# Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

Authors: M. De La Sen; Asier Ibeas
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper presents some results concerning the properties of distances and existence and uniqueness of best proximity points ofp-cyclic proximal, weak proximal contractions, and some of their generalizations for the non-self-mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i
(
p
≥
2
), where A
i and B
i, ∀
i
∈
p
-
=
{
1,2
,
…
,
p
}, are nonempty subsets of X which satisfy T
A
i
⊆
B
i
,
∀
i
∈
p
-, such that (
X
,
d
) is a metric space. The boundedness and the convergence of the sequences of distances in the domains and in their respective image sets of the cyclic proximal and weak cyclic proximal non-self-mapping, and of some of their generalizations are investigated. The existence and uniqueness of the best proximity points and the properties of convergence of the iterates to such points are also addressed.


---

## Body

## 1. Introduction

The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive, weak-contractive mappings, and cyclic mappings and also in related problems of proximal contractions, weak proximal contractions, and approximation results and methods [1–15]. The application of the theory of fixed points in stability issues of dynamic systems, [16–21] has been proved to be a very useful tool. See, for instance, [22–26] and references therein. Some best approximation problems in semiconvex and locally convex structures and Hyers-Ulam type stability in multivalued functions and in additive-quadratic functional equations are investigated in [27–30] and some of the references therein. Recent trends concerning best proximity points and related problems are dealt with in [31–35] and some references therein. In particular, the problem of best proximity points of two mappings in a cyclic disposal is investigated in [31] under a nonlinear contractive condition. In [32], several results are obtained for proximal and weak proximal contractions of several types as well as for generalized proximal nonexpansive mappings. A modified Suzuki α
-
ψ proximal contraction is proposed and discussed in [33] and “ad hoc” best proximity and fixed point results are obtained. Generalizations of proximal contractions of first and second kinds are given in [34, 35] for non-self-mappings and related optimal approximate solution theorems are obtained.This paper is devoted to formulating and proving some results being concerned with the boundedness and convergence properties of distances and the convergence of the built iterated sequences to unique existing best proximity points ofp-cyclic proximal and weak proximal contractions of the form T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i (p
≥
2) where A
i and B
i, for all i
∈
p
-
=
{
1,2
,
…
,
p
}, are nonempty subsets of X which satisfy T
(
A
i
)
⊆
B
i, for all i
∈
p
-, with (
X
,
d
) being a metric space. In the most general case, all the A
i and B
i pairs of subsets, for all i
∈
p
-, are assumed to be pairwise disjoint. The results are also extended to a class of generalized p-cyclic proximal and weak proximal contractions in the sense that the contractiveness constraints are referred to finite sets of consecutive iterations rather than to each iteration. The boundedness and convergence of the sequences of distances in the domains and image sets of the cyclic proximal and weak cyclic proximal non-self-mappings are investigated. The existence and uniqueness of the best proximity points and their allocation as limit points, or limit cycles of best proximity points, are also addressed. These last properties are achieved if the metric space is complete under approximative compactness’ assumptions of the image subsets of the cyclic mapping with respect to the domain subsets.


## 2.p-Cyclic Proximal Contractions, Extensions, Boundedness, and Convergence of Distances

Consider the metric space(
X
,
d
) and subsets A
i and B
i of X for i
∈
p
-, where p
-
=
{
1,2
,
…
,
p
} with p
≥
2. Consider also a non-self-mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i, satisfying T
(
A
i
)
⊆
B
i,  for all i
∈
p
-. Assume that D
i
=
d
(
A
i
+
1
,
B
i
),  D
A
=
d
(
A
i
,
A
i
+
1
), and D
B
=
d
(
B
i
,
B
i
+
1
), for all i
∈
p
- by assuming also that A
n
p
+
i
=
A
i and B
n
p
+
i
=
B
i, for all i
∈
p
¯, for all n
∈
Z
0
+. If the pair (
a
i
,
a
i
+
1
)
∈
A
i
×
A
i
+
1 satisfies d
(
a
i
+
1
,
T
a
i
)
=
D
i for any i
∈
p
-, then a
i
+
1
∈
A
i
+
1 and T
a
i
∈
B
i are best proximity points in A
i
+
1 and B
i with respect to T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i.In the following, the fact that the best proximity points are best proximity points with respect to the mapping is not mentioned explicitly.A
0
i
⊆
A
i is the set of best proximity points of A
i and B
0
i
⊆
B
i is the set of best proximity points of B
i. Through the paper, it is assumed that A
0
i
≠
⌀ and B
0
i
≠
⌀, for all i
∈
p
-. An important remark is that the above statement can be considered for the particular case that B
i
≡
A
i
+
1 which is well known in the context of p-cyclic self-mappings T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i with T
(
A
i
)
⊆
B
i for all i
∈
p
-. However, the proposed statement is more general in the sense of the following illustrative example.Example 1.
Consider a metric space(
X
,
d
) and T
:
A
1
∪
A
2
→
B
1
∪
B
2 such that A
i
,
B
i
⊂
X are nonempty with T
(
A
0
i
)
⊆
B
0
i, D
i
=
d
(
A
i
+
1
,
B
i
)
=
d
(
A
0
,
i
+
1
,
B
0
i
), D
A
=
d
(
A
i
,
A
i
+
1
)
=
d
(
A
0
i
,
A
0
,
i
+
1
), and D
B
=
d
(
B
i
,
B
i
+
1
)
=
d
(
B
0
i
,
B
0
,
i
+
1
) for i
=
1,2. Assume also that A
01
=
{
x
1
,
x
3
}
⊂
A
1, B
01
=
{
T
x
1
,
T
x
3
}
⊂
B
1, A
02
=
{
x
2
}
⊂
A
2, and B
02
=
{
T
x
2
}
⊂
B
2. Then, we can formulate the following simple 2-cyclic proximal-type problem. Fix x
1
∈
A
01 as a best proximity point of A
1 and then compute x
2
∈
A
02 and x
3
∈
A
01, best proximity points of A
2 and A
1, such thatd
(
x
2
,
T
x
1
)
=
d
(
A
2
,
B
1
)
=
d
(
A
02
,
B
01
)
=
D
1 (2-cyclic proximal constraint, first step);
d
(
x
3
,
T
x
2
)
=
d
(
A
1
,
B
2
)
=
d
(
A
01
,
B
02
)
=
D
2 (2-cyclic proximal constraint, second step);
d
(
A
1
,
A
2
)
=
d
(
A
01
,
A
02
)
=
d
(
x
1
,
x
2
)
=
d
(
x
2
,
x
3
)
=
D
A (2-cyclic best proximity constraints);
d
(
T
x
2
,
T
x
1
)
=
d
(
T
x
3
,
T
x
2
)
=
D
B
=
d
(
B
1
,
B
2
)
=
d
(
B
01
,
B
02
) (2-cyclic associate best proximity constraints for the images).
Note that there are four potentially distinct constraints related toD
1, D
2, D
A, and D
B which can be distinct so that the problem is more general than the simple use of D
=
d
(
A
i
,
A
i
+
1
) for i
=
1,2 for the 2-cyclic self-mapping T
:
A
1
∪
A
2
→
A
1
∪
A
2. A variant proximal-type problem arises if A
1
=
B
2 and A
2
=
B
1 and the best proximity points are taken as follows:
x
1
∈
A
01, T
x
1
∈
A
02, x
2
∈
A
01, T
x
2
∈
A
02, x
3
∈
A
01, and then d
(
x
2
,
T
x
1
)
=
d
(
x
3
,
T
x
2
)
=
D
A
=
d
(
A
1
,
A
2
).The following definitions will be then used through the paper.Definition 2.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a p-cyclic proximal contraction with respect to its domain (CPD
p) if there are real constants α
i
∈
[
0,1
), for all i
∈
p
-, such that any two sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j, for all i
∈
p
-, satisfy the constraints
(1)
d
(
x
n
p
+
i
+
1
,
x
-
n
p
+
i
)
≤
α
i
d
(
x
n
p
+
i
,
x
-
n
p
+
i
-
1
)
+
(
1
-
α
i
)
D
A
,
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
,
(2)
d
(
x
n
p
+
i
+
1
,
x
-
n
p
+
i
+
1
)
≤
α
i
d
(
x
n
p
+
i
,
x
-
n
p
+
i
)
,
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
,
provided that x
0
,
x
-
0
∈
A
j, for any given j
∈
p
- with A
i
+
j
=
A
i
+
j
-
p if i
>
p
-
j and that d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
d
(
A
i
+
1
,
B
i
)
=
D
i, for all i
∈
p
-.Definition 3.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a weak p-cyclic proximal contraction with respect to its domain (WCPD
p) if there are p real constants α
i
≥
0, for all i
∈
p
-, subject to α
=
∏
i
=
1
p
[
α
i
]
∈
[
0,1
), such that any two sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j, for all i
∈
p
-, satisfy the constraints (1) and (2) provided that x
0
,
x
-
0
∈
A
j for any given j
∈
p
- and that d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all i
∈
p
-.Definition 4.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a generalized p-cyclic proximal contraction with respect to its domain (GCPD
p) if there are p bounded real functions α
i
:
A
i
→
R
0
+, for all i
∈
p
-, such that any sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j, for all i
∈
p
-, satisfy the constraints (1) and (2) with the replacements α
i
→
sup
⁡
x
∈
A
i
α
i
(
x
), for all i
∈
p
-, provided that x
0
,
x
-
0
∈
A
j for any given j
∈
p
- and that d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all i
∈
p
-.Definition 5.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a generalized weak p-cyclic proximal contraction with respect to its domain (GWCPD
p) if there are p bounded real functions α
i
:
A
i
→
R
0
+, for all i
∈
p
-, and a strictly increasing sequence of integers {
n
k
}, subject to n
0
≤
N
-
0
<
+
∞, limsup
⁡
k
→
∞
(
n
k
+
1
-
n
k
)
≤
N
-
<
+
∞, and α
-
=
sup
⁡
k
∈
Z
0
+
α
(
n
k
,
n
k
+
1
)
∈
[
0,1
), where
(3)
α
(
n
k
,
n
k
+
1
)
=
∏
j
=
n
k
n
k
+
1
-
1
∏
i
=
1
p
[
sup
⁡
x
j
+
i
∈
A
i
α
j
p
+
i
(
x
j
p
+
i
)
]
,
∀
k
∈
Z
0
+
,
such that any two sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j, for all i
∈
p
-, provided that x
0
,
x
-
0
∈
A
j for any given j
∈
p
-, satisfy the constraints
(4)
d
(
x
n
k
+
1
p
,
x
-
n
k
+
1
p
-
1
)
≤
α
(
n
k
,
n
k
+
1
)
d
(
x
n
k
p
+
1
,
x
-
n
k
p
)
+
(
1
-
α
(
n
k
,
n
k
+
1
)
)
D
A
,
∀
k
∈
Z
0
+
,
(5)
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
-
1
)
≤
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
A
i
α
i
(
x
)
]
)
d
(
x
n
k
+
1
p
,
x
-
n
k
+
1
p
-
1
)
+
∑
j
=
1
i
-
1
(
∏
k
=
j
+
1
i
-
1
[
sup
⁡
x
∈
A
k
α
k
(
x
)
]
)
(
1
-
sup
⁡
x
∈
A
j
α
j
(
x
)
)
D
A
,
ii
∀
k
∈
Z
0
+
,
(6)
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
)
≤
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
A
i
α
i
(
x
)
]
)
α
(
n
k
,
n
k
+
1
)
d
(
x
n
k
p
,
x
-
n
k
p
)
,
ii
∀
k
∈
Z
0
+
,
and the constraints (1) and (2) provided that x
0
,
x
-
0
∈
A
j for any given j
∈
p
- and that d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all i
∈
p
-.The following assertions are obvious without proof from Definitions2–5.Assertions 1.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is CPD
p, then it is WCPD
p.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is WCPD
p, then it is GCPD
p.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is GCPD
p, then it is GWCPD
p.Note that the converse implications of those in Assertions1 are not true in general. The relevant distances satisfy the following convergence and boundedness result.Lemma 6.
Consider a metric space(
X
,
d
) with subsets A
i
,
B
i
⊂
X and a p-cyclic mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i which is G
W
C
P
D
p, subject to D
i
=
d
(
A
i
+
1
,
B
i
), D
A
=
d
(
A
i
,
A
i
+
1
), and D
B
=
d
(
B
i
,
B
i
+
1
), with T
(
A
0
i
)
⊆
B
0
i, for all i
∈
p
- such that A
0
i
⊆
A
i and B
0
i
⊆
B
i are nonempty, for all i
∈
p
-. Consider also any sequences {
x
n
p
+
j
}
,
{
x
-
n
p
+
j
}
⊆
⋃
i
∈
p
-
A
i which satisfy d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all i
∈
p
-. Then, the following properties hold.
(i) The sequences of distances{
d
(
x
n
,
x
-
n
+
1
)
}
→
D
A, {
d
(
x
n
,
x
-
n
)
}
→
0 and they are bounded for any given initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i, for any given j
∈
p
-.
If, furthermore,T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
T
(
A
0
i
), for all i
∈
p
-, then {
d
(
T
x
n
,
T
x
-
n
)
}
→
0 and {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B and both sequences of distances are bounded for any given initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i, for any given j
∈
p
-.
If the sets of best proximity pointsA
0
i and B
0
i, for all i
∈
p
-, are bounded, then the sequences {
d
(
x
n
,
x
-
n
+
1
)
}, {
d
(
x
n
,
x
-
n
)
}, {
d
(
T
x
n
,
T
x
-
n
)
}, and {
d
(
T
x
n
,
T
x
-
n
+
1
)
} are uniformly bounded for any initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i  for some j
∈
p
-.
(ii) The sequences{
x
n
p
+
i
}
⊆
cl
A
i
+
j, for all i
∈
p
- (note that A
i
+
j
=
A
i
+
j
-
p and B
i
+
j
=
B
i
+
j
-
p for i
>
p
-
j) are Cauchy sequences for any initial points any given initial point x
0
∈
⋃
i
∈
p
-
A
i for any arbitrary given j
∈
p
-. The corresponding image sequences {
T
x
n
p
+
i
}
⊆
cl
B
i
+
j, for all i
∈
p
-, are also Cauchy sequences if T
(
A
0
i
)
⊆
B
0
i, for all i
∈
p
-, and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
T
(
A
0
i
), for all i
∈
p
-.Proof.
Note that, for anyi
∈
p
-, ⌀
≠
A
0
i
⊆
A
i implies A
i
≠
⌀ and ⌀
≠
B
0
i
⊆
B
i implies B
i
≠
⌀. Take sequences with initial points x
0
,
x
-
0
⊂
⋃
i
∈
p
-
A
i such that x
0
∈
A
j and x
-
0
∈
A
i for some i
,
j
∈
p
-. The problem of boundedness and convergence of distances can be described equivalently from initial points x
0
,
x
-
0
′
∈
A
j (i.e., both initial conditions at the same set), with x
-
0
′
=
T
l
x
0 and denoting x
-
0
′
→
x
-
0 for some l
∈
p
-
1
¯
∪
{
0
} (in particular, l
=
0 if i
=
j for the set of both initial points) since l
≤
p
-
1. One has from (1)-(2) and (3)–(6) that, for any sequences {
x
n
p
+
j
+
i
}
⊆
A
j
+
i and {
x
-
n
p
+
j
+
i
}
⊆
A
j
+
i fulfilling {
x
0
}
∈
A
j
,
{
x
-
0
}
∈
A
j and d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all j
,
i
∈
p
-, such that A
j
+
i
=
A
j
+
i
-
p if i
>
p
-
j and
(7)
D
A
≤
d
(
x
n
p
+
j
+
1
,
x
-
n
p
+
j
)
≤
(
sup
⁡
x
∈
A
j
α
j
(
x
)
)
d
(
x
n
p
+
j
,
x
-
n
p
+
j
-
1
)
+
(
1
-
sup
⁡
x
∈
A
j
α
j
(
x
)
)
D
A
,
∀
j
∈
p
-
,
∀
n
∈
Z
0
+
,
D
A
≤
d
(
x
n
p
+
j
+
2
,
x
-
n
p
+
j
+
1
)
≤
(
sup
⁡
x
∈
A
j
+
1
α
j
+
1
(
x
)
)
×
(
(
sup
⁡
x
∈
A
j
α
j
(
x
)
)
d
(
x
n
p
+
j
,
x
-
n
p
+
j
-
1
)
+
(
1
-
sup
⁡
x
∈
A
j
α
j
(
x
)
)
D
A
)
+
(
1
-
sup
⁡
x
∈
A
j
α
j
+
1
(
x
)
)
D
A
=
D
A
+
(
sup
⁡
x
∈
A
j
+
1
α
j
+
1
(
x
)
)
(
sup
⁡
x
∈
A
j
α
j
(
x
)
)
×
(
d
(
x
n
p
+
j
,
x
-
n
p
+
j
-
1
)
-
D
A
)
,
D
A
≤
d
(
x
n
p
+
j
+
3
,
x
-
n
p
+
j
+
2
)
≤
D
A
+
(
sup
⁡
x
∈
A
j
+
2
α
j
+
1
(
x
)
)
×
(
sup
⁡
x
∈
A
j
+
1
α
j
+
1
(
x
)
)
(
sup
⁡
x
∈
A
j
α
j
(
x
)
)
×
(
d
(
x
n
p
+
j
,
x
-
n
p
+
j
-
1
)
-
D
A
)
,
(8a)
D
A
≤
d
(
x
n
k
+
1
p
,
x
-
n
k
+
1
p
-
1
)
≤
α
(
n
k
,
n
k
+
1
)
d
(
x
n
k
p
+
1
,
x
-
n
k
p
)
+
(
1
-
α
(
n
k
,
n
k
+
1
)
)
D
A
≤
D
A
+
α
(
n
k
,
n
k
+
1
)
α
(
n
k
-
1
,
n
k
-
2
)
×
(
d
(
x
n
k
-
1
p
+
1
,
x
-
n
k
-
1
p
)
-
D
A
)
≤
D
A
+
(
∏
j
=
0
k
[
α
(
n
j
,
n
j
+
1
)
]
)
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
≤
D
A
+
α
-
k
+
1
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
(8b)
<
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
<
+
∞
,
∀
k
∈
Z
0
+
,
(9a)
D
A
≤
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
-
1
)
≤
D
A
+
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
⋃
j
∈
p
-
A
j
α
j
(
x
)
]
)
×
(
∏
j
=
0
k
[
α
(
n
j
,
n
j
+
1
)
]
)
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
≤
D
A
+
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
⋃
j
∈
p
-
A
j
α
j
(
x
)
]
)
×
α
-
k
+
1
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
≤
D
A
+
α
-
k
+
1
(
1
+
N
-
+
N
-
′
)
α
^
M
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
(9b)
<
D
A
+
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
⋃
j
∈
p
-
A
j
α
j
(
x
)
]
)
×
(
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
-
D
A
)
≤
max
⁡
(
1
,
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
⋃
j
∈
p
-
A
j
α
j
(
x
)
]
)
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
≤
(
1
+
N
-
+
N
-
′
)
α
^
M
d
(
x
n
0
p
+
1
,
x
-
n
0
p
)
,
∀
k
∈
Z
0
+
,for all i
∈
(
1
,
max
⁡
(
n
k
+
1
-
n
k
)
)
∩
Z
0
+ since α
-
∈
[
0,1
), limsup
⁡
k
→
∞
(
n
k
+
1
-
n
k
)
≤
N
-
<
+
∞ and α
i
∈
B
(
A
i
;
R
0
+
) with max
⁡
i
∈
p
-
(
sup
⁡
x
∈
A
i
α
i
(
x
)
)
≤
α
^
M
<
+
∞.
Thus,∃
lim
⁡
k
→
∞
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
-
1
)
=
lim
⁡
k
→
∞
d
(
x
n
k
p
+
i
,
x
-
n
k
+
1
p
+
i
-
1
)
=
lim
⁡
k
→
∞
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
p
+
i
-
1
)
=
D
A, for all i
∈
(
1
,
max
⁡
(
n
k
+
1
-
n
k
)
)
∩
Z
0
+ from (9a) and the distance subsequence {
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
-
1
)
}, for all i
∈
(
1
,
max
⁡
(
n
k
+
1
-
n
k
)
)
∩
Z
0
+ is bounded from (9b) for any given initial points x
0
,
x
-
0
∈
⋃
i
∈
p
-
A
i. Also, one gets from (6), subject to (3), that
(10)
∃
lim
⁡
k
→
∞
d
(
x
n
k
+
1
p
+
i
,
x
-
n
k
+
1
p
+
i
)
=
lim
⁡
k
→
∞
d
(
x
n
k
p
+
i
,
x
-
n
k
+
1
p
+
i
)
=
lim
⁡
k
→
∞
d
(
x
-
n
k
p
+
i
,
x
n
k
+
1
p
+
i
)
=
0
,
for all i
∈
(
1
,
max
⁡
(
n
k
+
1
-
n
k
)
)
∩
Z
0
+. Those results also imply that the sequences of distances {
d
(
x
n
,
x
-
n
+
1
)
}
→
D
A
,
{
d
(
x
n
,
x
-
n
)
}
→
0. It is now proved by contradiction that {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B and {
d
(
T
x
n
,
T
x
-
n
)
}
→
0. Assume that, for each given x
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
- and any ε
∈
R
+, there are some δ
∈
R
+, some i
∈
p
-, some k
0
=
k
0
(
ε
,
x
0
)
∈
Z
0
+, sequences of integers {
n
k
l
}
⊆
{
n
k
}
⊆
Z
0
+, and sequences of best proximity points {
x
n
k
p
+
i
*
}
⊆
cl
⁡
(
A
0
,
i
+
j
) and {
T
x
n
k
p
+
i
*
}
⊆
cl
⁡
(
B
0
,
i
+
j
) for k
>
k
0, such that {
d
(
T
x
n
,
T
x
-
n
)
} does not converge to zero so that it has some subsequence which does not converge either:
(11)
d
(
A
i
,
A
i
+
1
)
=
d
(
A
0
i
,
A
0
,
i
+
1
)
=
D
A
,
d
(
T
(
A
0
i
)
,
T
(
A
0
,
i
+
1
)
)
=
d
(
B
i
,
B
i
+
1
)
=
d
(
B
0
i
,
B
0
,
i
+
1
)
=
D
B
,
∀
i
∈
p
-
,
d
(
x
n
k
p
+
i
,
x
n
k
p
+
i
-
1
*
)
<
D
A
+
ε
,
d
(
T
x
n
k
l
p
+
i
,
T
x
n
k
l
p
+
i
-
1
*
)
≥
δ
+
D
B
,
∀
k
>
k
0
,
d
(
x
n
k
p
+
i
,
x
n
k
p
+
i
*
)
<
ε
,
d
(
T
x
n
k
l
p
+
i
,
T
x
n
k
l
p
+
i
*
)
≥
δ
,
∀
k
>
k
0
,
T
x
n
k
p
+
i
*
∈
cl
⁡
B
0
,
i
+
j
⊆
cl
⁡
B
i
+
j
,
T
x
n
p
+
i
-
1
∈
cl
⁡
B
0
,
i
+
j
-
1
⊆
cl
⁡
B
i
+
j
-
1
,
∀
j
∈
p
-
since T
(
A
0
i
)
⊆
B
0
i, for all i
∈
p
-. This implies that {
x
n
k
l
p
+
i
-
x
n
k
l
p
+
i
*
}
→
0 while {
T
x
n
k
l
p
+
i
-
T
x
n
k
l
p
+
i
*
} does not converge to zero.
Since{
x
n
k
l
p
+
i
-
x
n
k
l
p
+
i
*
}
→
0, {
x
n
k
p
+
i
*
}
⊆
cl
⁡
(
A
0
,
i
+
j
)
,
{
T
x
n
k
p
+
i
*
}
⊆
cl
⁡
(
T
(
A
0
,
i
+
j
)
)
⊆
cl
⁡
(
B
0
,
i
+
j
), and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
⁡
(
A
0
,
i
+
j
), for all j
∈
p
- and any given j
∈
p
-, then {
T
x
n
k
l
p
+
i
-
T
x
n
k
l
p
+
i
*
}
→
0. Also, {
x
n
k
p
+
i
-
1
*
}
⊆
cl
⁡
(
A
0
,
i
+
j
-
1
)
,
{
T
x
n
k
p
+
i
-
1
*
}
⊆
cl
⁡
(
T
(
A
0
,
i
+
j
-
1
)
)
⊆
cl
⁡
(
B
0
,
i
+
j
-
1
), for all i
∈
p
- and any given j
∈
p
-; then {
d
(
T
x
n
k
l
p
+
i
,
T
x
n
k
l
p
+
i
-
1
*
)
}
→
D
B, {
d
(
T
x
n
k
l
p
+
i
-
1
,
T
x
n
k
l
p
+
i
*
)
}
→
D
B. Thus, {
d
(
T
x
n
,
T
x
-
n
)
}
→
0 and {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
0 if x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i, for any given j
∈
p
-.
On the other hand,{
d
(
T
x
n
,
T
x
-
n
)
} and {
d
(
T
x
n
,
T
x
-
n
+
1
)
} are bounded, since {
d
(
x
n
,
x
-
n
)
} is bounded from (6) because T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is GWCPD
p and since one has for some positive real constant M
=
M
(
d
(
x
0
,
x
-
0
)
) that
(12)
d
(
T
x
n
,
T
x
-
n
)
≤
d
(
T
x
n
,
x
n
)
+
d
(
x
-
n
,
T
x
-
n
)
+
d
(
x
n
,
x
-
n
)
i
≤
2
max
⁡
1
≤
i
≤
p
D
i
+
M
,
∀
n
∈
Z
0
+
,
d
(
T
x
n
,
T
x
-
n
+
1
)
≤
d
(
T
x
n
,
x
n
+
1
)
+
d
(
x
-
n
+
1
,
T
x
-
n
+
1
)
+
d
(
x
n
+
1
,
x
-
n
+
1
)
≤
2
max
⁡
1
≤
i
≤
p
D
i
+
M
,
∀
n
∈
Z
0
+
.
If the sets of best proximity points A
0
i for all i
∈
p
- are bounded, then the sequences of distances {
d
(
x
n
,
x
-
n
+
1
)
}, {
d
(
x
n
,
x
-
n
)
}, {
d
(
T
x
n
,
T
x
-
n
)
}, and {
d
(
T
x
n
,
T
x
-
n
+
1
)
} are uniformly bounded for any initial best proximity points x
0
,
x
-
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
- which follows by taking M
=
M
sup
⁡
x
0
,
x
-
0
∈
⋃
i
∈
p
-
A
i
(
d
(
x
0
,
x
-
0
)
). Property (i) has been fully proved.
To prove Property (ii), take any sequences{
x
n
p
+
i
}
⊆
A
0
,
i
+
j and {
x
-
n
p
+
i
}
⊆
A
0
,
i
+
j, for all i
∈
p
-, for given initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
-. Note, from (6) for {
x
-
n
k
+
1
p
+
i
}
=
{
x
n
k
p
+
i
}, for all i
∈
p
-, that(13)
d
(
x
n
k
+
1
p
+
i
,
x
n
k
p
+
i
)
≤
H
d
(
x
n
k
p
,
x
n
k
-
1
p
)
≤
h
0
α
-
d
(
x
n
k
p
,
x
n
k
-
1
p
)
,
∀
k
∈
Z
0
+
,
∀
i
∈
p
-
,
(14)
d
(
x
n
k
+
1
p
+
i
+
l
h
,
x
n
k
p
+
i
)
≤
h
0
α
-
(
1
+
α
-
+
⋯
+
α
-
l
)
d
(
x
n
k
p
,
x
n
k
-
1
p
)
d
(
x
n
k
+
1
p
+
i
+
l
h
,
x
n
k
p
+
i
)
≤
h
0
α
-
(
1
-
α
-
l
+
1
)
1
-
α
-
d
(
x
n
k
p
,
x
n
k
-
1
p
)
≤
h
0
α
-
1
-
α
-
d
(
x
n
k
p
,
x
n
k
-
1
p
)
,
∀
k
,
l
∈
Z
0
+
,
∀
i
∈
p
-
,
with the given upper bound being independent of the integers l and i. Thus, one has for any Z
0
+ that
(15)
d
(
x
n
k
+
m
+
1
p
+
i
+
l
j
,
x
n
k
+
m
p
+
i
)
≤
α
-
m
h
0
α
-
1
-
α
-
d
(
x
n
k
p
,
x
n
k
-
1
p
)
,
∀
k
∈
Z
0
+
,
∀
i
∈
p
-
,
∀
j
(
≤
h
)
,
l
∈
Z
0
+
,
where H
=
sup
⁡
k
∈
Z
0
+
(
∏
j
=
1
p
[
sup
⁡
x
∈
A
i
α
i
(
x
)
]
α
(
n
k
,
n
k
+
1
)
)
≤
h
0
α
-, with h
0
≥
sup
⁡
k
∈
Z
0
+
(
∏
j
=
1
p
[
sup
⁡
x
∈
A
i
α
i
(
x
)
]
) and 1
≤
j
≤
h
=
max
⁡
(
1
,
sup
⁡
k
∈
Z
0
+
(
n
k
+
1
-
n
k
)
), since Definition 5 holds for α
-
=
sup
⁡
k
∈
Z
0
+
α
(
n
k
,
n
k
+
1
)
∈
[
0,1
). Thus, one gets from (15) that, for any given real ε
∈
R
+, d
(
x
n
k
+
m
+
1
p
+
i
+
l
j
,
x
n
k
+
m
p
+
i
)
<
ε, for all  k
,
l
,
j
(
≤
h
)
∈
Z
0
+ for any i
∈
p
- and any given integers n
k
-
1
,
n
k
∈
Z
0
+ if m
(
∈
Z
0
+
)
>
ln
⁡
(
h
0
d
(
x
n
k
p
,
x
n
k
-
1
p
)
/
(
1
-
α
-
)
ε
)
/
|
ln
⁡
α
|
-
1. Thus, the sequences {
x
n
p
+
i
}
⊆
cl
⁡
A
0
i, for all  i
∈
p
-, are Cauchy sequences for any given initial point x
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i and any j
∈
p
-. This implies also that the sequences of images of the above points are also Cauchy sequences since T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is contractive and then continuous.From Assertions1, we also have the subsequent parallel result to Lemma 6.Lemma 7.
Assume thatT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is either G
C
P
D
p or W
C
P
D
p or C
P
D
p with the assumptions of Lemma 6, and consider any sequences {
x
n
p
+
j
}
,
{
x
-
n
p
+
j
}
⊆
⋃
i
∈
p
-
A
i which satisfy d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all  i
∈
p
-. Then, the following properties hold.
(i) The sequences of distances{
d
(
x
n
,
x
-
n
+
1
)
}
→
D
A, {
d
(
x
n
,
x
-
n
)
}
→
0 and they are bounded for any given initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i for any given j
∈
p
-.
If, furthermore,T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-, and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl T
(
A
0
i
), for all  i
∈
p
-, then {
d
(
T
x
n
,
T
x
-
n
)
}
→
0 and {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B and it is bounded for any initial points any given initial points x
0
,
x
-
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i, for some j
∈
p
-.
If the sets of best proximity pointsA
0
i and B
0
i, for all  i
∈
p
-, are bounded, then the sequences {
d
(
x
n
,
x
-
n
+
1
)
}, {
d
(
x
n
,
x
-
n
)
}, {
d
(
T
x
n
,
T
x
-
n
)
}, and {
d
(
T
x
n
,
T
x
-
n
+
1
)
} are uniformly bounded for any initial best proximity points x
0
,
x
-
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
-.
(ii) The sequences{
x
n
p
+
i
}
⊆
cl
A
0
,
i
+
j, for all  i
∈
p
-, are Cauchy sequences for any given initial point x
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for any given j
∈
p
-. The corresponding image sequences {
T
x
n
p
+
i
}
⊆
cl
B
0
,
i
+
j, for all  i
∈
p
-, are also convergent, then Cauchy sequences if T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-, and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
T
(
A
0
i
), for all  i
∈
p
-.Definition 8.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a p-cyclic proximal contraction with respect to its image (CPI
p) if there are real constants β
i
∈
[
0,1
), for all  i
∈
p
-, such that any two sequences {
T
x
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j and {
x
-
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j, for all  i
∈
p
-, being point-to-point images of sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j for any given j
∈
p
- which satisfy d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all  i
∈
p
-, where B
i
+
j
=
B
i
+
j
-
p if i
>
p
-
j such that the initial points T
x
0
,
T
x
-
0
∈
T
(
A
j
)
⊆
B
j are the images of points x
0
,
x
-
0
∈
A
j, for any given j
∈
p
-, satisfy the constraints
(16)
d
(
T
x
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
≤
β
i
d
(
T
x
n
p
+
i
,
T
x
-
n
p
+
i
-
1
)
+
(
1
-
β
i
)
D
B
,
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
,
(17)
d
(
T
x
n
p
+
i
+
1
,
T
x
-
n
p
+
i
+
1
)
≤
β
i
d
(
T
x
n
p
+
i
,
T
x
-
n
p
+
i
)
,
k
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
.Definition 9.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a weak p-cyclic proximal contraction with respect to its image (WCPI
p) if there are p real constants β
i
≥
0, for all  i
∈
p
-, subject to β
=
∏
i
=
1
p
[
β
i
]
∈
[
0,1
), such that any two sequences {
T
x
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j and {
T
x
-
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j, for all  i
∈
p
-, being point-to-point images of sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j for any given j
∈
p
-, where B
i
+
j
=
B
i
+
j
-
p for i
>
p
-
j, for all  i
∈
p
-, such that the initial points T
x
0
,
T
x
-
0
∈
T
(
A
j
)
⊆
B
j are the images of points x
0
,
x
-
0
∈
A
j, for any given j
∈
p
-, satisfy constraints (16) and (17).Definition 10.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a generalized p-cyclic proximal contraction with respect to its image (GCPI
p) if there are p bounded real functions β
i
:
A
i
→
R
0
+, for all  i
∈
p
-, such that any sequences {
T
x
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j and {
T
x
-
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j, for all  i
∈
p
-, being point-to-point images of sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j for any given j
∈
p
-, where B
i
+
j
=
B
i
+
j
-
p for i
>
p
-
j,  for all  i
∈
p
-, such that the initial points T
x
0
,
T
x
-
0
∈
T
(
A
j
)
⊆
B
j are the images of points x
0
,
x
-
0
∈
A
j, satisfy the constraints (16) and (17) with the replacements β
i
→
sup
⁡
x
∈
A
i
β
i
(
T
x
), for all  i
∈
p
-.Definition 11.
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is said to be a generalized weak p-cyclic proximal contraction with respect to its image (GWCPI
p) if there are p bounded real functions β
i
:
A
i
→
R
0
+, for all  i
∈
p
-, and a strictly increasing sequence of integers {
n
k
}, subject to n
0
≤
N
-
0
<
+
∞, limsup
⁡
k
→
∞
(
n
k
+
1
-
n
k
)
≤
N
-
<
+
∞, and α
-
=
sup
⁡
k
∈
Z
0
+
β
(
n
k
,
n
k
+
1
)
∈
[
0,1
), where
(18)
β
(
n
k
,
n
k
+
1
)
=
∏
j
=
n
k
n
k
+
1
-
1
∏
i
=
1
p
[
sup
⁡
x
j
+
i
∈
A
i
β
j
p
+
i
(
T
x
j
p
+
i
)
]
,
∀
k
∈
Z
0
+
,
such that any two sequences {
T
x
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j and {
x
-
n
p
+
i
}
⊆
T
(
A
i
+
j
)
⊆
B
i
+
j, for all  i
∈
p
-, being point-to-point images of sequences {
x
n
p
+
i
}
⊆
A
i
+
j and {
x
-
n
p
+
i
}
⊆
A
i
+
j for any given j
∈
p
-, where B
i
+
j
=
B
i
+
j
-
p for i
>
p
-
j, for all  i
∈
p
-, such that the initial points T
x
0
,
T
x
-
0
∈
T
(
A
j
)
⊆
B
j are the images of points x
0
,
x
-
0
∈
A
j, for any given j
∈
p
-, satisfy the following constraints:
(19)
d
(
T
x
n
k
+
1
p
,
T
x
-
n
k
+
1
p
-
1
)
≤
β
(
n
k
,
n
k
+
1
)
d
(
T
x
n
k
p
+
1
,
T
x
-
n
k
p
)
+
(
1
-
β
(
n
k
,
n
k
+
1
)
)
D
B
,
∀
k
∈
Z
0
+
,
(20)
d
(
T
x
n
k
+
1
p
+
i
,
T
x
-
n
k
+
1
p
+
i
-
1
)
≤
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
A
i
β
i
(
T
x
)
]
)
d
(
T
x
n
k
+
1
p
,
T
x
-
n
k
+
1
p
-
1
)
+
∑
j
=
1
i
-
1
(
∏
k
=
j
+
1
i
-
1
[
sup
⁡
x
∈
A
k
β
k
(
T
x
)
]
)
(
1
-
sup
⁡
x
∈
A
j
β
j
(
x
)
)
D
B
,
∀
k
∈
Z
0
+
,
(21)
d
(
T
x
n
k
+
1
p
+
i
,
T
x
-
n
k
+
1
p
+
i
)
≤
(
∏
j
=
1
i
-
1
[
sup
⁡
x
∈
A
i
β
i
(
T
x
)
]
)
β
(
n
k
,
n
k
+
1
)
d
(
T
x
n
k
p
,
T
x
-
n
k
p
)
,
∀
k
∈
Z
0
+
,
and constraints (16) and (17) with the replacements β
i
→
sup
⁡
x
∈
A
i
β
i
(
T
x
), for all  i
∈
p
-.The following assertions are obvious without proof from Definitions8–11 and are a parallel result to Assertions 1.Assertions 2.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is CPI
p, then it is WCPI
p.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is WCPI
p, then it is GCPI
p.
IfT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is GCPI
p, then it is GWCPI
p.Note that the converse implications of those in Assertions1 are not true in general.The relevant distances satisfy the following convergence and boundedness result which is a counterpart of Lemmas6 and 7. Its proof is close to that of Lemma 6 and Assertions 2 by using (16) and (17) for Definition 11 and their variants for Definitions 8–10.Lemma 12.
Assume thatT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is either G
W
C
P
I
p or G
C
P
I
p or W
C
P
I
p or C
P
I
p under the assumptions of Lemma 6, and consider any sequences {
x
n
p
+
j
}
,
{
x
-
n
p
+
j
}
⊆
⋃
i
∈
p
-
A
i which satisfy d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i, for all  i
∈
p
-. Then, the following properties hold.
(i) The sequences of distances{
d
(
x
n
,
x
-
n
+
1
)
}
→
D
A, {
d
(
x
n
,
x
-
n
)
}
→
0 and they are bounded for any given initial points x
0
,
x
-
0
∈
A
j
⊂
⋃
i
∈
p
-
A
i, for some j
∈
p
-.
If, furthermore,T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-, and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
T
(
A
0
i
), for all  i
∈
p
-, then {
d
(
T
x
n
,
T
x
-
n
)
}
→
0 and {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B and it is bounded any given initial points x
0
,
x
-
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
-.
If the sets of best proximity pointsA
0
i and B
0
i, for all  i
∈
p
-, are bounded, then the sequences {
d
(
x
n
,
x
-
n
+
1
)
}, {
d
(
x
n
,
x
-
n
)
}, {
d
(
T
x
n
,
T
x
-
n
)
}, and {
d
(
T
x
n
,
T
x
-
n
+
1
)
} are uniformly bounded for any initial best proximity points x
0
,
x
-
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for some j
∈
p
-.
(ii) The sequences{
x
n
p
+
i
}
⊆
cl
A
0
,
i
+
j, for all  i
∈
p
-, are Cauchy sequences for any given initial point x
0
∈
A
0
j
⊂
⋃
i
∈
p
-
A
i for any given j
∈
p
-. The corresponding image sequences {
T
x
n
p
+
i
}
⊆
cl
B
0
,
i
+
j, for all  i
∈
p
-, are also convergent; then Cauchy sequences if T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-, and T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i are continuous in cl
T
(
A
0
i
), for all  i
∈
p
-.Remark 13.
The result{
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B of Lemma 12, as well as Lemma 12 (ii), obtained under the assumption that T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous in cl
⁡
T
(
A
0
i
) and also holds without such a continuity assumption if the contractive conditions (16) and (17) become modified to the right limits as follows:
(22)
d
(
T
x
n
p
+
i
+
1
+
,
T
x
-
n
p
+
i
+
)
≤
β
i
d
(
T
x
n
p
+
i
+
,
T
x
-
n
p
+
i
-
1
+
)
+
(
1
-
β
i
)
D
B
,
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
,
(23)
d
(
T
x
n
p
+
i
+
1
+
,
T
x
-
n
p
+
i
+
1
+
)
≤
β
i
d
(
T
x
n
p
+
i
+
,
T
x
-
n
p
+
i
+
)
,
∀
i
∈
p
-
,
∀
n
∈
Z
0
+
,
provided that any discontinuity points in ⋃
i
∈
p
cl
⁡
T
(
A
0
i
), if any, are of first-class finite-jump type under right best proximity constraints
(24)
d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
+
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
+
)
=
D
i
,
∀
i
∈
p
-
.
In the same way, the result {
d
(
T
x
n
,
T
x
-
n
+
1
)
}
→
D
B of Lemmas 6 and 7, as well as their properties (ii) obtained under the assumption that T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i, is continuous in cl
⁡
T
(
A
0
i
) and also holds under finite-jump discontinuities in cl
⁡
T
(
A
0
i
) for sequences {
x
n
}
,
{
x
-
n
}, {
T
x
n
}, and {
T
x
-
n
} satisfying the contractive proximal conditions (1) and (2) if Definition 2, or their counterparts of Definitions 3–5 for right values T
x
n
+ and T
x
-
n
+ under right best proximity constraints (24).


## 3. Best Proximity Points and Related Convergence Results

We first recall the subsequent useful definition [2–4, 7] as follows.Definition 14.
LetA and B be two nonempty subsets of a metric space (
X
,
d
) and let d
(
y
,
A
)
=
inf
⁡
{
d
(
y
,
x
)
:
x
∈
A
} for y
∈
X. A is said to be approximately compact with respect to B if each sequence {
x
n
}
⊂
A satisfying {
d
(
y
,
x
n
)
}
→
d
(
y
,
A
) for some y
∈
B has a convergent subsequence.Note that if the sets of best proximity pointsA
0
⊆
A and B
0
⊆
B are nonempty if Definition 14 holds, then A is approximately compact with respect to B if every sequence {
x
n
}
⊂
A such that {
d
(
y
,
x
n
)
}
→
D for some y
∈
B
0 has a convergent subsequence {
x
n
k
}
⊆
{
x
n
} since D
=
d
(
y
,
A
)
=
d
(
B
0
,
A
)
=
d
(
B
0
,
A
0
). Note that every set is approximately compact with respect to itself and that every compact set is approximately compact with respect to any nonempty subset of a metric space. Also, if B is compact and A is approximately compact with respect to B, each sequence {
x
n
}
⊂
A has a convergent sequence. If A and B are nonempty and closed and A is approximately compact with respect to B, then B
0 is closed. See, for instance, [2–4, 7]. A result on existence and uniqueness of best proximity points follows for p-cyclic proximal contraction fulfilling Definitions 2–5 under Lemmas 6 and 7 follows.Theorem 15.
Consider a complete metric space(
X
,
d
) with nonempty closed subsets A
i
,
B
i
⊂
X and a p-cyclic mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i being either G
W
C
P
D
p or G
C
P
D
p or W
C
P
D
p or C
P
D
p, subject to set distances D
i
=
d
(
A
i
+
1
,
B
i
), D
A
=
d
(
A
i
,
A
i
+
1
), and D
B
=
d
(
B
i
,
B
i
+
1
), for all  i
∈
p
- such that A
0
i is nonempty and B
i is approximately compact with respect to A
i and T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-. The following properties hold.(i)
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i has a unique best proximity point x
i
* at A
i such that d
(
x
i
*
,
x
i
+
1
*
)
=
D
A, for all  i
∈
p
-, and all the sequences {
x
n
}
⊂
⋃
i
∈
p
-
A
i converge to a unique limit cycle {
x
1
*
,
x
2
*
,
…
,
x
p
*
}.
(ii)
Furthermore, ifT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i either is continuous, respectively, has eventual finite-jump discontinuity points, then z
i
*
=
T
x
i
*, for all  i
∈
p
-, respectively, z
i
*
+
=
(
T
x
i
*
)
+, for  all  i
∈
p
-, are unique best proximity points such that d
(
T
x
i
*
,
T
x
i
+
1
*
)
=
D
B, for all  i
∈
p
-, respectively, d
(
T
x
i
*
+
,
T
x
i
+
1
*
+
)
=
D
B, for all  i
∈
p
-, and all the sequences {
x
n
}
⊂
⋃
i
∈
p
-
A
i converge to a unique limit cycle {
T
x
1
*
+
,
T
x
2
*
+
,
…
,
T
x
p
*
+
}.Proof.
SinceA
0
i is nonempty and T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-, then T
(
A
0
i
) and B
0
i are nonempty, for all  i
∈
p
-. Also, A
0
i is closed since B
i is approximately compact with respect to A
i. Consider any sequences {
x
n
p
+
j
}
,
{
x
-
n
p
+
j
}
⊆
⋃
i
∈
p
-
A
i which satisfy
(25)
d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i
,
∀
i
∈
p
-
.
One gets, from Lemma 6(i), if the mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is GWCPD
p and, from Lemma 7(i), if the mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is either GCPD
p or WCPD
p or CPD
p that, since T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-,
(26)
{
d
(
x
-
p
(
n
+
1
)
+
i
+
j
,
x
p
n
+
i
+
j
)
}
⟶
0
,
{
d
(
x
-
p
(
n
+
1
)
+
i
+
j
+
1
,
x
p
n
+
i
+
j
+
1
)
}
⟶
0
,
(27)
{
d
(
x
-
p
n
+
i
+
j
,
x
p
n
+
i
+
j
+
1
)
}
⟶
D
A
=
d
(
y
i
+
j
,
A
i
+
1
)
=
d
(
y
i
+
j
,
A
0
,
i
+
1
)
=
d
(
y
i
+
j
,
x
p
n
k
+
i
+
j
+
1
)
,
(28)
{
d
(
x
-
p
n
+
i
+
j
+
1
,
T
x
p
n
+
i
+
j
)
}
⟶
D
i
=
d
(
y
i
+
j
+
1
,
B
i
)
=
d
(
y
i
+
j
+
1
,
T
(
A
i
)
)
=
d
(
y
i
+
j
+
1
,
T
(
A
0
i
)
)
=
d
(
y
i
+
j
+
1
,
B
0
i
)
=
d
(
y
i
+
j
+
1
,
T
x
p
n
k
+
i
+
j
)
,
∀
i
∈
p
-
,
for some y
i
+
j
∈
A
0
,
i
+
j and y
i
+
j
+
1
∈
A
0
,
i
+
j
+
1 since D
A
=
d
(
A
i
,
A
i
+
1
)
=
d
(
A
0
i
,
A
0
,
i
+
1
) and D
i
=
d
(
A
i
+
1
,
B
i
)
=
d
(
A
0
i
+
1
,
B
0
i
), for all  i
∈
p
-, and some subsequences
(29)
{
x
-
p
n
k
+
i
+
j
+
1
}
,
{
x
p
n
k
+
i
+
j
+
1
}
⊆
A
0
,
i
+
j
+
1
,
{
x
-
p
n
k
+
i
+
j
}
,
{
x
p
n
k
+
i
+
j
}
⊆
A
0
,
i
+
j
of the sequences {
x
-
p
n
+
i
+
j
+
1
}
,
{
x
p
n
+
i
+
j
+
1
}
⊆
A
i
+
j
+
1 and {
x
-
p
n
+
i
+
j
}
,
{
x
p
n
+
i
+
j
}
⊆
A
i
+
j, for all  i
∈
p
-, respectively, for any given initial points x
0
,
x
-
0
∈
A
j for any given j
∈
p
-. The following results hold.
(1) From (26) and by taking {
x
-
}
n
≡
{
x
n
} and {
d
(
x
p
(
n
+
1
)
+
i
+
j
,
x
p
n
+
i
+
j
)
}
→
0, for all  i
∈
p
- for x
0
∈
A
j for any given j
∈
p
-, one gets {
x
p
n
+
i
+
j
}
→
x
i
*, since A
0
i is closed, for all  i
∈
p
- and, from (27), d
(
x
i
*
,
x
i
+
1
*
)
=
D
A, for all  i
∈
p
-.
(2) Again, from (26) {
d
(
x
-
p
(
n
+
1
)
+
i
+
j
,
x
-
p
n
+
i
+
j
)
}
→
0, for all  i
∈
p
- for x
0
∈
A
j for any given j
∈
p
-, {
x
-
p
n
+
i
+
j
}
→
x
-
i
*, for all  i
∈
p
-.
(3) Combining results (1) and (2) with (26), it follows that x
-
i
*
=
x
i
*, for all  i
∈
p
-.
(4) Results (1)–(3) hold irrespective of the subsetA
j for j
∈
p
- where the initial conditions of the sequences belong to, so for any x
0
,
x
-
0
⊂
⋃
i
∈
p
-
A
i (see the beginning of the proof of Lemma 6). Thus, from result (
3
), there are unique limit points x
i
* at each subset A
i of all the sequences {
x
n
}
⊂
⋃
i
∈
p
-
A
i such that any such sequence converges to a unique limit cycle {
x
1
*
,
x
2
*
,
…
,
x
p
*
} consisting of best proximity points of adjacent subsets A
i, for all  i
∈
p
-.
(5) SinceB
i is closed and approximately compact with respect to A
i, for all  i
∈
p
-, one gets from (28) that a subsequence of {
T
x
p
n
+
i
+
j
} is convergent for each i
∈
p
-; say {
T
x
p
n
k
+
i
+
j
}
→
z
i
*
∈
T
(
A
i
)
⊆
B
0
i
⊂
B
i, for all  i
∈
p
-. Since {
x
p
n
+
i
+
j
}
→
x
i
*, all its subsequences converge to the same limit so that {
x
p
n
k
+
i
+
j
}
→
x
i
* and then z
i
*
=
T
x
i
* is unique, since each x
i
* is unique, within each B
i, for all  i
∈
p
- and, again, from (28), D
i
=
d
(
x
i
+
1
*
,
T
x
i
*
), for all  i
∈
p
- if T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous at x
i
* and D
i
=
d
(
x
i
+
1
*
,
T
x
i
*
+
) if T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i has a finite-jump discontinuity at x
i
*, then z
i
*
+
=
T
x
i
*
+ (see Remark 13). The result has been proved.A further result on the existence and uniqueness of best proximity points follows forp-cyclic proximal contractions subject to Definitions 8–11 under Lemma 12 and whose proof is very close to that of Theorem 15.Theorem 16.
Consider a complete metric space(
X
,
d
) with nonempty subsets A
i
,
B
i
⊂
X and a p-cyclic mapping T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i being either G
W
C
P
I
p or G
C
P
I
p or W
C
P
I
p or C
P
I
p, subject to set distances D
i
=
d
(
A
i
+
1
,
B
i
), D
A
=
d
(
A
i
,
A
i
+
1
), and D
B
=
d
(
B
i
,
B
i
+
1
), for all  i
∈
p
-, such that A
0
i is nonempty and closed and B
0
i is nonempty and A
i is approximately compact with respect to B
i and T
(
A
0
i
)
⊆
B
0
i, for all  i
∈
p
-. The following properties hold.(i)
T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i has a unique best proximity point x
i
* at A
i such that d
(
x
i
*
,
x
i
+
1
*
)
=
D
A, for all  i
∈
p
-, and all the sequences {
x
n
}
⊂
⋃
i
∈
p
-
A
i converge to a unique limit cycle {
x
1
*
,
x
2
*
,
…
,
x
p
*
}.
(ii)
Furthermore, ifT
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i either is continuous, respectively, has eventual finite-jump discontinuity points, then z
i
*
=
T
x
i
*, for all  i
∈
p
-, respectively, z
i
*
+
=
(
T
x
i
*
)
+, for all  i
∈
p
-, are unique best proximity points such that d
(
T
x
i
*
,
T
x
i
+
1
*
)
=
D
B, for  all  i
∈
p
-, respectively, d
(
T
x
i
*
+
,
T
x
i
+
1
*
+
)
=
D
B, for  all  i
∈
p
-, and all the sequences {
x
n
}
⊂
⋃
i
∈
p
-
A
i converge to a unique limit cycle {
T
x
1
*
+
,
T
x
2
*
+
,
…
,
T
x
p
*
+
}.Proof.
SinceT
(
A
0
i
)
⊆
B
0
i, for  all  i
∈
p
-, then T
(
A
0
i
) and B
0
i are nonempty, since A
0
i is nonempty, for  all  i
∈
p
-. B
0
i is, furthermore, closed since A
i is approximately compact with respect to B
i. Thus, A
0
i, T
(
A
0
i
), and B
0
i are nonempty and closed, for
all
i
∈
p
-. Consider any sequences {
x
n
p
+
j
}
,
{
x
-
n
p
+
j
}
⊆
⋃
i
∈
p
-
A
i which satisfy
(30)
d
(
x
n
p
+
i
+
1
,
T
x
n
p
+
i
)
=
d
(
x
-
n
p
+
i
+
1
,
T
x
-
n
p
+
i
)
=
D
i
,
∀
i
∈
p
-
.
One gets from Lemma 12 that, since T
(
A
0
i
)
⊆
B
0
i, for
all
i
∈
p
-,
(31)
{
d
(
T
x
-
p
(
n
+
1
)
+
i
+
j
,
T
x
p
n
+
i
+
j
)
}
⟶
0
,
{
d
(
T
x
-
p
(
n
+
1
)
+
i
+
j
+
1
,
T
x
p
n
+
i
+
j
+
1
)
}
⟶
0
,
(32)
{
d
(
T
x
-
p
n
+
i
+
j
,
T
x
p
n
+
i
+
j
+
1
)
}
⟶
D
B
=
d
(
y
i
+
j
,
B
i
+
1
)
=
d
(
y
i
+
j
,
B
0
,
i
+
1
)
=
d
(
y
i
+
j
,
x
p
n
k
+
i
+
j
+
1
)
,
(33)
{
d
(
x
-
p
n
+
i
+
j
+
1
,
T
x
p
n
+
i
+
j
)
}
⟶
D
i
=
d
(
y
i
+
j
+
1
,
B
i
)
=
d
(
y
i
+
j
+
1
,
T
(
A
i
)
)
=
d
(
y
i
+
j
+
1
,
T
(
A
0
i
)
)
=
d
(
y
i
+
j
+
1
,
B
0
i
)
=
d
(
y
i
+
j
+
1
,
T
x
p
n
k
+
i
+
j
)
,
∀
i
∈
p
-
,
for some y
i
+
j
∈
B
0
,
i
+
j, y
i
+
j
+
1
∈
B
0
,
i
+
j
+
1 since D
B
=
d
(
B
i
,
B
i
+
1
)
=
d
(
B
0
i
,
B
0
,
i
+
1
) and D
i
=
d
(
A
i
+
1
,
B
i
)
=
d
(
A
0
i
+
1
,
B
0
i
), for
all
i
∈
p
-, and some subsequences
(34)
{
T
x
-
p
n
k
+
i
+
j
+
1
}
,
{
T
x
p
n
k
+
i
+
j
+
1
}
⊆
T
(
A
0
,
i
+
j
+
1
)
⊆
B
0
,
i
+
j
+
1
,
{
T
x
-
p
n
k
+
i
+
j
}
,
{
T
x
p
n
k
+
i
+
j
}
⊆
T
(
A
0
,
i
+
j
+
1
)
⊆
B
0
,
i
+
j
of the sequences {
T
x
-
p
n
+
i
+
j
+
1
}
,
{
T
x
p
n
+
i
+
j
+
1
}
⊆
B
i
+
j
+
1 and {
T
x
-
p
n
+
i
+
j
}
,
{
T
x
p
n
+
i
+
j
}
⊆
B
i
+
j, for
all
i
∈
p
-, respectively, for any given initial points x
0
,
x
-
0
∈
A
j for any given j
∈
p
-. The following results hold.
(6) From (31) and by taking {
x
-
}
n
≡
{
x
n
}, {
d
(
T
x
p
(
n
+
1
)
+
i
+
j
,
T
x
p
n
+
i
+
j
)
}
→
0, for
all
i
∈
p
- for x
0
∈
A
j for any given j
∈
p
-, one gets {
T
x
p
n
+
i
+
j
}
→
z
i
*, since B
0
i is closed, for
all
i
∈
p
- and, from (32), d
(
z
i
*
,
z
i
+
1
*
)
=
D
B, for
all
i
∈
p
-.
(7) Again from (31), {
d
(
T
x
-
p
(
n
+
1
)
+
i
+
j
,
T
x
-
p
n
+
i
+
j
)
}
→
0, for
all
i
∈
p
- for x
0
∈
A
j for any given j
∈
p
-, so that {
T
x
-
p
n
+
i
+
j
}
→
z
-
i
*, for
all
i
∈
p
-.
(8) Combining results (6) and (7) with (31), it follows that z
-
i
*
=
z
i
*, for
all
i
∈
p
-.
(9) Results (6)–(9) hold irrespective of the subsetA
j for j
∈
p
- where the initial conditions of the sequences belong to then for any x
0
,
x
-
0
⊂
⋃
i
∈
p
-
A
i. Thus, considering result (
8
), there are unique limit points z
i
* at each subset B
i of all the sequences {
T
x
n
}
⊂
⋃
i
∈
p
-
T
(
A
i
) such that any such sequence converges to a unique limit cycle {
z
1
*
,
z
2
*
,
…
,
z
p
*
} consisting of best proximity points of adjacent subsets A
i, for
all
i
∈
p
-.
SinceB
i is closed and approximately compact with respect to A
i, for
all
i
∈
p
-, one gets from (33) that a subsequence of {
x
p
n
+
i
+
j
} is convergent for each i
∈
p
-; say {
x
p
n
k
+
i
+
j
}
→
x
i
*
∈
A
0
i, for
all
i
∈
p
-. Since {
T
x
p
n
+
i
+
j
}
→
z
i
*, all its subsequences converge to the same limit so that {
T
x
p
n
k
+
i
+
j
}
→
z
i
* and x
i
*
∈
A
0
i fulfilling z
i
*
=
T
x
i
* which is unique. Assume not so that there are x
i
*
,
x
-
i
*
(
≠
x
i
*
)
∈
A
0
i for some i
∈
p
- such that z
i
*
=
T
x
i
*
=
T
x
-
i
*. Assume a sequence {
x
p
n
k
+
i
+
j
}
→
x
i
*
∈
A
0
i with x
0
∈
A
j and a sequence {
x
p
n
k
+
i
+
j
}
→
x
-
i
*
∈
A
0
i with initial point x
-
00
∈
A
l and some x
-
0
=
T
δ
x
-
00
∈
A
j for some j
,
l
∈
p
- and some nonnegative integer δ
<
p. But then {
d
(
T
x
-
p
(
n
+
1
)
+
i
+
j
,
T
x
-
p
n
+
i
+
j
)
} does not converge to zero so that x
-
i
*
=
x
i
*
∈
A
0
i is unique, for
all
i
∈
p
-. The distance convergence properties are independent of the fact that for the initial condition l is as equal or distinct as j, as discussed in Lemma 6. If and another sequence. Since each z
i
* is unique, within each A
i, for
all
i
∈
p
- and, again, from (33), D
i
=
d
(
x
i
+
1
*
,
T
x
i
*
), for
all
i
∈
p
- if T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i is continuous at x
i
* and D
i
=
d
(
x
i
+
1
*
,
T
x
i
*
+
) if T
:
⋃
i
∈
p
-
A
i
→
⋃
i
∈
p
-
B
i has a finite-jump discontinuity at x
i
*, then z
i
*
+
=
T
x
i
*
+ (see Remark 13). The result has been proved.Example 17.
Consider a2-cyclic proximal contraction with respect to its domain: T
:
A
1
∪
A
2
→
B
1
∪
B
2, where A
i and B
i for i
=
1,2 are nonempty closed subsets of R. Take any sequences {
x
n
}
⊂
A
1, {
y
n
}
⊂
A
2 being subsequences of {
z
n
}
⊂
A
1
∪
A
2 defined either by z
2
n
=
x
n, z
2
n
+
1
=
y
n or by z
2
n
=
y
n, z
2
n
+
1
=
x
n, for
all
n
∈
Z
0
+, and subject to the constraints below under the Euclidean metric d
(
x
,
y
)
=
|
x
-
y
|, for
all
x
,
y
∈
R for some contractive real constant α
∈
[
0,1
) such that α
1
=
α
2
=
α and (
R
,
d
) is a complete metric space and also a Banach space. Assume that A
1
=
[
a
1
_
,
a
-
1
] and A
2
=
[
a
2
_
,
a
-
2
] with a
-
1
<
0 and a
2
_
>
0 and B
1
=
[
b
1
_
,
b
-
1
] and B
2
=
[
b
2
_
,
b
-
2
] with a
-
1
≤
0 and a
2
_
≥
0, b
1
_
≤
a
1
_, 0
≥
b
-
1
≥
a
-
1, 0
≤
b
2
_
≤
a
2
_, and b
-
2
≥
a
-
2, so that T
(
A
1
)
⊆
B
1, T
(
A
2
)
⊆
B
2 with
(35)
D
A
=
d
(
A
1
,
A
2
)
=
|
a
1
¯
|
+
a
2
_
,
D
B
=
d
(
B
1
,
B
2
)
=
|
b
1
¯
|
+
b
2
_
,
(36)
D
1
=
d
(
A
2
,
B
1
)
=
a
2
_
+
|
b
-
1
|
,
D
2
=
d
(
A
1
,
B
2
)
=
|
a
-
1
|
+
b
2
_
,
and T
:
A
1
∪
A
2
→
B
1
∪
B
2 is a CPD
2 (Definition 2 with p
=
2) if the subsequent constraints hold for all n
∈
Z
0
+:
(37)
y
n
+
1
+
|
T
x
n
|
=
D
1
,
T
y
n
+
|
x
n
+
1
|
=
D
2
,
(38)
|
x
n
+
2
-
x
n
+
1
|
=
|
|
x
n
+
1
|
-
|
x
n
+
2
|
|
=
|
T
y
n
-
T
y
n
+
1
|
≤
α
|
x
n
+
1
-
x
n
|
,
(39)
|
y
n
+
2
-
y
n
+
1
|
=
|
|
T
x
n
|
-
|
T
x
n
+
1
|
|
≤
α
|
y
n
+
1
-
y
n
|
,
(40a)
|
y
n
+
1
-
x
n
+
1
|
=
y
n
+
1
+
|
x
n
+
1
|
=
D
1
+
D
2
-
|
T
x
n
|
-
T
y
n
≤
α
|
y
n
-
x
n
|
+
(
1
-
α
)
D
A
(40b)
≤
α
|
y
n
-
x
n
|
+
(
1
-
α
)
D
A
=
α
(
y
n
+
|
x
n
|
)
+
(
1
-
α
)
D
A
.In particular, (37)–(40a) and (40b) are satisfied if, for all n
∈
Z
0
+,
(41)
T
y
n
≥
D
2
-
α
y
n
-
1
-
α
2
D
A
,
(42)
T
x
n
≤
-
D
1
-
α
x
n
+
1
-
α
2
D
A
(
equivalently
,
D
1
-
α
|
x
n
|
-
1
-
α
2
D
A
≤
|
T
x
n
|
)
,
(43)
y
n
+
1
≤
α
|
x
n
|
-
1
-
α
2
D
A
,
|
x
n
+
1
|
≤
α
y
n
+
1
-
α
2
D
A
.
Parallel results for the case when T
:
A
1
∪
A
2
→
B
1
∪
B
2 is WCPD
2, GCPD
2, or GWCPD
2 (Definitions 3–5 with p
=
2) can be discussed in the same way with the appropriate extensions for the contractive constant or function. It follows that {
d
(
y
n
,
y
n
+
1
)
}
→
0, {
d
(
x
n
,
x
n
+
1
)
}
→
0, {
d
(
z
n
,
z
n
+
1
)
}
→
D
A, {
y
n
}
→
a
2
_, {
x
n
}
→
a
-
1, {
d
(
T
y
n
,
T
y
n
+
1
)
}
→
0, {
d
(
T
x
n
,
T
x
n
+
1
)
}
→
0 and, according to (40a), (40b), and (35)-(36), {
d
(
T
z
n
,
T
z
n
+
1
)
}
→
D
B, since {
|
T
(
y
n
-
|
x
n
|
)
|
}
→
D
B
=
D
1
+
D
2
-
D
A, {
T
y
n
}
→
T
a
2
_
=
b
2
_, and {
T
x
n
}
→
T
a
-
1
=
b
-
1.Example 18.
Consider Example17 in the case that T
:
A
1
∪
A
2
→
B
1
∪
B
2 is CPI
2, WCPI
2, GCPI
2, or GWCPI
2 (Definitions 8–11 with p
=
2); (40a) using (37) can be reformulated accordingly. In particular, if it is CPI
2, then one gets for some real constant β
∈
[
0,1
)
(44)
|
T
x
n
+
1
|
+
T
y
n
+
1
=
T
y
n
+
1
-
T
x
n
+
1
=
D
1
+
D
2
-
|
x
n
+
2
|
-
y
n
+
2
=
D
A
+
D
B
-
|
x
n
+
2
|
-
y
n
+
2
≤
β
(
|
x
n
|
+
y
n
)
+
(
1
-
β
)
D
B
,
∀
n
∈
Z
0
+
.
Then, {
|
T
(
y
n
-
|
x
n
|
)
|
}
≡
{
d
(
T
z
n
,
T
z
n
+
1
)
}
→
D
B
=
D
1
+
D
2
-
D
A, {
|
x
n
|
+
y
n
}
≡
{
d
(
z
n
,
z
n
+
1
)
}
→
D
A, {
d
(
y
n
,
y
n
+
1
)
}
→
0, {
d
(
x
n
,
x
n
+
1
)
}
→
0, {
T
y
n
}
→
T
a
2
_
=
b
2
_, {
T
x
n
}
→
T
a
-
1
=
b
-
1, {
y
n
}
→
a
2
_, and {
x
n
}
→
a
-
1.



---

*Source: 914915-2014-10-15.xml*