Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces

Gunaseelan Mani; Arul Joseph Gnanaprakasam; Jung Rye Lee; Choonkil Park

doi:10.1515/math-2021-0075

Article Open Access

Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces

Gunaseelan Mani , Arul Joseph Gnanaprakasam , Jung Rye Lee and Choonkil Park

Published/Copyright: November 22, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

The concept of coupled 𝔉-orthogonal contraction mapping is introduced in this paper, and some coupled fixed point theorems in orthogonal metric spaces are proved. The obtained results generalize and extend some of the well-known results in the literature. An example is presented to support our results. Furthermore, we apply our result to obtain the existence theorem for a common solution of the integral equations:

ζ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ζ ( β ) , ξ ( β ) ) d β , v ∈ [ 0 , H ] , ξ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ξ ( β ) , ζ ( β ) ) d β , v ∈ [ 0 , H ] ,

where

ð : M → R and Ω : M × R × R → R are continuous;
Ξ : M × M is continuous and measurable at β ∈ M , ∀ v ∈ M ;
Ξ ( v , β ) ≥ 0 , ∀ v , β ∈ M and ∫ 0 H Ξ ( v , β ) d β ≤ 1 , ∀ v ∈ M .

Keywords: orthogonal set; orthogonal metric space; orthogonal continuous; orthogonal preserving; orthogonal 𝔉-contraction; coupled fixed point

MSC 2010: 47H10; 54H25

1 Introduction and preliminaries

The concept of an orthogonal set has many applications in several branches in mathematics and it has many types of orthogonality. Eshaghi Gordji et al. [1] introduced the new concept of orthogonality in metric spaces and proved the fixed point result for contraction mappings in metric spaces endowed with the new orthogonality. Eshaghi Gordji and Habibi [2] proved fixed point theory in generalized orthogonal metric space. Sawangsup et al. [3] introduced the new concept of orthogonal F -contraction mappings and proved the fixed point theorems on orthogonal-complete metric spaces. The (orthogonal) contractive-type mappings have been studied and important results have been obtained by many authors [4,5, 6,7,8, 9,10,11, 12,13,14, 15,16].

In this paper, we introduce the new concept coupled F -orthogonal contraction mapping and prove the coupled fixed point theorem on orthogonal complete metric space.

Throughout this paper, we denote by W , R + and ι the non-void set, the set of positive real numbers and the set of positive integers, respectively.

First, we recall the concept of a control function which was introduced by Wardowski [17]. Let ℑ denote the family of all functions F : R + → R satisfying the following properties:

F is strictly increasing;
for each sequence { κ ι } of positive numbers, we have
lim ι → ∞ κ ι = 0 ⇔ lim ι → ∞ F ( κ ι ) = − ∞ ;
there exists i ∈ ( 0 , 1 ) such that lim κ → 0 + κ i F ( κ ) = 0 .

Example 1.1

[17] Let F : R + → R be defined by F ( l ) = ln l for l > 0 . Then F ∈ ℑ .

Definition 1.1

[8] Let W be a nonempty set and L : W × W → W be a mapping. A point ( u , v ) ∈ W × W is said to be a coupled fixed point of L if L ( u , v ) = u and L ( v , u ) = v .

Ozkan [9] proved the following theorem.

Theorem 1.2

[9] Let ( W , d ) be a complete metric space and L : W × W → W be a mapping. If there exist F ∈ ℑ and ℓ ∈ ( 0 , ∞ ) such that the following condition holds

d ( L ( a , b ) , L ( u , v ) ) > 0 ⇒ ℓ + F ( d ( L ( a , b ) , L ( u , v ) ) ) ≤ F ( κ d ( a , u ) + l d ( b , v ) )

for all a , b , u , v ∈ W , κ , l ∈ R + , L ( a , b ) ≠ L ( u , v ) , where κ + l < 1 , then L has a unique coupled fixed point.

On the other hand, Eshaghi Gordji et al. [1] introduced the basic concept as follows:

Definition 1.2

[1] Let W ≠ ϕ and ⊢ ⊆ W × W be a binary relation. If ⊢ satisfies the following condition:

∃ w 0 ∈ W : ( ∀ w ∈ W , w ⊢ w 0 ) or ( ∀ w ∈ W , w 0 ⊢ w ) ,

then it is called an orthogonal set (briefly O -set). We denote this O -set by ( W , ⊢ ) .

Example 1.3

[1] Let W = [ 0 , ∞ ) and define w ⊢ u if w u ∈ { w , u } . Then, by setting w 0 = 0 or w 0 = 1 , ( W , ⊢ ) is an O -set.

Definition 1.3

[1] Let ( W , ⊢ ) be an O -set. A sequence { w ι } is called an orthogonal sequence (briefly, O -sequence) if

( ∀ ι ∈ ι , w ι ⊢ w ι + 1 ) or ( ∀ ι ∈ ι , w ι + 1 ⊢ w ι ) .

Definition 1.4

[1] A triplet ( W , ⊢ , d ) is called an orthogonal metric space if ( W , ⊢ ) is an O -set and ( W , d ) is a metric space.

Definition 1.5

Let ( W , ⊢ ) be an O -set. A mapping L : W × W → W is said to be ⊢ -preserving if L ( w , u ) ⊢ L ( s , r ) whenever w ⊢ s and u ⊢ r . Also, L : W × W → W is said to be weakly ⊢ -preserving if L ( w , u ) ⊢ L ( s , r ) or L ( s , r ) ⊢ L ( w , u ) whenever w ⊢ s and u ⊢ r .

2 Main results

In this section, inspired by Theorem 1.2 and an orthogonal set, we introduce a new coupled F -orthogonal contraction mapping and prove some coupled fixed point theorem for this contraction mapping in an orthogonal metric space.

Definition 2.1

Let ( W , ⊢ , d ) be an orthogonal metric space. A mapping L : W × W → W is said to be a coupled F -orthogonal contraction mapping on ( W , ⊢ , d ) if there are F ∈ ℑ and ℓ > 0 such that for all w , u , s , r ∈ W with w ⊢ s and u ⊢ r [ d ( L ( w , u ) , L ( s , r ) ) ] > 0 ,

ℓ + F ( d ( L ( w , u ) , L ( s , r ) ) ≤ F ( a d ( w , s ) + b d ( u , r ) ) ) ,

where a + b = 1 , a , b ≥ 0 .

Theorem 2.1

Let ( W , ⊢ , d ) be an O -complete metric space with an orthogonal element w 0 and L : W × W → W be a mapping. Suppose that there exist F ∈ ℑ and ℓ > 0 such that

L is ⊢ -preserving;
L is a coupled F -orthogonal contraction mapping.

Then L has a unique coupled fixed point.

Proof

Since ( W , ⊢ ) is an O -set, there exists w 0 ∈ W such that

( ∀ w ∈ W , w ⊢ w 0 ) or ( ∀ w ∈ W , w 0 ⊢ w ) ,

and there exists z 0 ∈ W such that

( ∀ w ∈ W , w ⊢ z 0 ) or ( ∀ w ∈ W , z 0 ⊢ w ) .

It follows that w 0 ⊢ L ( w 0 , z 0 ) or L ( w 0 , z 0 ) ⊢ w 0 and z 0 ⊢ L ( z 0 , w 0 ) or L ( z 0 , w 0 ) ⊢ z 0 .

Let

w 1 ≔ L ( w 0 , z 0 ) , w 2 ≔ L ( w 1 , z 1 ) = L 2 ( w 0 , z 0 ) , … , w ι + 1 ≔ L ( w ι , z ι ) = L ι + 1 ( w 0 , z 0 )

and

z 1 ≔ L ( z 0 , w 0 ) , z 2 ≔ L ( z 1 , w 1 ) = L 2 ( z 0 , w 0 ) , … , z ι + 1 ≔ L ( z ι , w ι ) = L ι + 1 ( z 0 , w 0 ) .

If w ι = w ι + 1 , z ι = z ι + 1 for any ι ∈ ι ∪ { 0 } , then it is clear that ( w ι , z ι ) is a coupled fixed point of L . Assume that w ι ≠ w ι + 1 or z ι ≠ z ι + 1 for all ι ∈ ι ∪ { 0 } . Then we have

d ( L ( w ι , z ι ) , L ( w ι + 1 , z ι + 1 ) ) > 0 or d ( L ( z ι , w ι ) , L ( z ι + 1 , w ι + 1 ) ) > 0

for all ι ∈ ι ∪ { 0 } . Since L is ⊢ -preserving, we have

w ι ⊢ w ι + 1 or w ι + 1 ⊢ w ι

and

z ι ⊢ z ι + 1 or z ι + 1 ⊢ z ι

for all ι ∈ ι ∪ { 0 } . This implies that { w ι } and { z ι } are O -sequences. Since L is a coupled F -orthogonal contraction mapping, we have

ℓ + F ( d ( w ι , w ι + 1 ) ) = ℓ + F ( d ( L ( w ι − 1 , z ι − 1 ) , L ( w ι , z ι ) ) ≤ F ( a d ( w ι − 1 , w ι ) + b d ( z ι − 1 , z ι ) ) , ∀ ι ∈ ι

and

ℓ + F ( d ( z ι , z ι + 1 ) ) = ℓ + F ( d ( L z ι − 1 , w ι − 1 ) , L ( z ι , w ι ) ) ≤ F ( a d ( z ι − 1 , z ι ) + b d ( w ι − 1 , w ι ) ) , ∀ ι ∈ ι .

By the property ( F 1 ) , we obtain

d ( w ι , w ι + 1 ) < a d ( w ι − 1 , w ι ) + b d ( z ι − 1 , z ι )

and

d ( z ι , z ι + 1 ) < a d ( z ι − 1 , z ι ) + b d ( w ι − 1 , w ι ) .

Set

h ι − 1 ≔ d ( w ι , w ι + 1 ) + d ( z ι , z ι + 1 ) .

Then we have

h ι − 1 < ( a + b ) d ( w ι − 1 , w ι ) + ( a + b ) d ( z ι − 1 , z ι ) = ( a + b ) h ι − 2 , ∀ ι ∈ ι .

Since a + b = 1 , we get h ι − 1 < h ι − 2 , ∀ ι ∈ ι . Consequently,

ℓ + F ( h ι − 1 ) ≤ F ( h ι − 2 ) , ∀ ι ∈ ι .

Therefore, we get

(2.1) F ( h ι ) ≤ F ( h ι − 1 ) − ℓ ≤ ⋯ ≤ F ( h 0 ) − ι ℓ , ∀ ι ∈ ι .

As ι → ∞ , we obtain lim ι → ∞ F ( h ι ) = − ∞ .

By the property ( F 2 ) , we have

lim ι → ∞ h ι = 0 .

By the property ( F 3 ) , there exists k ∈ ( 0 , ∞ ) such that

lim ι → ∞ h ι k F ( h ι ) = 0 .

By (2.1), we have

(2.2) h ι k F ( h ι ) − h ι k F ( h 0 ) ≤ − h ι k ι ℓ ≤ 0 .

Taking the limit as n → ∞ in (2.2), we get

lim ι → ∞ ι h ι k = 0 .

Then there exists ι 1 ∈ ι such that ι h ι k ≤ 1 for all ι ≥ ι 1 and so

(2.3) h ι ≤ 1 ι 1 k

for all ι ≥ ι 1 . Now, we claim that { w ι } ι = 1 ∞ and { z ι } ι = 1 ∞ are Cauchy O -sequences. Using (2.3), we obtain that, for all m > ι ≥ ι 1 ,

d ( w m , w ι ) + d ( z m , z ι ) ≤ d ( w m , w m − 1 ) + d ( z m , z ι ) + ⋯ + d ( w ι − 1 , w ι ) + d ( z ι − 1 , z ι ) = h m − 1 + h m − 2 + ⋯ + h ι = ∑ i = ι m − 1 h i ≤ ∑ i = 1 ∞ h i ≤ ∑ i = 1 ∞ 1 i 1 k .

Since ∑ i = ι ∞ 1 i 1 k < ∞ , { w ι } ι = 1 ∞ and { z ι } ι = 1 ∞ are Cauchy O -sequences. Since W is O -complete, there exist w , z ∈ W such that lim ι → ∞ w ι = w and lim ι → ∞ z ι = z . By the property of O -metric, we obtain

(2.4) d ( L ( w , z ) , w ) ≤ d ( L ( w , z ) , w ι + 1 ) + d ( w ι + 1 , w ) , d ( L ( w , z ) , w ) − d ( w ι + 1 , w ) ≤ d ( L ( w , z ) , w ι + 1 ) .

By choice of w and z , we have

w ⊢ w ι or w ι ⊢ w

and

z ⊢ z ι or z ι ⊢ z .

Since L is a coupled F -orthogonal contraction mapping, we get

F ( d ( L ( w , z ) , L ( w ι , z ι ) ) ) < ℓ + F ( d ( L ( w , z ) , L ( w ι , z ι ) ) ) ≤ F ( a d ( w , w ι ) + b d ( z , z ι ) ) .

By the property of ( F 1 ) , we have

(2.5) d ( L ( w , z ) , L ( w ι , z ι ) ) < a d ( w , w ι ) + b d ( z , z ι ) .

From (2.4) and (2.5), we obtain

d ( L ( w , z ) , w ) − d ( w ι + 1 , w ) ≤ a d ( w , w ι ) + b d ( z , z ι ) .

As ι → ∞ , we get

lim ι → ∞ d ( L ( w , z ) , w ) = 0 .

Therefore, L ( w , z ) = w .

Similarly, we can prove that L ( z , w ) = z . Then ( w , z ) is a coupled fixed point of L . Assume that ( w ∗ , z ∗ ) is another coupled fixed point of L such that ( w , z ) ≠ ( w ∗ , z ∗ ) . Then d ( L ( w , z ) , L ( w ∗ , z ∗ ) ) = d ( w , w ∗ ) > 0 and d ( L ( z , w ) , L ( z ∗ , w ∗ ) ) = d ( z , z ∗ ) > 0 . Since L is ⊢ -preserving, we get

w ⊢ w ∗ or w ∗ ⊢ w

and

z ⊢ z ∗ or z ∗ ⊢ z .

Since L is a coupled F -orthogonal contraction mapping, we get

(2.6) F ( d ( w , w ∗ ) ) = F ( d ( L ( w , z ) , L ( w ∗ , z ∗ ) ) ) ≤ F ( a d ( w , w ∗ ) + b d ( z , z ∗ ) ) − ℓ

and

(2.7) F ( d ( z , z ∗ ) ) = F ( d ( L ( z , w ) , L ( z ∗ , w ∗ ) ) ) ≤ F ( a d ( z , z ∗ ) + b d ( w , w ∗ ) ) − ℓ .

By the property of ( F 1 ) , (2.6) and (2.7), we get

d ( w , w ∗ ) ≤ a d ( w , w ∗ ) + b d ( z , z ∗ )

and

d ( z , z ∗ ) ≤ a d ( z , z ∗ ) + b d ( w , w ∗ ) .

Thus, we have

d ( w , w ∗ ) + d ( z , z ∗ ) ≤ ( a + b ) ( d ( w , w ∗ ) + d ( z , z ∗ ) ) .

Since a + b = 1 , we obtain

d ( w , w ∗ ) + d ( z , z ∗ ) = 0 .

Therefore, ( w , z ) = ( w ∗ , z ∗ ) , which is a contradiction. So L has a unique coupled fixed point.□

In Theorem 2.1, if a = 1 , b = 0 , then we obtain the following corollary.

Corollary 2.2

Let ( W , ⊢ , d ) be an O -complete metric space with an orthogonal element w 0 and L : W × W → W be a mapping. Suppose that there exist F ∈ ℑ and ℓ > 0 such that

L is ⊢ -preserving;
for all w , u , s , r ∈ W with w ⊢ u and s ⊢ r [ d ( L ( w , u ) , L ( s , r ) ) > 0 ,
ℓ + F ( d ( L ( w , u ) , L ( s , r ) ) ) ≤ F ( d ( w , s ) ) ] .

Then L has a unique coupled fixed point.

In Theorem 2.1, a = b = 1 2 , then we obtain the following corollary.

Corollary 2.3

Let ( W , ⊢ , d ) be an O -complete metric space with an orthogonal element w 0 and L : W × W → W be a mapping. Suppose that there exist F ∈ ℑ and ℓ > 0 such that

L is ⊢ -preserving;
for all w , u , s , r ∈ W with w ⊢ u and s ⊢ r [ d ( L ( w , u ) , L ( s , r ) ) > 0 ,
ℓ + F ( d ( L ( w , u ) , L ( s , r ) ) ) ≤ F ( d ( w , s ) + d ( u , r ) ) ] .

Then L has a unique coupled fixed point.

Example 2.4

Let W = R and d ( u , v ) = ∣ u − v ∣ for all u , v ∈ W . Define a relation ⊢ on K by

u ⊢ v iff u , v ≥ 0 .

Then ( W , ⊢ , d ) is an O -complete metric space. Define a mapping L : W × W → W by L ( u , v ) = u + v e ℓ for ℓ > 0 . It is easy to see that L is ⊢ -preserving. Next, let F ∈ ℑ be a function defined by F ( c ) = ln c for c > 0 . Then for all u , v , w , s ∈ W , L ( u , v ) ≠ L ( w , s ) and a = 5 2 , we get

ℓ + F ( d ( L ( u , v ) , L ( w , s ) ) ) = ℓ + ln u + v e ℓ − w + s e ℓ ≤ ℓ + ln u − w e ℓ + v − s e ℓ = ℓ + ln ( ∣ u − w ∣ + ∣ v − s ∣ ) − ln e ℓ = F ( d ( u , w ) + d ( v , s ) ) .

Therefore, all the conditions of Corollary 2.3 are satisfied and so L has a unique coupled fixed point ( 0 , 0 ) ∈ R × R .

3 Application

Let M = [ 0 , H ] . Let W = C ( M , R ) be the set of all real valued continuous functions with domain M . Consider the integral equations

(3.1) ζ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ζ ( β ) , ξ ( β ) ) d β , v ∈ [ 0 , H ] , ξ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ξ ( β ) , ζ ( β ) ) d β , v ∈ [ 0 , H ] ,

where

ð : M → R and Ω : M × R × R → R are continuous;
Ξ : M × M is continuous and measurable at β ∈ M , ∀ v ∈ M ;
Ξ ( v , β ) ≥ 0 , ∀ v , β ∈ M and ∫ 0 H Ξ ( v , β ) d β ≤ 1 , ∀ v ∈ M .

Theorem 3.1

Assume that the conditions (a)–(c) hold. Suppose that there exists ≀ > 0 such that

∣ Ω ( v , ζ ( v ) , ξ ( v ) ) − Ω ( v , ξ ( v ) , ζ ( v ) ) ∣ ≤ e − ≀ ∣ ζ ( v ) − ξ ( v ) ∣

for each v ∈ M and for all ζ , ξ ∈ C ( M , R ) . Then the integral equation (3.1) has a unique solution in C ( M , R ) .

Proof

Let W = { w ∈ C ( I , R ) : w ( ß ) > 0 for all ß ∈ I } . Define the orthogonality relation ⊢ on W by

ζ ⊢ ξ ⇔ ζ ( ß ) ξ ( ß ) ≥ ζ ( ß ) or ζ ( ß ) ξ ( ß ) ≥ ξ ( ß ) for all ß ∈ I .

Define a mapping d : W × W → [ 0 , ∞ ) by

d ( ζ , ξ ) = sup ß ∈ I ∣ ζ ( v ) − ξ ( v ) ∣

for all ζ , ξ ∈ W . Thus, ( W , d ) is a O -complete metric space. Define L : C ( M , R ) × C ( M , R ) → C ( M , R ) by

L ( ζ , ξ ) ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ζ ( β ) , ξ ( β ) ) d β , v ∈ [ 0 , H ] .

For each ζ , ξ ∈ W with ζ ⊢ ξ and v ∈ I , we have

L ( ζ , ξ ) ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ζ ( β ) , ξ ( β ) ) d β ≥ 1 .

It follows that [ L ( ζ , ξ ) ( v ) ] [ L ( ξ , ζ ) ( v ) ] ≥ L ( ζ , ξ ) ( v ) and so L ( ζ , ξ ) ( v ) ⊢ L ( ξ , ζ ) ( v ) . Then L is ⊢ -preserving.

Let ζ , ξ ∈ W with ζ ⊢ ξ . Suppose that L ( ζ , ξ ) ( v ) ≠ L ( ξ , ζ ) ( v ) . For every v ∈ [ 0 , H ] , we have

∣ L ( ζ , ξ ) ( v ) − L ( ξ , ζ ) ( v ) ∣ = ∫ 0 H ∣ Ξ ( v , β ) ( Ω ( β , ζ ( β ) , ξ ( β ) ) − Ω ( β , ξ ( β ) , ζ ( β ) ) ) ∣ d β ≤ ∫ 0 H Ξ ( v , β ) ∣ Ω ( β , ζ ( β ) , ξ ( β ) ) − Ω ( β , ξ ( β ) , ζ ( β ) ) ∣ d β ≤ ∫ 0 H Ξ ( v , β ) e − ≀ ∣ ζ ( v ) − ξ ( v ) ∣ d β ≤ e − ≀ ∣ ζ ( v ) − ξ ( v ) ∣ ∫ 0 H Ξ ( v , β ) d β ≤ e − ≀ ∣ ζ ( v ) − ξ ( v ) ∣ ,

which implies that

d ( L ( ζ , ξ ) , L ( ξ , ζ ) ) ≤ e − ≀ d ( ζ , ξ ) .

Therefore,

≀ + ln ( d ( L ( ζ , ξ ) , L ( ξ , ζ ) ) ) ≤ ln ( d ( ζ , ξ ) ) .

Taking F ( v ) = ln ( v ) , we obtain

≀ + F ( d ( L ( ζ , ξ ) , L ( ξ , ζ ) ) ) ≤ F ( d ( ζ , ξ ) )

for all ζ , ξ ∈ W . Therefore, by Corollary 2.2, L has a unique coupled fixed point. Hence, there is a unique solution for (3.1).□

4 Open problem

In this paper, we started the study of coupled ℑ -orthogonal contraction and established coupled fixed point results in O -complete metric space. Recently, Khalehoghli, Rahimi and Eshaghi Gordji [18,19] introduced R -metric spaces and obtained a generalization of Banach fixed point theorem. It is an interesting open problem to study the relation R instead of orthogonal relation and obtain coupled fixed point results on R -complete metric spaces.

Funding information: Not applicable.
Author contributions: The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment and read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Not applicable.

References

[1] M. Eshaghi Gordji , M. Ramezani , M. De La Sen , and Y. Cho , On orthogonal sets and Banach fixed point theorem, Fixed Point Theory 18 (2017), no. 2, 569–578, https://doi.org/10.24193/fpt-ro.2017.2.45. Search in Google Scholar

[2] M. Eshaghi Gordji and H. Habibi , Fixed point theory in generalized orthogonal metric space, J. Linear Topol. Algebra 6 (2017), no. 3, 251–260, http://jlta.iauctb.ac.ir/article_533328.html. Search in Google Scholar

[3] K. Sawangsup , W. Sintunavarat , and Y. Cho , Fixed point theorems for orthogonal F -contraction mappings on O -complete metric spaces, J. Fixed Point Theory Appl. 22 (2020), 10, https://doi.org/10.1007/s11784-019-0737-4. Search in Google Scholar

[4] M. Eshaghi Gordji and H. Habibi , Fixed point theory in ϵ -connected orthogonal metric space, Sahand Commun. Math. Anal. 16 (2019), no. 1, 35–46, https://doi.org/10.22130/scma.2018.72368.289. Search in Google Scholar

[5] M. Gunaseelan , L. N. Mishra , and V. N. Mishra , Generalized coupled fixed point results on complex partial metric space using contractive condition, J. Nonlinear Model. Anal. 3 (2021), 93–104, https://doi.org/10.12150/jnma.2021.93. Search in Google Scholar

[6] M. Gunaseelan , L. N. Mishra , V. N. Mishra , I. A. Baloch , and M. De La Sen , Application to coupled fixed point theorems on complex partial b -metric space , J. Math. 2020 (2020), 8881859, https://doi.org/10.1155/2020/8881859. Search in Google Scholar

[7] N. B. Gungor and D. Turkoglu , Fixed point theorems on orthogonal metric spaces via altering distance functions, AIP Conference Proceedings 2183 (2019), 040011, https://doi.org/10.1063/1.5136131. Search in Google Scholar

[8] D. Guo and V. Lakshmikantham , Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), no. 5, 623–632, https://doi.org/10.1016/0362-546X(87)90077-0 . 10.1016/0362-546X(87)90077-0Search in Google Scholar

[9] K. Ozkan , Some coupled fixed point theorems for F -contraction mappings, J. Sci. Tech. 13 (2020), no. 13, 97–105, https://doi.org/10.18185/erzifbed.637670. Search in Google Scholar

[10] H. Piri and P. Kumam , Some fixed point theorems concerning F -contraction in complete metric spaces, Fixed Point Theory Appl. 2014 (2014), 210, https://doi.org/10.1186/1687-1812-2014-210. Search in Google Scholar

[11] T. Rasham and A. Shoaib , Common fixed point results for two families of multivalued A -dominated contractive mappings on closed ball with applications, Open Math. 17 (2019), 1350–1360, https://doi.org/10.1515/math-2019-0114. Search in Google Scholar

[12] K. Sawangsup and W. Sintunavarat , Fixed point results for orthogonal Z -contraction mappings in O -complete metric space Int. J. Appl. Phys. Math. 10 (2020), 33–40, https://doi.org/10.17706/ijapm.2020.10.1.33-40. Search in Google Scholar

[13] T. Senapati , L. K. Dey , B. Damjanović , and A. Chanda , New fixed results in orthogonal metric spaces with an application, Kragujevac J. Math. 42 (2018), no. 4, 505–516, https://doi.org/10.5937/KgJMath1804505S. Search in Google Scholar

[14] F. Uddin , C. Park , K. Javed , M. Arshad , and J. Lee , Orthogonal m -metric spaces and an application to solve integral equations, Adv. Diff. Equ. 2021 (2021), 159, https://doi.org/10.1186/s13662-021-03323-x. Search in Google Scholar

[15] O. Yamaod and W. Sintunavarat , On new orthogonal contractions in b -metric spaces, Int. J. Pure Math. 5 (2018), 37–40, https://doi.org/10.3934/math.2021481. Search in Google Scholar

[16] K. Zoto , S. Radenović , and A. H. Ansari , On some fixed point results for (s,p,α) -contractive mappings in b -metric-like spaces and applications to integral equations, Open Math. 16 (2018), 235–249, https://doi.org/10.1515/math-2018-0024. Search in Google Scholar

[17] D. Wardowski , Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012 (2012), 94, https://doi.org/10.1186/1687-1812-2012-94. Search in Google Scholar

[18] S. Khalehoghli , H. Rahimi , and M. Eshaghi Gordji , Fixed point theorems in R -metric spaces with applications, AIMS Math. 5 (2020), no. 4, 3125–3137, https://doi.org/10.3934/math.2020201. Search in Google Scholar

[19] S. Khalehoghli , H. Rahimi , and M. Eshaghi Gordji , R -topological spaces and SR -topological spaces with their applications, Math. Sci. 14 (2020), 249–255, https://doi.org/10.1007/s40096-020-00338-5. Search in Google Scholar

Received: 2021-05-14

Revised: 2021-06-30

Accepted: 2021-07-05

Published Online: 2021-11-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0075

Keywords for this article

orthogonal set; orthogonal metric space; orthogonal continuous; orthogonal preserving; orthogonal 𝔉-contraction; coupled fixed point

Creative Commons

BY 4.0