1. Introduction
The classical Banach contraction principle (abbreviated as: BCP) established by Banach [
1] was widely accepted due to its simplicity as well as applicability. Ran and Reurings [
2] and Nieto and Rodríguez-López [
3] extended and improved BCP in the framework of partially ordered metric spaces. Later, Alam and Imdad [
4] investigated a novel generalization of BCP utilizing an arbitrary relation instead of partial order.
On the other hand, several diversified extensions of BCP are obtained by employing more general contractivity conditions. A self-map
on a metric space
is referred to as a linear contraction if there exists
satisfying
In inequality (
1), the constant
k plays a crucial role. Several researchers replaced the constant
k by a test function (say,
) depending upon the contractivity conditions. Any map
satisfying
for each
is referred as a control function. Any self-map
on a metric space
is known as
if it satisfies
Indeed, the notion of
-contraction was introduced by Browder [
5] in 1968 and was later generalized by Boyd and Wong [
6] and Mukherjea [
7] and Jotic [
8].
Following Khan et al. [
9], a map
is referred to as an altering distance map if it satisfies the following axioms:
- (i)
iff ,
- (ii)
remains increasing and continuous.
Dutta and Choudhury [
10] obtained a novel generalization of BCP employing a pair of auxiliary functions.
Theorem 1 ([
10]).
Assume that remains a complete metric space and remains a map from to itself. If ϕ and ψ remain two altering distance functions satisfyingfor all , then admits a unique fixed point. Later on, various authors improved Theorem 1, e.g., Doric [
11], Popescu [
12], Luong and Thuan [
13], Alam et al. [
14] and similar others. Shahzad et al. [
15] slightly modified the
-contractions by introducing the following one:
For a recent development of metric fixed point theory, refer to [
16,
17,
18] and the reference therein. The intent of the present article is to establish fixed point results for
-contractions in the sense of Shahzad et al. [
15] involving a locally finitely
-transitive relation. An example is also presented, which attests to the credibility of my results.
2. Preliminaries
This section involves some relevant notions and auxiliary results needed in main results. and denote, respectively, the set of natural numbers and that of whole numbers. Any subset of is referred to as a binary relation (or simply, a relation) on . Throughout this section, we assume that remains a set, remains a relation on and remains a map.
Definition 1 ([
4]).
Two elements are termed as -comparative (often denoted by ) if either or . Definition 2 ([
19]).
is termed as a complete relation if , for every . Definition 3 ([
19]).
is termed as inverse (transpose) of . Definition 4 ([
19]).
The relation is termed as symmetric closure of . Proposition 1 ([
4]).
Definition 5 ([
19]).
Given , the restriction of on is defined asClearly, is a relation on .
Definition 6 ([
4]).
is known as -closed if for every with , we have Example 1 ([
20]).
Consider the set equipped with a relation . Moreover, assume that is a map defined byThen, it remains to easily check that is -closed.
Proposition 2 ([
21]).
For each , is -closed whenever is -closed. Definition 7 ([
4]).
Any sequence satisfying , is known as -preserving. Definition 8 ([
22]).
is known as -complete if each -preserving Cauchy sequence in converges. It is clear that any complete metric space remains an -complete whatever . Moreover, in case , these two concepts coincide.
Example 2 ([
20]).
Consider equipped with standard metric ϱ. On , endow a relation . Then, remains -complete, however, it is not complete. Definition 9 ([
22]).
is known as -continuous at if for every -preserving sequence of elements of satisfying , we haveMoreover, if remains -continuous at each point of , then it is known as -continuous.
It is clear that any continuous map remains an -continuous whatever . Moreover, in case , these two concepts coincide.
Example 3 ([
20]).
Consider equipped with standard metric ϱ. On , endow a relation . If remains a map from to given byThen, remains -continuous, however, it is not continuous.
Definition 10 ([
4]).
is known as a ϱ-self-closed relation if every -preserving sequence (of elements of ) satisfying (for some ) admits a subsequence satisfying , for all Definition 11 ([
23]).
Any subset of is termed as -directed if for every pair , satisfying and . Definition 12 ([
24]).
A path of length k in a relation from r to t (whereas ) remains a finite sequence ⊂ satisfying- (i)
and ,
- (ii)
∈.
Definition 13 ([
21]).
A subset of is termed as an -connected set if each pair of elements of admits a path between them. Definition 14 ([
19]).
is termed as transitive if and implies . Definition 15 ([
21,
25]).
is termed as -transitive if for every triplets satisfying , we have Definition 16 ([
26]).
Given , , a relation is termed as m-transitive if for any satisfying , for we have Thus, a 2-transitive relation means the transitive relation.
is known as a finitely transitive relation if it remains
m-transitive for some
(cf. [
27]).
Definition 17 ([
27]).
is termed as locally finitely transitive if for every countably infinite subset of , , such that remains m-transitive. To make the two independent concepts (
-transitivity and locally finitely transitivity) compatible, Alam et al. [
28] initiated yet a new concept of transitivity as follows:
Definition 18 ([
28]).
is termed as locally finitely -transitive if for every every countably infinite subset of of , , such that remains m-transitive. In lieu of above the definitions, it is clear that the class of locally -transitive binary relation includes the classes of other types of transitive binary relations.
The statement of the relation-theoretic analogue of BCP established by Alam and Imdad [
4] is given as:
Theorem 2 ([
4,
20,
22]).
Assume that remains a metric space, remains a relation on and remains a map. Moreover,- (i)
remains-complete;
- (ii)
remains-closed;
- (iii)
eitherremains-continuous orremains ϱ-self-closed;
- (iv)
such that;
- (v)
such that
Then,admits a fixed point. Moreover, ifremains-connected, thenadmits a unique fixed point.
Finally, the following two known results are stated.
Lemma 1 ([
29]).
If the sequence , in a metric space , is not a Cauchy, then we are able to find an and two subsequences and of satisfying- (i)
, ;
- (ii)
;
- (iii)
.
Moreover, if , then Lemma 2 ([
27]).
Assume that is a set equipped with a relation and remains -preserving sequence. Furthermore, suppose that remains m-transitive on , then 3. Main Results
In what follows, denotes the class of all pair of auxiliary functions , wherein enjoy the following properties:
If the sequence verifies , , then .
Whenever two convergent sequences have a common limit L such that , and , then .
The above family of pair of functions is suggested by Shahzad et al. [
15].
Theorem 3. Assume that remains a metric space, remains a relation on and remains a map. Moreover,
- (a)
remains -complete;
- (b)
remains -closed as well as locally finitely -transitive;
- (c)
remains -continuous;
- (d)
such that ;
- (e)
such that
then admits a fixed point.
Proof. By assumption
, choose
, then we have
. Construct the sequence
such that
As
, assumption
and Proposition 2, we have
which, in lieu of (
2), becomes
This means that remains -preserving.
If there exists
such that
, then by (
2)
remains a fixed point of
. Otherwise, in case
for all
, we use assumption
to obtain
so that
Making use of the axiom
, we obtain
Suppose that
is not a Cauchy sequence. Consequently, Lemma 1 guarantees the existence of
and two subsequences
and
of
satisfying
,
and
, where
. Moreover, due to the availability of (
4), we have
Since , the range remains a denumerable subset of . Therefore, using locally finitely -transitivity of , we are able to find a natural number , such that remains m-transitive.
Since
and
, therefore applying division algorithm, one obtains
It can be noticed that
and
remain suitable numbers, so that the value of
may be considered finitely. Consequently, we are able to choose subsequences
and
of
(verifying (
5)) in such a way that
becomes a constant (say,
). We have
whereas
remains constant. Making use of (
5) and (
6), one obtains
Using triangular inequality, we have
and
or
Letting
in (
8) and (
9) and using (
4) and (
7), we obtain
Due to the availability of (
6) and Lemma 2, we obtain
. Furthermore, by assumption
, we obtain
Applying condition
to
,
and
, we find that
, which remains a contradiction. Hence,
is Cauchy. By
-completeness of
,
such that
. The
-continuity of
gives rise to
, which in view of (
2), reduces to
Finally, by uniqueness of limit, we obtain
. □
Theorem 4. Theorem 3 remains valid if assumption is replaced by the following condition:
is ϱ-self-closed, and the pair verifies the following property:
If verifies and , , then .
Proof. Similar to previous result, it can be shown that
Using the assumption
, we show that
r remains a fixed point of
. Since
remains an
-preserving sequence satisfying
, therefore by
-self-closedness of
, there exists a subsequence of
of
satisfying
for all
. Therefore, using assumption
, one obtains
Due to the fact that
, and the continuity of
, we have
as
. Therefore, making use of the property of the pair
, we have
so that
By uniqueness of limit, one obtains . □
Finally, the following uniqueness result is presented.
Theorem 5. Under the hypotheses of Theorem 3 (or Theorem 4), if remains -connected, then admits a unique fixed point.
Proof. In view of Theorem 3 (or Theorem 4), if
r and
t remain two fixed points of
, then
Clearly
. By the
-connectedness of
, we can find a path
in
from
r to
t so that
As
is
-closed, we have
Assume that
which gives rise
. Now, applying (
2), one obtains
. Hence,
. Therefore, by mathematical induction, one obtains
, ∀
, thereby implying t
.
As either case, one can assume that
∀
. Making use of (
12) together with condition
, one obtains
so that
Applying the property
, the above inequality yields that
Hence, in both cases, (
13) is proved. Consequently, one obtains
so that
. □
Corollary 1. Theorem 5 is valid if “-connectedness of ” is replaced by one of the following:
(i) remains -directed;
(ii) remains complete.
Proof. Assume that condition (i) holds. Take . Then, by assumption (i), one can find satisfying and . This implies that remains a path of length 2 in from r to t. Therefore, remains -connected and hence by Theorem 5, the conclusion holds.
If the assumption (ii) holds, then for each , we have . This implies that remains a path of length 1 in from r to t. Consequently, remains -connected and hence by Theorem 5, the conclusion holds. □
For , Theorem 5 reduces to:
Corollary 2. Assume that remains a complete metric space and remains a map from to itself. If there exists satisfyingfor all , then admits a unique fixed point. 4. An Illustrative Example
Now, I give an example in support of Theorem 3.
Example 4. Consider with a metric ϱ and a relation defined bythen is a -complete metric space. Define as follows:and Clearly, . Assume that is a map defined by Take with , then . Then, we have the following cases:
Case-1: When , then we have Case-2: When . If , then we have Otherwise, if , then we have Therefore, is -continuous and satisfies assumption of Theorem 3. Notice that here remains locally finitely -transitive. Moreover, the relation remains -closed. The rest of the conditions of Theorems 3 and 5 are also satisfied. Consequently, admits a unique fixed point (namely: ).
5. Conclusions
In this manuscript, the fixed point results in the framework of natural structure, namely, metric space (abbreviated as: MS) endowed with a locally finitely -transitive relation employing a pair of auxiliary functions, were proved. For future works, the analogues of these results can be proved in generalized metrical structure (such as, semi MS, quasi MS, pseudo MS, multiplicative MS, dislocated space, D-MS, 2-MS, S-MS, G-MS, b-MS, partial MS, cone MS, complex-valued MS, fuzzy MS, -MS, modular space and rectangular MS) endowed with locally finitely -transitive relations.