1. Introduction
Throughout this manuscript, the following notations and abbreviations will be used:
: the set of natural numbers;
: the set of real numbers;
;
;
: the space of continuous; functions from into ;
: the space of continuously differentiable functions from into ;
BCP: Banach contraction principle;
MS: metric space;
CMS: complete metric space;
BVP: boundary value problem.
Given a map , we have the following:
: fixed-point set of a self-map ;
;
.
The classical BCP [
1] serves as a key concept in nonlinear analysis. The BCP has been expanded and generalized in numerous ways. There is a number of extrapolations of the BCP by means of
-contraction in the extant literature, e.g., [
2,
3,
4,
5,
6]. A function
that verifies for every
that
is mentioned to as a control function. A self-map
on an MS
is termed as a nonlinear contraction relative to a control mapping
(in short:
-contraction) if
. Historically, Browder [
2] presented a first generalization for a class of nonlinear contractions. The finding of Browder [
2] was further strengthened by Boyd and Wong [
3], Mukherjea [
4], Jotić [
5] and Matkowski [
6]. Inspired by Suzuki [
7], Pant [
8] presented a recent finding under nonlinear contraction, which is presented as follows.
Theorem 1 ([
8])
. If is a CMS, remains a map and ϕ remains a strictly increasing, right continuous control function verifying where . Then, ζ admits a unique fixed point. During the past few years, several researchers have developed and broadened the BCP to ordered metric spaces, c.f. [
9,
10,
11,
12]. Subsequently, in 2015, the BCP was extended to a MS imposed with an arbitrary relation (rather than a partial order) by Alam and Imdad [
13]. The outcomes of this type can be found in [
14,
15,
16,
17,
18,
19,
20,
21] and references throughout. Such findings are used to solve elastic beam equations, nonlinear elliptic problems, ordinary differential equations, integral equations, fractional differential equations and matrix equations verifying certain auxiliary conditions, e.g., [
22,
23,
24,
25,
26,
27,
28,
29,
30].
This paper’s objective is to provide a relation-theoretic version of Suzuki–Ćirić-type nonlinear contractions through making use of comparison functions and implemented the same to determine the outcomes on fixed points employing a locally
-transitive relation. Our established outcomes generalize various existing findings, notably those reported by Alam and Imdad [
13], Pant [
8], Agarwal et al. [
12], Arif et al. [
17] and others. We offer an illustrative example that demonstrates the reliability of established findings. Employing our outcomes, we investigate the presence of a unique solution to an order-one periodic BVP.
As was previously noted, in comparison to conditions in the current literature, a considerably sharp contractive condition is used. Due to this limitation, the results presented here and in subsequent studies can be used for the BVP and the areas of nonlinear elliptic issues, fractional differential equations, matrix equations, Fredholm integral equations and delayed hematopoiesis models.
2. Preliminaries
Given a nonempty set , a subset of is termed as a binary relation (or simply, a relation) on . In the following, let be an MS endued with a relation and a map . We collect the following.
Definition 1 ([
13])
. Elements w and are ϱ-comparative if either or . We denote this by . Definition 2 ([
31])
. The relation is an inverse of ϱ. Definition 3 ([
31])
. being a symmetric relation is a symmetric closure of ϱ. Proposition 1 ([
13])
. Definition 4 ([
31])
. ϱ is complete if for every , we have Definition 5 ([
32])
. Given , the set (a relation on ) remains a restriction of on . Inspired by Roldán-López-de-Hierro et al. [
33] and Turinici [
34], Alam and Imdad [
14] introduced the next concept.
Definition 6 ([
14])
. ϱ is locally ζ-transitive if for every enumerable set , the restriction is transitive. Definition 7 ([
13])
. is ϱ-preserving if Definition 8 ([
15])
. remains ϱ-complete if every ϱ-preserving Cauchy sequence in converges. Definition 9 ([
13])
. ϱ remains ζ-closed if for every , Proposition 2 ([
14])
. For all , ϱ is -closed when ϱ is ζ-closed. Definition 10 ([
15])
. ζ is ϱ-continuous if for each and for any ϱ-preserving sequence such that , we have . Definition 11 ([
13])
. ϱ is ς-self-closed if any ϱ-preserving sequence with admits a subsequence Remark 1. Evidently, we have .
Proposition 3 ([
35])
. Every comparison function is a control function. 3. Main Results
We begin this section with the idea of Suzuki–Ćirić-type nonlinear contraction mappings via comparison functions as follows.
Definition 12. Suppose that is an MS imposed with a relation ϱ and ζ is a self-map on . If ϕ is a comparison function that verifies for every thatthen ζ is called a Suzuki–Ćirić-type nonlinear ϱ-contraction map. Metric symmetricity proposes the following fact.
Proposition 4. ζ is a Suzuki–Ćirić-type nonlinear ϱ-contraction⟺ζ is a Suzuki–Ćirić-type nonlinear -contraction.
Proof. Since every Suzuki–Ćirić-type nonlinear -contraction ⟹ Suzuki–Ćirić-type nonlinear -contraction (due to Proposition 1). To assert that is a Suzuki–Ćirić-type nonlinear -contraction ⟹ is a Suzuki–Ćirić-type nonlinear -contraction, assume that (I) is a Suzuki–Ćirić-type nonlinear -contraction. Choose such that . Then, the conclusion follows from assumption (I). Otherwise, if , then by assumption (I) and the symmetricity of d, we obtain the same conclusion again. □
The following notation will be utilised in our finding:
We present the outcome on existence for fixed points as follows.
Theorem 2. Suppose that is an MS imposed with a relation ϱ and ζ is a self-map on . Then, we also have the following:
- (a)
is ϱ-complete;
- (b)
;
- (c)
ϱ is ζ-closed and locally ζ-transitive;
- (d)
ζ is a Suzuki–Ćirić-type nonlinear ϱ-contraction relative to some comparison function ϕ;
- (e)
either ζ remains ϱ-continuous or ϱ serves as ς-self-closed.
Then, ζ has a fixed point.
Proof. Owing to
, let
. Then,
. Set
. As
is
-closed and using Proposition 2, we have
Therefore, the sequence is -preserving. If for some , then we have so that and so we are done.
If
then
. By (
1),
and the monotone property of
, one can find that for each
,
so that
In the case that
, by the control property of
and (
2), we obtain
, which raises a contradiction, and so (
2) becomes
which, in employing (
1),
and the monotone property of
, becomes
Taking
in (
4) and by the property of
, we find
For a given
due to (
5), we can determine
verifying
To verify that
is Cauchy, the monotone-property of
, (
1) and (
5) are employed so that
Again, owing to monotone property
, (
1) and the local
-transitivity of
we obtain
An easy induction on the above yields
which verifies that
is Cauchy. By the
-completeness of
,
with
.
If is -continuous, then . So, we find . Alternately, when remains -self-closed, ∃ a subsequence of with .
We assert that (for all
)
On the contrary, for some
, let us assume that
Applying the triangle inequality, we obtain
which is a contradiction. Consequently, (
7) holds for all
.
By
(in view of (
7)), Proposition 4 and
, we find
Set
. On the contrary, let us assume that
Employing
, Proposition 4 and
for all
we obtain
where in
If
, then (
8) reduces to
which, in using
, reduces to
which raises a contradiction.
then due to the fact that
, we can determine
with
By the monotone property of
, we have
By (
8) and (
9), we obtain
Making
and employing Proposition 3, we find
which raises a contradiction implying that
. Thus, we obtain
Hence, in both cases, , which ends the proof. □
Corollary 1. The conclusion of Theorem 2 remains true if ϕ is a strictly increasing, right continuous and control function instead of a comparison function.
Corollary 2. Theorem 2 remains true if ϱ remains a transitive relation instead of locally ζ-transitive.
The following is an outline of our uniqueness result.
Theorem 3. Under the assumptions of Theorem 2, if ϱ is a complete relation then ζ owns a unique fixed-point.
Proof. Since
(in lieu of Theorem 2), we can therefore choose
, i.e.,
. We prove that
If
, then the completeness of
implies that
. As
, employing
, we find
which contradicts our supposition. Hence, we have
. □
Remark 2. We conclude certain special demonstrations in the lines that follow. These are enhanced varieties of many famous fixed-point statements.
- (1)
If we choose to be , (universal relation) and ϕ to be an increasing continuous control function, then Theorem 3 reduces to the corresponding result of Pant [8]. - (2)
In taking to be and ϱ to be a partial order relation ⪯ in Theorem 3, we determine a sharpened variety of the main result of Agarwal et al. [12] in the context of the Suzuki condition. - (3)
In taking to be in Theorem 2, we obtain sharpened version of the main result of Arif et al. [17] in the context of Suzuki condition. - (4)
Under the Suzuki condition and the Ćirić contractive condition, Theorem 3 offers an enhanced version of Theorem 3.1 by Alam and Imdad [13] for (where ). The locally ζ-transitivity criterion on the involved relation is relaxed, still in this case. - (5)
If we choose to be , (universal relation) and (where ), then Theorem 3 reduces to the BCP [1] (without using the Suzuki condition).
4. An Illustrative Example
Now, we provide an example to illustrate the significance of our findings.
Example 1. Let be provided with the relation and the standard metric ς. Although ϱ is locally ζ-transitive, it is not transitive. Define a self-mapping ζ on byClearly, ϱ is ζ-closed, and is a ϱ-complete MS. Define a comparison function by . For any , the contraction condition of Theorem 2 is easily verified, with the exception of . So, we need to check the condition at . For , we have , and hence, Secondly, if we choose , thenand Taking any ϱ-preserving sequence such that As , for each one can determine verifying ; This indicates that ς-self closed is ϱ. Theorem 2 holds in all of its assumptions, and as an outcome, ζ possesses a fixed point. Since is not a complete MS in this case and ϱ is not a partial order, the findings of Pant [8] and Agarwal et al. [12] do not apply to this situation. This shows the utility of our findings. 5. An Application to BVP
This section concludes with studying the existence and uniqueness of the solution for following one-order periodic BVP:
where
, and
remains a known continuous function.
Definition 13 ([
10])
. is termed as a lower solution of (
10)
if Definition 14 ([
10])
. is termed an upper solution of (
10)
if The primary outcome of this section is now presented.
Theorem 4. Together with (
10)
, if ∃ and a comparison function ϕ verifying and , then Moreover, if there is a lower solution for (
10)
, then there is a unique solution for (
10).
Proof. Problem (
10) can be rearranged as
which is equivalent to the integral equation
where the Green function
is defined as
Set
. Define the function
as
Thus,
remains a fixed point of
iff
remains a solution of (
13) and hence of (
10).
Define a metric
on
by
Consider the following relation
on
:
Now, we verify all conditions of Theorem 2:
(a) Obviously, remains a -complete MS.
(b) Let
be a lower solution of (
10). Then, one has
Multiplying with
, we obtain
thereby yielding
Owing to
, one obtains
so that
By (
17) and (
18), one finds
so that
which yields that
so that
remains nonempty.
(c) Take
. Using (
11), one obtains
By (
14), (
19) and
, one finds
which, in utilizing (
16), yields that
; so,
is
-closed.
(d) Take
with
. Then, by (
11), (
14) and (
15), one has
Now,
. By the monotone property of
, we find
, and hence, (
20) reduces to
so that
(e) Let
be an
-preserving sequence converging to
. Then, for every
,
remains an increasing sequence in
that converges to
. So,
and
, we have
. As before,
, is the result of (
16), and as such,
is
-self-closed.
Let
be arbitrary. Then, for every
, we have
or
. Consequently, we have
, and hence,
is complete. As of right now,
possesses a unique fixed point according to Theorem 3, which preserves it a unique solution of problem (
10). □
6. Conclusions
In this work, we have proven the existence as well as uniqueness of the fixed point of a self-map under a Suzuki–Ćirić-type nonlinear -contraction map utilizing a locally -transitive relation . Also, an illustrative example is offered to demonstrate the usefulness of our present work over corresponding known findings. We also used our results to determine a unique solution of certain BVP when a lower solution is provided. Analogously, one can find similar application in the presence of an upper solution. Moreover, the development of our findings is in generalizing our findings under various types of nonlinear contractions, namely, weak contractions, -contractions, weaker versions of Matkoski contractions and several auxiliary functions. In the future, these can be extended to utilize our findings through different types of frameworks such as, symmetric spaces, partial MS, b-MS, cone MS, etc.
Author Contributions
All authors contributed equally in preparing the manuscript. The earlier version of the article was read and approved by the authors. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project, project number PNURSP2024R174, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Data Availability Statement
No new data were created or analyzed in this study.
Acknowledgments
The first author acknowledges the Princess Nourah bint Abdulrahman University Researchers Supporting Project, project number (PNURSP2024R174), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare no conflicts of interest.
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