A Triple Fixed-Point Theorem for Orthogonal ℓ-Compatible Maps in Orthogonal Complete Metric Space

In 1922, the seminal work of Banach [1] on contraction mappings led to the foundational fixed point (Fp) theorems in metric spaces (MS). These theorems have since been instrumental in various fields, extending the scope of Banach's principle to a broader context within MS. Fixed-point theorems have become a cornerstone in developing mathematical methods pivotal for tackling complex problems in applied mathematics and various scientific disciplines. The exploration of algorithms, particularly their convergence and divergence properties within optimization, and the incorporation of both ordinary and fractional differential equations, alongside integral equations, represent some of the intriguing applications of fixed-point theory.

Fixed-point theory is bifurcated into two distinct streams of inquiry: one that addresses metric spaces, which is integral to computational disciplines such as computing, computational biology, and bioinformatics, and another that focuses on topological problems, capturing the interest of topologists and theoretical computer scientists. The integration of fixed-point theory into these fields underscores its versatility and wide-ranging applications. Particularly, fractional differential equations have gained recognition for their efficacy in modeling the intricate dynamics of systems characterized by non-locality and memory effects.

Nadler's extension [2] of the Banach contraction principle to multi-valued mappings in 1969 marked a significant advancement, laying the groundwork for the exploration of Fps within the realm of Hausdorff metric spaces. Subsequent research has yielded a multitude of Fp results for various classes of multi-valued contractive mappings, as documented extensively in the literature [3-6].

The evolution of fixed-point theory continued with the introduction of the concept of coupled Fps by Guo and Lakshmikantham [7] in 1987 within the framework of partially ordered MS. This notion was further expanded to multi-valued mappings by Hussain and Alotaibi [8], with numerous authors contributing to the establishment of coupled Fp theorems across diverse MS settings.

Building on this foundation, Aydi et al. [9] in 2012 put forward the concept of coupled coincidence points for hybrid pairs of mappings, adhering to a monotone property. This was followed by the pioneering work of Berinde and Brocut [10], who presented the novel idea of tripled Fps in partially ordered MS, and Brocut [11], who explored tripled Fps for nonlinear mappings within the same context, elucidating both existence and uniqueness theorems for contractive type mappings. Amini-Haandi [12] further refined these ideas, proposing a simple and unified approach to the theories of coupled and tripled fixed points in partially ordered complete MS. For more details refer in references [13-17].

More recently, Rashwan et al. [18] broadened the scope of tripled Fps to Geraghty-type contractions in standard MSs endowed with binary relations. In the latest development, Etemad et al. [19] in 2022 validated the existence of solutions to certain problems by utilizing contraction mapping principles alongside measures of non-compactness, tripled Fp results, and the modulus of continuity.

In recent developments within the realm of fixed-point theory, the year 2017 marked a notable advancement with the work of Eshaghi Gordji et al. [20], who introduced a novel perspective on orthogonality in metric spaces, thereby providing a means to expand upon existing theorems. Expanding upon their own preliminary findings, Eshaghi Gordji and Habibi [21] furthered this line of inquiry by establishing fixed-point theorems pertinent to generalized orthogonal metric spaces. Additional contributions that delve into orthogonal concepts within this mathematical area can be found in references [22-26].

Building upon this conceptual framework, Rad et al. [27] ventured into uncharted territory by proposing the idea of orthogonal coupled fixed points within orthogonal metric spaces (O-MS). In our current study, we delve into the interplay between hybrid pairs and orthogonal ℓ-compatible mappings. We present a theorem that not only asserts the existence but also the uniqueness of tripled fixed points (TFp) in orthogonal metric spaces. This theorem we propose serves as a synthesis and expansion of numerous established results, thereby enriching the existing body of literature.

To illustrate the practical implications of our findings, we incorporate an example that underscores the effectiveness of these results. We culminate our discussion by demonstrating how the main theorem we have established can be effectively applied to ascertain the existence of non-negative solutions to integral equations.

We modify the notion of hybrid pair of map contraction to orthogonal sets in this article. To illustrate our results, we also give some examples and application.

Now, we begin with the definition of an orthogonal ℓ-compatible.

Definition 3.1. Let D: Υ³→2^Υ be a multi-valued mapping and ≀ be a self-map on Υ. The Hybrid pair {D, ≀} is said to be orthogonal ℓ-compatible if ≀(D(φ, ς, ω))⊆D(≀φ, ≀ς, ≀ω) whenever (φ, ς, ω) is a TCp of D and ≀.

Our main theorem is to present tripled Fp theorem via the orthogonal ℓ-compatible for an orthogonal complete MS.

Theorem 3.1. Let (Υ, ⊥, p) be an orthogonal complete MS and let D: Υ³→C_B(Υ) and ≀: Υ→Υ maps with D is O_pres satisfying:

(3.1) H(D(φ, ς, ω), D(ı, ȷ, ℓ))≤⊺p(≀φ, ≀ı)+ℜp(≀ς, ≀ȷ)+kp(≀ω, ≀ℓ), ∀φ, ς, ω, ı, ȷ, ℓ∈Υ and ⊺, ℜ, k∈[0, 1) with ⊺+ℜ+k≤κ<1, where κ is a fixed number.

(3.2) D(Υ³)⊆≀(Υ) and ≀(Υ) is a complete subspace of Υ.

Then the mappings D and ≀ have a TCp.

Further, D and ≀ have a unique tripled common Fp if one of the following conditions hold:

(3.3) (a) {Φ, ≀} is orthogonal ℓ-compatible, ∃ ı, ȷ, ℓ∈Υ s.t. $\lim _{i \rightarrow \infty}$≀ⁱφ =ı, $\lim _{i \rightarrow \infty}$≀ⁱς=ȷ and $\lim _{i \rightarrow \infty}$≀ⁱω=ℓ, whenever (φ, ς, ω) is a TCp of {Φ, ≀} and ≀ is orthogonal continuous at ı, ȷ, ℓ.

(b) ∃ ı, ȷ, ℓ∈Υ s.t. $\lim _{i \rightarrow \infty}$≀ⁱı=φ, $\lim _{i \rightarrow \infty}$≀ⁱȷ=ς and $\lim _{i \rightarrow \infty}$≀ⁱℓ=ω, whenever (φ, ς, ω) is a TCp of {Φ, ≀} is orthogonal continuous at φ, ς and ω.

Proof. By the definition of O_set, we can find φ₀∈Υ satisfying (∀ φ ∈ Υ, φ ⊥ φ₀) or (∀ φ ∈ Υ, φ₀ ⊥ φ)

We can find ς₀ ∈ Υ is satisfying (∀ φ ∈ Υ, φ ⊥ ς₀) or (∀ φ ∈ Υ, ς₀ ⊥ φ), and we can find ω₀ ∈ Υ is satisfying (∀φ∈Υ, φ⊥ω₀) or (φ∈Υ, ω₀⊥φ).

It follows that φ₀⊥D(φ₀, ς₀, ω₀) or D(φ₀, ς₀, ω₀) ⊥ φ₀, ς₀⊥D(ς₀, φ₀, ς₀) or D(ς₀, φ₀, ς₀) ⊥ ς₀ and ω₀⊥D(ω₀, ς₀, φ₀) or D(ω₀, ς₀, φ₀) ⊥ ω₀.

Let φ₁=D(φ₀, ς₀, ω₀), φ₂=D²(φ₀, ς₀, ω₀), ......, φ_i+1=Dⁱ⁺¹(φ₀, ς₀, ω₀); ς₁=D(ς₀, φ₀, ς₀), ς₂=D²(ς₀, φ₀, ς₀), .........,ς_i+1=Dⁱ⁺¹(ς₀, φ₀, ς₀), ω₁=D(ω₀, ς₀, φ₀), ω₂=D²(ω₀, ς₀, φ₀), ......, ω_i+1=Dⁱ⁺¹(ω₀, ς₀, φ₀).

If φ_i=φ_i+1, ς_i=ς_i+1 and ω_i=ω_i+1 for each I∈N∪{0}, then φ_i, ς_i, ω_i is a TFp of D. Suppose that φ_i≠φ_i+1, ς_i ≠ς_i+1 and ω_i≠ω_i+1 for all i∈N∪{0}. Then p D(φ_i, ς_i, ω_i), D(φ_i+1, ς_i+1, ω_i+1)>0, p D(ς_i, φ_i, ς_i), D(ς_i+1, φ_i+1, ς_i+1)<0, p D(ω_i, ς_i, φ_i), D(ω_i+1, ς_i+1,, φ_i+1)>0, for all i∈N∪{0}. Since D is O_pres, we have φ_i⊥φ_i+1 or φ_i+1⊥φ_i, ς_i⊥ς_i+1 or ς_i+1⊥ς _i, ω_i⊥ω_i+1 or ω_i+1⊥ω_i, ∀i∈N∪{0}.

Therefore, from (3.2), there exists an O_seq{φ_i}, {ς_i} and {ω_i} in Υ s.t.

≀φ_i+1 ∈ D(φ_i, ς_i, ω_i), ≀ς_i+1 ∈ D(ς_i, φ_i, ς_i) and ≀ω_i+1∈D(ω_i, ς_i, φ_i), i=0,1,2, ……

For deduced that, we get

p_i^φ=p(≀φ_i-1, ≀φ_i), p_i^ς(≀ς_i-1, ≀ς_i), p_i^ω=p(≀ω_i−1, ≀ω_i)            (1)

From (3.1), we have

p₂^φ=p(≀φ₁, ≀φ₂)≤H(D(φ₀, ς₀, ω₀), D(φ₁, ς₁, ω₁))+κ≤⊺p(≀φ₀, ≀φ₁)+ℜ(≀ς₀, ≀ς₁)+եp(≀ω₀, ≀ω₁)+κ=⊺p₁^φ +ℜp₁^ς+եp₁^ω+κ            (i)

p₂^ς=p(≀ς₁, ς₂)≤H(D(ς₀, φ₀, ς₀), D(ς₁, φ₁, ς₁))+κ≤⊺p(≀ς₀, ≀ς₁)+ℜp(≀φ₀, ≀φ₁)+եp(≀ς₀, ≀ς₁)+κ=ℜp₁^φ+(⊺+ե)+κ                 (ii)

p₂^ω=p(≀ω₁, ≀ω₂)≤H(D(ω₀, ς₀, φ₀), D(ω₁, ς₁, φ₁))+κ≤⊺p(≀ω₀, ≀ω₁)+ℜp(≀ς₀, ≀ς₁)+եp(≀φ₀, ≀φ₁)+κ=եp₁^φ+ℜp₁^ς+⊺p₁^ω

p₃^φ=p(≀φ₂, ≀φ₃)≤H(D(φ₁, ς₁, ω₁), D(φ₂, ς₂, ω₂))+κ²≤⊺p(≀φ₁, ≀φ₂)+ℜp(≀ς₁, ≀ς₂)+եp(≀ω₁, ≀ω₂)+κ²=⊺p₂^φ+ℜp₂^ς+եp₂^ω+κ²

≤⊺(⊺p₁^φ+ℜp₁^ς+եp₁^ω+κ)+ℜ(ℜp₁^φ+(⊺+ե)p₁^ς+κ)+ե(եp₁^φ+ℜp₁^ς+⊺p₁^ω+κ)+κ²=(⊺²+ℜ²+ե²)p₁^φ+(2⊺ℜ+2եℜ)p₁^ς+(2⊺ե)p₁^ω+κ²+(⊺+ℜ+ե)κ=(⊺²+ℜ²+ե²)p₁^φ+(2⊺ℜ+2եℜ)p₁^ς+(2⊺ե)p₁^ω+2κ²              (iii)

p₃^ς=p(≀ς₂, ≀ς₃)≤H(D(ς₁, φ₁, ς₁), D(ς₂, φ₂, ς₂))+κ²≤⊺p(≀ς₁, ≀ς₂)+ℜp(≀φ₁, ≀φ₂)+եp(≀ς₁, ≀ς₂)+κ²=ℜp₁^φ+(⊺+ե)p₁^ς+κ²≤ℜ(⊺p₁^φ+ℜp₁^ς+եp₁^ω+κ)+(⊺+ե)(ℜp₁^φ+(⊺+ե)p₁^ς+κ)+κ²

=(2⊺ℜ+2եℜ)p₁^φ+[(⊺+ե)²+ℜ²]p₁^ς+ℜեp₁^ω+κ²+(⊺+ℜ+ե)κ=(2⊺ℜ+2եℜ)p₁^φ+[(⊺+ե)²+ℜ²]p₁^ς+ℜեp₁^ω+2κ²              (iv)

p₃^ω=p(≀ω₂, ≀ω₃)≤H(D(ω₁, ς₁, φ₁), D(ω₂, ς₂, φ₂))+κ²≤⊺p(≀ω₁, ≀ω₂)+ℜp(≀ς₁, ≀ς₂)+եp(≀φ₁, ≀φ₂)+κ²=⊺p₂^ω+ℜp₂^ς+եp₂^φ+κ²=եp₂^φ+ℜp₂^ς+⊺p₂^ω+κ²=ե(⊺p₁^φ+ℜp₁^ς+ℜp₁^ς+κ)+ℜ(ℜp₁^φ+(⊺+ե)p₁^ς+κ]

+⊺(եp₁^φ+ℜp₁^ς+⊺p₁^ω+κ)+κ²=(2⊺ե+ℜ²)p₁^φ+2(⊺ℜ+եℜ)p₁^ς+(⊺²+ℜ²)p₁^ω+κ²+(⊺+ℜ+ե)κ=(2⊺ե+ℜ²)p₁^φ+2(⊺ℜ+եℜ)p₁^ς+(⊺²+ℜ²)p₁^ω+2κ²              (v)

Let

$\mathrm{F}=\left[\begin{array}{ccc}\mathrm{⊺} & \Re & \mathrm{ե} \\ \Re & \mathrm{⊺}+\mathrm{ե} & 0 \\ \mathfrak{ե} & \Re & \mathrm{⊺}\end{array}\right]$ denoted $\mathrm{F}^2$ by $\left[\begin{array}{ccc}\tau_2 & b_2 & c_2 \\ \mathrm{p}_2 & e_2 & ≀_2 \\ \mathrm{~g}_2 & b_2 & \kappa_2\end{array}\right]$             (2)

It is obviously τ₂+b₂+c₂=p₂+e₂+≀₂=g₂+b₂+κ₂ =(⊺+ℜ+k)²≤κ²<1.

Now, we prove by induction:

$\mathrm{F^i}=\left[\begin{array}{lll}\tau_i & b_i & c_i \\ \mathrm{p}_i & e_i & ≀_i \\ \mathrm{~g}_i & b_i & \kappa_i\end{array}\right]$               (3)

where, τ_i+b_i+c_i=p_i+e_i+≀_i=g_i+b_i+κ_i=(⊺+ℜ+ե)ⁱ≤κⁱ<1.

From Eq. (3) is true for i=1, 2.

Assume Eq. (3) is true for some i. Consider

$\mathrm{F}^{\mathrm{i}+1}=\mathrm{F}^i \cdot \mathrm{F}=\left[\begin{array}{ccc}\tau_i & b_i & c_i \\ \mathrm{p}_i & e_i & \imath_i \\ \mathrm{~g}_i & b_i & \kappa_i\end{array}\right] \cdot\left[\begin{array}{ccc}\mathrm{⊺} & \Re & \mathrm{ե} \\ \Re & \mathrm{⊺}+\mathrm{ե} & 0 \\ \mathrm{ե} & \mathfrak{R} & \mathrm{⊺}\end{array}\right]$

=$\left[\begin{array}{ccc}& \mathrm{⊺} \tau_i+\Re b_i+\mathrm{եc_i} \quad \tau_i+(\mathrm{⊺}+\mathrm{ե}) \mathrm{b}_i+\Re \mathrm{c}_i \quad \mathrm{k} \tau_i+\mathrm{⊺} \mathrm{c}_i \\ & \mathrm{⊺} p_i+\mathfrak{R} \mathrm{e}_i+\mathrm{ե≀}_i \quad \Re p_i+(\mathrm{⊺}+\mathrm{ե}) \mathrm{e}_i+\mathfrak{R} ≀_i \quad \mathrm{k} p_i+\mathrm{⊺} \iota_i \\ & \text {⊺} \mathrm{g}_i+\mathfrak{R} \mathrm{b}_i+\mathrm{ե\kappa}_i \quad \Re \mathrm{g}_i+(\mathrm{⊺}+\mathrm{ե}) \mathrm{b}_i+\Re \kappa_i \quad \mathrm{kg}_i+\mathrm{⊺} \kappa_i\end{array}\right]$

We obtain τ_i+1+b_i+1+c_i+1=(⊺+ℜ+ե)(τ_i+b_i+c_i) =(⊺+ℜ+ե)ⁱ⁺¹≤κⁱ⁺¹<1.

Similarly, we get p_i+1+e_i+1+≀_i+1=g_i+1+b_i+1+κ_i+1 =(ե+ℜ+ե)ⁱ⁺¹≤κⁱ⁺¹<1.

Then (3) is true for all positive integer value of i.

Now from (i)− (v) and proceeding this way, we obtain

$\begin{gathered}{\left[\begin{array}{l}\mathrm{p}_{i+1}^{\varphi} \\ \mathrm{p}_{i+1}^{\varsigma} \\ \mathrm{p}_{i+1}^\omega\end{array}\right]=\left[\begin{array}{ccc}\tau_i & b_i & c_i \\ \mathrm{p}_i & e_i & ≀_i \\ \mathrm{~g}_i & b_i & \kappa_i\end{array}\right]\left[\begin{array}{l}\mathrm{p}_1^{\varphi} \\ \mathrm{p}_1^{\varsigma} \\ \mathrm{p}_1^\omega\end{array}\right]+\left[\begin{array}{c}i \kappa^i \\ i \kappa^i \\ i \kappa^i\end{array}\right],} \\ \forall \mathrm{i}=1,2,3, \ldots .\end{gathered}$

That is, $\mathrm{p}_{i+1}^{\varphi} \leq \tau_i \mathrm{p}_1^{\varphi}+b_i \mathrm{p}_1^{\varsigma}+c_i \mathrm{p}_1^\omega+i \kappa^i ; \mathrm{p}_{i+1}^{\varsigma} \leq \mathrm{p}_i \mathrm{p}_1^{\varphi}+$ $e_i \mathrm{p}_1^{\varsigma}+≀ \mathrm{p}_1^\omega+i \kappa^i ; \mathrm{p}_{i+1}^\omega \leq \mathrm{g}_i \mathrm{p}_1^{\varphi}+b_i \mathrm{p}_1^{\varsigma}+\kappa_i \mathrm{p}_1^\omega+i \kappa^i, \forall \mathrm{i}=1,2$, $3, \ldots$

For j>i, we get p(≀φ_j, ≀φ_i)≤p(≀φ_j, ≀φ_j−1)+p(≀φ_j−1, ≀φ_j−2)+....... +p(≀φ_i+2, φ_i+1)+p(≀φ_i+1, ≀φ_i)=p_j^φ+p_j-1^φ+··· +p_i+2^φ+p_i+1^φ, ≤τ_j−1p₁^φ +b_j−1p₁^ς+c_j−1p₁^ω+(j−1)κ^j−1+τ_j−2p₁^φ+b_j−2p₁^ς+c_j−2p₁^ω+(j−2)κ^j−2+...., ≤τ_i+1p₁^φ+b_i+1p₁^ς+c_i+1p₁^ω+(i+1) κⁱ⁺¹+τ_ip₁^φ+b_ip₁^ς+c_ip₁^ω+iκⁱ, ≤(τ_j−1+τ_j−2+…+τ_i+1+τ_i) p₁^φ+(b_j−1+b_j−2+…+b_i+1+b_i)p₁^ς+(c_j−1+c_j−2 +c_i+1+c_i)p₁^ω+[(j−1)κ^j−1+(j−2)κ^j−2+.....+(i+1)κⁱ⁺¹+i κⁱ], ≤(κ^j−1+κ^j−2+…+κⁱ⁺¹+κⁱ)+(p₁^φ  p₁^ς+p₁^ω)+ $\sum_{\mathrm{T}=i}^{j-1} \mathrm{~\top} \kappa^{\top}$, ≤$\frac{\kappa^i}{1-\kappa}$(p₁^φ+p₁^ς+p₁^ω)+ $\sum_{\top=i}^{j-1} \mathrm{~T} \kappa^{\top}$→0, as i→∞, because 0≤κ<1.

Hence {≀φ_i} is a Cauchy O_seq. Similarly, we can show that {≀ς_i} and {≀ω_i} are Cauchy O_seq.

Suppose ≀(Υ) is an orthogonal complete, an O_seq {≀φ_i}, {≀ς_i} and {≀ω_i} are convergent to some α, β, γ∈≀(Υ), respectively. There exists φ, ς, ω∈Υ s.t. α=≀φ, β=≀ς and γ=≀ω.

Now, we get p(D(φ, ς, ω), α)≤p(D(φ, ς, ω), ≀φ_i+1)+p(≀φ_i+1, α)≤H(D(φ, ς, ω), D(φ_i, ς_i, ω_i))+p(≀φ_i+1, α)≤⊺p(≀φ, ≀φ_i)+ℜp(≀ς, ≀ς_i)+kp(≀ω, ≀ω_i)+p(≀φ_i+1, α)=⊺p(α, ≀φ_i)+ℜp(β, ≀ς_i)+kp(γ, ≀ω_i)+p(≀φ_i+1, α).

Taking i→∞, we get p(D(φ, ς, ω), α)≤0 so that α∈D(φ, ς, ω). i.e., ≀φ∈D(φ, ς, ω). Similarly, we prove ≀ς∈D(ς, φ, ς) and ≀ω∈D(ω, ς, φ).

Thus (φ, ς, ω) is a TCp of D and ≀. Suppose (3.3) (a) holds.

Since (φ, ς, ω) is a TCp of D and ≀, there exists ı, ȷ, ℓ∈Υ

s.t. $\lim _{i \rightarrow \infty}$≀ⁱφ=ı, $\lim _{i \rightarrow \infty}$≀ⁱς=ȷ and $\lim _{i \rightarrow \infty}$≀ⁱω=ℓ.

Since ≀ is an orthogonal continuous at ı, ȷ and ℓ, we have ≀ı=ı, ≀ȷ=ȷ and ≀ℓ=ℓ.

Since ≀φ∈D(φ, ς, ω), we have ≀²φ∈≀(D(φ,ς,ω))⊆D(≀φ,≀ς,≀ω).

Since ≀ς∈D(ς,φ,ς), we have ≀²ς∈≀(D(ς,φ,ς))⊆D(≀ς,≀φ,≀ς).

Since ≀ω∈D(ω,ς,φ), we have ≀²ω∈≀(D(ω,ς,φ))⊆D(≀ω,≀ς,≀φ).

Then (≀φ, ≀ς, ≀ω) is TCp of D and ≀.

Similarly, we prove that (≀ⁱφ, ≀ⁱς, ≀ⁱω) is a TCp D and ≀.

Also, it is clear ≀ⁱφ∈D(≀ⁱ⁻¹φ, ≀ⁱ⁻¹ς, ≀ⁱ⁻¹ω), ≀ⁱς∈D(≀ⁱ⁻¹ς, ≀ⁱ⁻¹φ, ≀ⁱ⁻¹ς), ≀ⁱω∈D(≀ⁱ⁻¹ω, ≀ⁱ⁻¹ς, ≀ⁱ⁻¹φ).

From condition (3.1), we get p(≀ı, ≀ȷ, ≀ℓ)≤p(≀ı, ≀ⁱφ)+p(≀ⁱφ, D(ı,ȷ,ℓ))≤ p(≀ı, ≀ⁱφ)+H(D(≀ⁱ⁻¹φ, ≀ⁱ⁻¹ς, ≀ⁱ⁻¹ω), D(ı, ȷ, ℓ))≤p(≀ı, ≀ⁱ)+⊺p(≀ⁱφ, ≀ı)+ℜp(≀ⁱς, ≀ȷ)+եp(≀ⁱω, ≀ℓ).

Taking i→∞, we get p(≀ı, D(ı, ȷ, ℓ))≤0, ⇒≀ı∈D(ı, ȷ, ℓ).

Hence, ı=≀ı∈D(ı, ȷ, ℓ). Similarly, we can prove that ȷ=≀ȷ∈D(ȷ, ı, ȷ) and ℓ=≀ℓ∈D(ℓ, ȷ, ı). Thus, (ı, ȷ, ℓ) is a tripled common Fp of D and ≀. Suppose condition (3.3) (b) holds.

Since (φ, ς, ω) is a coincidence point of {D, ≀}, there exists ı, ȷ, ℓ∈Υ s.t. $\lim _{i \rightarrow \infty}$≀ⁱı=φ, $\lim _{i \rightarrow \infty}$≀ⁱȷ=ς and $\lim _{i \rightarrow \infty}$≀ⁱℓ=ω.

Since ≀ is continuous at φ, ς, and ω, we get φ=≀φ, ς=≀ς and ω=≀ω.

Hence (φ, ς, ω) is a tripled common Fp of {D, ≀}.

Now, we prove uniqueness of tripled common Fp. Assume that (φ^∗, ς^∗, ω^∗)∈Υ is another TFp of D satisfying (φ, ς, ω)≠(φ^∗, ς^∗, ω^∗). Then p (D(φ, ς, ω), D(φ^*, ς^*, ω^*))=p(φ, φ^*)>0, p(D(ς, φ, ς), D(ς^*, φ^*, ς^*))=p(ς, ς ^*)<0, p(D(ω, ς, φ) D(ω^*, ς^*, φ^*))=p(ω, ω^*)>0.

Since {D, ≀} is O_pres, we get Dφ⊥Dφ^∗ and ≀φ⊥≀φ^∗ or Dφ^∗⊥Dφ and ≀φ^∗⊥≀φ, Dς⊥Dς^∗ and ≀ς⊥≀ς^∗ or Dς^∗⊥Dς and ≀ς^∗⊥≀ς, Dω⊥Dω^∗ and ≀ω⊥≀ω^∗ or Dω^∗⊥Dω and ≀ω^∗⊥≀ω.

From the condition of H(D(φ, ς, ω), D(φ^∗, ς^∗, ω^∗))≤⊺p(≀φ, ≀φ^∗)+ℜp(≀ς, ≀ς^∗)+kp(≀ω, ≀ω^∗), ∀φ, ς, ω, φ^∗, ς^∗, ω^∗∈Υ and ⊺, ℜ, ե∈[0, 1) with ⊺+ℜ+ե≤κ<1, we have H(D(φ, ς, ω), D(φ^∗, ς^∗, ω^∗))=0.

Therefore φ=φ^∗, ς=ς^∗ and ω=ω^∗.

So, {D, ≀} has a unique tripled common Fp.

The following example illustrates Theorem 3.1.

Example 3.2. Let Υ= [0, 1], and define a relation $\perp$ on Υ by φ$\perp$ς, ς$\perp$ω, and ω$\perp$φ, if φ, ω≥0, ς≤ 0.

Then (Υ, $\perp$, p) is an orthogonal complete MS. Define a maps D: Υ³→C_B(Υ) and ≀: Υ→Υ defined as D(φ, ς, ω)=[0, $\left(\frac{1}{8}\right) \sin (\varphi)+\left(\frac{1}{4}\right) \sin (\varsigma)+\left(\frac{1}{3}\right) \sin (\omega)$] and ≀φ=$\frac{7}{8}$φ.

H (D(φ, ς, ω), D(ı, ȷ, ℓ))=| $\left(\frac{1}{8} \sin \varphi+\frac{1}{4} \sin \varsigma+\frac{1}{3} \sin \omega\right)$–($\frac{1}{8}$sinı+ $\frac{1}{4}$sinȷ+ $\frac{1}{3}$sinℓ)|=|($\frac{1}{8}$sinφ- $\frac{1}{8}$sinı)+($\frac{1}{4}$sinς- $\frac{1}{4}$sinȷ)+($\frac{1}{3}$sinω-$\frac{1}{3}$sinℓ)|≤$\frac{1}{8}$|sinφ-sinı|+ $\frac{1}{4}$|sinς–sinȷ|+ $\frac{1}{3}$|sinω-sinℓ|≤$\frac{1}{8}$|φ-ı|+$\frac{1}{4}$|ς–ȷ|+$\frac{1}{3}$|ω-ℓ|≤$\frac{7}{56}$|φ-ı|+$\frac{14}{56}$|ς–ȷ|+$\frac{56}{3 X 56}$|ω-ℓ|≤$\frac{1}{7}$|$\frac{7}{8}$φ-$\frac{7}{8}$ı|+$\frac{2}{7}$|$\frac{7}{8}$ς–$\frac{7}{8}$ȷ|+$\frac{8}{21}$|$\frac{7}{8}$ω-$\frac{7}{8}$ℓ|≤$\frac{1}{7}$|≀φ-≀ı|+$\frac{2}{7}$|≀ς–≀ȷ|+$\frac{8}{21}$|≀ω-≀ℓ|.

It is clear that all axioms of Theorem 3.1 are fulfilled and (0, 0, 0) is tripled common Fp of D and ≀.

Let (Y, ⊥) be an orthogonal set. Consider on the product space Y ³ → Y, the following condition: (i, j, ℓ) ≤ (φ, ς, ω) if and only if i ≤ φ, j ≤ ς, ℓ ≤ ω, for (i, j, ℓ), (φ, ς, ω) ∈ Y ³.

Definition 3.3. Let Y be a nonempty set and D: Y ³ → Y be a map. An element (φ, ς, ω) is called a TFp on D if D(φ, ς, ω) = φ,  D(ς, ω, φ) = ς,   D(ω, φ, ς) = ω [28].

From, Hille and Phillips [29], Theorem 7.2.5 extended to concept of orthogonal complete MS as follows:

Definition 3.4. A function η: R+ → R+ is said to be superadditive if η(s) + η(t) ≤ η (s + t), ∀ s, t ∈ R+. It is well known that every nondecreasing, convex function ψ: R+ → R+ with ψ (0) = 0 is superadditive.

Theorem 3.5. Let (Y, ⊥, d) be an orthogonal complete MS. Assume there exists nondecreasing functions ψ_n: R+ → R+, n = 1,2,3 such that ψ = ψ₁ + ψ₂ + ψ₃ is convex, ψ (0) = 0, and ψⁿ(t) → 0 as n → for each t > 0. Let D: Y ³ → Y be a mapping which is nondecreasing in each of its variables and satisfying

≀ (D (φ, ς, ω), D(i, j, ℓ)) ≤ ψ₁(≀(φ, i)) + ψ₁(≀(ς, j)) + ψ₁(≀(ω,ℓ))           (4)

for each φ ≥ i, ς ≥ j, ς ≥ j. Suppose either

(a) D is an orthogonal continuous or;

(b) If a nondecreasing orthogonal sequence (φ_k, ς _k,ω_k) → (φ, ς,ω), then (φ_k, ς _k, ω_k) ≤ (φ, ς, ω), for all k ∈ N, if there exists φ₀,ζ₀,ω₀ ∈ Y with φ₀ ≤ D(φ₀, ς ₀,ω₀), ς ₀ ≤ D(ω₀,φ₀, ς ₀),ω₀ ≤ D(ς ₀,ω₀,φ₀),

(c) D is O preserving;

(d) If for each (φ, ς, ω), (i, j, ℓ) ∈ Y³, then D has a unique TFp.

Proof: By the definition of O_set, we can find φ₀∈Υ satisfying (∀ φ ∈ Υ, φ ⊥ φ₀) or (∀ φ ∈ Υ, φ₀ ⊥ φ)

We can find ς₀ ∈ Υ is satisfying (∀ φ ∈ Υ, φ ⊥ ς₀) or (∀ φ ∈ Υ, ς₀ ⊥ φ), and we can find ω₀ ∈ Υ is satisfying (∀φ∈Υ, φ⊥ω₀) or (φ∈Υ, ω₀⊥φ).

It follows that φ₀⊥D(φ₀, ς₀, ω₀) or D(φ₀, ς₀, ω₀) ⊥ φ₀, ς₀⊥D(ς₀, φ₀, ς₀) or D(ς₀, φ₀, ς₀) ⊥ ς₀ and ω₀⊥D(ω₀, ς₀, φ₀) or D(ω₀, ς₀, φ₀) ⊥ ω₀.

Let φ₁=D(φ₀, ς₀, ω₀), φ₂=D²(φ₀, ς₀, ω₀), ......, φ_i+1=Dⁱ⁺¹(φ₀, ς₀, ω₀); ς₁=D(ς₀, φ₀, ς₀), ς₂=D²(ς₀, φ₀, ς₀), .........,ς_i+1=Dⁱ⁺¹(ς₀, φ₀, ς₀), ω₁=D(ω₀, ς₀, φ₀), ω₂=D²(ω₀, ς₀, φ₀), ......, ω_i+1=Dⁱ⁺¹(ω₀, ς₀, φ₀).

If φ_i=φ_i+1, ς_i=ς_i+1 and ω_i=ω_i+1 for each I∈N∪{0}, then φ_i, ς_i, ω_i is a TFp of D. Suppose that φ_i≠φ_i+1, ς_i ≠ς_i+1 and ω_i≠ω_i+1 for all i∈N∪{0}. Then p D(φ_i, ς_i, ω_i), D(φ_i+1, ς_i+1, ω_i+1)>0, p D(ς_i, φ_i, ς_i), D(ς_i+1, φ_i+1, ς_i+1)<0, p D(ω_i, ς_i, φ_i), D(ω_i+1, ς_i+1,, φ_i+1)>0, for all i∈N∪{0}. Since D is O_pres, we have φ_i⊥φ_i+1 or φ_i+1⊥φ_i, ς_i⊥ς_i+1 or ς_i+1⊥ς _i, ω_i⊥ω_i+1 or ω_i+1⊥ω_i, ∀ i ∈ N∪{0}.

Let (M,ϱ) be a MS which is defined by ϱ((φ, ς, ω),(i,j,ℓ)) = ≀(φ,i) + ≀( ς,j) + ≀(ω,ℓ). Then it is straightforward to show that (M,⊥,ϱ) is a orthogonal complete MS. Let ϱ: M → M be defined by

ϱ(φ, ς, ω) = (D(φ, ς, ω),D(ς, ω, φ),D(ω, φ, ς)).

Then from Eq. (1), we have ≀(ϱ(φ,ζ,ω),ϱ(i,j,ℓ)) = ≀(D(φ, ς, ω),D(i, j,ℓ))+ ≀(D(ς, ω, φ),D(j,ℓ,i)) + ≀(D(ω, φ, ς),

D(ℓ,i,j))≤ ψ₁(≀(φ,i)) + ψ₂(≀(ς,j)) + ψ₃(≀(ω,ℓ)) + ψ₃(≀(φ,i)) + ψ₁(≀(ς, j))+ ψ₂(≀(ω, ℓ)) + ψ₂(≀(φ, i)) + ψ₃(≀(ς, j)) + ψ₁(≀(ω, ℓ))

= ψ(≀(φ, i)) + ψ(≀(ς, j)) + ψ(≀(ω,ℓ)) ≤ ψ(≀(φ, i)) + ≀( ς, j) + ≀(ω, ℓ) = ψ(≀((φ, ς, ω),(i, j,ℓ))).

Since D is nondecreasing in each of its variables then ϱ is non-decreasing. From our assumptions, ϱ is either orthogonal continuous or if a nondecreasing orthogonal sequence i_k → i, for i_k,i ∈ M then i_k ≤ i for each k ∈ N. Also, (φ₀, ς ₀, ω₀) ≤ ϱ(φ₀, ς ₀, ω₀).

Then all the assumptions of Theorem 3.5. are satisfied. Thus, ϱ has a Fp (φ₀, ς ₀, ω₀) and so (φ₀, ς ₀, ω₀) is a TFp of D.

Now suppose that condition (c) holds. Then for each i, j ∈ M, there exists ℓ ∈ M which is comparable to φ and ζ. Thus, by Theorem 3.5., the Fp of ϱ is unique and so (φ, ς, ω) is the unique tripled fixed point of D. Since (ς, ω, φ) and (ω, φ, ς) are TFp’s of ϱ too then, by the uniqueness, we get φ = ς = ω.

The application of fixed-point methods is a cornerstone of mathematical analysis due to their broad utility across numerous areas. This has drawn the attention of numerous researchers to the potential of these techniques. Among the most notable areas of application are the analysis of algorithmic behavior, particularly the phenomena of convergence and divergence in optimization, game theory, as well as in the study of both ordinary and fractional differential equations, alongside differential and integral equations, among others.

In our study, we have formulated and proven theorems related to tripled fixed points (TFps) for mappings that are orthogonal and ℓ-compatible within the context of orthogonally complete metric spaces. We have supplemented our principal findings with illustrative examples that demonstrate their application. Additionally, we have deduced the existence of solutions for a class of complex tripled integral equations. Inspired by the pioneering efforts of Rad et al. [27], which explored the interconnections between n-tuple fixed point theorems and single fixed points, and the advances made by Roldan et al. [30] in establishing fixed point theorems within ordered metric spaces, we envision future research to extend the exploration of TFps to additional structural formations within metric spaces.