1. Introduction
The Banach fixed-point theorem [
1], also known as the contraction mapping principle, is a fundamental result in functional analysis. It states that if a self-map on a complete metric space satisfies the contraction condition, then it possesses a unique fixed point within that space.
A mapping on a metric space is called a Banach contraction if there exists a constant such that for all , we have .
The Banach fixed-point theorem, named after the Polish mathematician Stefan Banach, has broad applications across various mathematical domains. It plays a crucial role in solving ordinary and partial differential equations, optimization problems, integral equations, and variational analysis. Moreover, its significance extends beyond mathematics into fields like game theory, economics, biology, and more.
The theorem ensures the existence and uniqueness of fixed points for self-maps that satisfy contraction criteria within a complete metric space. Over the years, the Banach fixed-point theorem has been extensively studied using a variety of methodological techniques, contributing significantly to the advancement of mathematical theory and its applications. As part of these studies, the mapping’s scope has been expanded; Nadler [
2] was the first to propose the generalization of the Banach contraction theorem for multi-valued contractions. Kikkawa and Suzuki [
3] obtained three fixed-point approaches to generalized contractions, while Din et al. [
4] investigated single- and multi-valued
F-contractions under binary relations. A new kind of contractive multi-valued operator was presented by Moţ and Petruşel [
5], whereas Sintunavarat and Kumam [
6] provided a common fixed-point theorem for cyclic generalized multi-valued mappings. Multi-valued contractions on b-metric spaces were explained by Petre and Bota [
7], and multi-valued fixed-point theorems in dislocated b-metric spaces were discussed by Rasham et al. [
8] with applications to nonlinear integral equations.
The criteria for contraction mappings have been broadened in various ways, thereby increasing the versatility of the Banach contraction theorem. Browder [
9] examined nonexpansive nonlinear operators on Banach spaces and derived fixed-point results. Kannan introduced the concept of a contraction that does not require the continuity of the self-operator but still ensures the existence of a fixed point [
10]. Wardowski and Dung discussed weak F-contractions and related fixed-point theorems [
11]. Karapinar defined interpolative Kannan contractions and presented corresponding fixed-point results [
12]. Another notable contribution was made by Jleli and Samet [
13], who introduced the concept of
-contraction to establish fixed-point theorems, representing a fascinating and profound extension of the Banach fixed-point theorem. This achievement was further advanced by Altun et al. [
14], who expanded upon it to explore fixed-point results for Perov-type
-contractions. Additionally, Alam and Imdad [
15] initiated the idea of binary relations on metric spaces for single-valued Banach contraction, Lipschutz [
16] and Agarwal et al. [
17] discussed the ordered fixed-point results for Banach spaces with applications in nonlinear integral equations, Hussain et al. [
18] studied the Krasnoselskii and Ky Fan-type ordered fixed points over Banach spaces, and Ran and Reurings [
19] explored fundamental concepts related to binary relations and partial ordered theoretic fixed-point theorems. While Berzig [
20] dealt with a class of matrix equations using the Bhaskar–Lakshmikantham coupled fixed-point theorem, Berzig and Samet [
21] explored the systems of nonlinear matrix equations involving Lipshitzian mappings, and Long et al. [
22] focused on determining the conditions for the existence of a solution of the nonlinear matrix equation
. Vetro and Radenović [
23] discussed some Perov-type results in rectangular cone metric spaces, while Guran et al. [
24] explored some multi-valued results in the metric spaces of Perov’s type.
The notion of an improved vector-valued metric space using a binary relation was recently presented by Almalki et al. [
25]. They used the
F-contraction in the context of a generalized metric space that has a binary connection to extend the Perov and Filip–Petrusel fixed-point theorems [
26,
27] to single-valued and multi-valued mappings. Compared to the typical contractive inequality, the contractive inequality in this case is relatively weaker. In this case, rather than throughout the entire space, the contractive inequality must only be satisfied among components that are related to one another according to the binary relation.
In this work, we will explore fixed points for single- and multi-valued mappings by employing the notion of a binary relation and stressing -contraction. Additionally, we will offer examples to show the validity of our findings and an existence condition for the solutions of a system of matrix equations as well.
2. Preliminaries
We provide a summary of the key concepts required to establish our findings in this section. Let be a non-empty set. The set of all non-negative real numbers is denoted by . The set of all real matrices is represented by , which is the set of all matrices with entries greater than . In the event that , they have the following forms: and , where T stands for matrix transposition. We denote (or ) to indicate that (or .
Definition 1 ([
26]).
A function is defined as a vector-valued metric on if , the following hold:1: ;
2: if and only if ;
3: ;
4: .
In this case, the zero matrix of order is denoted as . Consequently, a generalized metric space or vector-valued metric space is defined as the pair .
It is important to know that the concepts of completeness, Cauchy sequences, and convergent sequences are similar to those in a usual metric space. The zero matrix of order is represented as , the identity matrix as , and the set of all square matrices of order with non-negative entries as . Notably, we have for every .
Definition 2 ([
26]).
Consider . We say that N is a matrix converging to zero if approaches as n tends to infinity. Example 1. , where and , is a matrix convergent to zero in .
Example 2. If , then converges to zero.
From Petrusel–Filip [
27], some other comparable conditions for Definition 2 are as follows.
Proposition 1 ([
27]).
Let ; then, the following statements are comparable:- (a)
The matrix as ;
- (b)
Every complex number λ with is contained in disc ;
- (c)
Det and
- (d)
as , for all .
Example 3. If and , thenis not convergent to zero. Definition 3. Assume that the set is not empty. The Cartesian product on is then given by the following definition: A binary relation on is defined as any subset of .
Observe that one of the following two circumstances must apply to each pair :
- (1)
indicates that and are related under or that is related to under . Additionally, we can write as .
- (2)
indicates that either does not relate to under or is not -related to . As , we can alternatively write .
Trivial binary relations on are defined as and , which are two trivial subsets of .
Definition 4 ([
15]).
Consider a binary relation, denoted as , defined over the non-empty set . Then, if either or , then any two elements are -Comparative. We set if are -Comparative. By establishing appropriate conditions, it is possible to categorize a binary relation into various types. Various widely recognized binary relations, along with their significant properties, are detailed in [
15,
16]. The following is a well-known proposition in binary relations.
Proposition 2 ([
16]).
Assume that represents the universal relation established on a non-empty set . In this case, is a full equivalence relation. Definition 5 ([
25]).
Assume a binary relation on . A sequence is termed -preserving if Alam and Imdad introduced the idea of
-self-closedness for any
defined on some
, as elaborated in [
15]. This concept was further elaborated by Almaliki et al. in [
25] as follows.
Definition 6 ([
25]).
In a generalized metric space a sequence that converges to while preserving the relation is called -self-closed with a binary relation if and only if is a subsequence of such that, for each , . Definition 7 ([
15]).
Let be a mapping, where . Then, is called -closed if Lemma 1 ([
25]).
Let , be a binary relation on , and be a mapping. Then, being -closed implies that is -closed, where . Definition 8 ([
16]).
Given and a binary relation defined on , a subset of is considered -directed if for all , there exists such that both and . The concept of a path between two points within a set furnished with a binary relation in a vector-valued metric space was introduced by Almaliki et al. in [
25] as follows.
Definition 9 ([
25]).
Let be a binary relation on . A path of length in from to is said to be a path of length in for iff:(1): and ;
(2): holds for every .
Note that while not always distinct, every path of length contains members of .
For our inquiry, we next need the following ideas from the work of Almaliki et al. [
25].
Definition 10 ([
25]).
A compound structure is defined as the pair , which consists of an arbitrary binary relation and a single-valued mapping over a vector-valued metric space such that the following are true:- (i):
;
- (ii):
is -self-closed;
- (iii):
is -closed.
Definition 11 ([
25]).
Let and have their usual meanings as discussed earlier. Consider a multi-valued mapping . Then, a binary relation over is termed -closed if for every pair , Definition 12 ([
25]).
Denote any vector-valued metric space with a binary relation by denoting . Let . The class of all non-empty subsets of is denoted by . Then, for multi-valued mappings, the pair is considered to be a compound structure if the following requirements are satisfied:- 1.
is -closed;
- 2.
;
- 3.
is strongly -self-closed; that is, for every sequence in with for all natural numbers and , we obtain for all , where a positive integer is used.
Theorem 1 ([
15]).
Consider a self mapping and a binary relation on a complete vector-valued metric space such that the following are satisfied:(i) The pair forms a compound structure.
(ii) For all with , the conditionholds, where converges to zero. Then, possesses a fixed point. (iii) Moreover, if , then the fixed point of is unique.
The term
-contraction was first used by Altun et al. [
14], who defined it as follows.
Definition 13 ([
14]).
Let be a function, where denotes the set of all real matrices with entries exceeding j. The function satisfies the following properties:- Θ1
For any , , if , then .
- Θ2
For each sequence of , where .
- Θ3
There exist and , such that , , where
The set of all functions θ that satisfy to is denoted as .
Example 4. Let be given by Then, .
Example 5. Let be given by Then, .
Altun et al. [
14] initiated the concept of Perov-type
-contraction by employing the family
and defining the notion
, where
.
Definition 14 ([
14]).
A mapping on a vector-valued metric space is regarded as a Perov-type θ-contraction if and , with each , such that For a Perov-type
-contraction in vector-valued metric spaces, the fixed-point theorem proved by Altun et al. [
14] is as follows.
Theorem 2. Let be a Perov-type θ-contraction and be any complete vector-valued metric space. Then, in , the fixed point of is unique.
3. Main Results
Before discussing our findings, we define a Perov-type -contraction enriched with a binary relation and state a lemma that will be useful in the proof. This section presents a fixed-point theorem for single-valued theoretic-order Perov-type -contraction. We also generalize this theorem to multi-valued mappings.
Definition 15. Assume any vector-valued metric space and any arbitrary binary relation . Then, if there exists with each and , then this single-valued self-mapping of on is referred to as a Perov-type θ-contraction enriched with binary relation if with .
Lemma 2. Let represent any vector-valued metric space with a binary relation . If is a Perov-type θ-contraction enriched with the binary relation , where with each and , then the following statements are equivalent (given ):
- 1.
with
- 2.
with
Proof. This lemma’s proof is straightforward and limited to the use of the metric’s symmetric condition. □
We now present the following initial result for a single-valued Perov-type -contraction.
Theorem 3. In any complete vector-valued metric space with an arbitrary binary relation , let be a Perov-type θ-contraction. Assuming that on forms a compound structure with , then .
Also, if , then is a singleton set.
Proof. Let
be any element. We define an iterative sequence
. According to the definition of
, we have
. Since
is
-closed, we can observe the following chain of relations:
Therefore, the sequence preserves the binary relation . If for some , we have , then , showing that is a fixed point of in . Otherwise, if for all natural numbers n, then for all .
Thus, we obtain
and using Equation (
2), we obtain
Using this information consistently, we obtain
Since each
is less than 1,
Therefore, based on
, we can deduce that for each
,
According to
, there exist
in the interval
and
in the interval
such that for each
,
In the case where
is finite, if we set
and using the definition of a limit, we can find an
for which this yields that for all
and
,
or
After rearranging the expression, we obtain that
and
,
Now, if
, for
such that for all
and
,
Using (
5), (
6), and (
4), we obtain
By considering the limit as
n approaches infinity in Equation (
7), we have for each
,
So, for
, there exists
such that for all
,
Consequently, for any
, we derive the following expression: for each
,
or
For
, we now claim that
is a Cauchy sequence. To demonstrate this, we use the triangular inequality with inequality (
8) and
to obtain
This reveals that in is Cauchy. It follows that such that for every , .
Applying the definitions of
and
, we find that for all
where
,
According to the definition of
,
is
-self-closed. Then, for a sequence
that preserves
and converges to
, there must exist a subsequence
of
such that
and
for all
. Lemma 2 and inequality (
9) therefore allow us to obtain, for
,
which yields
Hence,
which shows that
.
To demonstrate that has a cardinality of one, we begin by assuming that for all . This implies that there exists a path between every pair of points in . Now, suppose to the contrary that with (i.e., . Thus, for , there exists satisfying the following:
- 1.
and
- 2.
,
So, by letting
, we obtain
Thus, . □
Theorem 4. Let represent a vector-valued metric space that is complete, along with a binary relation , and a multi-valued mapping . Additionally, suppose that the following are satisfied:
- 1.
The pair forms a compound structure;
- 2.
and , s.t. where with each and .
Then, has a fixed point.
Proof. Given any element
, there exists a
such that
. From the definition of a compound structure, we can say that for
and
, there exists a
such that
Since
is
-closed,
. Moreover, by assumption, for
and
,
for which
and
, which implies, by inequality (
11), that
where
. We obtain a sequence
by repeating the same procedure. This sequence is defined by
,
, with the property that
(i.e.,
is
-preserving ) and
Then, using methodology similar to that used in Theorem 3, yields that for all
and
,
Now, in accordance with inequality (
13) and the triangular inequality, for
,
Consequently,
in
is a Cauchy sequence. We obtain
such that
using the completeness property of
. Using
and given assumption, we determine that for each
implies that
We obtain
, since
is strongly
-self-closed, where
is any natural number and
for all
. In the context of a given assumption and inequality (
14),
such that for all
, for all
, and
,
Hence, . □
Remark 1. It is crucial to note that holds true for all if represents a full order or if is -directed.
Proof. If forms a complete order, then every pair is -comparative, implying that for all . This indicates that constitutes a path from to with a length of 1 in . Consequently, for all .
If is an -directed set, then for every , there exists such that and . This demonstrates that for each , we have a path from to with a length of 2 in . Thus, is non-empty for every . □
Corollary 1. If is an -directed set or represents a full order, then all the hypotheses of Theorem 3 hold. Then, there exists a unique fixed point for .
By setting the full relation, that is, taking
in Theorem 3, we obtain the main result of Jleli and Samet [
13] as follows.
Corollary 2. In any complete vector-valued metric space , let be a Perov-type θ-contraction. Then, has a unique fixed point in .
Example 6. For , let be equipped with the standard vector-valued metric, that is, . Consider the sequence in defined by Define the binary relation as follows: Now, define the mapping by Then, is continuous, . Next, we will show that satisfies the contraction condition for for . Now, let be such that and . It must be the case that and for some . To prove contraction condition (2), it is enough to show that for some ; that is, we have to show that for some . Now, observe that Hence, inequality (2) holds for . Therefore, all the hypotheses of Theorem 3 are satisfied. Therefore, has a fixed point in . Remark 2. The theorem referenced in [14] is not appropriate in the circumstances of Example 6 due to the fact that , which means that their contraction conditions are not satisfied. Therefore, our findings represent a suitable extension of the work by Altun et al. [14]. 4. Applications Associated with Nonlinear System of Matrix Equations
Fixed-point theorems have undergone thorough explorations for a variety of functions within ordered metric spaces, resulting in numerous applications spanning across different fields of science and mathematics. In particular, these theorems find significance in contexts involving differential, integral, and matrix equations. These extensive studies and their applications are well-documented in various references such as [
17,
18,
19], along with additional sources referenced therein.
Let
represent the set of all
complex matrices,
the set of all Hermitian matrices in
,
the collection of all positive definite matrices in
, and
the class of all positive semidefinite matrices in
. To indicate that a matrix
N in
(or
) is positive definite (or positive semidefinite), we use the notation
(or
). Furthermore,
(or
) implies that
(or
). The symbol
represents the vector-valued spectral norm of matrix
, defined as
, where
is the largest eigenvalue of
, with
being the conjugate transpose of
. Moreover,
is defined as
, where
(for
) denotes the singular values of
. The pair
constitutes a complete vector-valued metric space (for further details, refer to References [
19,
20,
21]). Furthermore, the binary relation ⪯ on
is defined as follows:
if and only if
holds for all
.
In this section, we utilize our findings to provide a solution to the system comprising two nonlinear matrix equations as presented below.
Here, represents a continuous, order-preserving mapping with , denotes a Hermitian positive definite matrix, and are arbitrary matrices, with denoting their conjugates.
Now, we present the following lemmas, which prove to be beneficial in the subsequent results.
Lemma 3 ([
19]).
Let such that and . Then, Lemma 4 ([
22]).
If such that , then . Theorem 5. Take into account the system of Matrix Equation (15), along with the parameters L and being positive real numbers such that we have the following: - 1.
For with and , we have - 2.
and .
Then, the system of Equation (15) has a solution. Proof. Define the operators
by
In such a case, ’s elements are properly defined, and the ordering ⪯ on is closed under both and .
Now, define a mapping
by
Then, the system of Equation (
15) can be transformed to the following fixed-point problem:
Given that both and maintain closure under the binary relation ⪯, this consequently implies that the operator also upholds closure under ⪯. Furthermore, since for all , it follows that for both .
Next, we show that (
2) holds for the mapping
. For this purpose, take
with
as given by
Consider
such that
and
, which implies that
. Given that
is order-preserving, it follows that
for
. Therefore,
which further implies that
This shows that
satisfies all the assumptions of Theorem 3, so it has a fixed-point
such that
This implies that
and
. Thus, the system of Matrix Equation (
15) has a solution in
. □