# Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

Authors: Erdal Karapınar; V. Pragadeeswarar; M. Marudai
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.


---

## Body

## 1. Introduction and Preliminaries

A self-mappingT, defined on a metric space (
X
,
d
), is said to be a contraction if there exists a constant k
∈
[
0,1
) such that the inequality d
(
T
x
,
T
y
)
≤
k
d
(
x
,
y
) holds for all x
,
y
∈
X. Moreover, a self-mapping T is called a contractive mapping if d
(
T
x
,
T
y
)
<
d
(
x
,
y
) holds for all x
,
y
∈
X with x
≠
y.The celebrated Banach contraction principle says that ifX is complete, then every contraction has a unique fixed point. In fact, the fixed point of a contraction mapping T is obtained as a limit of repeated iteration of the mapping for any (initial) point of X. Let Φ be the class of continuous, nondecreasing mapping ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) such that ϕ is positive on (
0
,
∞
) and ϕ
(
0
)
=
0. A function ϕ
∈
Φ is called an altering distance function.A mappingT
:
X
→
X is called a weak-ϕ contraction if there exists a ϕ
∈
Φ such that d
(
T
x
,
T
y
)
≤
d
(
x
,
y
)
-
ϕ
(
d
(
x
,
y
)
) for each x
,
y
∈
X. The notion of weak-ϕ contraction was defined by Alber and Guerre-Delabriere [1] to generalize the well-known Banach contraction principle in the setting of Hilbert spaces. Later, Rhoades [2] noticed that most of the results of Alber and Guerre-Delabriere [1] are valid for any Banach space. Rhoades also proved the following generalization of the Banach contraction principle (see also [3–7]).Theorem 1.
Let(
X
,
d
) be a nonempty complete metric space and let T
:
X
→
X be a weak-ϕ contraction on X; then T has a unique fixed point.Recently, Dutta and Choudhury [8] proved the following generalization of Theorem 1 by using (
ψ
,
ϕ
)-weak contraction map.Theorem 2.
Let(
X
,
d
) be a nonempty complete metric space and let T
:
X
→
X be a self-mapping satisfying the inequality
(1)
ψ
(
d
(
T
x
,
T
y
)
)
≤
ψ
(
d
(
x
,
y
)
)
-
ϕ
(
d
(
x
,
y
)
)
,
for all x
,
y
∈
X, where of all function. Then T has a unique fixed point.LetΓ be the class of all function;   φ
:
[
0
,
∞
)
→
[
0
,
∞
) is a lower semicontinuous with φ
(
t
)
=
0 if and only if t
=
0. In [9] Dorić proved the following generalization of Theorem 2 by using generalized (
ψ
,
ϕ
)-weak contractions which contains the (
ψ
,
ϕ
)-weak contractions as a subclass.Theorem 3.
Let(
X
,
d
) be a nonempty complete metric space and let T
:
X
→
X be a generalized (
ψ
,
ϕ
)-weak contraction map; that is, T satisfies the following inequality:
(2)
ψ
(
d
(
T
x
,
T
y
)
)
≤
ψ
(
m
(
x
,
y
)
)
-
φ
(
m
(
x
,
y
)
)
,
where  ψ
∈
Φ,  φ
∈
Γ, and m
(
x
,
y
)
=
max
⁡
{
d
(
x
,
y
)
,
d
(
T
x
,
x
)
,
d
(
T
y
,
y
)
,
(
1
/
2
)
[
d
(
y
,
T
x
)
+
d
(
x
,
T
y
)
]
} for all x
,
y
∈
X. Then T has a unique fixed point.One of the aims of this paper is to extend Theorem3 via best proximity point. For this purpose, we recollect the basic definitions and fundamentals results as follows.


### 1.1. Best Proximity Point Theorems

We first recall the notion of best proximity point for nonself-mappings.Definition 4.
Let(
A
,
B
) be a pair of two nonempty subsets of a metric space X. An element x
∈
A is said to be a best proximity point of the nonself-mappings T
:
A
→
B if it satisfies the condition that d
(
x
,
T
x
)
=
d
(
A
,
B
) where A and B are nonempty subsets of a metric space.Best proximity point theorems have been studied to find necessary condition such that the minimization problemmin
⁡
x
∈
A
d
(
x
,
T
x
) has at least one solution.Existence and convergence of best proximity point is an interesting topic of optimization theory which recently attracted the attention of many authors [10–15]. A best proximity point theorem for nonself-proximal contractions has been investigated in [16–18].In this paper, let us consider the mappingsT
:
A
→
B, where A and B are nonempty subsets of a metric space X with generalized proximal weak contraction on T which ensure the existence of a unique point x
∈
A which satisfies d
(
x
,
T
x
)
=
d
(
A
,
B
). When the map T is considered to be self-map, then our result reduces to Theorem 3.Given nonempty subsetsA and B of a metric space X, the following notions are used subsequently:
(3)
d
(
A
,
B
)
:
=
inf
⁡
{
d
(
x
,
y
)
:
x
∈
A
,
y
∈
B
}
,
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
}
.
In [13], the authors discussed sufficient conditions which guarantee the nonemptiness of A
0 and B
0. Also, in [14], the authors proved that A
0 is contained in the boundary of A.Definition 5.
The setB is said to be approximatively compact with respect to A if every sequence {
y
n
} in B satisfying the condition that d
(
x
,
y
n
)
→
d
(
x
,
B
) for some x in A has a convergent subsequence.Note that every set is approximatively compact with respect to itself and that every compact set is approximatively compact. Further,A
0 and B
0 are nonempty if A is compact and B is approximatively compact with respect to A.Let us define the notion of generalized proximal weak contraction maps as follows. For this goal, we first introduce the following class of the mapping. LetΩ be the set of all function;  ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is nondecreasing with the following property: ϕ
(
t
)
=
0 if and only if t
=
0.Definition 6.
A mappingT
:
A
→
B is said to be a generalized proximal weak contraction on A if there exists functions ψ
,
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) satisfying the following condition:
(4)
d
(
u
,
T
x
)
=
d
(
A
,
B
)
d
(
v
,
T
y
)
=
d
(
A
,
B
)
}
⇒
ψ
(
d
(
u
,
v
)
)
≤
ψ
(
m
(
x
,
y
)
)
-
ϕ
(
m
(
x
,
y
)
)
,
where m
(
x
,
y
)
=
max
⁡
{
d
(
x
,
y
)
,
d
(
x
,
u
)
,
d
(
y
,
v
)
,
(
d
(
x
,
v
)
+
d
(
y
,
u
)
)
/
2
} for all u
,
v
,
x
,
y in A, where ψ
∈
Φ and ϕ
∈
Ω.Remark 7.
Definition6 guarantees that if a mapping T has a best proximity point then it should be unique. Indeed, we can prove our claim easily. Let x
∈
A be a best proximity point of T. Suppose, on the contrary, that there is another element y such that d
(
y
,
T
y
)
=
d
(
A
,
B
). Since T is a generalized proximal weak contraction on A, we have
(5)
ψ
(
d
(
x
,
y
)
)
≤
ψ
(
m
(
x
,
y
)
)
-
ϕ
(
m
(
x
,
y
)
)
,
where m
(
x
,
y
)
=
max
⁡
{
d
(
x
,
y
)
,
d
(
x
,
x
)
,
d
(
y
,
y
)
,
(
d
(
x
,
y
)
+
d
(
y
,
x
)
)
/
2
}
=
d
(
x
,
y
).
From (5), we obtain ψ
(
d
(
x
,
y
)
)
≤
ψ
(
d
(
x
,
y
)
)
-
ϕ
(
d
(
x
,
y
)
), which implies ϕ
(
d
(
x
,
y
)
)
=
0, and by our assumption about ϕ, we get d
(
x
,
y
)
=
0, or equivalently, x
=
y.Definition 8.
A mappingT
:
A
→
B is said to be a generalized proximal weak contraction on B if there exist functions ψ
,
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) satisfying the following condition:
(6)
d
(
u
,
T
x
)
=
d
(
A
,
B
)
d
(
v
,
T
y
)
=
d
(
A
,
B
)
}
⇒
ψ
(
d
(
T
u
,
T
v
)
)
≤
ψ
(
m
(
T
x
,
T
y
)
)
-
ϕ
(
m
(
T
x
,
T
y
)
)
,
where m
(
T
x
,
T
y
)
=
max
⁡
{
d
(
T
x
,
T
y
)
,
d
(
T
x
,
T
u
)
,
d
(
T
y
,
T
v
), (
d
(
T
x
,
T
v
)
+
d
(
T
y
,
T
u
)
)
/
2
} for all u
,
v
,
x
,
y in A, where ψ
∈
Φ and ϕ
∈
Ω.For self-mappings, it is clear that every generalized proximal weak contraction onA is a generalized proximal weak contraction on B. An operator T is said to be a generalized proximal weak contraction if it is both generalized proximal weak contraction on A and generalized proximal weak contraction on B.


## 1.1. Best Proximity Point Theorems

We first recall the notion of best proximity point for nonself-mappings.Definition 4.
Let(
A
,
B
) be a pair of two nonempty subsets of a metric space X. An element x
∈
A is said to be a best proximity point of the nonself-mappings T
:
A
→
B if it satisfies the condition that d
(
x
,
T
x
)
=
d
(
A
,
B
) where A and B are nonempty subsets of a metric space.Best proximity point theorems have been studied to find necessary condition such that the minimization problemmin
⁡
x
∈
A
d
(
x
,
T
x
) has at least one solution.Existence and convergence of best proximity point is an interesting topic of optimization theory which recently attracted the attention of many authors [10–15]. A best proximity point theorem for nonself-proximal contractions has been investigated in [16–18].In this paper, let us consider the mappingsT
:
A
→
B, where A and B are nonempty subsets of a metric space X with generalized proximal weak contraction on T which ensure the existence of a unique point x
∈
A which satisfies d
(
x
,
T
x
)
=
d
(
A
,
B
). When the map T is considered to be self-map, then our result reduces to Theorem 3.Given nonempty subsetsA and B of a metric space X, the following notions are used subsequently:
(3)
d
(
A
,
B
)
:
=
inf
⁡
{
d
(
x
,
y
)
:
x
∈
A
,
y
∈
B
}
,
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
}
.
In [13], the authors discussed sufficient conditions which guarantee the nonemptiness of A
0 and B
0. Also, in [14], the authors proved that A
0 is contained in the boundary of A.Definition 5.
The setB is said to be approximatively compact with respect to A if every sequence {
y
n
} in B satisfying the condition that d
(
x
,
y
n
)
→
d
(
x
,
B
) for some x in A has a convergent subsequence.Note that every set is approximatively compact with respect to itself and that every compact set is approximatively compact. Further,A
0 and B
0 are nonempty if A is compact and B is approximatively compact with respect to A.Let us define the notion of generalized proximal weak contraction maps as follows. For this goal, we first introduce the following class of the mapping. LetΩ be the set of all function;  ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is nondecreasing with the following property: ϕ
(
t
)
=
0 if and only if t
=
0.Definition 6.
A mappingT
:
A
→
B is said to be a generalized proximal weak contraction on A if there exists functions ψ
,
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) satisfying the following condition:
(4)
d
(
u
,
T
x
)
=
d
(
A
,
B
)
d
(
v
,
T
y
)
=
d
(
A
,
B
)
}
⇒
ψ
(
d
(
u
,
v
)
)
≤
ψ
(
m
(
x
,
y
)
)
-
ϕ
(
m
(
x
,
y
)
)
,
where m
(
x
,
y
)
=
max
⁡
{
d
(
x
,
y
)
,
d
(
x
,
u
)
,
d
(
y
,
v
)
,
(
d
(
x
,
v
)
+
d
(
y
,
u
)
)
/
2
} for all u
,
v
,
x
,
y in A, where ψ
∈
Φ and ϕ
∈
Ω.Remark 7.
Definition6 guarantees that if a mapping T has a best proximity point then it should be unique. Indeed, we can prove our claim easily. Let x
∈
A be a best proximity point of T. Suppose, on the contrary, that there is another element y such that d
(
y
,
T
y
)
=
d
(
A
,
B
). Since T is a generalized proximal weak contraction on A, we have
(5)
ψ
(
d
(
x
,
y
)
)
≤
ψ
(
m
(
x
,
y
)
)
-
ϕ
(
m
(
x
,
y
)
)
,
where m
(
x
,
y
)
=
max
⁡
{
d
(
x
,
y
)
,
d
(
x
,
x
)
,
d
(
y
,
y
)
,
(
d
(
x
,
y
)
+
d
(
y
,
x
)
)
/
2
}
=
d
(
x
,
y
).
From (5), we obtain ψ
(
d
(
x
,
y
)
)
≤
ψ
(
d
(
x
,
y
)
)
-
ϕ
(
d
(
x
,
y
)
), which implies ϕ
(
d
(
x
,
y
)
)
=
0, and by our assumption about ϕ, we get d
(
x
,
y
)
=
0, or equivalently, x
=
y.Definition 8.
A mappingT
:
A
→
B is said to be a generalized proximal weak contraction on B if there exist functions ψ
,
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) satisfying the following condition:
(6)
d
(
u
,
T
x
)
=
d
(
A
,
B
)
d
(
v
,
T
y
)
=
d
(
A
,
B
)
}
⇒
ψ
(
d
(
T
u
,
T
v
)
)
≤
ψ
(
m
(
T
x
,
T
y
)
)
-
ϕ
(
m
(
T
x
,
T
y
)
)
,
where m
(
T
x
,
T
y
)
=
max
⁡
{
d
(
T
x
,
T
y
)
,
d
(
T
x
,
T
u
)
,
d
(
T
y
,
T
v
), (
d
(
T
x
,
T
v
)
+
d
(
T
y
,
T
u
)
)
/
2
} for all u
,
v
,
x
,
y in A, where ψ
∈
Φ and ϕ
∈
Ω.For self-mappings, it is clear that every generalized proximal weak contraction onA is a generalized proximal weak contraction on B. An operator T is said to be a generalized proximal weak contraction if it is both generalized proximal weak contraction on A and generalized proximal weak contraction on B.


## 2. Main Results

We start this section with our main result.Theorem 9.
Let(
A
,
B
) be a pair of two nonempty closed subsets of a complete metric space X such that A
0 is nonempty. Let T
:
A
→
B be a map satisfying the following conditions:(i)
Tis a generalized proximal weak contraction,
(ii)
T
(
A
0
)
⊆
B
0.Then, there exists a unique x
∈
A such that d
(
x
,
T
x
)
=
d
(
A
,
B
). Further, for any fixed element x
0
∈
A
0, the sequence {
x
n
}, defined by d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
), converges to the element x.Proof.
We prove the theorem in several steps.
Step
1. Let x
0
∈
A
0. Since T
x
0
∈
T
(
A
0
)
⊆
B
0, there exists x
1
∈
A
0 such that d
(
x
1
,
T
x
0
)
=
d
(
A
,
B
). Due to the fact that T
x
1
∈
T
(
A
0
)
⊆
B
0, there exists x
2
∈
A
0 such that d
(
x
2
,
T
x
1
)
=
d
(
A
,
B
). Recursively, we find a sequence {
x
n
} in A
0 such that
(7)
d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
)
,
∀
n
∈
N
.
If there exists n
0 such that x
n
0
=
x
n
0
+
1, then d
(
x
n
0
+
1
,
T
x
n
0
)
=
d
(
x
n
0
,
T
x
n
0
)
=
d
(
A
,
B
); that is, x
n
0 is a best proximity point of T. Thus, the proof is finished. Hence, we suppose that x
n
≠
x
n
+
1 for all n. Since T is a generalized proximal weak contraction on A, it follows that
(8)
ψ
(
d
(
x
n
,
x
n
+
1
)
)
≤
ψ
(
m
(
x
n
-
1
,
x
n
)
)
-
ϕ
(
m
(
x
n
-
1
,
x
n
)
)
≤
ψ
(
m
(
x
n
-
1
,
x
n
)
)
.
Using the monotone property of the ψ-function, we get d
(
x
n
,
x
n
+
1
)
≤
m
(
x
n
-
1
,
x
n
).
Now from the triangle inequality ford, we have
(9)
m
(
x
n
-
1
,
x
n
)

=
max
⁡
{
d
(
x
n
-
1
,
x
n
+
1
)
+
d
(
x
n
,
x
n
)
2
d
(
x
n
-
1
,
x
n
)
,
d
(
x
n
-
1
,
x
n
)
,
d
(
x
n
,
x
n
+
1
)
,
d
(
x
n
-
1
,
x
n
+
1
)
+
d
(
x
n
,
x
n
)
2
}
=
max
⁡
⁡
{
d
(
x
n
-
1
,
x
n
+
1
)
2
d
(
x
n
-
1
,
x
n
)
,
d
(
x
n
,
x
n
+
1
)
,
d
(
x
n
-
1
,
x
n
+
1
)
2
}
≤
max
⁡
⁡
{
d
(
x
n
-
1
,
x
n
)
+
d
(
x
n
,
x
n
+
1
)
2
d
(
x
n
-
1
,
x
n
)
,
d
(
x
n
,
x
n
+
1
)
,
d
(
x
n
-
1
,
x
n
)
+
d
(
x
n
,
x
n
+
1
)
2
}
=
max
⁡
{
d
(
x
n
-
1
,
x
n
)
,
d
(
x
n
,
x
n
+
1
)
}
.
If d
(
x
n
,
x
n
+
1
)
≥
d
(
x
n
-
1
,
x
n
), then m
(
x
n
-
1
,
x
n
)
=
d
(
x
n
,
x
n
+
1
)
>
0. From (8), we obtain ψ
(
d
(
x
n
,
x
n
+
1
)
)
≤
ψ
(
d
(
x
n
,
x
n
+
1
)
)
-
ϕ
(
d
(
x
n
,
x
n
+
1
)
), which is a contradiction. So, we have
(10)
d
(
x
n
,
x
n
+
1
)
≤
m
(
x
n
-
1
,
x
n
)
≤
d
(
x
n
-
1
,
x
n
)
.
Hence, the sequence {
d
(
x
n
,
x
n
+
1
)
} is monotone nonincreasing and bounded. Thus, there exists r
≥
0 such that
(11)
lim
⁡
n
→
∞
d
(
x
n
,
x
n
+
1
)
=
lim
⁡
n
→
∞
m
(
x
n
-
1
,
x
n
)
=
r
≥
0
.
Suppose that lim
⁡
n
→
∞
d
(
x
n
,
x
n
+
1
)
=
lim
⁡
n
→
∞
m
(
x
n
-
1
,
x
n
)
=
r
>
0. Then the inequality
(12)
ψ
(
d
(
x
n
,
x
n
+
1
)
)
≤
ψ
(
m
(
x
n
-
1
,
x
n
)
)
-
ϕ
(
m
(
x
n
-
1
,
x
n
)
)
≤
ψ
(
m
(
x
n
-
1
,
x
n
)
)
implies that
(13)
lim
⁡
n
→
∞
ϕ
(
m
(
x
n
-
1
,
x
n
)
)
=
0
.
But as 0
<
r
≤
d
(
x
n
,
x
n
+
1
)
≤
m
(
x
n
-
1
,
x
n
) and ϕ is nondecreasing function,
(14)
0
<
ϕ
(
r
)
≤
ϕ
(
m
(
x
n
-
1
,
x
n
)
)
,
and this gives us lim
⁡
n
→
∞
ϕ
(
m
(
x
n
-
1
,
x
n
)
)
≥
ϕ
(
r
)
>
0 which contradicts to (13). Hence,
(15)
lim
⁡
n
→
∞
m
(
x
n
-
1
,
x
n
)
=
0
=
lim
⁡
n
→
∞
d
(
x
n
,
x
n
+
1
)
.
Step
2. We will show that {
x
n
} is a Cauchy sequence. Suppose, on the contrary, that {
x
n
} is not a Cauchy sequence. Thus, there exists ϵ
>
0 for which we can find subsequences {
x
m
(
k
)
} and {
x
n
(
k
)
} of {
x
n
} such that n
(
k
) is the smallest index for which n
(
k
)
>
m
(
k
)
>
k
,
d
(
x
m
(
k
)
,
x
n
(
k
)
)
≥
ϵ.
This means that(16)
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
<
ϵ
,
ϵ
≤
d
(
x
m
(
k
)
,
x
n
(
k
)
)
ϵ
≤
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
+
d
(
x
n
(
k
)
-
1
,
x
n
(
k
)
)
ϵ
<
ϵ
+
d
(
x
n
(
k
)
-
1
,
x
n
(
k
)
)
.
Letting k
→
∞ and by using (15), we conclude that
(17)
lim
⁡
k
→
∞
d
(
x
m
(
k
)
,
x
n
(
k
)
)
=
ϵ
.
Again,
(18)
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
≤
d
(
x
m
(
k
)
,
x
n
(
k
)
)
+
d
(
x
n
(
k
)
,
x
n
(
k
)
-
1
)
,
d
(
x
m
(
k
)
,
x
n
(
k
)
)
≤
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
+
d
(
x
n
(
k
)
,
x
n
(
k
)
-
1
)
.
Therefore,
(19)
|
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
-
d
(
x
m
(
k
)
,
x
n
(
k
)
)
|
≤
d
(
x
n
(
k
)
,
x
n
(
k
)
-
1
)
.
Letting k
→
∞ and by using (17) together with (15), it follows that
(20)
lim
⁡
k
→
∞
d
(
x
m
(
k
)
,
x
n
(
k
)
-
1
)
=
ϵ
.
Similarly, we derive that
(21)
lim
⁡
k
→
∞
d
(
x
m
(
k
)
-
1
,
x
n
(
k
)
)
=
lim
⁡
k
→
∞
d
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
=
lim
⁡
k
→
∞
d
(
x
m
(
k
)
+
1
,
x
n
(
k
)
)
=
lim
⁡
k
→
∞
d
(
x
m
(
k
)
,
x
n
(
k
)
+
1
)
=
ϵ
.
Then, we have
(22)
lim
⁡
k
→
∞
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
=
ϵ
.
Using the fact that T is generalized proximal weak contraction on A for d
(
x
m
(
k
)
,
T
x
m
(
k
)
-
1
)
=
d
(
A
,
B
) and d
(
x
n
(
k
)
,
T
x
n
(
k
)
-
1
)
=
d
(
A
,
B
), we obtain
(23)
0
<
ψ
(
ϵ
)
≤
ψ
(
d
(
x
m
(
k
)
,
x
n
(
k
)
)
)
≤
ψ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
)
-
ϕ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
)
≤
ψ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
)
.
By using (22) and the continuity of ψ in the above inequality, we find that
(24)
lim
⁡
k
→
∞
ϕ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
)
=
0
.
But from lim
⁡
k
→
∞
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
=
ϵ, we can find k
0
∈
ℕ such that for any k
≥
k
0
(25)
ϵ
2
≤
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
,
and consequently,
(26)
0
<
ϕ
(
ϵ
2
)
≤
ϕ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
)
for
k
≥
k
0
.
Therefore, 0
<
ϕ
(
ϵ
/
2
)
≤
ϕ
(
m
(
x
m
(
k
)
-
1
,
x
n
(
k
)
-
1
)
) and this contradicts to (24). Thus, {
x
n
} is a Cauchy sequence in A and hence converges to some element x in A. Analogously, by using the fact that T is generalized proximal weak contraction on B, we conclude that {
T
x
n
} is a Cauchy sequence in B. Hence, {
T
x
n
} converges to some element y in B.
Step
3. Let us now prove that x is best proximity point for T.
Recall thatx
n
→
x in A and T
x
n
→
y in B. Therefore, from (7), we get d
(
x
,
y
)
=
d
(
A
,
B
) and hence x is a member of A
0. Since T
(
A
0
)
⊆
B
0, we get T
x
∈
B
0; hence there exists z
∈
A such that
(27)
d
(
z
,
T
x
)
=
d
(
A
,
B
)
.
Since T is a generalized proximal weak contraction on A, we obtain
(28)
ψ
(
d
(
x
n
+
1
,
z
)
)
≤
ψ
(
m
(
x
n
,
x
)
)
-
ϕ
(
m
(
x
n
,
x
)
)
≤
ψ
(
m
(
x
n
,
x
)
)
,
where m
(
x
n
,
x
)
=
max
⁡
{
d
(
x
n
,
x
)
,
d
(
x
n
,
x
n
+
1
)
,
d
(
x
,
z
)
,
(
d
(
x
n
,
z
)
+
d
(
x
,
x
n
+
1
)
)
/
2
}.
By using the fact thatx
n
→
x, we get
(29)
lim
⁡
n
→
∞
m
(
x
n
,
x
)
=
d
(
x
,
z
)
.
Regarding (29) and continuity of ψ in (28), we can obtain
(30)
lim
⁡
n
→
∞
ϕ
(
m
(
x
n
,
x
)
)
=
0
.
But from lim
⁡
n
→
∞
m
(
x
n
,
x
)
=
d
(
x
,
z
) we find n
0
∈
ℕ such that for any n
≥
n
0
(31)
d
(
x
,
z
)
2
≤
m
(
x
n
,
x
)
and consequently, since ϕ is nondecreasing, we get
(32)
ϕ
(
d
(
x
,
z
)
2
)
≤
ϕ
(
m
(
x
n
,
x
)
)
∀
n
≥
n
0
.
By using (30) in the inequality above, we get ϕ
(
d
(
x
,
z
)
/
2
)
=
0 and from the property of ϕ, we obtain d
(
x
,
z
)
/
2
=
0 or equivalently, x
=
z. Hence, from (27), we have d
(
x
,
T
x
)
=
d
(
A
,
B
).Example 10.
Consider the complete spaceX
=
ℝ
2 with usual metric.Suppose thatA
:
=
{
(
x
,
0
)
:
x
∈
[
-
2
,
-
1
]
} and B
:
=
{
(
1
,
x
)
:
x
∈
[
3
/
2,2
]
}. Then A and B are nonempty closed subsets of X and A
0
=
{
(
-
1,0
)
} and B
0
=
{
(
1,3
/
2
)
}. Note that d
(
A
,
B
)
=
5
/
2. Let T
:
A
→
B be defined as
(33)
T
(
x
,
0
)
=
(
1,1
-
x
2
)
,
∀
(
x
,
0
)
∈
A
.
Suppose that (
u
1
,
0
), (
u
2
,
0
), (
x
1
,
0
), (
x
2
,
0
) are elements in A such that
(34)
d
(
(
u
1
,
0
)
,
T
(
x
1
,
0
)
)
=
d
(
(
u
2
,
0
)
,
T
(
x
2
,
0
)
)
=
d
(
A
,
B
)
.
Then, (
u
1
,
0
) and (
u
2
,
0
) become the members of A
0. Consequently, we have
(35)
d
(
(
u
1
,
0
)
,
(
u
2
,
0
)
)
=
d
(
T
(
u
1
,
0
)
,
T
(
u
2
,
0
)
)
=
0
.
By assuming that ψ, ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) such that ψ
(
t
)
=
2
t and ϕ
(
t
)
=
t
/
2, we get
(36)
ψ
(
d
(
(
u
1
,
0
)
,
(
u
2
,
0
)
)
)
=
ψ
(
d
(
T
(
u
1
,
0
)
,
T
(
u
2
,
0
)
)
)
=
0
,
ψ
(
m
(
(
x
1
,
0
)
,
(
x
2
,
0
)
)
)
-
ϕ
(
m
(
(
x
1
,
0
)
,
(
x
2
,
0
)
)
)
=
3
2
m
(
(
x
1
,
0
)
,
(
x
2
,
0
)
)
,
ψ
(
m
(
T
(
x
1
,
0
)
,
T
(
x
2
,
0
)
)
)
-
ϕ
(
m
(
T
(
x
1
,
0
)
,
T
(
x
2
,
0
)
)
)
=
3
2
m
(
T
(
x
1
,
0
)
,
T
(
x
2
,
0
)
)
.Therefore,T is both generalized proximal weak contraction on A and generalized proximal weak contraction on B. Hence T is generalized proximal weak contraction such that T
(
A
0
)
⊆
B
0. So, all the hypotheses of Theorem 9 are satisfied. Further, it is easy to see that (
-
1,0
) is the unique element satisfying the conclusion of Theorem 9.It is easy to see that a self-mapping that is a generalized proximal weak contraction reduces to a generalized(
ψ
,
ϕ
)-weak contraction. Hence the above Theorem 9 gives rise to the following fixed point theorem, due to Dorić [9], which in turn extends the famous contraction principle.Corollary 11.
Let(
X
,
d
) be a nonempty complete metric space and let T
:
X
→
X be a generalized proximal weak contraction map. Then T has a unique fixed point.Theorem 12.
Let(
A
,
B
) be a pair of two nonempty closed subsets of a complete metric space X such that B is approximatively compact with respect to A and A
0 is nonempty. Let T
:
A
→
B be a map satisfying the following conditions:(i)
T is a generalized proximal weak contraction on A,
(ii)
T
(
A
0
)
⊆
B
0.Then, there exists a unique x
∈
A such that d
(
x
,
T
x
)
=
d
(
A
,
B
). Further, for any fixed element x
0
∈
A
0, the sequence {
x
n
}, defined by d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
), converges to the element x.Proof.
Following the proof of Theorem9, there exists a sequence {
x
n
} in A satisfying the following conditions:
(37)
d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
)
,
∀
n
∈
ℕ
and x
n converges to x in A. Let us now prove that x is best proximity point for T.
Note that from (37), we have
(38)
d
(
x
,
B
)
≤
d
(
x
,
T
x
n
)
≤
d
(
x
,
x
n
+
1
)
+
d
(
x
n
+
1
,
T
x
n
)
≤
d
(
x
,
x
n
+
1
)
+
d
(
A
,
B
)
≤
d
(
x
,
x
n
+
1
)
+
d
(
x
,
B
)
.
Since x
n
→
x, we get d
(
x
,
T
x
n
)
→
d
(
x
,
B
). Since B is approximatively compact with respect to the set A, it follows that the sequence {
T
x
n
} has a subsequence converging to some y in B. Now arguing like Step 3 and Step 4 of Theorem 9, we get the required result.In what follows we prove that Theorem12 is still valid for B not necessarily approximatively compact with respect to A, assuming that A
0 is closed.Theorem 13.
Assume that “A
0 is closed” instead of “the statement B is  approximatively compact with respect to A” in Theorem 12.Proof.
Following the proof of Theorem9, there exists a sequence {
x
n
} in A satisfying the following condition:
(39)
d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
)
,
and x
n converges to x in A. Note that the sequence {
x
n
} in A
0 and A
0 is closed. Therefore, x
∈
A
0. Since T
(
A
0
)
⊆
B
0, we get T
x
∈
B
0. Since T
x
∈
B
0, there exists z
∈
A such that d
(
z
,
T
x
)
=
d
(
A
,
B
). Since T is a generalized proximal weak contraction on A, we have
(40)
ψ
(
d
(
x
n
+
1
,
z
)
)
≤
ψ
(
m
(
x
,
x
n
)
)
-
ϕ
(
m
(
x
,
x
n
)
)
,
where m
(
x
,
x
n
)
=
max
⁡
{
d
(
x
n
,
x
)
,
d
(
x
n
,
x
n
+
1
)
,
d
(
x
,
z
)
,
(
d
(
x
n
,
z
)
+
d
(
x
,
x
n
+
1
)
)
/
2
}.
Now arguing like Step 3 and Step 4 of Theorem9, we get the required result.Theorem 14.
Assume that “T is continuous” instead of the statement “B is approximatively compact with respect to A” in the Theorem 12.Proof.
Following the proof of Theorem9, there exists a sequence {
x
n
} in A satisfying the following condition:
(41)
d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
)
,
and x
n converges to x in A. Since T are continuous, we have
(42)
d
(
x
,
T
x
)
=
lim
⁡
n
→
∞
d
(
x
n
+
1
,
T
x
n
)
=
d
(
A
,
B
)
.
Uniqueness follows the same as in Theorem 9.



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*Source: 150941-2014-01-16.xml*