Fixed-Point Results of Generalized ( ϕ , Ψ ) -Contractive Mappings in Partially Ordered Controlled Metric Spaces with an Application to a System of Integral Equations

: In this manuscript, we prove numerous results concerning fixed points, common fixed points, coincidence points, coupled coincidence points, and coupled common fixed points for ( ϕ , Ψ ) - contractive mappings in the framework of partially ordered controlled metric spaces. Our findings introduce a novel perspective on this mathematical context, and we illustrate the uniqueness of our findings through various explanatory examples. Also, we apply the main result to find the existence and uniqueness of the solution of the system of integral equations as an application


Introduction
The fixed-point (FP) theory is a pivotal branch in mathematics and has found extensive applications across various disciplines, ranging from functional analysis and topology to physics, economics, and beyond.The essence of FP theory is the investigation of mappings that retain certain points during transformation, which serves as a foundational tool for understanding equilibrium and stability in various systems.In 1993, Czerwik [1] introduced the notion of the b-metric space (BMS) and proved the Banach contraction principle (BCP) in the framework of the complete BMS.This pioneering work established the groundwork for following research on endeavors in BMSs, establishing a diverse field of study.Further, in 2019, Mlaiki et al. [2] extended this preliminary work by including (Ω, ω)-admissible mappings and generalized quasi-contraction in the setting of BMSs, unveiling deeper insights into the FP results.In 2019, Faraji et al. [3] delved into Geraghtytype contractive mappings, utilizing BMSs to not only present the BCP, but also give the solutions for nonlinear integral equations and highlighting the real-life significance of these theoretical developments.In 2020, subsequent advancements by Abbas et al. [4] presented the generalization of the BCP by introducing the Suzuki-type multi-valued mapping and examining coincident and common FPs in the context of the BMS.These findings acted as accelerators for other research efforts, resulting in a series of consequences and insights throughout the area of the BMS, as indicated by the works [5][6][7][8].
In 2018, Mlaiki et al. [9] incorporated controlled functions in the triangle inequality.This novel concept paved the way for a more generalized form of the Banach FP theorem (BFPT), offering a broader scope for applications and theoretical investigations in the FP theory.In 2003, Ran and Reurings [10] used the notion of a partially ordered metric space, and their formulation of the BFPT imposed contractivity conditions exclusively on elements comparable within a partial order, as well as imposed the contractivity condition on the nonlinear map exclusively for elements that can be compared within the partial order.Later, in 2010, Amini-Harandi and Emami [11] investigated the existence and uniqueness of solutions for periodic and boundary-value problems using partially ordered complete metric spaces and the Banach contraction principle (BCP), showcasing the applicability of the FP theory in addressing real-world problems in various domains.In 2022, Farhan et al. [12] discussed Reich-type and (α, )-contractions in partially ordered double controlled metric-type spaces (PODCMSs), illuminating the solution of nonlinear fractional differential equations through a monotonic iterative approach.
The emergence of coupled FPs (CFPs), initially introduced by Bhaskar and Lakshmikantham [13], was utilized to investigate and analyze the presence and exclusivity of solutions for boundary-value problems.Further, in 2009, Lakshmikantham and Ćirić [14] were the pioneers in introducing the concept of the coupled coincidence FP (CCFP) and coupled common FP for nonlinear contractive mappings with a monotone property in partially ordered complete metric spaces (POCMSs).In 2011, Choudhury et al. [15] with their results applied a control function to extend the coupled contraction mapping theorem (CCMT) developed by Gnana Bhaskar and Lakshmikantham in partially ordered metric spaces to a coupled coincidence point conclusion for two compatible mappings.Additionally, it was assumed that the mappings satisfy a weak contractive inequality.In 2020, Mitiku et al. [16] unified fundamental metrical FP theorems, establishing coincidence points, coupled coincidences, and the CCFP for generalized (ϕ, ψ)-contractive mappings in partially ordered b-metric spaces.For more on this, see the related literature [17][18][19][20].Brzdęk et al. [21] proved a fixed point theorem and the Ulam stability in generalized dq-metric spaces.Antón-Sancho [22,23] presented fixed points of principal E six-bundles over a compact algebraic curve and of the automorphisms of the vector bundle moduli space over a compact Riemann surface.
In this study, our aim is to go deeper into the realm of coincidence points, coupled coincidences, and CCFPs within the context of generalized (ϕ, ψ)-contractive mappings.These results are developed within the framework of partially ordered controlled-type metric spaces.

Preliminaries
In this section, we explain some core concepts that will be helpful for the proof of our main results.

Definition 1 ([1]
). Assume a non-empty set Ω and the function s ≥ 1 to be a given real number.A mapping Θ : Ω × Ω −→ [0, ∞) is said to be a b-metric space if the following axioms hold: Then, the pair (Ω, Θ) is called a b-metric space.

Definition 4 ([14]
). Assume that (Ω, Θ, ⪯) is a POS, and consider two mappings h : Ω × Ω −→ Ω and g : Ω −→ Ω such that we have the following: 1. h has the mixed g-monotone property if h is non-decreasing g-monotone in its first argument and is non-increasing g-monotone in its second argument, that is, for any u, v ∈ Ω, 2. An ordered pair element (u, v) ∈ Ω × Ω is said to be a coupled coincidence point (CCP) of h and g if the following relation holds: Also, if g is an identity mapping, then (u, v) is a CFP (CFP) of h. 3.An element u ∈ Ω is said to have a common FP of g and h if h(u, u) = gu = u.

g and h are commutative, if
∀ u, v ∈ Ω, h(gu, gv) = g(hu, hv) g and h are compatible if whenever {u n } and {v n } are two sequences in Ω such that, for all u, v ∈ Ω, The results presented here can be utilized for the convergence of a sequence in the controlled metric space (CMS).
Then, they are alternating distance functions.
Here, ψ : R + −→ R + is ψ(l) = 0 if and only if l = 0.The set of all lower semicontinuous functions is denoted by Ψ.
is called a generalized (Φ, Ψ)-contractive mapping if it satisfies the inequality given below: for any u, v ∈ Ω with u ⪯ v.
Lemma 1. Assume (Ω, Θ, ⪯, α) to be a POCMS with control function α and {u n } and {v n } be two sequences that are α-convergent to u and v, respectively.Then, In a special case, if u = v, then Θ(u p , v p ) = 0.

Main Results
In this section, we formulate the outcomes concerning the existence of coincidence points, coupled coincidences, and CCFPs in the realm of generalized (ϕ, ψ)-contractive mappings.These findings are developed within the specific setting of the POCMS.Theorem 1. Assume (Ω, Θ, ⪯, α) to be a CPOCMS with metric Θ and α : Ω × Ω −→ [1, ∞) to be a controlled function.Assume a mapping h : Ω −→ Ω, which is an almost generalized (ϕ, ψ)-contractive mapping and a continuous, non-decreasing mapping with partial order ⪯ .If there exists a u 0 ∈ Ω with u 0 ⪯ hu 0 , then h have the FP in Ω.
Proof.Assume u 0 ∈ Ω to be an arbitrary point in Ω such that u 0 = hu 0 , then we have a result.Assume u 0 ⪯ hu 0 , and define the sequence {u p } by u p+1 = hu p , for all p ≥ 0. As h is non-decreasing, so by induction, we obtain If there exists p o ∈ N such that u p o = u p o +1 , then from (4), u p o is an FP of h, then we have nothing to prove.Next, we assume that u p ̸ = u p+1 for all p ≥ 1.Since u p > u p−1 for n ≥ 1 and then from the contractive condition (3), we have then, from (5), we obtain where for some p ≥ 1.So, from (6), it follows that a contradiction.This implies that for p ≥ 1.Hence, from (6), we obtain Since, 1 α ∈ (0, 1), then the sequence u p is a Cauchy sequence by [6-9].As Ω is complete, so there exists some element ü ∈ Ω such that u p −→ ü.Moreover, the continuity of h implies that Hence, ü is an FP of h in Ω.
Proof.Using the proof of the above theorem, we construct a non-decreasing Cauchy sequence u p , which converges to ü in Ω.So, we have u p ⪯ ü for any p ∈ N, which implies that sup u p = ü.Now, we have to prove that ü is an FP of h, i.e., hu = u.Assume that hu ̸ = u.Let and Letting p −→ +∞ and by utilizing lim and lim We know that, for all p, u p ⪯ u, then from the contractive condition (3), we obtain Letting p −→ +∞ and using ( 13) and ( 14), we obtain which is a contradiction, by the above inequality (16).Thus, h ü = ü.That is, ü is an FP of Ω.Now, we provide the essential condition for the uniqueness of the FP in Theorems 1 and 2.
Condition 1.Every pair of elements has a lower bound or an upper bound.
The above condition states that, ∀ u, v ∈ Ω, there exist an element w ∈ Ω such that w is comparable to u and v. Theorem 3. In addition, the hypothesis of Theorem 1 (or Theorem 2) and Condition 1 gives the uniqueness of an FP of h in Ω.
Proof.By applying Theorems 1 and 2, we deduce that h has a non-empty set of FPs.Assume that u * and u * * are two FPs of h in Ω.We want to prove that u * = u * * .Assume, on the contrary, u * ̸ = u * * , then by the hypothesis, we have As a consequence, we obtain where From inequality (18), we conclude that which is a contradiction.By deduction, we obtain u * = u * * .This completes the proof.
Theorem 4. Assume (Ω, Θ, ⪯, α) to be a POCMS with metric Θ and controlled function α.We define a generalized (ϕ, ψ)-contraction mapping h : Ω −→ Ω with respect to g : Ω −→ Ω; here, h and g are continuous such that h is a monotone g-non-decreasing mapping, compatible with g and hΩ ⊆ gΩ.If, for some u o ∈ Ω, such that gu o ⪯ hp, then h and g have a coincidence point in Ω.
This contradicts the inequality, if h ü ̸ = hϱ.Hence, The above relation shows that ϱ is a common FP of h and g.
By (50), we deduce that δ p ≤ λ p δ o .As a result, Thus, according to Lemma 3.1 of [5], the sequences gu p and gv p are Cauchy sequences in Ω.We can demonstrate that h and g have a coincidence point in Ω by applying the proof of Theorem 2.2 of [10].
Corollary 1. Assume (Ω, Θ, ⪯, α) to be a POCMS with metric Θ and controlled function α; also, h : Ω × Ω −→ Ω is a continuous mapping, where h satisfies the mixed monotone condition.Assume there exist ϕ ∈ Φ and ψ ∈ Ψ such that Proof.Choose g = I p in Theorem 3.7; we obtain the required proof.
Proof.From Theorem 5, we have at least one coupled coincidence point in Ω for h and g.Suppose that (u, v), (s, t) are two CFPs of h and g, i.e., h(u, v) = gu, h(v, u) = gv and h(s, t) = gs, h(t, s) = gt.

Assume that a
By repeating the procedure performed above, we obtain two sequences ga * p and gb * p in Ω such that In the same manner, we define a sequence gu p , gv p and gs p , gt p as above in Ω by setting Additionally, we have .
Proof.By Theorem 6, h and g have a unique common FP (u, v) ∈ Ω.It is sufficient to demonstrate that u = v.Then, by the hypothesis, gu o and gv o are comparable.Now, we assume that gu o ⪯ gv o .So, by induction, we deduce that gu p ⪯ gv p for all p ≥ 0. We take the sequence gu p and gv p from Theorem 5. Now, by Lemma 1, we obtain which is a contradiction.Hence, u = v, i.e., h and g have a unique common FP in Ω.
Corollary 3. Assume (Ω, Θ, ⪯, α) to be a CPOCMS with metric Θ and controlled function α; also, h : Ω −→ Ω is a continuous non-decreasing mapping with partial order ⪯ such that there exists u o ∈ Ω with u o ⪯ hu o .Assume that Here, the conditions upon M(u, v) and ϕ, ψ are similar to Theorem 1.Then, h has a unique FP in Ω.
Proof.Setting M(u, v) = N(u, v) in a contractive condition (3) and by utilizing Theorem 1, we obtain the proof.Corollary 4. Assume (Ω, Θ, ⪯, α) to be a CPOCMS with metric Θ and controlled function α; also, h : Ω −→ Ω is a continuous non-decreasing mapping with partial order ⪯ .Now, for any u, v ∈ Ω with partial order u ⪯ v, there exists k ∈ [0, 1) such that if there exists u o ∈ Ω with u o ⪯ hu o , then h has a unique FP in Ω.

Application
In this section, we explore the existence of solutions for a set of nonlinear integral equations by manipulating the findings established in the preceding sections.
Consider the following system of integral equations: Now, the system of integral equations will be examined under the following assumptions: (iv) There exists a w > 0 such that, for all x, y ∈ R, Assume that ℑ = C([0, T], ℜ) is a space of all continuous functions defined on [0, T] provided with the controlled metric space given by for all u, v ∈ ℑ, where α = 2 q−1 and q ≥ 1.Now, we endow ℑ with partial order ⪯ given by u ⪯ v ⇐⇒ u(t) ⪯ v(t) for all t ∈ [0, T].

Conclusions
In this work, we proved several concrete theorems concerning FPs, common FPs, coincidence points, coupled coincidence points, and coupled common fixed points satisfying (ϕ, Ψ)-contractive mappings in the context of the POCMS.Furthermore, we provided several non-trivial examples and an application to the system of nonlinear integral equations.This work is extendable in the framework of partially ordered double controlled metric spaces, partially ordered fuzzy metric spaces, partially ordered intuitionistic fuzzy metric spaces, and many others.