1. Introduction
When viewing the topic of n-sets methodology, it can be seen that many authors will often create contractions that consider only a sequential pattern in their published findings.
However, Kirk et al. (2003) [
1], and Eldred and Veeramani [
2] (2006)’s 2-sets methodologies which both use a sequential pattern also allows for this paper’s newly proposed “alternative n-sets” contraction to be defined as a special case in which we may now consider that the map of point does not need to follow a sequential pattern. Additionally, this alternative n-sets contraction in conjunction with Suzuki et al. [
3] (2009)’s UC Property and Fan [
4] (1969)’s Best Proximity Points allows for the defining of a “New UC Condition” and non-sequential best proximity points.
It is this paper’s purpose to explore this idea of a non-sequential n-sets methodology and inspire new possibilities with said idea.
Definition 1 (Kirk et al. [
1])
. Let be a metric space, be nonempty subsets of , . A map is a cyclic map if for and . The map proposed by Kirk et al. may follow a certain fixed path; that is, the point always is sent by an operator
T from one set to another sequentially. Moreover, Eldred and Veeramani gives a contraction on the two subsets of
in 2006 as follows:
Definition 2 (Eldred and Veeramani [
2])
. Let be a metric space, and let and be nonempty subsets of . A map is a cyclic map if and . A map is a cyclic contraction if there exists satisfyingfor all , . The contribution of Eldred and Veeramani was to create the cyclic contraction. Many authors have created several different contractions by modifying Eldred and Veeramani; see e.g., [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. In this paper, we define a new map which is concerned with the alternative
n-sets methodology instead of 2-sets or sequentially
n-sets as follows:
Definition 3. Let be a metric space, , be nonempty subsets of . A map is called an alternative map if , for . Next, T is defined as an alternative contraction (AC) if T is an alternative map and there exists a constant such that for any , for some , , the following condition holds: Now, consider Suzuki et al. [
3]’s UC condition:
Definition 4 (Suzuki et al. [
3])
. Let A and B be nonempty subsets of a metric space . Then is said to satisfy the property UC if the following holds: If and are sequences in A and is a sequence in B such that and , then holds. The UC condition only takes into consideration Eldred and Veeramani [
2]’s 2-sets model. Taking into account the above alternative contraction, an alternative UC Condition may be created:
Definition 5 (Alternative UC condition). Let be a metric space and , be nonempty subsets of . If is said to satisfy the UC condition if the following holds. Let , and , , for some positive integer . If for some , then .
One can easily see that a cyclic map is a special case of an alternative map.
Due to Fan [
4], many authors publish their papers with their Best Proximity Points in [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34]:
Definition 6. Let be a metric space and , be nonempty subsets of . If is called a best proximity point of a cyclic map T if Using the alternative UC condition and the alternative contraction, we may define new best proximity points:
Definition 7. Let be a metric space and , , be nonempty subsets of . If is called a best proximity point of an alternative map T if there exists a positive integer , for some and , , and satisfying the following three conditions:
- 1.
,
- 2.
, (Here ),
- 3.
The traditional cyclic map can be used to deal with the best proximity points of a model using two mirrors. Based on Eldred and Veeramani [
2]’s cyclic map, the cyclic contraction is formulated to specify the path that the light will follow from one mirror to next. Suzuki et al. [
32] proves that under cyclic contraction and the UC condition, there will be certain points reflected many times between the two mirrors; these points, called the best proximity points, are bounced infinite times, creating particularly bright points. This paper’s alternative map changes the model from a two-mirror map into one concerning numerous mirrors. In this model, after the light is reflected from a mirror, the light will not necessarily return to said mirror and its path may diverge to any given point. According to the variance of the angle of refraction, it is impossible to know to which mirror the light will be reflected to next. Nevertheless, in this article we prove when light is reflected many times under the conditions of the proposed alternative contraction and the UC condition, in some mirrors, these points will be become particularly shining points, or the best proximity points of the alternative map.
Theorem 5 proves that if
T is an alternative contraction, some proximity points exist. Some preparations presented in
Section 2 and
Section 3 explain the reasoning behind this proof. Finally, we may assert that the main results are not unusual since it can be applied to both compact space and complete metric space.
3. Best Proximity Points
In this section, in order to give the definition of best proximity points of alternative maps, we would remind the definition of a best proximal point of a cyclic map which is proposed by Fan and we rewrite to be applied in the metric space below.
Definition 9 (Fan [
4])
. Let be a metric space and , be nonempty subsets of . If is called a best proximity point of a cyclic map T if Next, we would like to extend the above definitions to the alternative map as follows:
Definition 10. Let be a metric space and , , be nonempty subsets of . If is called a best proximity point of an alternative map T if there exists a positive integer , for some and , , and satisfying the following three conditions:
- 1.
,
- 2.
,
- 3.
From the above definitions, we know that the number
ℓ is not unique. We give an example as follows:
Example 1. Let , , , . Let the map T aswhere τ is a random variable in . At second, we will complete our works step by step as follows.
Theorem 2. Let be a metric space and , , be nonempty subsets of . Let be an alternative contraction, then for any , we have Proof. Assume that
, for some
, this would imply that
for any
n because
T is an alternative map. Moreover, since
T is an alternative contraction, there exists a constant
such that
Taking “lim sup” on both sides of inequality
and let
by Theorem 1, it follows that
and then
dividing
on both sides of
, we have
For any
, since
and
, we have
and then
By
and
, we have
Since the cardinal number of
is “
m”, the choice of
is at most
and we have
for some
. Hence
☐
Lemma 3. Let be a metric space, , , be nonempty subsets of and S be a subsequence of with cardinal number . Consider an alternative map and for any , we can find a subsequence and some subsets , , such thatfor some .(Please note that some of may be the same.) Proof. Let , , then . By the pigeonhole principle, there exists with and such that for any .
Therefore, we have for any . Since the cardinal number of is also infinitely many and the cardinal number of is only “”, there exists and with such that for any .
Similarly, we have for any . Since the cardinal number of is also infinitely many and the cardinal number of is only “”, there exists and with such that for any . If , let , the proof is completed. Else, if , will continue the process.
By induction, there exists
,
and
with
for
such that
for any
,
. Continuing the process until
, then stopping, say
for some
. In this way, we reset the index of
,
and let
, then
Next, we would check that this process would finish by at most
steps. Consider the
-th step, since the steps of algorithms is already
, that is, we produce
sets. However, the cardinal number of
is only
m, and so, by the pigeonhole principle, there exists two different
,
such that
. WLOG (Without loss of generalization), we may assume that
. Reset
as
and reset
as
, reset
as
, reset
as
reset
as
, reset
as
such that
The proof is completed. ☐
Lemma 4. Let be a metric space, , , be nonempty subsets of and T be an alternative contraction. Assume that there exists and a subsequence such that exists. Let for some and for some for any , then Proof. Let
. Since
and
, we have
for any
j. Thus,
Since
T is an alternative contraction, by Theorem 1,
exists. It follows that
The proof is completed by and . ☐
Next, we would use the alternative UC condition as follows:
Definition 11 (Alternative UC condition). Let be a metric space and , be nonempty subsets of . If is said to satisfy the UC condition if the following holds. Let , and , , for some positive integer . If for some , then .
It is clear that the UC condition of the cyclic map is a special case of the alternative UC Condition.
Lemma 5. Let be a metric space, , be nonempty subsets of and be an alternative contraction. If there exists such that exists, for some and for some subsequence , then the following four conditions are equivalent
- (1)
,
- (2)
,
- (3)
,
- (4)
.
Proof. ,
,
and
are obvious. The remainder part is to prove
. By Corollary
,
is a decreasing sequence, we have
Since
exist,
and then taking lim on inequality
. It follows that
Combining
and
, we have
☐
Theorem 3. Let be a metric space with UC condition, , be nonempty subsets of and be an ADC with the UC condition. If there exists and a subsequence of such that , with , then there exist different k, such that .
Proof. Since
is an ADC and there exists a subsequence
such that
,
with
, by Lemma 5, we have
Since
,
, the cardinal number of
is
and the cardinal number of
is
m, by the pigeonhole principle, there exist two different
k,
such that
,
for some
j. WLOG, we assume that
, then
By Theorem 2, we have for some , and it then by and the UC condition. ☐
Theorem 4. Let be a metric space, , be nonempty subsets of . Suppose is a ALC, if for any , there exists a subsequence of with existing for , then , .
Proof. For any
,
By induction, we have . ☐
Theorem 5. Let be a metric space, , be nonempty subsets of . Suppose is an alternative contraction, if there exists and a subsequence of such that exists, , then there exist some best proximity points of T.
Proof. Since
T is an AC and by Theorem 4, we have
Moreover, by the definitions of ALC,
We can derive by
and it leads
Moreover, by Lemma 5, we observe
Let
,
and
,
, it then
by
. By
above, we can derive that
One has
by Lemma 5, this implies
Then by
and
,
Let
for some
,
. Hence
Moreover, since
and
, we have
Combining
and
, it follows that
for
. By
,
and
, we have
p is a best proximity point of
T. ☐
Theorem 6. Let be a compact metric space and , be nonempty closed subsets of . Suppose is an alternative map, then there exists a best proximity point of T.
Proof. By Lemma 3, for any integer
k, there exists a subsequence
such that
for all
i. For any
, since
is compact, we have
exists. Moreover, since
and
is closed,
, it follows that
exists and
By Theorem 5, there exists a best proximity point of
T. ☐
Corollary 2. Let be a compact(or weakly countable compact) metric space, , be nonempty closed subsets of . Suppose is an AC, then there exists a best proximity point of T.
Proof. This corollary can be derived by Theorem 5 and Theorem 6 immediately. ☐
Theorem 7. Let be a complete metric space, , be nonempty closed subsets of . Suppose is an AC with UC condition, then there exists a best proximity point of T.
Proof. Since
,
, by the pigeonhole principle, there exist
such that
,
for any
and some
. Moreover, since
T is an alternative contraction, by Theorem 1, we have
By UC condition,
. This can imply lead to a Cauchy sequence. That is
for some
. By Theorem 5, there exists a best proximity point of
T. ☐