1. Introduction and Preliminaries
Fixed point theory (FPT) and its applications provide an important framework for the study of symmetry in mathematics [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17]. The literature contains many extensions of the concept of FPT in metric spaces (MSs) and its topological structure. Matthews [
18] introduced the notion of partial metric space (PMS) and proved that the Banach contraction theorem (BCT) (or contraction theorem) can be generalized to the partial metric context for applications in program verification. The concept of
b-metric space (BMS) was introduced and studied by Czerwik [
19]. Recently, Amini-Harandi [
20] introduced the notion of metric-like space (MLS). Afterward, Alghamdi et al. [
21] introduced the notion of BMLS, which is an interesting generalization of PMS and MLS. While examining this with the PMS, they ascertain that every PMS is an BMLS, but the converse does not need to be true, showing that a BMLS is more general structure than the PMS and MLS.
The contraption of DS is a strong formalistic apparatus, associated with a large-spectrum analysis of multistage decision-making problems (MDMP). Such problems appear and are congruent in essentially all human activities. Unfavorably, for explicit reasons, the analysis of MDMP is difficult. MDMP are characteristic of all DS in which the associated variables are state and decision variables (see more, [
22,
23]). In recent years, Klim and Wardowski [
24], discuss the idea of DS instead of the Picard iterative sequence in the context of fixed-point theory. Their objective was further exploited by numerous researchers in many ways (see more details in [
25]).
Recently, Gordji et al. [
26] established the new idea of an orthogonality behavior in the context of metric spaces (MSs) and provided some new fixed-point theorems for the Banach contraction theorem (BCT) in the MSs class that is endowed with this new type of orthogonal binary relation ⊥.
The main objective of this manuscript is to introduce and investigate a new concept of OBMLS and ODS for hybrid pairs of mappings. Some new, related, multi-valued -ST fixed-point theorems are established with respect to . Our investigation is completed by tangible examples and applications in ordinary differential equations and nonlinear fractional differential equations.
Hereinafter, we recall the definition of the orthogonal set (briefly -set) and some related fixed-point results.
Definition 1. [26] A -set is a pair where form a binary relation and is a non-empty set; therefore, we have Example 1. Define ⊥ on by , where we consider to be the collection of all people in the world. Let provide blood to . Based on Figure 1, if is a body in so that his/her blood group type is O-negative, we can write ∀. This implies that is an -set. According to this fashion, in the set, is not unique. Note that, in this logical example, may be a body with the blood group type -positive. Therefore, we write ∀. Definition 2. [26] A sequence on is known as orthogonal sequence (-sequence) if the following condition holds true Definition 3. [26] The triplet pair is known as orthogonal MS (-MS) if is an -set and is an MS. Theorem 1. [26] Let triplet pair be an -complete MS (complete MS not needed) and . Let , such that the following conditions hold true: - (i)
—continuous;
- (ii)
—contraction mapping endowed with Lipschitz constant;
- (iii)
—preserving. Then, μ possesses a unique fixed point. Moreover, ∀.
Alghamdi et al. [
21] introduced the notion of
b-metric-like space as follows:
Definition 4. A BMLS on a non-empty set is a function such that, for each with , we have
- ()
if implies ;
- ()
;
- ()
.
The pair is known as a BMLS.
Definition 5. Let be a BMLS. Then, we have
- (i)
a sequence of converges to a point iff - (ii)
a sequence of BMLS is known as a Cauchy-sequence, iff exists (and is finite).
- (iii)
a BMLS is known as complete if every Cauchy-sequence converges and is endowed with to , such that .
Nadler [
27] developed the concept of Hausdorff metric (HM) and improved the BCT for multi-valued operators instead of single-valued operators. Herein, we investigate the concept of HM-like in light of HM, as follows. Let
be a BMLS. For
and
, let
Define
by
for each
denotes the family of all non-empty closed and bounded-subsets of
and
denotes the family of all non-empty closed-subsets of
.
Theorem 2. Let be a complete MS and is known as Nadler contraction mapping, if exists in such a way thatThen, μ possesses at least one fixed point (see more details in [27]). In 2012, Wardowski [
28] developed the concept of a contraction operator known as an
F-contraction and improved the Banach contraction theorem (BCT) via
F-contraction, which is the real generalization of BCP. Then, the concept of
F-contraction was advanced to the case of non-linear
F-contractions with a dynamic system, justifying that
F-contractions with a dynamic system have a more general structure than the
F-contraction (see more details in [
24]).
Definition 6. [28] Let be the set of mapping , satisfying each of the following axioms , and : - ()
∀
such that- ()
∀
we have- ()
exists such that . Let be an MS and is known as F-contraction, if exists such that
We provide some related examples of mappings belonging to as follows:
Example 2. [28] Let be the set of mappings defined by: - (1)
∀;
- (2)
∀;
- (3)
∀;
- (4)
∀.
Now, we recall the following basic concept of the dynamic system (DS):
Definition 7. Let be a mapping. A setknown as DS of μ with respect to the starting point . is arbitrary and fixed. In light of onward has the form (see more in [24]). We now recall some basic concepts of F-contraction with respect to the dynamic system (DS), as follows:
Theorem 3. [24] Let be a multi-valued F-contraction with respect to a , if there is a function , such thatAssume that there are , such that for each and a mapping is a dynamic lower semi-continuous mapping. Then, μ has a fixed point. 2. Main Results
First, to provide our new definition as a generalization of B-metric-like spaces:
Definition 8. An OBMLS on a non-empty set with is a function such that, for each with respect to orthogonal relation (, if the following conditions hold:
- ()
if implies
- ()
- ()
The pair is known as an OBMLS.
Example 3. Let and is given by with respect to and if, and only if, Then, is an OBMLS with The above example is not BMLS since Remark 1. Every BMLS is OBMLS, but the converse does not generally hold true.
The fashions of convergence, Cauchy sequence and completeness criteria are same as in BMLS. The term Hausdorff metric can easily be amplified to the case of an OBMLS.
Let
be an OBMLS. For
and
, let
Define
as by
for each
In the following, the concept of ODS, ODS for a hybrid pair of mappings and its ⊥-preservation are introduced, and some elementary facts about these concepts are discussed.
Definition 9. Let , be a fixed point. A setis known as the ODS of μ with . Herein, ODS and onward has the form . Definition 10. Let and let , be a fixed point. A setis known as the ODS of ρ and μ with . Moreover, ODS and onward has the form . Example 4. Define and as by and , respectively. The sequence , as given by , is an ρ iterative sequence of μ with a starting point of
Definition 11. Let be an -set and are called the ⊥-preserving of μ if whenever for .
Definition 12. Let be an -set and and are called the ⊥-preserving of a hybrid pair of mappings if whenever for .
The first main result of this exposition is given as follows.
Definition 13. Let be an OBMLS. Let and are called a hybrid pair of mappings of -ST-I contraction with ODS , if for some and such thatimplying that∀
(or , where such that and . Remark 2. In our investigation, we examine that the following property does not hold true:then, there exist some , such that and . Therefore, we obtainwhich implies the existence of a common fixed point of the pair Therefore, considering the , satisfying Equation (2) does not depress the generality of our investigation. Theorem 4. Let be an -complete OBMLS. Let and are called a hybrid pair of mappings of the -ST-I contraction with respect to ODS Assume that:
- (O1)
: If, in addition, the hybrid pair of mapping are ⊥-preserving;
- (O2)
: There is such that, for each
- (O3)
: If, moreover, is super-additive, i.e., for , we have
Then, ρ and μ have a common fixed point in
Proof. Owing to the fact that the pair
is an
-set, then there is
, such that
(or
)
. It follows that
(or
). Upon setting
In case there is
for some
, then our proof of 4 proceeds as follows. Therefore, without loss, we may assume that
; thus, we have
for each
Since the hybrid pair of mapping
are ⊥-preserving, we can write
Thus,
is an ODS
. Since
-ST-I is a contraction operator, we have
Therefore, (
1), results in the following:
Next, we certify the following inequality
Based on the contrary, we assume that there exist
in such a way that
. Then, in view of (
4) we have:
Since
is super-additive, we can obtain
By appealing to the above fashion, we have
which, by virtue of
, implies that
, which contradicts this. Hence, (
5) holds true. In light of the above observations,
is a decreasing sequence in
R and and is bounded from below. Assuming that there is
, such that
We now need to prove that
. We assume, based on the contrary, that
For a given
, there exist a number of
, in such a way that
By virtue of
we can write:
By referencing (
3), we have
Since the hybrid pair of mappings
of
-ST-I provide a contraction operatror, we can obtain:
Owing to the above hypothesis, this, in turn, yields:
Since
is super-additive, we can write
By given condition
, we get
Again, in light of
-ST-I contraction, we obtain
Continuing these steps, we can write
Following
, along with
, we have
. Additionally, in view of
, we have
Therefore, there is
such that
∀
, which is a contradiction. Therefore, we can write:
Supposing, on the contrary, that for
there are sequences
and
in
, we have
By (
13), there is
, such that
which, together with (
16), yields,
implies
In view of (
15) and (
17), we have
Applying triangle inequality, we find that
Next, we proceed to the limitas
in (
21) and make use of (
13); then, we have
There also exist
such that
for each
Further, since the hybrid pair of mappings
are ⊥-preserving, in addition to being ⊥-transitive, we can write
Following the
-ST-I contraction, we find that:
In the light of (
17), (
18) and (
19), implies (
22)
for
Taking the limit as
in Equation (
23), we have
which, by virtue of (
), implies that
. Therefore, from (
21), we have
which implies that
is a contradiction. Hence, (
14) holds true. Therefore,
is a
-Cauchy sequence in
Since
is a
-complete OBMLS, there is a point
, such that
Now, we further prove that for each
Assuming that there is
, such that
From (
5) and (
26), we have
a contradiction. Hence, (
25) holds true. Furthermore, we can see that
or
Let us now consider (
27) that holds true, From (
27), we have
From (
13) and (
24), there exists
for some
, such that
Therefore, (
29) and (
30)
. As a consequence of these facts, the lateral limit as
, which implies
Furthermore,
. Following
and using (
24) and (
32), we can obtain
Thus, clearly,
has a common fixed point of a hybrid pair of mappings
□
From the above developments, we have found some important corollaries:
Corollary 1. Let be an -complete OBMLS. Let and be known as a hybrid pair of mappings of -ST-II contractions with ODS. If, for some and , such thatimplies∀
(or , , where and , such that . Assume that (O1), (O2) and (O3) holds true. Then, ρ and μ has a common fixed point in Corollary 2. Let be an -complete OBMLS. Let and be called a hybrid pair of mappings of -ST-III contraction via ODS. If, for some and such thatimplies∀
and . Assume that (O1) and (O2) hold true. Then, ρ and μ have a common fixed point in Corollary 3. Let be an -complete OBMLS. Let be a -ST-1V contraction mapping involving ODS. If, for some and , such thatthen this implies∀
, , where , such that and . Assume that (O1), (O2) and (O3) hold true. Then, μ has a fixed point in Corollary 4. Let be an -complete OBMLS. Let be known as -ST-V contraction mapping via ODS. If, for some and such thatimplies∀
and . Assume that (O1) and (O2) hold true. Then, μ has a fixed point in Corollary 5. Let be an -complete OBMLS. Let be called an -type-VI contraction mapping with ODS. If, for some and such that∀
and . Assume that (O1) and (O1) hold true. Then, μ has a fixed point in Corollary 6. Let be an -complete OBMLS. Let be called an -ST-VII contraction mapping via ODS. If, for some , and form a non-negative Lebesgue integrable operator, which is summable on each compact subset of κ such thatthis implies∀
and ∀ given , so that . Suppose that (O1) and (O2) hold true. Then, μ has a fixed point in Corollary 7. Let be an -complete OBMLS. Let and are called a hybrid pair of mappings of -ST-VIII contraction via ODS. If, for some , and a non-negative Lebesgue integrable operator, which is summable on each compact subset of κ such thatimplies∀
and ∀ given so that . Suppose that (O1) and (O2) hold true. Then, ρ and μ has a common fixed point in In the following, the first main tangible example of this exposition is given.
Example 5. Let , and let be define ∀ Define the binary relation ⊥ on by if for some or Then, clearly, is a -complete OBMLS. The mappings , and to be considered here are as follows:Define and by andLet and ⊥-sequence be as defined by in then, the following cases hold true: Case 1: Let ∀ then, and
Case 2: Let for some then, there exists , such that and Therefore, the hybrid pair of mappings ρ and μ are ⊥-continuous on but not continuous on . Taking , thenwhich implies∀
Hence, Theorem (4) is not satisfied. Clearly, the hybrid pair of mapping is ⊥-preserving. Taking and with respect to (or , we can easily obtainimplying that∀
with respect to or and Therefore, all the required conclusions of Theorem (4) are fulfilled and 0 is a common fixed point of ρ and μ. In the following, some applications in the context of ordinary differential equations and nonlinear fractional differential equations are designed with respect to the integral boundary value conditions, which are given to highlight the usability and validity of the theoretical results.
3. Application to Ordinary Differential Equations
In this section, we investigate an application of Corollary (5) to establish the existence of solutions to ordinary differential equations (ODE) under the influence of complex valued mearurable functions and orthogonal binary relations ⊥. This is in effect for our purpose. First, we recall that the space
consists of of all complex valued measurable functions
underlying space
for each
, such that
where
A is called the
-algebra of mearurable sets and
is the measure scale. Taking
, the space
consists of all integrable functions
on
and defines the
norm of
by
Now, we consider the following differential equations:
where
and
is an integrable functions satisfying the following axioms:
- ()
: for each and
- ()
: ∀ with respect to or for each there exists and such that
and
where
Define a mappings
by
Thus,
is an OBMLS with
. The ⊥-continuous mapping
to be considered here are as follows. Let us say that
In addition, the following relation of such objects is useful. Let us define the orthogonality binary relation ⊥ on
by
Further, we are now in the position to state the second main result of this exposition as follows:
Theorem 5. If assumptions (40)–(46) and are satisfied, then the differential Equation (42) has a solution, as follows: Proof. Since
, we have
for almost everywhere
. Now, we can see that
is ⊥-preserving.
with
and
, we have
Therefore, it follows that
and so
Hence,
is ⊥-preserving. Next, we prove that
is
-contraction. Let
with
and
we have
and, therefore,
Owing to (
49) and (
), we have
Thus, it follows that
which implies
Setting
it follows that
is an
-contraction. Thus, all the required hypotheses of Corollary (5) are satisfied and we have shown that Equation (
42) has at least one solution. □
4. Application to Nonlinear Fractional Differential Equations
In this section, we developed an application of Corollary (5) to establish the existence of solutions to nonlinear fractional differential equations (NFDE) via orthogonal binary relations ⊥ (see more [
29,
30,
31,
32]). First, we recall the existence of solutions for the NFDE
with the integral boundary value conditions (IBVC)
Where we can denote that the
Caputo fractional derivative (CFD) of the order
and
is a continuous function. Here,
is the Banach space of continuous functions from
I into R, endowed with the supremum norm
Mappings
are defined by
Thus,
is a BMLS with
. The Caputo fractional differential equation (CFDE) can be defined with respect to order
by
where
, the family
represents
represents the Gamma function and
is a continuous function. Moreover, the Riemann–Liouville fractional derivatives (RLFD) of order
, for a continuous function
, is defined by
We are now in the position to state the third main result of this exposition as follows:
Theorem 6. Suppose that there exists a function , such that
- ()
: ∀ with respect to or for each there exist such that - ()
: there exists such that for each where the ⊥-continuous operator is defined by for and
- ()
: for each and such that implies
Under the assumptions (53)–(59), and if - are satisfied, then the NFDE (53) possesses at least one solution. Proof. Define the orthogonality binary relation ⊥ on
by
Now, we prove that
is ⊥-preserving. ∀
with
and
, we have
Therefore, it follows that
, and so
Hence,
is ⊥-preserving. Next, we prove that
is an
-contraction. Let
with
and
For each
, and owing to ⊥-continuous operator
one can write:
This, in turn, yields (owing to the above hypothesis):
Thus,
with respect to
,
and
for each
we have
or
By applying this to logarithm, we can write
and, hence, we can easily obtain, with the setting of
Thus, due to assumptions (
53)–(
59), all the required hypotheses of Corollary (5) are satisfied. Therefore, Equation (
53) possesses at least one solution. □
Example 6. Let CFDE, with respect to order η and its IBVB, be written as follows:andwhere and Furthermore, and therefore, after setting forTherefore, we can apply Corollary (5). Hence, there is a solution to Equation (63) in Λ, along with the conditions (64).