# On Orthogonal Coupled Fixed Point Results with an Application

Authors: Gunaseelan Mani; Arul Joseph Gnanaprakasam; Khalil Javed; Santosh Kumar
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this manuscript, owing to the concept of orthogonal coupled contraction mappings type I and II, we prove coupled fixed point theorem in orthogonal metric spaces. In order to strengthen our main results, a suitable example is presented. Moreover, the results we obtained supplement and improve previous research findings. A fruitful application is also supplied to endorse our outcomes.


---

## Body

## 1. Introduction

One of the simple and widely applicable result in nonlinear analysis is Banach contraction principle and it is prolonged in360o in the last century. One ordinary approach to bolster the Banach contraction principle is to restore the metric by weird generalized metric spaces. In 2017, Eshaghi Gordji et al. [1] established the idea of orthogonality and offered a framework to enlarge the results. In the same year, Eshaghi Gordji and Habibi [2] extended this work and proved some fixed point theorem in generalized O-metric spaces. Later, in 2019, Gordji and Habibi [3] demonstrated fixed point theorems in ε-connected O-metric spaces. By applying altering distance functions, Gungor and Turkoglu [4], in 2019, established fixed point results on O-metric spaces. Using orthogonal F-contraction mappings, Sawangsup et al. [5] proved some fixed point theorem on O-complete metric spaces. In 2021, Beg et al. [6] proved fixed point theorems on O-complete b-metric spaces. In 2021, Arul et al. [7] proved fixed point theorems on O-metric spaces. The concept of O-triangular α-admissibility introduced by Arul et al. [8] proved fixed point theorems on O-metric spaces in 2022.In 1987, on the other extreme, Gue and Lakshmikantham [9] prompted the conception of a coupled fixed point in partially ordered metric spaces. Following that, Bhaskar and Lakshmikantham [10], in 2006, demonstrated presence of coupled fixed point theorems by utilizing the mixed monotone property. The same coupled fixed point theorems on complete cone metric spaces was exposed by Sabetghadam et al. [11], in 2009. Afterwards, Guneseelan et al. [12] analyzed these results on complex partial b-metric space. Motivated by the above work, here, we prove coupled fixed point theorems on O-complete metric spaces.


## 2. Preliminaries

In 2021, Gunaseelan et al. [13] proved the following theorem.Theorem 1.
Let∐,φ be a complete complex partial metric space and the mapping Φ:∐×∐⟶∐ such that
(1)φΦϑ,ℵ,Φσ,μ≤ζ1φΦϑ,ℵ,σ+ζ2φΦσ,μ,ϑ,where ζ1 and ζ2 are nonnegative constants with ζ1+ζ2<1. Then, there exists a unique coupled fixed point of Φ.Now, let us recall some basic concepts, which will be used in the sequel.Definition 2 (see [1]).
Let∐≠∅ and ⊥⊆∐×∐ be a binary relation such that or
(2)∃ϑ0∈∐:∀ϑ∈∐,ϑ⊥ϑ0∀ϑ∈∐,ϑ0⊥ϑ,then, it is called an orthogonal set (briefly O-set). We denote this O-set by ∐,⊥.Definition 3 (see [1]).
Let∐,⊥ be an O-set. A sequence ϑι is called an orthogonal sequence (briefly, O-sequence) if or
(3)∀ι∈ℕ,ϑι⊥ϑι+1,∀ι∈ℕ,ϑι+1⊥ϑι.Definition 4 (see [1]).
A triplet∐,⊥,φ is called an orthogonal metric space (briefly O-metric space) if ∐,⊥ is an O-set and ∐,φ is a metric space.Definition 5 (see [1]).
Let∐,⊥ be an O-set. A mapping Φ:∐×∐⟶∐ is said to be ⊥-preserving (briefly OP) if Φϑ,ℵ⊥Φσ,μ whenever ϑ⊥σ and ℵ⊥μ. Also, Φ:∐×∐⟶∐ is said to be weakly ⊥-preserving (briefly WOP) if Φϑ,ℵ⊥Φσ,μ or Φσ,μ⊥Φϑ,ℵ whenever ϑ⊥σ and ℵ⊥μ.Definition 6 (see).
Let∐ be a nonempty set and Φ:∐×∐⟶∐ be a mapping. A point ℵ,χ∈∐×∐ is said to be a coupled fixed point of Φ if Φℵ,χ=ℵ and Φχ,ℵ=χ.


## 3. Main Result

This section presents the new results motivated by Theorem1 and an O-set, we introduce new O-coupled contraction mappings of type I and II.Definition 7.
Let∐,⊥,φ be an O-metric space. A function Φ:∐×∐⟶∐ is called an O-coupled contraction mapping of type I (briefly OCCMT-I) on ∐,⊥,φ if for all ϑ,ℵ,σ,μ∈∐ with ϑ⊥σ and ℵ⊥μ,φΦϑ,ℵ,Φσ,μ>0,
(4)φΦϑ,ℵ,Φσ,μ≤ζ1φϑ,σ+ζ2φℵ,μ,where ζ1 and ζ2 are nonnegative constants with ζ1+ζ2<1.Definition 8.
Let∐,⊥,φ be an O-metric space. A function Φ:∐×∐⟶∐ is called an O-coupled contraction mapping of type II (briefly OCCMT-II) on ∐,⊥,φ if for all ϑ,ℵ,σ,μ∈∐ with ϑ⊥σ and ℵ⊥μ,φΦϑ,ℵ,Φσ,μ>0,
(5)φΦϑ,ℵ,Φσ,μ≤ζ1φΦϑ,ℵ,σ+ζ2φΦσ,μ,ϑ,where ζ1 and ζ2 are nonnegative constants with ζ1+ζ2<1.Theorem 9.
Let∐,⊥,φ be an O-complete metric space with an orthogonal element ϑ0,z0 and Φ:∐×∐⟶∐ be a mapping such that
(i)
Φ is OP(ii)
Φ is OCCMT-I
Then, there exists a unique coupled fixed point ofΦ.Proof.
By the definition of anO-set, we can find ϑ0∈∐ satisfying or
(6)∀ϑ∈∐,ϑ⊥ϑ0∀ϑ∈∐,ϑ0⊥ϑ,and we can find ϖ0∈∐ satisfying or
(7)∀ϑ∈∐,ϑ⊥ϖ0∀ϑ∈∐,ϖ0⊥ϑ.It follows thatϑ0⊥Φϑ0,ϖ0 or Φϑ0,ϖ0⊥ϑ0 and ϖ0⊥Φϖ0,ϑ0 or Φϖ0,ϑ0⊥ϖ0. Let
(8)ϑ1≔Φϑ0,ϖ0,ϑ2≔Φϑ1,ϖ1=Φ2ϑ0,ϖ0,⋯,ϑι+1≔Φϑι,ϖι=Φι+1ϑ0,ϖ0,ϖ1≔Φϖ0,ϑ0,ϖ2≔Φϖ1,ϑ1=Φ2ϖ0,ϑ0,⋯,ϖι+1≔Φϖι,ϑι=Φι+1ϖ0,ϑ0.Ifϑι=ϑι+1, ϖι=zι+1 for any ι∈ℕ∪0, then ϑι,ϖι is a coupled fixed point of Φ. Suppose that ϑι≠ϑι+1 or ϖι≠ϖι+1, ∀ι∈ℕ∪0. Then, or
(9)φΦϑι,ϖι,Φϑι+1,ϖι+1>0,φΦϖι,ϑι,Φzι+1,ϑι+1>0for allι∈ℕ∪0. Since Φ is OP, we have
(10)ϑι⊥ϑι+1orϑι+1⊥ϑι,ϖι⊥ϖι+1orϖι+1⊥ϖι,∀ι∈ℕ∪0. Therefore, ϑι and ϖι are O-sequences. Since Φ is OCCMT-I,
(11)φϑι,ϑι+1≤ζ1φϑι−1,ϑι+ζ2φϖι−1,ϖι,φϖι,ϖι+1≤ζ1φzι−1,ϖι+ζ2φϑι−1,ϑι.Set(12)hι≔φϑι,ϑι+1+φzι,ϖι+1.Then, we have(13)hι≤ζ1+ζ2φϑι−1,ϑι+ζ1+ζ2φϖι−1,ϖι=ζ1+ζ2hι−1,∀ι∈ℕ.Sincej≔ζ2/1−ζ2<1, we get
(14)hι≤jhι−1≤⋯≤jιh0.Ifh0=0, then
(15)φϑ0,ϑ1+φϖ0,ϖ1=0.Hence,ϑ0=ϑ1=Φϑ0,ϖ0 and ϖ0=ϖ1=Φϖ0,ϑ0, which implies that ϑ0,ϖ0 is a coupled fixed point of Φ. Let h0>0. For each ι≥θ,
(16)φϑι,ϑθ≤φϑι,ϑι−1+φϑι−1,ϑι−2+⋯φϑθ+1,ϑθ.Similarly, we can derive that(17)φϖι,ϖθ≤φϖι,ϖι−1+φϖι−1,ϖι−2+⋯φϖθ+1,ϖθ.Thus,(18)φϑι,ϑθ+φϖι,ϖθ≤hι−1+hι−2+⋯+hθ≤jι−1+jι−2+⋯+jθh0≤jθ1−jh0⟶0,ι⟶∞,which implies that ϑι and ϖι are Cauchy O-sequences. By the definition of an O-complete, we can find ϑ,ϖ∈∐ satisfying limι⟶∞ϑι=ϑ and limι⟶∞ϖι=ϖ. Now,
(19)φΦϑ,ϖ,ϑ≤φΦϑ,ϖ,ϑι+1+φϑι+1,ϑ,φΦϑ,ϖ,ϑ−φϑι+1,ϑ≤φΦϑ,ϖ,ϑι+1.By choice ofϑ and ϖ, we have
(20)ϑ⊥ϑιorϑι⊥ϑ,ϖ⊥ϖιorϖι⊥ϖ.SinceΦ is OCCMT-I, we get
(21)φΦϑ,ϖ,Φϑι,ϖι≤ζ1φϑ,ϑι+ζ2φϖ,ϖι.From (19) and (21), we obtain
(22)φΦϑ,ϖ,ϑ−φϑι+1,ϑ≤ζ1φϑ,ϑι+ζ2φϖ,ϖι.Asι⟶∞, we get
(23)limι⟶∞φΦϑ,ϖ,ϑ=0.Therefore,Φϑ,ϖ=ϑ.Similarly, we can prove thatΦϖ,ϑ=ϖ. Assume that ϑ∗,ϖ∗ is another coupled fixed point of Φ satisfying ϑ,ϖ≠ϑ∗,ϖ∗. Then, φΦϑ,ϖ,Φϑ∗,ϖ∗=φϑ,ϑ∗>0 and φΦϖ,ϑ,Φϖ∗,ϑ∗=φz,ϖ∗>0. Since Φ is OP, we get
(24)ϑ⊥ϑ∗orϑ∗⊥ϑ,ϖ⊥ϖ∗orϖ∗⊥ϖ.SinceΦ is an OCCMT-I, we get
(25)φϑ,ϑ∗=φΦϑ,ϖ,Φϑ∗,ϖ∗≤ζ1φϑ,ϑ∗+ζ2φϖ,ϖ∗,φϖ,ϖ∗=φΦϖ,ϑ,Φϖ∗,ϑ∗≤ζ1φϖ,ϖ∗+ζ2φϑ,ϑ∗.Thus, we have(26)φϑ,ϑ∗+φϖ,ϖ∗≤ζ1+ζ2φϑ,ϑ∗+φϖ,ϖ∗.Sinceζ1+ζ2<1, we obtain
(27)φϑ,ϑ∗+φϖ,ϖ∗=0.Therefore,ϑ=ϑ∗ and ϖ=ϖ∗, which is a absurdity. So, Φ has a unique coupled fixed point.Example 1.
Let∐=ℝ and φℵ,χ=ℵ−v for all ℵ,χ∈∐. Define a relation ⊥ on ∐ by
(28)ℵ⊥χiffℵ,χ≥0.
Then,∐,⊥,φ is an O-complete metric space. Define a mapping Φ:∐×∐⟶∐ by Φℵ,χ=ℵ+3χ/5. Let ℵ⊥ϑ and χ⊥σ. Then, ℵ,ϑ≥0 and χ,σ≥0. Now,
(29)Φℵ,χ=ℵ+3χ5≥0,Φϑ,σ=ϑ+3σ5≥0.It follows that(30)Φℵ,χ⊥Φϑ,σ.Therefore,Φ is OP. Then, for all ℵ,χ,ϑ,σ∈ϑ, Φℵ,χ≠Φϑ,σ, we get
(31)φΦℵ,χ,Φϑ,σ=ℵ+3χ5−ϑ+3σ5≤ℵ−ϑ5+3χ−σ5=ℵ−ϑ5+3χ−σ5=15φℵ,ϑ+35φχ,σ.Therefore, all the hypotheses of Theorem9 are fulfilled with ζ1=1/5+ζ2=3/5<1. Hence, Φ has a unique coupled fixed point 0,0∈ℝ×ℝ.Theorem 10.
Let∐,⊥,φ be an O-complete metric space with an orthogonal element ϑ0,z0 and Φ:∐×∐⟶∐ be a mapping such that
(i)
Φ is OP(ii)
Φ is OCCMT-II
Then, there exists a unique coupled fixed point ofΦ.Proof.
By the definition of anO-set, we can find ϑ0∈∐ satisfying
(32)∀ϑ∈∐,ϑ⊥ϑ0or∀ϑ∈∐,ϑ0⊥ϑ,and we can find ϖ0∈∐ satisfying
(33)∀ϑ∈∐,ϑ⊥ϖ0or∀ϑ∈∐,ϖ0⊥ϑ.It follows thatϑ0⊥Φϑ0,ϖ0 or Φϑ0,ϖ0⊥ϑ0 and ϖ0⊥Φϖ0,ϑ0 or Φϖ0,ϑ0⊥ϖ0. Let
(34)ϑ1≔Φϑ0,ϖ0,ϑ2≔Φϑ1,ϖ1=Φ2ϑ0,ϖ0,⋯,ϑι+1≔Φϑι,ϖι=Φι+1ϑ0,ϖ0,ϖ1≔Φϖ0,ϑ0,ϖ2≔Φϖ1,ϑ1=Φ2ϖ0,ϑ0,⋯,ϖι+1≔Φϖι,ϑι=Φι+1ϖ0,ϑ0.Ifϑι=ϑι+1, ϖι=zι+1 for any ι∈ℕ∪0, then, ϑι,ϖι is a coupled fixed point of Φ. Suppose that ϑι≠ϑι+1 or ϖι≠ϖι+1, ∀ι∈ℕ∪0. Then
(35)φΦϑι,ϖι,Φϑι+1,ϖι+1>0orφΦϖι,ϑι,Φzι+1,ϑι+1>0,for all ι∈ℕ∪0. Since Φ is OP, we have
(36)ϑι⊥ϑι+1orϑι+1⊥ϑι,ϖι⊥ϖι+1orϖι+1⊥ϖι,∀ι∈ℕ∪0. Therefore ϑι and ϖι are O-sequences. Since Φ is OCCMT-II, we derive that
(37)φϑι,ϑι+1=φΦϑι−1,ϖι−1,Φϑι,ϖι≤ζ1φϑι,ϑι+ζ2φϑι+1,ϑι−1≤ζ2φϑι+1,ϑι+φϑι,ϑι−1=ζ21−ζ2φϑι,ϑι−1.Similarly, we can derive that(38)φϖι,ϖι+1≤ζ21−ζ2φϖι,ϖι−1,From (37) and (38), we derive
(39)φϑι,ϑι+1+φϖι,ϖι+1≤ζ21−ζ2φϑι,ϑι−1+φϖι,ϖι−1.Set(40)hι≔φϑι,ϑι+1+φzι,ϖι+1.Then, we have(41)hι≤ζ21−ζ2φϑι−1,ϑι+φϖι−1,ϖι=ζ21−ζ2hι−1,∀ι∈ℕ.Sinceζ1+ζ2<1, then j=ζ2/1−ζ2<1, we get
(42)hι<jhι−1≤⋯≤jιh0.Ifh0=0, then
(43)φϑ0,ϑ1+φϖ0,ϖ1=0.Hence,ϑ0=ϑ1=Φϑ0,ϖ0 and ϖ0=ϖ1=Φϖ0,ϑ0, which implies that ϑ0,ϖ0 is a coupled fixed point of Φ. Let h0>0. For each ι≥θ,
(44)φϑι,ϑθ≤φϑι,ϑι−1+φϑι−1,ϑι−2+⋯φϑθ+1,ϑθ.Similarly, we can derive that(45)φϖι,ϖθ≤φϖι,ϖι−1+φϖι−1,ϖι−2+⋯φϖθ+1,ϖθ.Thus,(46)φϑι,ϑθ+φϖι,ϖθ≤hι−1+hι−2+⋯+hθ≤jι−1+jι−2+⋯+jθh0≤jθ1−jh0⟶0,ι⟶∞,which implies that ϑι and ϖι are Cauchy O-sequences. By the definition of an O-complete, we can find ϑ,ϖ∈∐ satisfying limι⟶∞ϑι=ϑ and limι⟶∞ϖι=. Now,
(47)φΦϑ,ϖ,ϑ≤φΦϑ,ϖ,ϑι+1+φϑι+1,ϑ,φΦϑ,ϖ,ϑ−φϑι+1,ϑ≤φΦϑ,ϖ,ϑι+1.By choice ofϑ and ϖ, we have
(48)ϑ⊥ϑιorϑι⊥ϑ,ϖ⊥ϖιorϖι⊥ϖ.SinceΦ is OCCMT-II, we get
(49)φΦϑ,ϖ,Φϑι,ϖι≤ζ1φΦϑ,ϖ,ϑι+ζ2φΦϑι,ϖι,ϑ.From (47) and (49), we obtain
(50)φΦϑ,ϖ,ϑ−φϑι+1,ϑ≤ζ1φΦϑ,ϖ,ϑι+ζ2φϑι+1,ϑ.Asι⟶∞, we get
(51)limι⟶∞φΦϑ,ϖ,ϑ=0.Therefore,Φϑ,ϖ=ϑ.Similarly, we can prove thatΦϖ,ϑ=ϖ. Assume that ϑ∗,ϖ∗ is another coupled fixed point of Φ satisfying ϑ,ϖ≠ϑ∗,ϖ∗. Then, φΦϑ,ϖ,Φϑ∗,ϖ∗=φϑ,ϑ∗>0 and φΦϖ,ϑ,Φϖ∗,ϑ∗=φz,ϖ∗>0. Since Φ is OP, we get
(52)ϑ⊥ϑ∗orϑ∗⊥ϑ,ϖ⊥ϖ∗orϖ∗⊥ϖ.SinceΦ is an OCCMT-II, we get
(53)φϑ,ϑ∗=φΦϑ,ϖ,Φϑ∗,ϖ∗≤ζ1φΦϑ,ϖ,ϑ∗+ζ2φΦϑ∗,ϖ∗,ϑand φϖ,ϖ∗=φΦϖ,ϑ,Φϖ∗,ϑ∗≤ζ1φΦϖ,ϑ,ϖ∗+ζ2φΦϖ∗,ϑ∗,ϖ.Thus, we have(54)φϑ,ϑ∗+φϖ,ϖ∗≤ζ1+ζ2φϑ,ϑ∗+φϖ,ϖ∗.Sincea+b<1, we obtain
(55)φϑ,ϑ∗+φϖ,ϖ∗=0.Therefore,ϑ=ϑ∗ and ϖ=ϖ∗, which is a absurdity. So, Φ has a unique coupled fixed point.Example 2.
Let∐=0,1 and φℵ,χ=ℵ−v for all ℵ,χ∈∐. Define a relation ⊥ on ∐ by
(56)ℵ⊥χiffℵ,χ≥0.
Then,∐,⊥,φ is an O-complete metric space. Define a mapping Φ:∐×∐⟶∐ by Φℵ,χ=ℵ1−χ. Let ℵ⊥ϑ and χ⊥σ. Then ℵ,ϑ≥0 and χ,σ≥0. Now
(57)Φℵ,χ=ℵ1−χ≥0,Φϑ,σ=ϑ1−σ≥0.It follows that(58)Φℵ,χ⊥Φϑ,σ.Therefore,Φ is OP. Then, for all ℵ,χ,ϑ,σ∈ϑ, Φℵ,χ≠Φϑ,σ, we get
(59)φΦℵ,χ,Φϑ,σ=ℵ1−χ−ϑ1−σ≤34ℵ1−χ−ϑ+15ϑ1−σ−ℵ=34φΦℵ,χ,ϑ+15φΦϑ,σ,ℵ.Therefore, all the hypotheses of Theorem10 are fulfilled with ζ1=3/4+ζ2=1/5<1. Hence, Φ has a unique coupled fixed point 0,0∈ℝ×ℝ.


## 4. Supportive Application

LetH=0,H and ∐=CH,ℝ=G:H⟶ℝGisacontinuousfunction. Consider the integral equations:
(60)ρχ=ðχ+∫0∐δv,αΩα,ρα,δαdα,χ∈0,H,δχ=ðχ+∫0∐δχ,αΩα,δα,ραdα,χ∈0,H,where
(a)
ð:∐⟶ℝ and Ω:∐×ℝ×ℝ⟶ℝ are continuous,(b)
δ:∐×∐ is continuous and measurable at α∈∐, ∀χ∈∐;(c)
δχ,α≥0, ∀χ,α∈∐ and ∫0Hδχ,αdα≤1 , ∀χ∈∐.Theorem 11.
Assume that the conditionsa–c hold. Suppose that
(61)Ωχ,ρχ,δχ−Ωv,θχ,qχ≤ρχ−θχ2+δχ−qχ3,for each χ∈∐ and ∀ρ,δ∈CH,ℝ. Then, equation (60) has a unique solution in ∐.Proof.
Let∐=κ∈CH,ℝ:κβ>0∀β∈H. Define a relation ⊥ on ∐ by
(62)ρ⊥δiffρβδβ≥ρβorρβδβ≥δβ∀β∈0,H.
Define a mappingφ:∐×∐⟶0,∞ by
(63)φρ,δ=supβ∈Iρχ−δχ,for all ρ,δ∈∐. Thus, ∐,φ is a O-complete metric space. Define Φ:∐×∐⟶∐ by
(64)Φρ,δχ=ðχ+∫0∐δχ,αΩα,ρα,δαdα,χ∈0,H.For eachρ,δ∈∐ with ρ⊥δ and χ∈0,H, we have
(65)Φρ,δχ=ðχ+∫0∐δχ,αΩα,ρα,δαdα≥1.It follows thatΦρ,δχΦδ,ρχ≥Φρ,δχ and so Φρ,δχ⊥Φδ,ρχ. Then, Φ is OP. Let ρ,δ,θ,q∈ϑ with ρ⊥δ and θ⊥q. Suppose that Φρ,δχ≠Φθ,qχ. For every χ∈0,H, we have
(66)Φρ,δχ−Φθ,qχ=∫0Hδχ,αΩα,ρα,δα−Ωα,θα,qαdα≤∫0Hδχ,αΩα,ρα,δα−Ωα,θα,qαdα≤∫0Hδχ,αρχ−θχ2+δχ−qχ3dα≤ρχ−θχ2+δχ−qχ3∫0Hδχ,αdα≤ρχ−θχ2+δχ−qχ3,which implies that
(67)φΦρ,δ,Φθ,q≤a=12φρ,θ+b=13φδ,q.Therefore, all the hypotheses of Theorem9 are fulfilled and hence equation (60) has a unique solution.


## 5. Open problem

In this article, we proved coupled fixed point theorems for orthogonal coupled contraction mappings of typeI and II in O-complete metric spaces. An illustrative example is presented to strengthen our obtained main results. Agarwal and Karapinar [14] proved coupled fixed point theorems in G-metric spaces in 2013. Here, the intriguing open problem is to investigate the coupled fixed point theorems on orthogonal G-metric spaces instead of coupled fixed point theorems on O-metric space. In 2014, Roldán et al. [15] introduced multidimensional fixed point theorems. It is an intriguing open problem to investigate the orthogonal multidimensional fixed point theorems instead of orthogonal coupled fixed point theorem. In 2016, Roldán, Sintunavarat [17] proved common coupled fixed point theorems in fuzzy metric spaces using the CLRg property. It is an intriguing open problem to investigate the orthogonal common coupled fixed point theorems instead of orthogonal coupled fixed point theorem.



---

*Source: 5044181-2022-06-14.xml*