1. Introduction and Preliminaries
Fixed-point theorems were used to indicate the presence and inimitableness of the solution of differential equation
where
F is a continuously differentiable function. In 1877, this proof was simplified by R. Lipschitz. In 1890, G. Peano generalized the idea of Cauchy by supposing only the continuity of
In 1922, Banach [
1] stated his famous theorem, the Banach contraction principle:
Every self-mapping
h on complete metric spaces
satisfying case
has a unique fixed point.
This theorem has been used to show the presence and inimitableness of solutions to (
1) after that time. This result is more powerful since it not only guarantees the presence and inimitableness of fixed points of exact self-maps of metric spaces, but it also ensures a constitutive technique to discover those fixed points. After Banach, many authors worked on fixed-point theory and gave generalizations of the Banach contraction principle on complete metrics.
In 1968, for example, Kannan proved the next theorem by adding a new contraction condition on h mapping:
Theorem 1 ([
2]).
Let be a complete metric space, and be a given mapping. Assume there exists such that, for all ,In this case, h has a unique fixed point.
A comparison of the Kannan and Banach fixed-point theorems shows that, although Inequality (
2) guarantees the continuity of the given transformation, Inequality (
3) does not guarantee the continuity of the given transformation. So, Inequalities (
2) and (
3) are independent conditions.
Kada, Suzuki, and Takahashi [
3] introduced the concept of
w-distance on metric spaces, and appointed different famous conclusions using this field in 1996. Then, Alber and Guerre-Delabriere [
4] presented fixed-point theorems for weakly contractive maps in Hilbert spaces in 1997. Results of [
4] were shown to be valuable in complete metric spaces by Rhoades [
5]. In 2019, Lakzian, Rakocevic, and Aydi [
6], inspired by [
2,
4,
5], introduced and studied fixed-point results for
-Kannan contractions on metric spaces by
w-distance.
Notions of orthogonal sets and orthogonal metric spaces were otherwise described by Gordji et al. [
7]. Then, Gordji and Habibi [
8] proved the presence and inimitableness theorem of fixed points for mappings on a generalized orthogonal metric space. They also applied this method on first-order differential equations. Moreover, Ramezani [
9] established generalized convex contractions on orthogonal metric spaces that might be called their definitive versions. Baghani et al. [
10] proved some fixed-point theorems on orthogonal spaces. These theorems improved the consequence of the paper by Eshaghi Gordji et al. [
7]. Then, Ramezani and Baghani [
11] presented the concept of strongly orthogonal sets, and obtained an actual generalization of Banach’s fixed-point theorem. They also obtained an illustrative example that highlights the importance of their main theorem.
In 2018, Senapati et al. [
12], inspired by [
3,
7], defined orthogonal lower semicontinuity and offered the notion
w-distance on an orthogonal metric space. They also presented the version of Banach’s fixed-point theorem on orthogonal metric spaces via
w-distance.
Then, the researchers acquired generalized fixed-point theorems in this field. Indeed, the concept of an orthogonal
F contraction mapping is defined, and many fixed-point results were produced for this type of contraction mapping on orthogonal metric spaces by Sawangsup et al. [
13]. Very recently, in 2021, Uddin et al. [
14] presented the notion of orthogonal
m-metric space and gave fixed-point theorems on orthogonal
m-metric space. Uddin et al. [
15] generalized the notion of control fuzzy metric spaces via presentation orthogonal control fuzzy metric spaces. Then, Beg et al. [
16] presented the notion of generalized orthogonal F-Suzuki contraction mapping and obtained fixed-point theorems on orthogonal
b-metric spaces. Furthermore, notions of generalized orthogonal
F-contraction and orthogonal
F-Suzuki contraction mappings were presented by Mani et al. [
17]. Thus, many results that were very common in the literature were generalized.
In this study, the notion of -Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance also does not need to be zero in w-distance mappings. The intent of this study is to enhance the new improvement of fixed-point theory on orthogonal metric spaces and related nonlinear problems via the notion of w-distance. On this basis, some fixed-point results are debated. Explanatory examples are delivered that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are explored through the main results in this paper.
Throughout this study, denote real numbers and integers, respectively.
Definition 1 ([
7]).
Let and be a binary relation. If ⊥ satisfies conditionthen is called an orthogonal set. In this case, is called an orthogonal element.
Example 1 ([
8]).
Let . Determine if there exists , such that . One can see that for all . Hence, is an orthogonal set. Definition 2 ([
7]).
Let be an orthogonal set, and d be a metric on Ω. Then, is called an orthogonal metric space. Definition 3 ([
7]).
Let be an orthogonal metric space. A sequence is called orthogonal ifIn the same way, a Cauchy sequence is orthogonal if Definition 4 ([
7]).
Orthogonal metric space is an orthogonal complete metric space if every orthogonal Cauchy sequence converges in Ω. Definition 5 ([
7]).
Let be an orthogonal metric space. Function is orthogonal continuous at k if for each orthogonal sequence converging to k implies as . h is orthogonal continuous on Ω if h is orthogonal continuous in each . Definition 6 ([
7]).
Let be an orthogonal metric space, and , . A function is an orthogonal contraction with Lipschitz constant θ iffor all whenever . Definition 7 ([
7]).
Let be an orthogonal metric space. A function is orthogonal preserving if whenever . Theorem 2 ([
7]).
Let be an orthogonal complete metric space and . Let be an orthogonal preserving mapping that is orthogonal continuous and orthogonal contraction mapping with Lipschitz constant θ. Then, h is a Picard operator, that is, h has a unique fixed point, and every Picard iteration in Ω converges to this fixed point. Lemma 1 ([
12]).
Let be an orthogonal metric space and be a w-distance. Assume that and are two orthogonal sequences in Ω and . Let and be sequences of positive real numbers converging to 0. Then, we have the following: - (i)
If and then . Moreover, if and , then .
- (ii)
If and , then as .
- (iii)
If for all , then is a orthogonal Cauchy sequence in Ω.
- (iv)
If , then is a orthogonal Cauchy sequence in Ω.
Definition 8 ([
12]).
Let be an orthogonal metric space, and be a w-distance. Mapping is an orthogonal p-contraction if there exists , such thatfor all with . Theorem 3 ([
12]).
Let be an orthogonal complete metric space with a w-distance p. If h is an orthogonal p-contractive, orthogonal preserving and orthogonal continuous self-mapping, then - (a)
h has a unique fixed point ;
- (b)
Picard sequence converges to for every .
2. Main Results
Definition 9. Let be an orthogonal set. Any two elements are orthogonally related if or .
Theorem 4. Let be an orthogonal complete metric space with a w-distance p, be a self-map, be continuous function, and . Assume that h is orthogonal preserving self-mapping satisfying inequalityfor all orthogonally related . Then, there exists a point , such that, for any orthogonal element , iteration sequence converges to this point.
either
h is orthogonal continuous at
or
if an orthogonal sequence converges to then or for all n,
in this case, is the unique fixed point of h.
.
Proof. Because
is an orthogonal set,
If for any orthogonal element
, since
h is self-mapping on
,
can be chosen to be
. Thus,
Then, if we continue in the same way,
so
is an iteration sequence. Since
h is orthogonal preserving,
is an orthogonal sequence. If
for some
n, then
is a fixed point of
h. So, we assume that
for each
n. By using (
9) with
, we obtain
So,
for all
. Thus, real sequence
is monotone nonincreasing and bounded below; so, there exists
, such that
. Letting
in (
13) and using the continuity of
, we obtain
, and so
by a feature of
. Thus,
is obtained.
On the other hand, by using (
9) with
, we obtain
So and letting in the last inequality, is obtained.
Next, we prove that
is an orthogonal Cauchy sequence. If
is not an orthogonal Cauchy sequence, by using Lemma 1
, there exists a sequence
of positive real numbers converging to 0, and corresponding subsequences
and
of
satisfying
, for which
Thus, there exists
that satisfies
If
is chosen as the smallest integer satisfying (
16), that is,
By the triangular inequality of
p and (
16), we easily derive that
Letting in the last inequality, since , a contradiction is obtained. So, is an orthogonal Cauchy sequence. By the orthogonal completeness of , there exists such that converges to this point.
Suppose that
h is orthogonal continuous mapping. In that case,
and so
is a fixed point of
h.
Suppose that, if a sequence
converges to
, then
or
for all
n. In that case, by using the existence of
, such that
converges to this point, then
or
for all
n. From
p being orthogonal lower semicontinuous, we obtain
and so
is obtained, and
Letting in the last inequality, since and , is obtained. Hence, by using Lemma 1, .
The uniqueness of the fixed point is shown as follows:
Assume that
is another fixed point of
h, such that, for any orthogonal element
, iteration sequence
converges to this point. Using the triangular inequality of
p and Inequality (
9),
Letting
in the last inequality, since
and
,
is obtained. In this case, we obtain
Letting in the last inequality, since , and , is obtained. Hence, by using Lemma 1, . □
Now, we can give an example for Theorem 4. The space given in this example is an orthogonal complete metric space, but not a complete metric space. Therefore, this example cannot be applied to Kannan or orthogonal weak Kannan-type contraction theorems given in complete metric spaces, so the concept of orthogonal metric space is very important in theory.
Example 2. Let be a set, and define , such that . Let binary relation ⊥ on Ω, such that . Then, is an orthogonal set, and d is a metric on Ω. So, is an orthogonal metric space. In this space, any orthogonal Cauchy sequence is convergent. Indeed, if is an arbitrary orthogonal Cauchy sequence in Ω, then there exists a subsequence of , for all or a subsequence of for all , . So, this subsequence is convergent in Ω. Every Cauchy sequence with a convergent subsequence is convergent, so is convergent in Ω. So, is an orthogonal complete metric space. Consider , , which is a w-distance on Ω. Let be defined as In this case, h is orthogonal preserving mapping. Indeed, suppose that . Without loss of generality, can be chosen. So, and . Thus, two cases are obtained:
Case (I): and ; then, .
Case (II): and ; then, .
These cases imply that .
Consider , for all . ϕ is a continuous function, and . h also satisfies Inequality (9). Indeed, for any orthogonally related , is obtained. Then, there are two cases: Case (I): Suppose that , and so . Then, ( or ; in both cases, Inequality (9) is satisfied. Case (II): Suppose that , and so . Then, ( or ; in both cases, Inequality (9) is satisfied. Therefore, all hypotheses of Theorem 4 are satisfied. For any orthogonal element , iteration sequence converges to . h is orthogonal continuous at , so this point is the unique fixed point of h and .
Theorem 5. Let be an orthogonal complete metric space with a w-distance p, be a self-map, be continuous function, and . Assume that f is orthogonal preserving self-mapping satisfying inequalityfor all orthogonally related . Then, there exists a point , such that, for any orthogonal element , iteration sequence converges to this point.
either
h is orthogonal continuous at
or
if an orthogonal sequence converges to , then or for all n and also are orthogonally related elements; in this case, is the unique fixed point of h.
.
Proof. Because
is an orthogonal set,
If for any orthogonal element
, since
h is a self-mapping on
,
can be chosen as
. Thus,
Then, if we continue in the same way,
so,
is an iteration sequence. Since
h is orthogonal preserving,
is an orthogonal sequence. If
for some
n, then
is a fixed point of
h. So, we assume that
for each
n. By using (
25) with
, we obtain
and so
for all
. Furthermore,
so,
for all
. Thus,
Letting
in (
34) and using the continuity of
, we obtain
; so,
by the property of
.
On the other hand, by using (
25),
so,
is obtained. Similarly,
and
are obtained by using (
25).
Now, using the way in the proof of Theorem 4, we conclude that is an orthogonal Cauchy sequence. By the orthogonal completeness of , there exists , such that converges to this point.
Suppose that
h is orthogonal continuous mapping. In that case,
so
is a fixed point of
h.
Suppose that, if a sequence
converges to
, then
or
for all
n. In that case, by using the existence
, such that
converges to this point, then
or
for all
n, and
are orthogonally related elements. From
p being orthogonal lower semicontinuous, we obtain
so,
is obtained, and
Thus,
is obtained. Hence, by using Lemma 1
,
, and
. Thus, ; so, by using Lemma 1, , .
The uniqueness of the fixed point is shown in the following:
Assume that
is another fixed point of
h, such that, for any orthogonal element
, iteration sequence
converges to this point. Using the triangular inequality of
p and (
25),
Letting
in the last inequality, since
and
,
is obtained. Similarly, using the triangular inequality of
p and (
25),
Letting in the last inequality, since , and , is obtained. Hence, by using Lemma 1, . □
Now, we give an example for Theorem 5. The space given in this example is an orthogonal complete metric space, but not a complete metric space. Therefore, the classical Kannan theorem is not applicable to the following example. In this example, h is not a Kannan contraction with the w-distance p for . Both cases are examined in detail at the end of the example.
Example 3. Let be a set, and define the discrete metric such that Let binary relation ⊥ on Ω, such that . Then, is an orthogonal set, and d is a metric on Ω. So, is an orthogonal metric space. In this space, any orthogonal Cauchy sequence is convergent. Indeed, if is an arbitrary orthogonal Cauchy sequence in Ω. Then, there exists a subsequence of , for all , where c is a constant element of Ω. So, this subsequence is convergent in Ω. Every Cauchy sequence with a convergent subsequence is convergent, so is convergent in Ω. So, is an orthogonal complete metric space. Consider , which is a w-distance on Ω. Let be defined as In this case, h is orthogonal preserving mapping. Without loss of generality, can be chosen. So, and . Thus, two cases are obtained:
Case (I): and , then .
Case (II): and , then
These cases imply that .
Consider , for all . ϕ is a continuous function, and . h also satisfies Inequality (25). Indeed, for any orthogonally related , is obtained. Then, there are two cases: Case (I): Suppose that , and so and . Then, or and in both cases, Inequality (25) is satisfied. Case (II): Suppose that and so and . Then, or and in both cases, Inequality (25) is satisfied. Therefore, all conditions of Theorem 5 are satisfied. Careful examination shows that, for any orthogonal element , theiteration sequence converges to . h is orthogonal continuous at , so this point is the unique fixed point of h, and .
Otherwise, h is not a Kannan contraction connected to metric d, so the classical Kannan theorem is not practicable to the metric d: h is not a Kannan contraction with the w-distance p for . Actually, Corollary 1. Let be an orthogonal complete metric space with a w-distance p, be a self-map, be a continuous function, and . Suppose that h is orthogonal preserving self-mapping satisfying inequalitiesfor all orthogonally related . Then, there exists a point , such that, for any orthogonal element , iteration sequence converges to this point.
either
h is ⊥-continuous at
or
if an orthogonal sequence converges to , then or for all n;
in this case, is the unique fixed point of h.
.
Proof. In Theorem 4, it is sufficient to choose □
Corollary 2. Let be an orthogonal complete metric space with a w-distance p, be a self-map, be a continuous function, and . Suppose that h is orthogonal preserving self-mapping satisfying inequalitiesfor all orthogonally related . Then, there exists a point , such that, for any orthogonal element , iteration sequence converges to this point.
either
h is orthogonal continuous at
or
if an orthogonal sequence converges to , then or for all n, and are orthogonally related elements.
In this case, is the unique fixed point of h.
.
Proof. In Theorem 5, it is sufficient to choose □
In the following, another consequence of Theorem 4 corresponds to the contraction, which we can call the orthogonal p-Kannan or orthogonal weak Kannan-type contraction.
Corollary 3. Let be an orthogonal complete metric space with a w-distance p. Suppose that is orthogonal preserving self-mapping, there exists , such thatfor all orthogonally related . Then, there exists a point such that, for any orthogonal element , iteration sequence converges to this point.
either
h is orthogonal continuous at
or
if an orthogonal sequence converges to , then or for all n.
In this case, is the unique fixed point of h.
.
Proof. In Theorem 4, it is sufficient to choose □
Taking in Corollary 3, orthogonal Kannan fixed-point theorem is obtained, which is a generalization of Theorem 1 on orthogonal metric spaces.