Coupled Fixed Point Theory in Subordinate Semimetric Spaces

: The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples


Introduction and Preliminaries
One of the most important tools in nonlinear functional analysis is fixed point theory.It is very well known that most nonlinear analysis problems can be treated as fixed point problems.Banach proposed that each contraction on a complete metric space possesses a unique fixed point.In [1], J. Villa-Morales introduced the concept of subordinate semimetric spaces.A subordinate semimetric space is an extension of the concept of the RS-space introduced by Rolda'n and Shahzad in [2].Also, the notions of Jleli and Samet's metric space and Branciari's generalized metric space are special cases of an RS-space.The purpose of this article is to study the existence of coupled fixed points (CFPs) on complete subordinate semimetric spaces.We also aim to provide some applications and examples to illustrate our results.In this article, we operate on the set of extended real numbers using standard arithmetic operations, R = R ∪ {∞, −∞}, and the notations have their regular meanings.Let Γ be a nonempty set.We begin with an extension of the definition of a semimetric space.
We will use this notion to define several fundamental topological concepts.
(iii) The pair (Γ, Ψ) is called a complete semimetric space if each Cauchy sequence in Γ is convergent.
Our approach exhibits symmetry with the generalized metric space concept that Jleli and Samet established in their work [3].Rolda'n and Shahzad [2] promptly generalized this concept in the manner described below.
Remark 1.Note that each RS-space is a subordinate semimetric space (if we take ξ(ω) = cx), but the converse is not true.Examples 2, 3, and 5 of [1] are subordinate semimetric spaces but they are not RS-spaces.
The next proposition proves the uniqueness of the limit of a convergent sequence in a subordinate semimetric space, which is necessary for our main results.Proposition 1.Let (ω n ) be an infinite Cauchy sequence in a subordinate semimetric space (Γ, Ψ).Suppose that (ω n ) converges to ω and λ.Then, ω = λ.
Definition 6 ([4]).Assume (Γ, ≤) is a PO set and G : Γ 2 → Γ is a function.Then, we say that G has the MM property if the following hold: Given this notion, the authors of [4] established the next theorem, which shows the existence of the CFP of an operator with the MM property in the setting of a complete PO metric space.
Senapati and Dey in [6] improved and extended Berinde's CFP findings in [5] using the condition of contraction (2) for an MM operator on partially ordered complete JSmetric spaces.
In this work, motivated by the concepts of subordinate semimetric spaces, we extended and improved the CFP findings of Senapati and Dey [6] due to the condition of contraction (2) for an MM operator in PO complete subordinate semimetric spaces.In order to support our main finding, we constructed some examples.

Main Results
We will first provide some notions related to the structure before introducing our main results.
From the above, we see that there exists a function ξ ′ : [0, ∞] → [0, ∞] with the following conditions: In a similar fashion, we define a distance function on each n-tuple set Γ n for each n ≥ 2.
Thus, we define the function It is easy to check that Ψ m meets the axioms of a subordinate semimetric space.Furthermore, (Γ 2 , Ψ m ) is a Ψ m -subordinate semimetric space.Proceeding in this way, we may establish an n-tuple Ψ m -subordinate semimetric space for each n ≥ 2.
The following proposition will be necessary in order to state our main results.
Remark 2. By Proposition (1), the limit of a Cauchy convergent sequence is unique in the subordinate semimetric space By Proposition (2), we may derive the next argument.
Let (ω, ν) ∈ Γ 2 , and let there be a function of G : Γ 2 → Γ that has an MM operator.We have defined where Remember that the partial order ′ ≤ ′ on Γ 2 is defined in the following manner:

CFP Results
Throughout this part, we generalize the works of Senapati and Dey [6], improving the results of Berinde [5].

Remark 3.
Clearly, from the above, the CFP theorem for G simplifies to the common fixed point theorem for T G , as T G has a fixed point if and only if G has a CFP.
Let ω 1 = G(ω 0 , ν 0 ) and ν 1 = G(ν 0 , ω 0 ), and we denote the following: In a similar fashion, since G is an MM operator, we obtain Throughout Remark 3, to establish the presence of a CFP of G, it is enough to prove the presence of a fixed point of T G provided by Equation (4).To demonstrate this, let us assume Proceeding in this way, we obtain Thus, {σ n } is a Picard sequence that has the initial approximation σ 0 .Also, since G is an MM operator, one can easily check that for any n ≥ 0, ω n ≤ ω n+1 and ν n ≥ ν n+1 .Thus σ n ≤ σ n+1 , i.e., {σ n } is a non-decreasing sequence.Now, we can show that {σ n } is a Cauchy sequence due to the fact that G meets the condition of contraction (3), for each n ≥ 0 and i ≤ j.Therefore, we have the following: This holds for each n ∈ N such that for each i ≤ j, we obtain Also, we know that Since δ G (Ψ, (ω 0 , ν 0 )) < ∞ and δ G (Ψ, (ν 0 , ω 0 )) < ∞, then we have Using this in (6) , for all m ∈ N, we obtain Thus, {σ n } is a Cauchy sequence.As (Γ 2 , Ψ + ) is complete, the sequence {σ n } converges to σ for some σ = ( ω, ν) ∈ Γ 2 .
Finally, we need to prove that σ = ( ω, ν) is a fixed point of T G and that it is a CFP of G. Now, we have two cases to consider about the Cauchy sequence Thus, σ is a fixed point of T G .Case (2): If {σ n } = {(ω n , ν n )} is an infinite Cauchy sequence and we suppose that T G ( σ) ̸ = σ, then since the space (Γ 2 , Ψ + ) is subordinate with the function ξ ′ , we have This implies that σ = T G ( σ); that is, σ is a fixed point of T G .By using Remark 3, we can deduce that σ = ( ω, ν) is a CFP of G; that is, ω = G( ω, ν) and ν = G( ν, ω).
Theorem 4. Assume ρ = (λ, µ) and σ = ( ω, ν) are CFPs of G such that they are incomparable.Assume that there is a lower bound or upper bound σ * = (ω * , ν * ) ∈ Γ 2 of ρ and σ such that Proof.It is clear that, for every n ∈ N, T n G (σ * ) is comparable to ρ = T n G (ρ) as well as to σ = T n G ( σ).By using the contraction principle (5), we obtain and In a similar way, we obtain Employing the axioms of Ψ + -subordinate semimetric spaces and the above inequality, we obtain Thus, the sequence {T n G (σ * )} also converges to ρ.Similarly, it can be demonstrated that the sequence {T n G (σ * )} also converges to σ.Through Remark 2, we can conclude that σ = ρ; that is, ( ω, ν) = (λ, µ).
Theorem 5.If we add any of the preceding requirements to the hypothesis of Theorem 2, then the components of a CFP are equal.
Proof.The theorem is proved by the following cases.
Case I: Assume that requirement (Q1) is satisfied, together with the assumptions of Theorem 2. Let Σ = ( ω, ν) and ∆ = ( ν, ω).By using the contraction principal in Theorem 2, we obtain Case II: Assume that requirement (Q2) is satisfied, together with the assumptions of Theorem 2. We consider ( ω, ν) to be a CFP of G with ω, ν being incomparable.
The following corollary is a new form of Theorem (2.1.6)in [6].
Remark 4. To prove the presence of CFPs, the authors of [4] investigated two different assumptions.
The first assumption is that the function G is continuous and the second assumption is if {ω n } and {ν n } are non-increasing and non-decreasing sequences, respectively, such that {ω n } → ω and {ν n } → ν, it follows that ω n ≤ ω and ν n ≤ ν for all n ∈ N.However, Corollary 1 guarantees the presence of CFPs without requiring any of the preceding assumptions.
Remark 5. Since each b-metric space is a subordinate semimetric space such that ξ(ω) = s(ω) ≥ 1 in Definition 4, it is easy to prove the CFP results in a PO b-metric space based on this paper's findings.In particular, the CFP findings in a b-metric space can be deduced from Theorem (2.2) in [7] using Corollary 1.

Remark 6.
In Corollary 1, the quality of the components of a CFP and the uniqueness of a CFP of G are ensured using Theorems 3-5 as well.
Similarly, anyone can also prove the presence of a CFP of Ψ m on Γ 2 .The next theorem presented addresses this.Theorem 6. Assume that the mapping G : Γ 2 → Γ satisfies the MM property on Γ and there is a k ∈ (0, 1) with for all ω ≥ λ and ν ≤ µ.If there exist ω 0 , ν 0 ∈ Γ such that the following hold: Proof.The proof is essentially the same as the proof of Theorem 2. Hence, we will skip the proof.
We will now present examples to support our major conclusion.
Thus, G is monotonically non-increasing in its second component.
Thus, the sequence n m(n+1) n∈N is an infinite Cauchy sequence that is convergent to 1 m .Now, suppose there is a c > 0 such that , 0 = cm, then c ≥ m for all m ∈ N. Hence, (Γ, Ψ) is not an RS-space.Note that (Γ, Ψ) is subordinately semimetric to ξ(t) = t, 0 ≤ t ≤ 1; t 4 , t > 1.Let s be a real number such that s > 1. 1.
G has MM property.
G satisfies the contraction condition.
Note that (a − b) 2 ≤ 2(a 2 + b 2 ), a, b ∈ R. We then have Thus, G satisfies the contraction condition.Hence, the point (0, 0) is a coupled fixed point of G. Also, the point (∞, −∞) is a coupled fixed point of G as well.
is an infinite Cauchy sequence that is convergent to 1 m .Now, suppose there is a c > 0 such thatm 2 = Ψ 1 m , 0 ≤ c lim sup n→∞ Ψ n m(n + 1)