1. Introduction
Banach contraction principle (abbreviated as: BCP) and its applications are well-known. In subsequent years, various generalizations of this pivotal result were obtained by improving the underlying contraction condition. One of the remarkable generalized contractions is
-contraction, which is obtained from usual contraction by replacing the Lipschitzian constant
with an auxiliary function
. The concept of
-contraction is essentially investigated by Browder [
1] in 1968, wherein the author considered
to be increasing and right continuous control function and utilized the same to extend the BCP. Afterward, many researchers generalized Browder fixed point theorem by modifying the properties of control function
(e.g., Boyd–Wong contractions [
2] and Matkowski contractions [
3]). Indeed, Matkowski [
3] employed a class of control functions, which are later termed as comparison functions. Matkowski contractions have been further studied in [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13] besides several others.
In 2015, Alam and Imdad [
14] established a novel variant of the BCP in a metric space equipped with an amorphous relation (see also [
15,
16]). In the recent years, various fixed point results have been proved under different types of contractivity conditions in relational metric spaces. The contraction conditions utilized in such results are indeed desired to hold for the elements that are comparative with respect to the underlying relation. These fixed point theorems can be applied to compute the unique solutions of certain matrix equations and boundary value problems (abbreviated as: BVP).
Relation-theoretic variants of Boyd–Wong fixed point result and Matkowski fixed point result are obtained recently by Alam and Imdad [
17] and Arif et al. [
18], respectively. To ensure the existence of a fixed point of a mapping satisfying such types of
-contractions, transitivity condition on underlying relation is additionally required. Due to restrictive nature of transitivity requirement, the authors [
17,
18] used an optimal condition of transitivity (locally
-transitive relation). On the other hand, Ahmadullah et al. [
19] proved a fixed point theorem in a metric space endowed with an amorphous relation satisfying generalized
-contractions using the concept of (c)-comparison functions. The contractivity condition utilized in [
19] is stronger than Matkowski contraction. However, the authors succeed to prove the results for amorphous relation instead of a transitive relation. In the setting of a relational metric space, to ascertain the uniqueness of fixed point, usually the image of ambient space must be
-connected set. However, the results of Ahmadullah et al. [
19] hold for the
-directed set, which is a particular class of
-connected set. On the other hand, the results of Alam and Imdad [
17] and Arif et al. [
18] are proved for
-connected sets.
Motivated by the above existing works, we established a fixed point theorem for a natural version of
-contractions employing (c)-comparison functions in relational metric space. In our results, the underlying relation is amorphous, while the uniqueness part requires the image of ambient space to be
-connected. This substantiates the utility of our main result over the results of Alam and Imdad [
17], Arif et al. [
18] and Ahmadullah et al. [
19]. Some examples are constructed to substantiate the utility of our findings. To demonstrate the degree of applicability of our result, we studied the existence and uniqueness of solution of certain BVP of order one.
2. Preliminaries
Throughout this presentation, will denote the set of natural numbers. Any subset of is termed as a relation (or more precisely, a binary relation) on the set .
Definition 1. Assume that is a set, ϱ is a metric on , is a relation on and is a function. One says that:
- •
Ref. [14] A pair is -comparative, denoted by , if - •
Ref. [20] The relation is transpose of . - •
Ref. [20] The relation is symmetric closure of . - •
Ref. [21] A relation on a subset defined byis the restriction of on . - •
Ref. [14] is -closed if it satisfies verifying .
- •
Ref. [14] A sequence verifying , ∀, is -preserving. - •
Ref. [15] is -complete if each -preserving Cauchy sequence in is convergent. - •
Ref. [15] is -continuous at if for every -preserving sequence satisfying , one has - •
Ref. [15] is -continuous if it is -continuous at all points of . - •
Ref. [14] is ϱ-self-closed if every -preserving sequence verifying has a subsequence satisfying . - •
Ref. [21] Given , a subset ⊂ is a path of length in from x to y if , and ∈. - •
Ref. [17] A subset is -connected if each pair of elements of admits path in . - •
Ref. [8] A subset is -directed if for each pair , ∃ satisfying and .
Proposition 1 ([
14]).
Proposition 2 ([
17]).
is -closed provided is -closed. Remark 1 (see [
17]).
If a subset of is -directed, then each pair of elements of admits a path of length 2. Consequently, is -connected. Therefore, every -directed subset of is also -connected. The following notations will be utilized in future.
- •
: = the set of fixed points of ,
- •
.
The following result proved by Alam and Imdad [
14] is popular as relation-theoretic contraction principle.
Theorem 1 ([
14,
16]).
Assume that is metric space equipped with a relation while is a function. Additionally,- (i)
is -complete,
- (ii)
,
- (iii)
is -closed,
- (iv)
is -continuous or is ϱ-self-closed,
- (v)
verifying
Then, admits a fixed point. Additionally, if is -connected, then admits a unique fixed point.
3. (c)-Comparison Functions
Let us recall two families of auxiliary functions of existing literature.
Definition 2 ([
22]).
A mapping is termed as comparison function if it enjoys the following ones:- (i)
ϕ is monotonic increasing,
- (ii)
.
Definition 3 ([
22]).
A mapping is termed as (c)-comparison function if it enjoys the following ones:- (i)
ϕ is monotonic increasing,
- (ii)
.
Clearly, every (c)-comparison function is a comparison function.
Remark 2 ([
22]).
Let ϕ be a (c)-comparison function. Then- (i)
,
- (ii)
,
- (iii)
ϕ is right continuous at 0.
In 2019, Ahmadullah et al. [
19] proved the following result:
Theorem 2 (see Theorem 2.5 [
19]).
Assume that is metric space equipped with a relation while is a function. Additionally,- (i)
is -complete,
- (ii)
,
- (iii)
is -closed,
- (iv)
is -continuous or is ϱ-self-closed,
- (v)
∃
a (c)-comparison function ϕ verifying
Then, admits a fixed point. Additionally, if is -directed, then admits a unique fixed point.
The following consequence of Theorem 2 is immediate.
Corollary 1. Assume that is metric space equipped with a relation while is a function. Additionally,
- (i)
is -complete,
- (ii)
,
- (iii)
is -closed,
- (iv)
is -continuous or is ϱ-self-closed,
- (v)
∃
a (c)-comparison function ϕ verifying
Then, admits a fixed point. Additionally, if is -directed, then admits a unique fixed point.
Now, one proposes the following fact.
Proposition 3. Assume that is metric space equipped with a relation while is a function. If ϕ is a (c)-comparison function, then the following contractivity conditions are equivalent:
- (I)
,
- (II)
Proof. The conclusion (II)⇒(I) holds trivially. Conversely, let (I) holds. Assume that
with
. Then, in case
, (I) implies (II). Otherwise, in case
, by symmetry of
and (I), one obtains
It follows that (I)⇒(II). □
4. Main Results
In view of Remark 1,
-connectedness is weaker than
-directedness. However, Ahmadullah et al. [
19] could not succeed to extend Theorem 2 to
-connected sets. Our main result, which extends Corollary 1 to
-connected sets, runs as follows:
Theorem 3. Assume that is metric space equipped with a relation while is a function. Additionally,
- (i)
is -complete,
- (ii)
,
- (iii)
is -closed,
- (iv)
is -continuous or is ϱ-self-closed,
- (v)
∃
a (c)-comparison function ϕ verifying
Then, admits a fixed point. Additionally, if is -connected, then admits a unique fixed point.
Proof. Using assumption (ii), one can choose
. Define the sequence
verifying
Using the fact
,
-closedness of
and Proposition 2, one obtains
which, making use of (
1), becomes
Thus,
is an
-preserving sequence. Applying hypothesis (v) to (
2), one finds
which by induction and monotonicity of
gives rise to
with
, making use of (
3) and triangular inequality, one obtains
which yields that
is Cauchy. Hence,
is an
-preserving Cauchy sequence and hence by
-completeness of
, ∃
verifying
.
Now, one has to use condition (iv) to prove that
p is a fixed point of
. Let
be
-continuous. Since
is
-preserving verifying
, therefore by
-continuity of
, one concludes that
. Uniqueness property of limit gives rise to
. Alternately, suppose that
is
-self-closed. Since
is
-preserving and
, therefore
has a subsequence
satisfying
,
By assumption (v), Proposition 3 and
, one obtains
Using items (i) and (ii) of Remark 2 (whether
is non-zero or zero) and
, we obtain
so that
. Uniqueness property of limit gives rise to
. Hence in both the cases,
p is a fixed point of
.
To prove uniqueness, take
, one obtains
Since
and
is
-connected, therefore ∃ a path
in
from
p to
q so that
Using
-closedness of
and Proposition 2, one has
By (
6), hypothesis (v) and Proposition 3, one has
For each fixed
j (
), two cases arise. Firstly, suppose that
for some
, i.e.,
implying thereby
. Consequently, one finds
and thus by induction, we get
so that
. Secondly, suppose that
. Using induction on
n and monotonicity of
in (
7), one has
so that
Letting
in (
9) and owing to the definition of (c)-comparison functions, one has
Therefore, in each case, (
8) is verified. In lieu of (
4), (
5), (
8), and triangular inequality, one has
so that
. Hence
admits a unique fixed point. □
5. Illustrative Examples
To attest the credibility of Theorem 3, let us adopt the following examples.
Example 1. Consider equipped with standard metric ϱ and a relation . Define a function by Then is -complete metric space, is -continuous and is -closed. One can easily check that assumption (v) of Theorem 3 holds for an arbitrary (c)-comparison function ϕ. Therefore, the assumptions (i)–(v) of Theorem 3 hold. Consequently, admits a fixed point. Since there in no path between and 1, therefore is not -connected. Note that and are two fixed points of .
Example 2. Consider equipped with standard metric ϱ and a relation Define a function by Then is -complete metric space while is -closed. Let be an -preserving sequence verifying so that . Notice that yielding thereby so that . Since is closed, therefore one has . Thus, is ϱ-self-closed. One can verify contractivity condition(v)of Theorem 3 with . Rest of the hypotheses of Theorem 3 also hold. Consequently, has a unique fixed point .
6. Application to Boundary Value Problem
Consider the following BVP:
where
and
is a continuous function.
As usual, will denote the family of all real valued continuous functions on . Let us revisit to the following notions:
Definition 4 ([
23]).
One says that forms a lower solution of (10) if Definition 5 ([
23]).
One says that forms an upper solution of (10) if One presents the existence and uniqueness theorem to determine a solution of Problem (
10) as follows.
Theorem 4. In addition to Problem (10), if ∃ and ∃ a (c)-comparison function ϕ verifying with that Further, if (10) has a lower solution, then ∃ a unique solution of Problem (10). Proof. Equation (
12) is equivalent to the integral equation
where
is Green function defined by
Define a mapping
by
Thus,
is a fixed point of
if, and only if,
forms a solution of (
13) and, hence, of (
10).
Define a metric
on
by
Undertake a relation
on
defined by
Now, one will verify all the hypotheses of Theorem 3.
- (i)
Obviously, is -complete metric space.
- (ii)
Let
be a lower solution of (
10), then one has
Multiplying on both the sides by
, one obtains
yielding thereby
Owing to
, one obtains
so that
By (
17) and (
18), one finds
so that
which yields that
so that
is non-empty.
- (iii)
Take
verifying
. Using (
11), one obtains
By (
14), (
19), and due to
,
, one obtains
which by using (
16) yields that
and hence
is
-closed.
- (iv)
Let
be an
-preserving sequence converging to
. Then for each
,
is monotone increasing sequence in
converging to
. Consequently,
and
, one has
. Again due to (
16), it follows that
and hence
is
-self-closed.
- (v)
Take
verifying
. Then by (
11), (
14), and (
15), one has
Now,
. Using the monotonicity of
, one obtains
and, hence, (
20) becomes
so that
Let
be arbitrary. Then, one has
. As
and
,
is path in
between
and
. Thus,
is
-connected and hence by Theorem 3,
admits a unique fixed point, which forms the unique solution of Problem (
10). □
Intending to illustrate Theorem 4, one considers the following example.
Example 3. Let for , then Φ is a continuous functions. Define a function by Then ϕ is a (c)-comparison function. Additionally, for any arbitrary pair with , the inequality (11) holds. Moreover, is a lower solution for . Therefore, Theorem 4 can be applied for the given problem and, hence, forms the unique solution. 7. Conclusions
Alam and Imdad [
17] and Arif et al. [
18] established fixed point results under relation-theoretic Boyd–Wong contractions and Matkowski contractions employing a class of transitive relation, namely: locally
-transitive relation. In the process, they observed that their results cannot be extended to an amorphous relation. The contraction condition utilized in our main result is restrictive as compared to Boyd–Wong contractions and Matkowski contractions, but we succeed to extend the fixed point theorem up to an amorphous binary relation. The existence part of our main result can be deduced from the result of Ahmadullah et al. [
19] but the hypothesis of uniqueness part of our result is relatively more general, which concludes that our result is independent from that of Ahmadullah et al. [
19]. Thus Theorem 3 is different from the results of Ahmadullah et al. [
19], Alam and Imdad [
17], and Arif et al. [
18].
A relatively weaker contraction condition is utilized compared with those in the recent literature, as in this work, the contraction condition is desired to hold merely for comparative elements via underlying relation and not for the entire space. Owing to this restrictive nature, results employing relation-theoretic contractions are applicable in fields of non-linear matrix equations and BVP. In future, readers can extend our results to a pair of self-mappings by proving coincidence and common fixed point theorems, which are very influential and applicable areas by their own.