# A Note on Best Proximity Point Theorems under WeakP-Property

Authors: Angel Almeida; Erdal Karapınar; Kishin Sadarangani
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

In the very recent paper of  Akbar and Gabeleh (2013), by using the notion ofP-property, it was proved that some late results about the existence and uniqueness of best proximity points can be obtained from the versions of associated existing results in the fixed point theory. Along the same line, in this paper, we prove that these results can be obtained under a weaker condition, namely, weak P-property.


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## Body

## 1. Introduction and Preliminaries

LetA and B be two nonempty subsets of a metric space (
X
,
d
). By A
0 and B
0, we denote the following sets:
(1)
A
0
=
{
x
∈
A
:
d
(
x
,
y
)
=
d
(
A
,
B
)
for
some
y
∈
B
}
,
B
0
=
{
y
∈
B
:
d
(
x
,
y
)
=
d
(
A
,
B
)
for
some
x
∈
A
}
,
where d
(
A
,
B
)
=
inf
⁡
{
d
(
x
,
y
)
:
x
∈
A
and
y
∈
B
}.Definition 1.
LetA and B be a pair of nonempty subsets of a metric space (
X
,
d
) and let T
:
A
→
B be a mapping. One will say that x
*
∈
A is a best proximity of T if
(2)
d
(
x
*
,
T
x
*
)
=
d
(
A
,
B
)
.In [1], the authors introduced the notion of P-property as follows.Definition 2 (see [1]).
Let(
A
,
B
) be a pair of nonempty subsets of a metric space (
X
,
d
) with A
0
≠
∅. Then, the pair (
A
,
B
) is said to have the P-property if
(3)
d
(
x
1
,
y
1
)
=
d
(
A
,
B
)
d
(
x
2
,
y
2
)
=
d
(
A
,
B
)


}
⟹
d
(
x
1
,
x
2
)
=
d
(
y
1
,
y
2
)
,
where x
1, x
2
∈
A
0 and y
1, y
2
∈
B
0.Very recently, Zhang et al. [2] introduced the weak P-property.Definition 3 (see [2]).
Let(
A
,
B
) be a pair of nonempty subsets of a metric space (
X
,
d
) with A
0
≠
∅. Then, the pair (
A
,
B
) is said to have the P-property if
(4)
d
(
x
1
,
y
1
)
=
d
(
A
,
B
)
d
(
x
2
,
y
2
)
=
d
(
A
,
B
)


}
⟹
d
(
x
1
,
x
2
)
≤
d
(
y
1
,
y
2
)
,
where x
1, x
2
∈
A
0 and y
1, y
2
∈
B
0.It is evident that if a pair(
A
,
B
) has the P-property, then it has the weak P-property. In [2], the authors established some examples to prove that the converse of the statement above is false.In the literature, a great number of fixed point theorems have appeared to generalize, extend, and improve the celebrated Banach's contraction principle. Among them, we present two of these results as examples and we prove that their versions in the context of the best proximity point theory can be deduced as consequences of results in the setting of fixed point theory.LetΦ denote the set of all functions ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) which satisfy(
ϕ
1
)
ϕis continuous, and nondecreasing;
(
ϕ
2
)
ϕ is positive on (
0
,
∞
) and ϕ
(
0
)
=
0.Theorem 4 (see [3]).
Let(
X
,
d
) be a complete metric space and T
:
X
→
X an operator satisfying
(5)
d
(
T
x
,
T
y
)
≤
d
(
x
,
y
)
-
ϕ
(
d
(
x
,
y
)
)
f
o
r
a
n
y
x
,
y
∈
X
,
where ϕ
∈
Φ which satisfy lim
⁡
t
→
∞
ϕ
(
t
)
=
∞. Then, T has a unique fixed point.Remark 5.
Notice that Theorem4 remains valid if we remove the assumption that lim
⁡
t
→
∞
ϕ
(
t
)
=
∞.Definition 6.
LetΨ denote the family of all functions ψ
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) which satisfy (
ψ
1
)
lim
⁡
n
→
∞
ψ
n
(
t
)
=
0 for each t
>
0, where ψ
n is the nth iterate of ψ.
(
ψ
2
)
ψ is nondecreasing.Remark 7.
A function,ψ
∈
Ψ, is known as a comparison function or Bianchini-Grandolfi gauge function in the literature [4]. Moreover, it is easily seen that such functions satisfy the conditions
(6)
ψ
(
t
)
<
t
for
t
>
0
,
ψ
(
0
)
=
0
.In 2012, Romaguera [5] proved the analog of the following fixed point theorem in the context of partial metric spaces. It is evident that its metric version remains true.Theorem 8 (see [5]).
Let(
X
,
d
) be a complete metric space and T
:
X
→
X an operator satisfying
(7)
d
(
T
x
,
T
y
)
≤
ψ
(
M
T
(
x
,
y
)
)
f
o
r
a
n
y
x
,
y
∈
X
,
where ψ
∈
Ψ and
(8)
M
T
(
x
,
y
)
=
max
⁡
⁡
{
d
(
x
,
y
)
,
d
(
x
,
T
x
)
,
d
(
y
,
T
y
)
}
.
Then, T has a unique fixed point.


## 2. Main Results

We start this section with the following lemma which appeared implicitly in [2]. We prove this lemma for that the paper is self-contained.Lemma 9.
Let(
A
,
B
) be a pair of nonempty closed subsets of a complete metric space (
X
,
d
). Suppose that the following conditions hold: (i)
A
0
≠
∅;
(ii)
the pair(
A
,
B
) has the weak P-property. Then, the set B
0 is closed.Proof.
We haveB
0
≠
∅ due to the assumption that A
0
≠
∅. Let (
y
n
)
⊂
B
0 be a sequence such that y
n
→
y
∈
B. We shall prove that y
∈
B
0. In fact, since (
y
n
)
⊂
B
0, we find a sequence (
x
n
)
⊂
A
0 such that
(9)
d
(
x
n
,
y
n
)
=
d
(
A
,
B
)
for
any
n
∈
ℕ
.
By using the weak P-property, we infer that
(10)
d
(
x
n
,
x
m
)
≤
d
(
y
n
,
y
m
)
for
any
n
,
m
∈
ℕ
.
As (
y
n
) is a Cauchy sequence in B; from the inequality (10), we derive that the sequence (
x
n
) is a Cauchy sequence in A. Therefore, since A is closed, we deduce that x
n
→
x
∈
A for certain x
∈
A.
Finally, by the continuity of the metricd together with (9), it gives us d
(
x
,
y
)
=
d
(
A
,
B
). Hence, y
∈
B
0.Lemma 10.
Let(
A
,
B
) be a pair of nonempty closed subsets of a complete metric space (
X
,
d
). Suppose that the following conditions hold: (i)
A
0
≠
∅;
(ii)
the pair(
A
,
B
) has the weak P-property;
(iii)
T
:
A
→
B is a continuous mapping with T
(
A
0
)
⊂
B
0. Then, we have T
(
A
0
¯
)
⊂
B
0.Proof.
It is sufficient to prove that ifx
∈
A
0
¯
∖
A
0, then T
x
∈
B
0. In fact, since x
∈
A
0
¯
∖
A
0, we can find a sequence (
x
n
)
⊂
A
0 such that x
n
→
x. Due to (iii), we have T
(
x
n
)
⊂
B
0. Since the mapping T is continuous and B
0 is closed by Lemma 9, we conclude that T
x
∈
B
0.Before stating the main result of this paper, we need to recall the main result of Sankar Raj [1].Theorem 11.
Let(
A
,
B
) be a pair of nonempty closed subsets of a complete metric space (
X
,
d
) with A
0
≠
∅. Let T
:
A
→
B be a mapping satisfying
(11)
d
(
T
x
,
T
y
)
≤
d
(
x
,
y
)
-
ϕ
(
d
(
x
,
y
)
)
f
o
r
a
n
y
x
,
y
∈
X
,
where ϕ
∈
Φ. Suppose also that the pair (
A
,
B
) has the P-property and T
(
A
0
)
⊂
B
0. Then, T has a unique best proximity point.Now, we present the main results of the paper. If in Theorem11 we replace P-property by the weak P-property, we derive the following result.Theorem 12.
Theorem11 under the the assumption of the weak P-property instead of the P-property is a consequence of Theorem 4.Proof.
Suppose that all the conditions of Theorem11 are fulfilled under the the assumption of the weak P-property instead of the P-property. We first note that T is continuous, since
(12)
d
(
T
x
,
T
y
)
≤
d
(
x
,
y
)
-
ϕ
(
d
(
x
,
y
)
)
≤
d
(
x
,
y
)
for any x
,
y
∈
X. Hence, by Lemma 10, we have T
(
A
0
¯
)
⊂
B
0.
Next, we define an operatorP
:
T
(
A
0
¯
)
→
A
0 by P
y
=
x such that d
(
x
,
y
)
=
d
(
A
,
B
). As y
∈
T
(
A
0
¯
)
⊂
B
0, we can find x
∈
A
0 such that d
(
x
,
y
)
=
d
(
A
,
B
). Moreover, P is a well-defined mapping. Suppose, on the contrary, that there exists another x
0
∈
A
0 such that d
(
x
0
,
y
)
=
d
(
A
,
B
). By using the fact that the pair (
A
,
B
) has the weak P-property, we derive that
(13)
d
(
x
0
,
x
)
≤
d
(
y
,
y
)
=
0
,
and, consequently, x
=
x
0.
Now, we consider the operatorP
∘
T
:
A
0
¯
→
A
0. In the sequel, we shall prove that the operator P
∘
T satisfies all assumptions of Theorem 4. Notice that (
A
0
¯
,
d
) is a complete metric space since A
0
¯ is closed.
On the other hand, we have(14)
d
(
P
(
T
x
)
,
T
x
)
=
d
(
A
,
B
)
d
(
P
(
T
y
)
,
T
y
)
=
d
(
A
,
B
)

,
for x
,
y
∈
A
0
¯. As the pair (
A
,
B
) has the weak P-property, we infer from (14) that
(15)
d
(
P
(
T
x
)
,
P
T
y
)
≤
d
(
T
x
,
T
y
)
.
Regarding (11), we derive that
(16)
d
(
P
(
T
x
)
,
P
T
y
)
≤
d
(
T
x
,
T
y
)
≤
d
(
x
,
y
)
-
ϕ
(
d
(
x
,
y
)
)
,
for any x, y
∈
A
0
¯.
Therefore, by Theorem4, there exists a unique fixed point x
*
∈
A
0
¯ of P
∘
T, that is, P
(
T
x
*
)
=
x
*. By definition, we have
(17)
d
(
x
*
,
T
x
*
)
=
d
(
A
,
B
)
.
Therefore, x
* is a best proximity point of the mapping T and this completes the existence part of the proof.
We shall show thatx
* is the unique best proximity of the mapping T. Suppose, on the contrary, that z
* is another best proximity of the mapping T. Consequently, we have
(18)
d
(
z
*
,
T
z
*
)
=
d
(
A
,
B
)
,
and also z
*
∈
A
0
⊂
A
0
¯. Moreover, taking into account the definition of the operator P, this means that
(19)
P
(
T
z
*
)
=
z
*
.
Hence, z
* is a fixed point of P
∘
T. By the uniqueness of the fixed point of P
∘
T, we deduce that x
*
=
z
*, which completes the proof.Theorem4 imposes Theorem 11 because we have used a weaker condition, namely, the notion of the weak P-property.Theorem 13.
Let(
A
,
B
) be a pair of nonempty closed subsets of a complete metric space (
X
,
d
) with A
0
≠
∅. Let T
:
A
→
B be a continuous mapping satisfying
(20)
d
(
T
x
,
T
y
)
≤
ψ
(
M
T
(
x
,
y
)
)
f
o
r
a
n
y
x
,
y
∈
X
,
where ψ
∈
Ψ and
(21)
M
T
(
x
,
y
)
=
max
⁡
⁡
{
d
(
x
,
y
)
,
d
(
x
,
T
x
)
,
d
(
y
,
T
y
)
}
.
Suppose also that the pair (
A
,
B
) has the weak P-property and T
(
A
0
)
⊂
B
0. Then, T has a unique best proximity point.By using the same techniques used in Theorem12, we derive Theorem 13.Theorem 14.
Theorem13 is a consequence of Theorem 8.To avoid the repetition, we omit the proof Theorem14.Remark 15.
Notice that in Theorem14, we assume that T is continuous since the contractive condition (20) appeared in Theorem 13 does not imply the continuity of T.Corollary 16.
Let(
A
,
B
) be a pair of nonempty closed subsets of a complete metric space (
X
,
d
) with A
0
≠
∅. Let T
:
A
→
B be a continuous mapping satisfying
(22)
d
(
T
x
,
T
y
)
≤
ψ
(
M
T
(
x
,
y
,
A
,
B
)
)
f
o
r
a
n
y
x
,
y
∈
X
,
where ψ
∈
Ψ and
(23)
M
T
(
x
,
y
,
A
,
B
)
=
max
⁡
⁡
{
d
(
x
,
y
)
,
d
(
x
,
T
x
)
-
d
(
A
,
B
)
,
0000000
d
(
y
,
T
y
)
-
d
(
A
,
B
)
}
.
Suppose also that the pair (
A
,
B
) has the weak P-property and T
(
A
0
)
⊂
B
0. Then, T has a unique best proximity point.We skip the proof since the mappingψ is nondecreasing and hence we get
(24)
d
(
T
x
,
T
y
)
≤
ψ
(
M
T
(
x
,
y
,
A
,
B
)
)
≤
ψ
(
M
T
(
x
,
y
)
)
0000000000000
ai
00
for
any
x
,
y
∈
X
.
Therefore, Corollary 16 is a consequence of Theorem 13.Remark 17.
Corollary16 can be considered as a partial version of Theorem  20 of [6].



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*Source: 716825-2014-03-12.xml*