PT,S
a
diam PT,S
A, B sup d x, y : d Tx, Ty ≤ dA, B for some > 0 .
3.15
Example 3.6. Suppose A {x, 0 : 0 ≤ x ≤ 1}, B {x, 1 : 0 ≤ x ≤ 1}, Tx, 0 Tx, 1 a
A, B 1/2, 1, and Sx, 1 Sx, 0 1/2, 0. Then dTx, 0, Sy, 1 1 and diamPT,S
√
diamA × B 2.
Theorem 3.7. Let T : A ∪ B → A ∪ B and S : A ∪ B → A ∪ B be continuous maps such that
TA ⊆ B, SB ⊆ A. If there exists a k ∈ 0, 1,
dx, Tx d Sy, y ≤ kd x, y ,
3.16
then
a
diam PT,S
A, B ≤
dA, B
1−k
1−k
for some > 0.
3.17
a
Proof. If x, y ∈ PT,S
A, B, then
d x, y ≤ dx, Tx d Tx, Sy d Sy, y
≤ kd x, y dA, B.
3.18
a
Therefore, dx, y ≤ /1 − k dA, B/1 − k. Then diamPT,S
A, B ≤ /1 − k dA, B/1 − k.
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Abstract and Applied Analysis
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