# Solving an Integral Equation via Orthogonal Branciari Metric Spaces

Authors: Aiman Mukheimer; Arul Joseph Gnanaprakasam; Absar Ul Haq; Senthil Kumar Prakasam; Gunaseelan Mani; Imran Abbas Baloch
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we present an orthogonalL-contraction mapping concepts and prove a fixed point theorem on orthogonal complete Branciari metric spaces. As an application, we apply our major results to solving integral equations.


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## Body

## 1. Introduction

The general metric concept was introduced by Branciari [1] in 2000 and which is known as the Branciari metric. Later, many authors were interested to the Branciari metric space for extending the results of Branciarib-metric spaces (see [2–7]). The Θ¨-contraction concept was introduced by Jleli and Samet [8] in 2014. It is based on some fixed point results [9, 10]. An orthogonality concept in metric spaces was introduced by Gordji et al. [11, 12]. Several authors proved the fixed point results in the generalized orthogonal metric space of Branciari metric spaces (BMS) [13–17]. The L-contraction concept was introduced by Cho [17] in 2018. In this article, we present the new concepts of L-contractive orthogonal mapping and prove fixed point theorems in an orthogonal complete Branciari metric space (OCBMS). We also give an example to our current results for using the integral equation solved, respectively.


## 2. Preliminaries

The basic definitions and results are required in the next section as follows.Definition 1 (see [1]).
LetP be a non-empty set and S:P×P⟶ℝ+ a mapping such that for all I1,I2∈P and all I3≠I4∈P/I1,I2:
(BM1)SI1,I2=0,iffI1=I2
(BM2)SI1,I2=SI2,I1
(BM3)SI1,I2≤SI1,I3+SI3,I4+SI4,I2.
The metricS is called a Branciari metric, and the pair P,S is called a BMS.Definition 2 (see [1]).
LetP,S be a BMS. A self-map H:P⟶P is called Θ¨-contraction if there exist Θ¨∈Γ1,2,3 and ν∈0,1 such that ∀I1,I2∈P:
(1)SHI1,HI2>0⇒Θ¨SHI1,HI2≤Θ¨SI1,I2ν,where Γ1,2,3 is the family of all functions Θ¨:0,∞⟶0,∞ which satisfy the following conditions:
(Θ¨1) Θ¨ is increasing
(Θ¨2) For each sequence αı⊂0,∞,limi⟶∞Θ¨αı=1⇔limi⟶∞αı=0+.
(Θ¨3) Θ¨ is continuous.Remark 3.
We know that everyΘ¨-contraction mapping is continuous.The following notes are subsequently adopted:(1)
Γ1,2,3 is the class of all functions Θ¨ which satisfy Θ¨1−Θ¨3Definition 4 (see [17]).
LetP,S be a BMS. A mapping H:P⟶P is called L-contraction with respect to ς∈L if there exists Θ¨∈Γ1,2,3 such that (for all I1,I2∈P):
(2)SHI1,HI2>0⇒ςΘ¨SHI1,HI2,Θ¨SI1,I2≥1,where L is the class of all functions ς:1,∞×1,∞⟶ℝ in which the following conditions are satisfied (ς1∗):
(ς1∗) ς1,1=1
(ς2∗) ςρ.ρ1<ρ1/ρ, for all ρ,ρ1>1
(ς3∗) If ρı and ρ1ı are two sequence in 1,∞ with ρı<ρ1ı, such that limı⟶∞ρı=limı⟶∞ρ1ı>1, then limsupı⟶∞ςρı,ρ1ı<1.Example 1 (see [17]).
Letςν,ςψ:1,∞×1,∞⟶ℝ be two functions defined by
(a)
ςνρ,ρ1=ρ1ν/ρ, for all ρ,ρ1≥1, where ν∈0,1(b)
ςψρ,ρ1=ρ1/ρψρ1, for all ρ,ρ1≥1, where ψ:1,∞⟶1,∞ is a lower semicontinuous and increasing function with ψ−11=1
Then,ςν,ςψ∈L.Cho [17] proved the following theorem.Theorem 5 (see [17]).
LetP,S be a complete BMS and H:P⟶P an L-contraction mapping. Then, H has a unique fixed point.Remark 6.
Letaı,bı,cı are sequences of ℝ+ such that limı⟶∞aı=a,limı⟶∞bı=b and limı⟶∞cı=c. Then,
(3)limı⟶∞maxaı,bı,cı=maxa,b,climı⟶∞minaı,bı,cı=mina,b,c.Lemma 7 (see [7]).
LetI1ı be a Cauchy sequence in a BMS P,S such that limı⟶∞SI1ı,I1=0, for some I1∈P. Then, limı⟶∞SI1ı,I2=SI1,I2, for all I2∈P. In particular, I1ı diverge to I2 if I2≠I1.Definition 8 (see [11]).
LetP≠ϕ and ∇⊆P×P be a binary relation. If ∇ satisfies the following condition:
(4)∃I10∈P:∀I1∈P,I1∇I10or∀I1∈P,I10∇I1,then it is called an orthogonal set. We denote this O-set by P,∇.Example 2.
LetP=Z and define I2∇I1 if there exists ν∈Z such that I2=νI1. It is easy to see that 0∇I1 for all I1∈Z. Hence P,∇ is an O-set.Example 3 (see [11]).
A wheel graphWı is a graph (see, for example, Figure 1) with ı vertices for each ı≥4, a single vertex connect to all vertex to all vertices of an ı−1-cycle. Let P be the set of all vertices of Wı for each ı≥4. Define I1∇I2 if there is a connection from I1 to I2. Then, P,∇ is an O-set.Definition 9 (see [11]).
LetP,∇ be an O-set. A sequence I1ı is called an orthogonal sequence (shortly, O-sequence) if
(5)∀i∈ℕ,I1ı∇I1ı+1or∀ı∈ℕ,I1ı+1∇I1ı.Definition 10 (see [11]).
The tripletP,∇,S is called an orthogonal metric space if P,∇ is an O-set and P,S is a metric space.Definition 11 (see [11]).
LetP,∇,S be an orthogonal metric space. Then, a mapping H:P⟶P is said to be orthogonally continuous in I1∈P if for each O-sequence I1ı in P with I1ı⟶I1 as ı⟶∞, we have HI1ı⟶HI1 as ı⟶∞. Also, H is said to be ∇-continuous on P if H is ∇-continuous in each I1∈P.Definition 12 (see [11]).
LetP,∇,S be an orthogonal metric space. Then, P is said to be an orthogonally complete, if every Cauchy O-sequence is convergent.Definition 13 (see [11]).
LetP,∇ be an O-set. A mapping H:P⟶P is said to be ∇-preserving if HI1∇HI2 whenever I1∇I2. Also, H:P⟶P is said to be weakly ∇-preserving if HI1∇HI2 or HI2∇HI1 whenever I1∇I2 for all I1,I2∈P.Figure 1
An image of a wheel graph.


## 3. Major Results

In this section, we present the generalized orthogonalL-contraction notion.Definition 14.
LetP,∇ be an O-set and S:P×P⟶ℝ+ a mapping such that for all I1,I2,I3,I4∈P and all I3≠I4∈P/I1,I2:
(OBM1)SI1,I2=0, if and only if I1=I2
(OBM2)SI1,I2=SI2,I1
(OBM3)SI1,I2≤SI2,I3+SI3,I4+SI4,I2 for all I1∇I2,I1∇I3,I3∇I4,I4∇I2.
The metricS is an orthogonal Branciari metric (shortly OBM), and the pair P,∇,S is an orthogonal BMS (shortly OCBMS).Definition 15.
LetP,∇,S be a OCBMS and H:P⟶P. Then, H is said to be generalized orthogonal L-contraction with respect to ς∈L if there exist Θ¨∈Γ1,2,3 such that
(6)∀I1,I2∈PwithI1∇I2SHI1,HI2>0⇒ςΘ¨SHI1,HI2,Θ¨SI1,I2≥1.Theorem 16.
LetP,∇,S be a complete OCBMS with an orthogonal element I2 and a self-mapping H:P⟶P. Suppose that there exist ς∈L and l>0 such that the following conditions hold:
(i)
H is ∇-preserving;(ii)
H is generalized orthogonal L-contraction mapping;(iii)
H is ∇-continuous.
Then,H has a unique fixed point.Proof.
SinceP,∇ is an O-set,
(7)∃I2∈P:∀I1∈P,I1∇I2or∀I1∈P,I2∇I1.
It follows thatI2∇HI2 or HI2∇I2. Let
(8)I11=HI10,I12=HI11=H2I10,⋯⋯,I1ı+1=HI1ı=Hı+1I10,for all ı∈ℕ∪0. If I1ı0=I1ı0+1 for any ı0∈ℕ∪0, then it is clear that I1ı0 is a fixed point of H. Now, we consider I1ı0≠I1ı0+1 for all ı0∈ℕ∪0. Since H is ∇-preserving, we have
(9)I1ı0∇I1ı0+1orI1ı0+1∇I1ı0,for all ı0∈ℕ∪0. This implies I1ı is an O-sequence. Using contractive Condition (6) and ς2∗, we have
(10)1≤ςΘ¨SHI1n−1,HI1ı,Θ¨SI1ı−1,I1ı=ςΘ¨SI1ı,I1ı+1,Θ¨SI1ı−1,I1ı<Θ¨SI1ı−1,I1ıΘ¨SI1ı,I1ı+1,which is implies that
(11)Θ¨SI1ı,I1ı+1<Θ¨SI1ı−1,I1ı∀ı∈ℕ.
Hence, inequality (11) becomes (in view of (Θ¨1)) that
(12)SI1ı,I1ı+1<SI1ı−1,I1ı,∀ı∈ℕ.
Therefore, the sequenceSI1ı−1,I1ı is non-increasing and bounded below by 0. Then, ℓ≥0 such that limi⟶∞SI1ı−1,I1ı=ℓ. We can claim that ℓ≠0, then
(13)limı⟶∞Θ¨SI1ı−1,I1ı>1.
Settingρı=Θ¨SI1ı,I1ı+1 and ρ1ı=Θ¨SI1ı−1,I1ı. In view of (11),(13), and (Θ¨3), we have limı⟶∞ρı=limı⟶∞ρ1ı>1 and ρı<ρ1ı, for all ı∈ℕ. Therefore, applying the condition ς3∗, we deduce
(14)1≤limsupı⟶∞ςρı,ρ1ı<1,which is a contradiction, and therefore
(15)limı⟶∞SI1ı−1,I1ı=0.
Now, we considerI1ȷ=I1ı, for some ȷ>ı. Then, also I1ȷ+1=I1ı+1. Using (11), we get
(16)Θ¨SI1ȷ,I1ȷ+1<Θ¨SI1ȷ−1,I1ȷ<Θ¨SI1ȷ−2,I1ȷ−1<⋯<Θ¨SI1ı,I1ı+1=Θ¨SI1ȷ,I1ȷ+1,which is a contradiction. Hence, we conclude that I1ȷ≠I1ı,∀ı≠ȷ.Next, we show thatI1ı is a Cauchy sequence in P,∇,S. On the contrary, assume that it is not Cauchy, then there exists an ϵ>0 for which we can find two subsequences I1ıν and I1ȷν of I1ı such that ıν>ȷν≥ν, for all ν∈N and
(17)SI1ȷν,I1ıν≥ϵ.Suppose thatiν is the least integer exceeding ȷν satisfying inequality (17). Then,
(18)SI1ıν,I1ıν−1<ϵ.Using (17), (18), and the triangular inequality, we get
(19)ϵ<SI1ȷν,I1ıν≤SI1ȷν,I1ıν−2+SI1ıν−2,I1ıν−1+SI1ıν−1,I1ıν<ϵ+SI1ıν−2,I1ıν−1+SI1ıν−1,I1ıν.Asν⟶∞,
(20)limν⟶∞SI1ȷν,I1ıν=ϵ.Employing the triangular inequality once again, we get(21)SI1ȷν,I1ıν≤SI1ȷν,I1ȷν−1+SI1ȷν−1,I1ıν−1+SI1ıν−1,I1ıν≤2SI1ȷν,I1ȷν−1+SI1ȷν,I1ıν+2SI1ıν−1,I1ıν.On lettingν⟶∞ and using (15) as well as (20) we get
(22)limν⟶∞SI1ȷν−1,I1ıν−1=ϵ.Now, using (6) and ς2∗, we obtain
(23)1≤ςΘ¨SHI1ȷν−1,HI1ıν−1,Θ¨SI1ȷν−1,I1ıν−1=Θ¨SI1ȷν,I1ıν,Θ¨SI1ȷν−1,I1ıν−1<Θ¨SI1ȷν−1,I1ıν−1Θ¨SI1ȷν,I1ıν.Consequently, we deduce that(24)Θ¨SI1ȷν,I1ıν<Θ¨SI1ȷν−1,I1ıν−1,∀ν∈N.Letρν=Θ¨SI1ȷν,I1ıν and ρ1ν=Θ¨SI1ȷν−1,I1ıν−1. Then, in view of Remark 6 and (24), we have limν⟶∞ρν=limν⟶∞ρ1ν>1 and ρν<ρ1ν,∀ν∈N. So, on using ς3∗, we obtain
(25)1≤limsupν⟶∞ςρν,ρ1ν<1,which is a reductio ad absurdum. Therefore, I1ı must be a Cauchy sequence in P,∇,S. Since P,∇,S is a complete, then there exists ν1∈P such that limi⟶∞I1ı=ν1, then,
(26)limı⟶∞SI1ı,ν1=0.AsH is continuous, then we get that (due to (26))
(27)limı⟶∞SI1ı+1,Hν1=limı⟶∞SHI1ı,Hν1=0.Using Lemma7, we have ν1=Hν1 that is, ν1 is a fixed point of H. On the contrary, assume that there are two fixed points such that Sν1,I3=SHν1,HI3>0. From (6), since H is preserving, ∀Hν1∇HI3 we have
(28)1≤ςΘ¨SHν1,HI3,Θ¨Sν1,I3=ςΘ¨Sν1,I3,Θ¨Sν1,I3<Θ¨Sν1,I3Θ¨Sν1,I3.This is implies that(29)Θ¨Sν1,I3<Θ¨Sν1,I3,which is a reductio ad absurdum. Then, H has a unique fixed point.Example 4.
LetP=E∪G, where E=0,2 and G=1/i:n=2,3,4,5. Define the binary relation ∇ on P by I1∇I2 if I1,I2≥0. Define a mapping S:P×P⟶0,∞ defined by SI1,I2=I1−I2, for all I1,I2∈P.It is easy to see thatP,∇,S is an orthogonal complete BMS. Let H:P⟶P be defined as HI1=I1/6 for all I1∈P. Clearly, H is an orthogonal preserving and orthogonal continuous. Observe that H is an L-contraction with respect to ς:1,∞×1,∞⟶ℝ, where
(30)ςνρ,ρ1=ρ1νρ,∀ρ,ρ1∈1,∞,ν∈0,1,and Θ¨:0,∞⟶1,∞ such that Θ¨ρ=eρ,∀ρ>0.LetI1,I2∈P; then,
(31)ςΘ¨SHI1,HI2,Θ¨SI1,I2=Θ¨SI1,I2νΘ¨SHI1,HI2=eνI1−I2eI1−I2/6≥1.Hence, all the hypotheses of Theorem16 are satisfied, and I1=0 is the unique fixed point of H.


## 4. Applications

As an application of Theorem16, we find an existence and uniqueness of the solution of the following integral equation:
(32)I1ρ=gρ+∫0aȷρ,ν1fν1,I1ν1dν1,ρ∈0,a,a>0.LetU=C0,a,R be the set of real continuous functions defined on 0,a and the mapping H:U⟶U defined by
(33)HI1ρ=gρ+∫0aȷρ,ν1fν1,I1ν1dν1,ρ∈0,a.Obviously,I1ρ is a solution of integral Equation (32) iff I1ρ is a fixed point of H.Theorem 17.
Suppose that
(R1) The mappingsȷ:0,a×ℝ⟶0,∞, f:0,a×ℝ⟶ℝ, and g:0,a⟶ℝ are continuous functions;
(R2) there existτ>0 and ν∈0,1 such that
(34)fν1,I1ν1−fν1,I2ν1≤νI1ν1−I2ν1;
(R3)∫0aȷρ,ν1dν1≤1.Then, the integral Equation (32) has a unique solution in U.Proof.
Define the orthogonality relation∇ on U by
(35)I1∇I2⇔I1ρI2ρ≥I1ρorI1ρI2ρ≥I2ρ,∀ρ∈0,a.
DefineS:U×U⟶0,∞ given by
(36)SI1,I2=supρ∈0,aI1ρ−I2ρ,for all I1,I2∈U. It is easy to see that U,∇,S is complete orthogonal BMS. For each I1,I2∈U with I1∇I2 and ρ∈0,a, we have
(37)HI1ρ=gρ+∫0ajρ,ν1fν1,I1ν1dν1≥1.
Accordingly,HI1ρHI2ρ≥HI2ρ and so HI1ρ∇HI2ρ. Then, H is ∇-preserving. Let I1,I2∈U with I1∇I2. Suppose that HI1≠HI2. For each ρ∈0,a, we have
(38)HI1ρ−HI2ρ=∫0ajρ,ν1fν1,I1ν1−fν1,I2ν1du≤∫0ajρ,ν1νI1ν1−I2ν1du≤ν∫0ajρ,ν1dusupu∈0,aI1u−I2u≤νSI1,I2.
Thus,(39)SHI1,HI2≤νSI1,I2,which implies that
(40)eSHI1,HI2≤eνSI1,I2,for each I1,I2∈U. We consider L-simulation mapping ς:1,∞×1,∞⟶ℝ, where
(41)ςνρ,ρ1=ρ1νρ,∀ρ,ρ1∈1,∞,ν∈0,1,and Θ¨:0,∞⟶1,∞ such that Θ¨ρ=eρ,∀ρ>0. Then,
(42)ςΘ¨SHI1,HI2,Θ¨SI1,I2≥1.
Hence, all the conditions of Theorem16 are fulfilled. Therefore, the integral equation has a unique solution.


## 5. Conclusion

In this article, we proved the fixed point theorems for orthogonalL-contraction mapping on orthogonal complete BMS.



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