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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 287-295

Published online September 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.3.287

© The Korean Institute of Intelligent Systems

## Solution of Integral Equation via Orthogonally Modified ℱ-Contraction Mappings on -Complete Metric-Like Space

Senthil Kumar Prakasam1, Arul Joseph Gnanaprakasam1, Nasreen Kausar2 Gunaseelan Mani3, Mohammed Munir4, and Salahuddin

1Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Tamil Nadu, India
2Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University, Istanbul, Turkey
3Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Tamil Nadu, India
5Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia

Correspondence to :
Nasreen Kausar (kausar.nasreen57@gmail.com)

Received: May 9, 2021; Revised: April 2, 2022; Accepted: September 7, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study introduces the concepts of orthogonally modified F-contraction of type-I and type-II and certain fixed-point theorems for self-mapping in orthogonal metrics, such as space, have been proved. The proven results generalize and extend some well-known results in the literature. The application of the main results has been presented.

Keywords: Orthogonal set, Orthogonally modified F-contraction of type-I, Type-II, Fixed point

### 1. Introduction

As a generalization of the metric space, by introducing the concept of metric-like space in 2000, Hitzler and Seda [1] provided a valuable contribution to the fixed point theory, permitting self-distance to be non-zero, as it cannot be possible in metric space. They studied the metric-like space under the name of “Dislocated metric space,” and Amini-Harandi [2] reintroduced the dislocated metric space by a new name known as metric-like space.

The concept of an orthogonal set has many applications in several branches of mathematics and several types of orthogonalities [35]. Wardowski [6] introduced a new concept of contraction and proved the fixed-point theorem, which is the generalized Banach contraction principle. Nazam et al. [7] proved fixed-point problems for generalized contractions. Alsulami et al. [8] introduced fixed points of modified ℱ-contractive mapping in complete metric-like spaces. Orthogonal contractive-type mappings have been studied by many authors, and important results have been obtained by [911]. In this paper, we introduce a new orthogonally modified ℱ-contraction of type-I, type-II and prove the fixed point on an orthogonal-complete metric, such as space.

### 2. Preliminaries

Throughout this paper, we use , ℝ+, and to denote the non-empty set, the set of positive real numbers, the set of positive integers, and the set of non-negative integers, respectively.

First, we recall the concept of the control function introduced by Wardowski [6]. Let ϒ denote the family of all functions ℱ: ℝ+ → ℝ that satisfy the following properties:

(ℱ1) ℱ is strictly non-decreasing;

(ℱ2) for every sequence {ηι} of non-negative numbers, we have

limιηι=0limιF(ηι)=-α

In 2012, Amini-Harandi [12] introduced the concept of metric-type space as follows:

### Definition 2.1 ([12])

Let be a non-empty set. A function is said to be metric-like (dislocated) on , if for all , the following conditions hold:

• (⊤1) ⊤(ν,μ) = 0 ⇒ ν = μ,

• (⊤2) ⊤(ν,μ) = ⊤(μ,ν),

• (⊤3) ⊤(ν,μ) ≤ ⊤(ν,ß) + ⊤(ß,μ),

Then, pair is said to be a metric-like (or dislocated) space.

In 2015, Alsulami et al. [8] introduced the notions of modified ℱ-contraction of type-I, type-II and proved the existence and uniqueness of a fixed-point theorem. Furthermore, Gordji et al. [3] introduced the concept of an orthogonal set (or -set), and some examples and properties of the orthogonal sets are as follows:

### Definition 2.2 ([3])

Let and be binary relations. If ⊦ satisfies the following condition:

w0J:(wJ,ww0)         or         (wJ,w0w),

Thus, it is known as an orthogonal set. We denote this -set as .

### Example 2.1 ([3])

Let W = ℤ. We define the binary relation ⊦ on W as mn if there exists k ∈ ℤ such that m = kn. Furthermore, 0 ⊦ n for all n ∈ ℤ. Hence, (W, ⊦) is a -set.

### Example 2.2 ([3])

Let and define μν if μν ∈ {μ, ν}. Subsequently, by setting μ0 = 0 or μ0 = 1, is an -set.

Now, we provide the concepts of an -sequence, an -continuous mapping, an -complete metric like space, an -preserving mapping and a weakly -preserving mapping.

### Definition 2.3 ([3])

Let be an -set. A sequence {νι} is called an orthogonal sequence ( -sequence) if

(ιN,wιwι+1)         or         (ιN,wι+1wι).

### Definition 2.4 ([3])

The triplet is called an orthogonal metric, such as space, if is an -set and is a metric-like space.

### Definition 2.5 ([3])

Let be an orthogonal metric-like space. Then, a mapping is said to be orthogonally continuous in if, for each -sequence {νι} in with νιν as ι → ∞, we have D(νι) → D(ν) as ι → ∞. In addition, D is said to be ⊦-continuous in if D is ⊦-continuous in each .

### Example 2.3

Let with ⊤(μ, ν) = |μ – ν| for all μ, . Define orthogonal relation ⊦-by μ ⊦-νμν ∈ max{μ, ν}. We define as:

D(μ)={1,μ4μ4μ>4.

Thus, D is ⊦-contraction with (μ)=eμ.

### Definition 2.6 ([3])

Let be an orthogonal metric-like space. Then, is said to be orthogonally complete if every Cauchy -sequence converges.

### Definition 2.7 ([3])

Let be a -set. A mapping is said to be ⊦-preserving if whenever νμ. In addition, is said to be weakly ⊦-preserving if D(ν) ⊦ D(μ) or D(μ) ⊦ D(μ) when νμ.

In this study, we modify the concepts of modified ℱ-contraction mapping of type-I, type-II to orthogonal sets and prove certain fixed-point theorems for modified ℱ-contraction mapping of type-I, type-II in orthogonality complete metric-like spaces.

### 3. Main Results

In this section, inspired by the notions of modified ℱ-contraction of type-I, type-II and an orthogonal set, we introduce a new modified ℱ-contraction of type-I, type-II and prove the fixed points for these contraction mappings in an orthogonal metric-like space.

### Definition 3.1

Let be an orthogonal metric-like space. A map is said to be an orthogonal modified ℱ-contraction mapping of type-I on if there exists ℱ ∈ ϒ and l > 0 such that the following condition holds:

ν,μJwith         νμ[(Dν,Dμ)>0,12(ν,Dn)<(ν,μ)         l+((Dν,Dμ))q((ν,μ))         +r((ν,Dν))s((μ,Dμ))],

where s ∈ [0, 1) and q, r ∈ [0, 1] are real numbers, such that q + r + s = 1.

From the above Definition 3.1, if we consider q = 1 and r = s = 0; we obtain the following definition:

### Definition 3.2

Let be an orthogonal metric-like space. Mapping is said to be an orthogonally modified ℱ-contraction of type-II if there exist ℱ ∈ ϒ and l > 0, such that the following condition holds:

ν,μJwithνμ[(Dν,Dμ)>0,12(ν,Dν)<(ν,μ)l+((Dν,Dμ))((ν,μ))].

First, we provide the fixed-point theorem for orthogonally modified ℱ-contraction of type-I in an -complete metric-like space.

### Theorem 3.1

Let be an -complete metric-like space with an orthogonal element ν0 and a mapping . Suppose there exists ℱ ∈ ϒ and l > 0 such that the following conditions hold:

• (i) D is ⊦-preserving,

• (ii) D is orthogonal modified ℱ-contraction of type-I.

Thus, D has a unique fixed point .

Proof

Because is an -set,

ν0J:(νJ,νν0)         or         (νJ,ν0ν).

It follows that ν00 or 0ν0. Let

ν1:=Dν0,ν2:=Dν1=D2ν0,,νι+1:=Dνι=Dι+1ν0,

for all . If there exists such that ⊤(νι0, νι0+1) = 0, then ß = νι0 is the desired fixed point of D, which completes the proof. Consequently, we assume 0 < ⊤(νι, νι+1) for all . Because D is ⊦-preserving ν, we have

νινι+1         or         νι+1νι,

for all . This implies that {νι} is a -sequence. We have

12(νι,νι)=12(νι,Dνι+1)<(νι,νι+1)         for all ιN.

Because D is an orthogonally modified ℱ-contraction of type-I, we have

l+((Dνι,Dνι+1))q((νι,νι+1))+r((νι,Dνι))+s((νι+1,Dνι+1)),

and hence

l+(1-s)((νι+1,νι+2))(q+r)((νι),νι+1).

As q + r + s = 1. We obtain

((νι+1,νι+2))((νι,νι+1))-lq+r<((νι,νι+1)).

Thus, from (ℱ1), we conclude that

(νι+1,νι+2)<(νι,νι+1)ιN.

Therefore, ⊤(νι, νι+1) is a decreasing sequence of real numbers bounded below. This implies ⊤(νι, νι+1) converges and

limιηι=r=inf{((νι,νι+1):ιN}.

We demonstrate that r = 0. Suppose, contrarily r > 0. For every ε > 0 there exists such that

(νk,Dνk)<r+ɛ.

Hence, from (ℱ1) we obtain

((νk,Dνk)<(r+ɛ).

However, from Eq. (3), we have

12(νk,Dνk)<(νk,Dνk).

Because D is an orthogonally modified ℱ-contraction of type-I, we obtain

l+((Dνk,D2νk))q((νk,Dνk))+r((νk,Dνk))+s((Dνk,D2νk)),

which is equivalent to

l+(1-s)((Dνk,D2νk))<(q+r)((νk,Dνk)).

Consequently, we derive the following:

((Dνk,D2νk))<((νk,Dνk))-lq+r,

as q + r + s = 1. Based on Eq. (3), we have

12(νk,D2νk)<(νk,D2νk).

Because D is an orthogonally modified ℱ-contraction of type-I, we have

l+((D2νk,D3νk))q((Dνk,D2νk))+r((Dνk,D2νk))+s((D2νk,D3νk)),

which yields

l+(1-s)((D2νk,D3νk))(q+r)((Dνk,D2νk)).

Because q + r + s = 1, we have

((D2νk,D3νk))((Dνk,D2νk))-lq+r.

Now, using Eq. (6) and continuing the same way as in the derivation of Eqs. (7) and (8), we deduce that

((D2νk,Dι+1νk))((Dινk,Dι-1νk))-lq+r((Dι-1νk,Dι-2νk))-2lq+r((Dνk,νk))-ιlq+r<(r+ɛ)-ιlq+r.

This implies

limι((Dινk,Dι+1νk))=-.

From (ℱ2)

limι((Dινk,Dι+1νk))=0,

and there exists such that for all ιN1. Therefore, from Eq. (1), we obtain . This contradicts the definition of R given in Eq. (5). Next, we obtain r = 0 and from Eq. (5), we conclude that

limι(νι,νι+1)=0.

In the next step, we obtain

limι(νι,νk)=0.

Suppose there exists ε > 0 and sequences {a(ι)} and {b(ι)} of natural numbers such that

a(ι)>b(ι)>ι,   (νa(ι),νb(ι))ɛ,(νa(ι)-1,νb(ι))<ɛ,for all ιN.

Using triangular inequality, we have

ɛ(νa(ι),νb(ι))(νa(ι),νa(ι)-1)+(νa(ι)-1,νb(ι))(νa(ι),νa(ι)-1)<ɛ=(νa(ι)-1,Dνa(ι)-1)+ɛ,   ιN.

If follows from Eqs. (9), (12), and squeezing the theorem that

limt(νa(ι),νb(ι))=ɛ.

From Eqs. (9),(12), and (13), there exists such that

12(νa(ι),Dνa(ι))<12ɛ<ɛ(νa(ι),νb(ι))<2ɛ,ιN2.

Hence, from Eq. (14), (ℱ1) and the hypothesis of the theroem, we have

l+((Dνa(ι),Dνb(ι)))q((νa(ι),νb(ι)))+r((νa(ι),Dννa(ι)))+s((νb(ι),Dνb(ι)))q(2ɛ)+r((νa(ι),Dνa(ι)))+s((νb(ι),Dνb(ι)),ι>N2.

From Eq. (9) and (ℱ2) it follows that

limι((Dνa(ι),Dνb(ι)))=-,

and thus, we obtain

limι((Dνa(ι),Dνb(ι)))=0limι(νa(ι)+1,Dνb(ι)+1)=0.

However, this contradicts the relationship in Eq. (12). Hence,

limι,j(νι,νj)=0.

Therefore, {νι} is the Cauchy sequence in . From the completeness of there exists such that

(ß,ß)=limι(νι,ß)=limι,j(νι,νj)=0.

Subsequently, we prove that for every ,

12(νι,Dνι)<(νι,ß),or 12(Dνι,D2νι)<(Dνι,ß),

. By contradiction, we assume that there exists such that

12(νj,Dνj)(νj,ß),or 12(Dνj,D2νj)(Dνj,ß).

From Eq. (7) and (ℱ1), we obtain

(Dνj,D2νj)<(νj,Dνj).

It follows from Eqs. (17) and (18) that:

(νj,DνJ)(νj,ß)+(ß,Dνj)12(νj,Dνj)+12(Dνj,D2νj)<12(νj,Dνj)+12(νj,Dνj)=(νj,Dνj).

Evidently, this is a contradiction. Hence, inequality (16) is satisfied. Regarding the assumption of the theorem, Eq. (16) implies that

l+((Dνι,Dß))q((νι,ß))+r((νι,Dνι))+s((ß,Dß)),

or

l+((D2νι,Dß))q((Dνι,ß))+r((Dνι,D2νι))+s((ß,Dß)),ιN.

In the first case, because of (ℱ2) the limits in Eqs. (9) and (15) we get

limι((ν(ι),ß))=-,limι(ν(ι),Dν(ι))=-.

Thus, substituting ι → ∞ in Eq. (19), we obtain

limι((Dν(ι),Dß))=-.

Again, from (ℱ2), we observe that

limι(Dν(ι),Dß)=0.

However, from Eq. (1), we have

(ß,Dß)(ß,Dνι)+(Dνι,Dß)=(ß,νι+1)+(Dνι,Dß).

It follows from Eqs. (15) and (21) that ⊤(ß, Dß) = 0. Therefore ß = Dß.

In the second case from Eq. (1), we have

((D2νι,Dt))<l+((D2νι,Dß))q((Dνι,ß))+r((Dνι,D2νι))+s((ß,Dß))q((Dνι+1,ß))+r((Dνι+1,Dνι+1))+s((ß,Dß)).

Then, by employing Eqs. (9), (15), and (ℱ2), we conclude that

limι((D2νι,Dß))=-.

Similarly, from (ℱ2) we obtain

limι(D2νι,Dß)=0.

Using Eq. (9), we obtain

(ß,Dß)(ß,D2νι)+(D2νι,Dß)=(ß,νι+2)+(D2νι,Dß).

Finally, from Eqs. (15) and (22), it follows that ⊤(ß, Dß) = 0; thus, ß = Dß. Hence, ß is a fixed point on D.

We prove that D has a unique fixed point in .

Let be two fixed points of D and suppose that Dιß = ß ≠ ϕ = Dιϕ, . By selecting ν0 we obtain

(ν0ßand ν0ϕ)or(ν0ϕ         ν0ß).

Because D is ⊦-preserving ν, we have

(Dιν0Dιßand Dιν0Dιϕ),or (Dιν0Dιϕand Dιν0Dιß),ιN.

Now

(ß,ϕ)=(Dιß,Dιϕ)(Dιß,Dιν0)+(Dιν0,Dιϕ).

As ι → ∞, we obtain ⊤(ß, ϕ) ≤ 0. Therefore, ß = ϕ. Hence, D has a unique fixed point in .

Now, we provide the fixed-point theorem for orthogonally modified ℱ-contraction of type-II in an -complete metriclike space.

### Theorem 3.2

Let be an orthogonal complete metric-like space and D be an orthogonal modified ℱ-contraction of type-II. Then, D has a unique fixed point , that is, Dß = ß.

Proof

It is sufficient to consider that q = 1 and r = s = 0 in Theorem 3.1.

### Example 3.3

Let and let be defined by

(ı,ȷ)=max{ı,ȷ}

We define ⊦ on by ı ⊦ u if ı, ȷ ≥ 0. Then, it can be proved that is an -complete, metric-like space. We define mapping as

D(ı)={ı4,ı[0,1),16,ı=1.

Clearly, D is ⊦-preserving and ⊦-continuous. We define the function F(r) = lnr for r ∈ ℝ+ and obtain

+((Dı,Dȷ))((ı,ȷ))ln ((ı,ȷ)(Dı,Dȷ)),

for all ı, . First, we observe the following:

(Dı,Dȷ)>0,12(ı,Dı)<(ı,ȷ){(ı=1and ȷ[0,1])(ı<1and ȷ=1)(ı<ȷ<1)(ȷı<1)}.

For ı = 1 and ȷ ∈ [0, 1], we have ⊤(ı, ȷ) = ⊤(1, ȷ) = 1 and

(Dı,Dȷ)=(16,Dȷ)={16,ȷ[0,12],16,ȷ(12,1),16,ȷ=1.

Hence, we obtain

(ı,ȷ)(Dı,Dȷ)={6,ȷ[0,12],6ȷ,u(12,1,)6,ȷ=1.

For ı < 1 and ȷ = 1, we have ⊤(ı, ȷ) = ⊤ (ı, 1) = 1 and

(Dı,Dȷ)=(Dı,16)={16,ı[0,12],16,ı(12,1].

Hence, we get

(ı,ȷ)(Dı,Dȷ)={6,ı[0,12]6ı,ı(12,1).

For ı < ȷ < 1, we have ⊤(ı, ȷ) = ȷ and (Dı,Dȷ)=(ı4,ȷ4)=ȷ4. Hence, we get

(ı,ȷ)(Dı,Dȷ)=4.

For ȷ ≤ ı < 1, we have ⊤(ı, ȷ) = ı and (Dı,Dȷ)=(ı4,u4)=ı4. Hence, we get

(ı,ȷ)(Dı,Dȷ)=4.

From Eqs. (23)(26), we can obtain that if 0 < ℓ ≤ ln 4, then ln ((ı,ȷ)(Dı,Dȷ)). Thus,

+((Dı,Dȷ))((ı,ȷ)).

Therefore, D satisfies the conditions of Theorem 3.2 with 0 < ℓ ≤ ln 4. Hence, all required hypotheses of Theorem 3.2 are satisfied. Therefore, D has a unique fixed point.

### 4. Application

Let , Let be the set of all real-valued continuous functions in domain . The following integral equation is considered

ζ(v)=0Ξ(v,β)Ω(β,ζ(β))dβ,v[0,],

where

• (a) is continuous;

• (b) is continuous and measurable at

• (c) Ξ(v, β) ≥ 0, and 0Ξ(v,β)dβ1, .

### Theorem 4.1

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

Ω(v,ζ(v))+Ω(v,ξ(v))e-(ζ(v)+ξ(v)),

for each vM and for all . Thus, the integral equation (27) has a unique solution in .

Proof

Let . The orthogonality relation ⊦ on is defined as

ζξζ(ß)ξ(ß)ζ(ß)orζ(ß)ξ(ß)ξ(ß)ßM.

We define the mapping as

(ζ,ξ)=ζ(v)+ξ(v).

for all . Thus, is a metric-like space and a complete metric-like space. We define as:

Dζ(v)=0Ξ(v,β)Ω(β,ζ(v)),v[0,].

Now, we show that D is ⊦-preserving. For each with ζξ and ß ∈ I, we have

Dζ(v)=0Ξ(v,β)Ω(β,ζ(v))1.

It follows that [()(ß)][()(ß)]≥()(ß) and thus, ()(ß) ⊦ ()(ß). Then, D is ⊦-preserving.

Next, we claim that D is an orthogonally modified ℱ-contraction of type-II. Let with ζξ. We assume that D(ζ) ≠ D(ξ). For every v ∈ [0, ℋ], we have

(Dζ,Dξ)=Dζ(v)+Dξ(v)=0Ξ(v,β)(Ω(β,ζ(β))+Ω(β,ξ(β)))dβ0Ξ(v,β)(Ω(β,ζ(β))+Ω(β,ξ(β)))dβ0Ξ(v,β)e-(ζ(v)+ξ(v))dβe-(ξ(v)+ξ(v))0Ξ(v,β)dβe-(ξ(v)+ξ(v))=e-(ζ,ξ).

Therefore,

+ln((Dζ,Dξ))ln ((ζ,ξ)).

Considering ℱ(v) = ln(v), we obtain

+((Dζ,Dξ))((ζ,ξ)),

for all . Therefore, according to Theorem 3.2, D has a unique fixed point. Hence, there exists a unique solution for Eq. (27).

### 5. Conclusion

In this study, we proved fixed-point theorems for orthogonally modified ℱ-contraction of type-I, type-II on an -complete metric-like space.

### Conflict of Interest

1. Hitzler, P, and Seda, AK (2000). Dislocated topologies. Journal of Electrical Engineering. 51, 3-7.
2. Amini-Harandi, A (2012). Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications. 2012. article no 204
3. Gordji, ME, Ramezani, M, De La Sen, M, and Cho, YJ (2017). On orthogonal sets and Banach fixed point theorem. Fixed Point Theory. 18, 569-578.
4. Eshaghi Gordji, M, and Habibi, H (2017). Fixed point theory in generalized orthogonal metric space. Journal of Linear and Topological Algebra. 6, 251-260.
5. Sawangsup, K, Sintunavarat, W, and Cho, YJ (2020). Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces. Journal of Fixed Point Theory and Applications. 22. article no 10
6. Wardowski, D (2012). Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications. 2012. article no 94
7. Nazam, M, Park, C, and Arshad, M (2021). Fixed point problems for generalized contractions with applications. Advances in Difference Equations. 2021. article no 247
8. Alsulami, HH, Karapınar, E, and Piri, H (2015). Fixed points of modified-contractive mappings in complete metric-like spaces. Journal of Function Spaces. 2015. article no 270971
9. Yamaod, O, and Sintunavarat, W (2018). On new orthogonal contractions in b-metric spaces. International Journal of Pure Mathematics. 5, 37-40.
10. Senapati, T, Dey, LK, Damjanovic, B, and Chanda, A (2018). New fixed point results in orthogonal metric spaces with an application. Kragujevac Journal of Mathematics. 42, 505-516.
11. Beg, I, Mani, G, and Gnanaprakasam, AJ (2021). Fixed point of orthogonal F-Suzuki contraction mapping on O-complete b-metric spaces with applications. Journal of Function Spaces. 2021. article no 6692112
12. Amini-Harandi, A (2012). Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications. 2012. article no 204

### Biographies

Senthil Kumar Prakasam was born in Tamilnadu, India. He is full time research scholar in SRM IST under he guidance of Dr. Arul Joseph Gnanaprakasam. His research interests lie in fixed point theory and its applications to differential and integral equations.

E-mail: sp2989@srmist.edu.in

Arul Joseph Gnanaprakasam received the Ph.D. degree in Mathematics from the Madurai Kamaraj University, Madurai, Tamilnadu, India. He is currently an assistant professor of Mathematics in SRM Institute of Science and Technology, Kattankulathur, Kanchipuram, Tamilnadu, India. His research interests lie in fixed point theory and delay differential equations (DDEs).

E-mail: aruljoseph.alex@gmail.com

Nasreen Kausar received the Ph.D. degree in Mathematics from the Quaid-i-Azam University, Islamabad, Pakistan. She is an associate professor of Mathematics in Yildiz Technical University, Istanbul, Turkey. Her research interests include the numerical analysis and numerical solutions of the Ordinary Differential equations(ODEs), Partial Differential Equations (PDEs), Voltara integral equations. She has also research interests in associative and commutative, non-associative and non-commutative fuzzy algebraic structure and their applications.

Gunaseelan Mani received the Ph.D. degree in Mathematics from the Bharathidasan University of Trichy, Tamilnadu, India. He is currently an assistant professor of Mathematics in Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602 105, Tamil Nadu, India. His research interests lie in fixed point theory and best approximation theory.

E-mail: mathsguna@yahoo.com

Mohammad Munir possesses his Ph.D. degree in Applied Mathematics from the University of Graz, Graz, Austria. His research interests lie in mathematical modelling of the biological system in the fields of the glucose-insulin dynamics, solute kinetics and haemodialysis using the ordinary differential equations (ODEs). Parameter identification, sensitivity analysis and generalized sensitivity analysis are more concentrated areas of his research. His other interests include the applications of the fuzzy sets theory to the Multi Criteria Decision Making (MCDM) problems.

Salahuddin is a faculty member of the Department of Mathematics, Jazan University, Jazan, Saudi Arabia. He is working on fuzzy set, fuzzy group theory, fuzzy ring and fuzzy ideal theory, variational inequality, and optimization theory.

Email: drsalah12@hotmail.com

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 287-295

Published online September 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.3.287

## Solution of Integral Equation via Orthogonally Modified ℱ-Contraction Mappings on -Complete Metric-Like Space

Senthil Kumar Prakasam1, Arul Joseph Gnanaprakasam1, Nasreen Kausar2 Gunaseelan Mani3, Mohammed Munir4, and Salahuddin

1Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Tamil Nadu, India
2Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University, Istanbul, Turkey
3Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Tamil Nadu, India
5Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia

Correspondence to:Nasreen Kausar (kausar.nasreen57@gmail.com)

Received: May 9, 2021; Revised: April 2, 2022; Accepted: September 7, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This study introduces the concepts of orthogonally modified F-contraction of type-I and type-II and certain fixed-point theorems for self-mapping in orthogonal metrics, such as space, have been proved. The proven results generalize and extend some well-known results in the literature. The application of the main results has been presented.

Keywords: Orthogonal set, Orthogonally modified F-contraction of type-I, Type-II, Fixed point

### 1. Introduction

As a generalization of the metric space, by introducing the concept of metric-like space in 2000, Hitzler and Seda [1] provided a valuable contribution to the fixed point theory, permitting self-distance to be non-zero, as it cannot be possible in metric space. They studied the metric-like space under the name of “Dislocated metric space,” and Amini-Harandi [2] reintroduced the dislocated metric space by a new name known as metric-like space.

The concept of an orthogonal set has many applications in several branches of mathematics and several types of orthogonalities [35]. Wardowski [6] introduced a new concept of contraction and proved the fixed-point theorem, which is the generalized Banach contraction principle. Nazam et al. [7] proved fixed-point problems for generalized contractions. Alsulami et al. [8] introduced fixed points of modified ℱ-contractive mapping in complete metric-like spaces. Orthogonal contractive-type mappings have been studied by many authors, and important results have been obtained by [911]. In this paper, we introduce a new orthogonally modified ℱ-contraction of type-I, type-II and prove the fixed point on an orthogonal-complete metric, such as space.

### 2. Preliminaries

Throughout this paper, we use , ℝ+, and to denote the non-empty set, the set of positive real numbers, the set of positive integers, and the set of non-negative integers, respectively.

First, we recall the concept of the control function introduced by Wardowski [6]. Let ϒ denote the family of all functions ℱ: ℝ+ → ℝ that satisfy the following properties:

(ℱ1) ℱ is strictly non-decreasing;

(ℱ2) for every sequence {ηι} of non-negative numbers, we have

$limι→∞ ηι=0⇔limι→∞ F(ηι)=-∞α$

In 2012, Amini-Harandi [12] introduced the concept of metric-type space as follows:

### Definition 2.1 ([12])

Let be a non-empty set. A function is said to be metric-like (dislocated) on , if for all , the following conditions hold:

• (⊤1) ⊤(ν,μ) = 0 ⇒ ν = μ,

• (⊤2) ⊤(ν,μ) = ⊤(μ,ν),

• (⊤3) ⊤(ν,μ) ≤ ⊤(ν,ß) + ⊤(ß,μ),

Then, pair is said to be a metric-like (or dislocated) space.

In 2015, Alsulami et al. [8] introduced the notions of modified ℱ-contraction of type-I, type-II and proved the existence and uniqueness of a fixed-point theorem. Furthermore, Gordji et al. [3] introduced the concept of an orthogonal set (or -set), and some examples and properties of the orthogonal sets are as follows:

### Definition 2.2 ([3])

Let and be binary relations. If ⊦ satisfies the following condition:

$∃w0∈J:(∀w∈J,w⊢w0) or (∀w∈J,w0⊢w),$

Thus, it is known as an orthogonal set. We denote this -set as .

### Example 2.1 ([3])

Let W = ℤ. We define the binary relation ⊦ on W as mn if there exists k ∈ ℤ such that m = kn. Furthermore, 0 ⊦ n for all n ∈ ℤ. Hence, (W, ⊦) is a -set.

### Example 2.2 ([3])

Let and define μν if μν ∈ {μ, ν}. Subsequently, by setting μ0 = 0 or μ0 = 1, is an -set.

Now, we provide the concepts of an -sequence, an -continuous mapping, an -complete metric like space, an -preserving mapping and a weakly -preserving mapping.

### Definition 2.3 ([3])

Let be an -set. A sequence {νι} is called an orthogonal sequence ( -sequence) if

$(∀ι∈N,wι⊢wι+1) or (∀ι∈N,wι+1⊢wι).$

### Definition 2.4 ([3])

The triplet is called an orthogonal metric, such as space, if is an -set and is a metric-like space.

### Definition 2.5 ([3])

Let be an orthogonal metric-like space. Then, a mapping is said to be orthogonally continuous in if, for each -sequence {νι} in with νιν as ι → ∞, we have D(νι) → D(ν) as ι → ∞. In addition, D is said to be ⊦-continuous in if D is ⊦-continuous in each .

### Example 2.3

Let with ⊤(μ, ν) = |μ – ν| for all μ, . Define orthogonal relation ⊦-by μ ⊦-νμν ∈ max{μ, ν}. We define as:

$D(μ)={1,μ≤4μ4μ>4.$

Thus, D is ⊦-contraction with $ℱ(μ)=eμ$.

### Definition 2.6 ([3])

Let be an orthogonal metric-like space. Then, is said to be orthogonally complete if every Cauchy -sequence converges.

### Definition 2.7 ([3])

Let be a -set. A mapping is said to be ⊦-preserving if whenever νμ. In addition, is said to be weakly ⊦-preserving if D(ν) ⊦ D(μ) or D(μ) ⊦ D(μ) when νμ.

In this study, we modify the concepts of modified ℱ-contraction mapping of type-I, type-II to orthogonal sets and prove certain fixed-point theorems for modified ℱ-contraction mapping of type-I, type-II in orthogonality complete metric-like spaces.

### 3. Main Results

In this section, inspired by the notions of modified ℱ-contraction of type-I, type-II and an orthogonal set, we introduce a new modified ℱ-contraction of type-I, type-II and prove the fixed points for these contraction mappings in an orthogonal metric-like space.

### Definition 3.1

Let be an orthogonal metric-like space. A map is said to be an orthogonal modified ℱ-contraction mapping of type-I on if there exists ℱ ∈ ϒ and l > 0 such that the following condition holds:

$∀ν,μ∈J with ν⊢μ [⊤(Dν,Dμ)>0,12⊤(ν,Dn)<⊤(ν,μ) ⇒l+ℱ(⊤(Dν,Dμ))≤qℱ(⊤(ν,μ)) +rℱ(⊤(ν,Dν))≤sℱ(⊤(μ,Dμ))],$

where s ∈ [0, 1) and q, r ∈ [0, 1] are real numbers, such that q + r + s = 1.

From the above Definition 3.1, if we consider q = 1 and r = s = 0; we obtain the following definition:

### Definition 3.2

Let be an orthogonal metric-like space. Mapping is said to be an orthogonally modified ℱ-contraction of type-II if there exist ℱ ∈ ϒ and l > 0, such that the following condition holds:

$∀ν,μ∈J withν⊢μ [⊤(Dν,Dμ)>0,12⊤(ν,Dν)<⊤(ν,μ)⇒l+ℱ(⊤(Dν,Dμ))≤ℱ(⊤(ν,μ))].$

First, we provide the fixed-point theorem for orthogonally modified ℱ-contraction of type-I in an -complete metric-like space.

### Theorem 3.1

Let be an -complete metric-like space with an orthogonal element ν0 and a mapping . Suppose there exists ℱ ∈ ϒ and l > 0 such that the following conditions hold:

• (i) D is ⊦-preserving,

• (ii) D is orthogonal modified ℱ-contraction of type-I.

Thus, D has a unique fixed point .

Proof

Because is an -set,

$∃ ν0∈J:(∀ν∈J,ν⊢ν0) or (∀ν∈J,ν0⊢ν).$

It follows that ν00 or 0ν0. Let

$ν1:=Dν0, ν2:=Dν1=D2ν0, …,νι+1:=Dνι=Dι+1ν0,$

for all . If there exists such that ⊤(νι0, νι0+1) = 0, then ß = νι0 is the desired fixed point of D, which completes the proof. Consequently, we assume 0 < ⊤(νι, νι+1) for all . Because D is ⊦-preserving ν, we have

$νι⊢νι+1 or νι+1⊢νι,$

for all . This implies that {νι} is a -sequence. We have

$12⊤(νι,νι)=12⊤(νι,Dνι+1)< ⊤(νι,νι+1) for all ι∈N.$

Because D is an orthogonally modified ℱ-contraction of type-I, we have

$l+ℱ(⊤(Dνι,Dνι+1))≤qℱ(⊤(νι,νι+1))+rℱ( ⊤(νι,Dνι))+sℱ( ⊤(νι+1,Dνι+1)),$

and hence

$l+(1-s)ℱ(⊤(νι+1,νι+2))≤(q+r)ℱ(⊤(νι),νι+1).$

As q + r + s = 1. We obtain

$ℱ(⊤(νι+1,νι+2))≤ℱ(⊤(νι,νι+1))-lq+r<ℱ( ⊤(νι,νι+1)).$

Thus, from (ℱ1), we conclude that

$⊤(νι+1,νι+2)<⊤(νι,νι+1)∀ι∈N.$

Therefore, ⊤(νι, νι+1) is a decreasing sequence of real numbers bounded below. This implies ⊤(νι, νι+1) converges and

$limι→∞ ηι=r=inf{(⊤(νι,νι+1):ι∈N}.$

We demonstrate that r = 0. Suppose, contrarily r > 0. For every ε > 0 there exists such that

$⊤(νk,Dνk)

Hence, from (ℱ1) we obtain

$ℱ(⊤(νk,Dνk)<(r+ɛ).$

However, from Eq. (3), we have

$12⊤(νk,Dνk)<⊤(νk,Dνk).$

Because D is an orthogonally modified ℱ-contraction of type-I, we obtain

$l+ℱ(⊤(Dνk,D2νk))≤qℱ(⊤(νk,Dνk))+rℱ( ⊤(νk,Dνk))+sℱ( ⊤(Dνk,D2νk)),$

which is equivalent to

$l+(1-s)ℱ(⊤(Dνk,D2νk))<(q+r)ℱ(⊤(νk,Dνk)).$

Consequently, we derive the following:

$ℱ(⊤(Dνk,D2νk))<ℱ(⊤(νk,Dνk))-lq+r,$

as q + r + s = 1. Based on Eq. (3), we have

$12⊤(νk,D2νk)<⊤(νk,D2νk).$

Because D is an orthogonally modified ℱ-contraction of type-I, we have

$l+ℱ(⊤(D2νk,D3νk))≤qℱ(⊤(Dνk,D2νk))+rℱ( ⊤(Dνk,D2νk))+sℱ( ⊤(D2νk,D3νk)),$

which yields

$l+(1-s)ℱ(⊤(D2νk,D3νk))≤(q+r)ℱ(⊤(Dνk,D2νk)).$

Because q + r + s = 1, we have

$ℱ(⊤(D2νk,D3νk))≤ℱ(⊤(Dνk,D2νk))-lq+r.$

Now, using Eq. (6) and continuing the same way as in the derivation of Eqs. (7) and (8), we deduce that

$ℱ(⊤(D2νk,Dι+1νk))≤ℱ(⊤(Dινk,Dι-1νk))-lq+r≤ℱ(⊤(Dι-1νk,Dι-2νk))-2lq+r≤ℱ(⊤(Dνk,νk))-ιlq+r…<ℱ(r+ɛ)-ιlq+r.$

This implies

$limι→∞ ℱ(⊤(Dινk,Dι+1νk))=-∞.$

From (ℱ2)

$limι→∞(⊤(Dινk,Dι+1νk))=0,$

and there exists such that for all ιN1. Therefore, from Eq. (1), we obtain . This contradicts the definition of R given in Eq. (5). Next, we obtain r = 0 and from Eq. (5), we conclude that

$limι→∞ ⊤(νι,νι+1)=0.$

In the next step, we obtain

$limι→∞ ⊤(νι,νk)=0.$

Suppose there exists ε > 0 and sequences {a(ι)} and {b(ι)} of natural numbers such that

$a(ι)>b(ι)>ι, ⊤(νa(ι),νb(ι))≥ɛ, ⊤(νa(ι)-1,νb(ι))<ɛ, for all ι∈N.$

Using triangular inequality, we have

$ɛ≤⊤(νa(ι),νb(ι))≤⊤(νa(ι),νa(ι)-1)+⊤(νa(ι)-1,νb(ι))≤⊤(νa(ι),νa(ι)-1)<ɛ$$=⊤(νa(ι)-1,Dνa(ι)-1)+ɛ, ∀ι∈N.$

If follows from Eqs. (9), (12), and squeezing the theorem that

$limt→∞ ⊤(νa(ι),νb(ι))=ɛ.$

From Eqs. (9),(12), and (13), there exists such that

$12⊤(νa(ι),Dνa(ι))<12ɛ<ɛ≤⊤(νa(ι),νb(ι))<2ɛ,∀ι∈N2.$

Hence, from Eq. (14), (ℱ1) and the hypothesis of the theroem, we have

$l+ℱ(⊤(Dνa(ι),Dνb(ι)))≤qℱ(⊤(νa(ι),νb(ι)))+rℱ( ⊤(νa(ι),Dννa(ι)))+sℱ( ⊤(νb(ι),Dνb(ι)))≤qℱ(2ɛ)+rℱ(⊤(νa(ι),Dνa(ι)))+sℱ( ⊤(νb(ι),Dνb(ι)), ∀ι>N2.$

From Eq. (9) and (ℱ2) it follows that

$limι→∞ ℱ( ⊤(Dνa(ι),Dνb(ι)))=-∞,$

and thus, we obtain

$limι→∞ ℱ( ⊤(Dνa(ι),Dνb(ι)))=0⇔limι→∞ ⊤(νa(ι)+1,Dνb(ι)+1)=0.$

However, this contradicts the relationship in Eq. (12). Hence,

$limι,j→∞ ⊤(νι,νj)=0.$

Therefore, {νι} is the Cauchy sequence in . From the completeness of there exists such that

$⊤(ß,ß)=limι→∞ ⊤(νι,ß)=limι,j→∞ ⊤(νι,νj)=0.$

Subsequently, we prove that for every ,

$12 ⊤(νι,Dνι)<⊤(νι,ß),or 12 ⊤(Dνι,D2νι)<⊤(Dνι,ß),$

. By contradiction, we assume that there exists such that

$12 ⊤(νj,Dνj)≥⊤(νj,ß),or 12⊤(Dνj,D2νj)≥⊤(Dνj,ß).$

From Eq. (7) and (ℱ1), we obtain

$⊤(Dνj,D2νj)<⊤(νj,Dνj).$

It follows from Eqs. (17) and (18) that:

$⊤(νj,DνJ)≤⊤(νj,ß)+⊤(ß,Dνj)≤12⊤(νj,Dνj)+12⊤(Dνj,D2νj)<12⊤(νj,Dνj)+12⊤(νj,Dνj)=⊤(νj,Dνj).$

Evidently, this is a contradiction. Hence, inequality (16) is satisfied. Regarding the assumption of the theorem, Eq. (16) implies that

$l+ℱ(⊤(Dνι,Dß))≤qℱ(⊤(νι,ß))+rℱ( ⊤(νι,Dνι))+sℱ(⊤(ß,Dß)),$

or

$l+ℱ(⊤(D2νι,Dß))≤qℱ(⊤(Dνι,ß))+rℱ( ⊤(Dνι,D2νι))+sℱ(⊤(ß,Dß)),∀ι∈N.$

In the first case, because of (ℱ2) the limits in Eqs. (9) and (15) we get

$limι→∞ ℱ(⊤(ν(ι),ß))=-∞, limι→∞ ⊤(ν(ι),Dν(ι))=-∞.$

Thus, substituting ι → ∞ in Eq. (19), we obtain

$limι→∞ ℱ(⊤(Dν(ι),Dß))=-∞.$

Again, from (ℱ2), we observe that

$limι→∞ ⊤(Dν(ι),Dß)=0.$

However, from Eq. (1), we have

$⊤(ß,Dß)≤⊤(ß,Dνι)+⊤(Dνι,Dß)=⊤(ß,νι+1)+⊤(Dνι,Dß).$

It follows from Eqs. (15) and (21) that ⊤(ß, Dß) = 0. Therefore ß = Dß.

In the second case from Eq. (1), we have

$ℱ(⊤(D2νι,Dt))

Then, by employing Eqs. (9), (15), and (ℱ2), we conclude that

$limι→∞ ℱ(⊤(D2νι,Dß))=-∞.$

Similarly, from (ℱ2) we obtain

$limι→∞ ⊤(D2νι,Dß)=0.$

Using Eq. (9), we obtain

$⊤(ß,Dß)≤⊤(ß,D2νι)+⊤(D2νι,Dß)=⊤(ß,νι+2)+⊤(D2νι,Dß).$

Finally, from Eqs. (15) and (22), it follows that ⊤(ß, Dß) = 0; thus, ß = Dß. Hence, ß is a fixed point on D.

We prove that D has a unique fixed point in .

Let be two fixed points of D and suppose that Dιß = ß ≠ ϕ = Dιϕ, . By selecting ν0 we obtain

$(ν0⊢ß and ν0⊢ϕ) or (ν0⊢ϕ ν0⊢ß).$

Because D is ⊦-preserving ν, we have

$(Dιν0⊢Dιß and Dιν0⊢Dιϕ), or (Dιν0⊢Dιϕ and Dιν0⊢Dιß), ∀ι∈N.$

Now

$⊤(ß,ϕ)=⊤( Dιß,Dιϕ)≤⊤ (Dιß,Dιν0)+⊤(Dιν0,Dιϕ).$

As ι → ∞, we obtain ⊤(ß, ϕ) ≤ 0. Therefore, ß = ϕ. Hence, D has a unique fixed point in .

Now, we provide the fixed-point theorem for orthogonally modified ℱ-contraction of type-II in an -complete metriclike space.

### Theorem 3.2

Let be an orthogonal complete metric-like space and D be an orthogonal modified ℱ-contraction of type-II. Then, D has a unique fixed point , that is, Dß = ß.

Proof

It is sufficient to consider that q = 1 and r = s = 0 in Theorem 3.1.

### Example 3.3

Let and let be defined by

$⊤(ı,ȷ)=max{ı,ȷ}$

We define ⊦ on by ı ⊦ u if ı, ȷ ≥ 0. Then, it can be proved that is an -complete, metric-like space. We define mapping as

$D(ı)={ı4,ı∈[0,1),16,ı=1.$

Clearly, D is ⊦-preserving and ⊦-continuous. We define the function F(r) = lnr for r ∈ ℝ+ and obtain

$ℓ+ℱ(⊤(Dı,Dȷ))≤ℱ(⊤(ı,ȷ))⇔ln (⊤(ı,ȷ)⊤(Dı,Dȷ))≥ℓ,$

for all ı, . First, we observe the following:

$⊤(Dı,Dȷ)>0,12⊤(ı,Dı)<⊤(ı,ȷ)⇔{(ı=1 and ȷ∈[0,1])∨(ı<1 and ȷ=1)∨(ı<ȷ<1)∨(ȷ≤ı<1)}.$

For ı = 1 and ȷ ∈ [0, 1], we have ⊤(ı, ȷ) = ⊤(1, ȷ) = 1 and

$⊤(Dı,Dȷ)=⊤(16,Dȷ)={16,ȷ∈[0,12],16,ȷ∈(12,1),16,ȷ=1.$

Hence, we obtain

$⊤(ı,ȷ)⊤(Dı,Dȷ)={6,ȷ∈[0,12],6ȷ,u∈(12,1,)6,ȷ=1.$

For ı < 1 and ȷ = 1, we have ⊤(ı, ȷ) = ⊤ (ı, 1) = 1 and

$⊤(Dı,Dȷ)=⊤(Dı,16)={16,ı∈[0,12],16,ı∈(12,1].$

Hence, we get

$⊤(ı,ȷ)⊤(Dı,Dȷ)={6,ı∈[0,12]6ı,ı∈(12,1).$

For ı < ȷ < 1, we have ⊤(ı, ȷ) = ȷ and $⊤(Dı,Dȷ)=⊤(ı4,ȷ4)=ȷ4$. Hence, we get

$⊤(ı,ȷ)⊤(Dı,Dȷ)=4.$

For ȷ ≤ ı < 1, we have ⊤(ı, ȷ) = ı and $⊤(Dı,Dȷ)=⊤(ı4,u4)=ı4$. Hence, we get

$⊤(ı,ȷ)⊤(Dı,Dȷ)=4.$

From Eqs. (23)(26), we can obtain that if 0 < ℓ ≤ ln 4, then $ln (⊤(ı,ȷ)⊤(Dı,Dȷ))≥ℓ$. Thus,

$ℓ+ℱ(⊤(Dı,Dȷ))≤ℱ(⊤(ı,ȷ)).$

Therefore, D satisfies the conditions of Theorem 3.2 with 0 < ℓ ≤ ln 4. Hence, all required hypotheses of Theorem 3.2 are satisfied. Therefore, D has a unique fixed point.

### 4. Application

Let , Let be the set of all real-valued continuous functions in domain . The following integral equation is considered

$ζ(v)=∫0ℋΞ(v,β)Ω(β,ζ(β))dβ, v∈[0,ℋ],$

where

• (a) is continuous;

• (b) is continuous and measurable at

• (c) Ξ(v, β) ≥ 0, and $∫0ℋΞ(v,β)dβ≤1$, .

### Theorem 4.1

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

$Ω(v,ζ(v))+Ω(v,ξ(v))≤e-≀(ζ(v)+ξ(v)),$

for each vM and for all . Thus, the integral equation (27) has a unique solution in .

Proof

Let . The orthogonality relation ⊦ on is defined as

$ζ⊢ξ⇔ζ(ß)ξ(ß)≥ζ(ß) or ζ(ß)ξ(ß)≥ξ(ß) ∀ ß∈M.$

We define the mapping as

$⊤(ζ,ξ)=ζ(v)+ξ(v).$

for all . Thus, is a metric-like space and a complete metric-like space. We define as:

$Dζ(v)=∫0ℋΞ(v,β)Ω(β,ζ(v)), v∈[0,ℋ].$

Now, we show that D is ⊦-preserving. For each with ζξ and ß ∈ I, we have

$Dζ(v)=∫0ℋΞ(v,β)Ω(β,ζ(v))≥1.$

It follows that [()(ß)][()(ß)]≥()(ß) and thus, ()(ß) ⊦ ()(ß). Then, D is ⊦-preserving.

Next, we claim that D is an orthogonally modified ℱ-contraction of type-II. Let with ζξ. We assume that D(ζ) ≠ D(ξ). For every v ∈ [0, ℋ], we have

$⊤(Dζ,Dξ)=Dζ(v)+Dξ(v)=∫0ℋΞ(v,β)(Ω(β,ζ(β))+Ω(β,ξ(β))) dβ≤∫0ℋΞ(v,β)(Ω(β,ζ(β))+Ω(β,ξ(β))) dβ≤∫0ℋΞ(v,β)e-≀(ζ(v)+ξ(v)) dβ≤e-≀(ξ(v)+ξ(v))∫0ℋΞ(v,β)dβ≤e-≀(ξ(v)+ξ(v))=e-≀⊤(ζ,ξ).$

Therefore,

$≀+ln(⊤(Dζ,Dξ))≤ln (⊤(ζ,ξ)).$

Considering ℱ(v) = ln(v), we obtain

$≀+ℱ(⊤(Dζ,Dξ))≤ℱ(⊤(ζ,ξ)),$

for all . Therefore, according to Theorem 3.2, D has a unique fixed point. Hence, there exists a unique solution for Eq. (27).

### 5. Conclusion

In this study, we proved fixed-point theorems for orthogonally modified ℱ-contraction of type-I, type-II on an -complete metric-like space.

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