Certain New Development to the Orthogonal Binary Relations

: In this study, inspired by the concept of B -metric-like space (BMLS), we introduce the concept of orthogonal B -metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new reﬁnements of the F ⊥ -Suzuki-type ( F ⊥ -ST) ﬁxed-point results are presented. These include some tangible instances, and applications in the ﬁeld of nonlinear analysis are given to highlight the usability and validity of the theoretical results.


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 largespectrum 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 ODSĎ ⊥ (µ, ρ,ȇ 0 ) for hybrid pairs of mappings. Some new, related, multivalued F ⊥ -ST fixed-point theorems are established with respect toĎ ⊥ (µ, ρ,ȇ 0 ). Our investigation is completed by tangible examples and applications in ordinary differential equations and nonlinear fractional differential equations.
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 (C, ξ) be a BMLS.
. CB(C) denotes the family of all non-empty closed and boundedsubsets of C and CL(C) denotes the family of all non-empty closed-subsets of C. Theorem 2. Let (C, ð) be a complete MS and µ : C → CB(C) is known as Nadler contraction mapping, if σ ∈ [0, 1) exists in such a way that Then, 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 F :Ř + −→Ř, satisfying each of the following axioms (F i ), (F ii ) and (F iii ): We provide some related examples of mappings belonging to ∇ as follows: Example 2. [28] Let ∇ be the set of mappings F 1 , F 2 , F 3 , F 4 :Ř + −→Ř defined by: (4) F 4 (ȇ) = ln(ȇ 2 +ȇ) ∀ȇ > 0. Now, we recall the following basic concept of the dynamic system (DS): known as DSĎ(µ,ȇ 0 ) of µ with respect to the starting pointȇ 0 .ȇ 0 ∈ C is arbitrary and fixed.
We now recall some basic concepts of F-contraction with respect to the dynamic system (DS), as follows: Theorem 3. [24] Let µ : C → C(C) be a multi-valued F-contraction with respect to aĎ(µ,ȇ 0 ), if there is a function τ : Assume that there areȇ a ∈Ď(µ,ȇ 0 ), such that lim inf l→a + τ(l) > 0 for each a ≥ 0 and a mapping C ȇ a −→ d(ȇ a , µȇ a ) isĎ(µ,ȇ 0 ) a dynamic lower semi-continuous mapping. Then, µ has a fixed point.

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.
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.
The first main result of this exposition is given as follows.
Next, we certify the following inequality Based on the contrary, we assume that there exist r ∈ N in such a way that ζ(ȇ r+1 ,ȇ r+2 ) ζ(ȇ r ,ȇ r+1 ). Then, in view of (4) we have: Since F ⊥ is super-additive, we can obtain By appealing to the above fashion, we have which, by virtue of (F i ), implies that ζ(ȇ r+1 ,ȇ r+2 ) ζ(ρȇ r−1 , ρȇ r ), which contradicts this. Hence, (5) holds true. In light of the above observations, ζ(ȇ r ,ȇ r+1 ) is a decreasing sequence in R and and is bounded from below. Assuming that there is Ω 0, such that We now need to prove that Ω = 0. We assume, based on the contrary, that Ω > 0. For a given ε > 0, there exist a number of σ ∈ N, in such a way that By virtue of (F i ), we can write: By referencing (3), we have Since the hybrid pair of mappings (ρ, µ) of F ⊥ -ST-I provide a contraction operatror, we can obtain: ].
In the following, the first main tangible example of this exposition is given.
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.

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 L P (∆, A, κ) consists of of all complex valued measurable functions δ underlying space ∆ for each 1 ≤ P ≤ ∞, such that ∆ |δ(y)| P dκ(y), where A is called the σ-algebra of mearurable sets and κ is the measure scale. Taking P = 1, the space L 1 (∆, A, κ) consists of all integrable functions δ on ∆ and defines the L 1 norm of δ by Now, we consider the following differential equations: dµ dr = Υ(r, µ(r)), r ∈ I = [0, L]; where L > 0 and Υ : I ×Ř →Ř is an integrable functions satisfying the following axioms: (O a ) : Υ(h, P), for each P ≥ 0 and h ∈ I; In addition, the following relation of such objects is useful. Let us define the orthogonality binary relation ⊥ on ∆ by y ⊥ y iff y(r)y (r) ≥ y(r) or y(r)y (r) ≥ y (r) ∀ r ∈ I.
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 ηĉĎ η and its IBVB, be written as follows: Therefore, we can apply Corollary (5). Hence, there is a solution to Equation (63) in Λ, along with the conditions (64).

Open Problems
In this section, we pose some challenging questions for researchers. Problem I: Clearly, the limit of the convergent sequence is not necessarily unique in BMLS. Can the limit of a convergent sequence be unique in OBMLS? Problem II: Can Theorem (1) be proved by the Semi-F ⊥ -contraction?

Conclusions
In conclusion, this manuscript deals with the new concept of OBMLS to approximate the fixed-point results for a hybrid pair of mappings that are are introduced and studied. In addition, a new ODSĎ ⊥ (µ, ρ,ȇ 0 ) is provided and determined via a hybrid pair of mappings. A new multi-valued F ⊥ Suzuki-type contractive condition is proposed in an OBMLS under the influence of ODSĎ ⊥ (µ, ρ,ȇ 0 ). Finally, our investigation is completed with tangible examples and applications to ordinary differential equations and nonlinear fractional differential equations in the field of nonlinear analysis.
Some potential future works are as follows: (i) Discuss the possibility of applying multi-valued F ⊥ Suzuki-type fixed-point results with respect to ODSĎ ⊥ (µ, ρ,ȇ 0 ) to the context of fuzzy mapping; (ii) Discuss the possibility of an orthogonal B-metric-like space with respect to ODŠ D ⊥ (µ, ρ,ȇ 0 ) in the context of an orthogonal fuzzy B-metric-like space. Funding: This research received no external funding.