# On Orthogonal Partialb-Metric Spaces with an Application

Authors: Khalil Javed; Hassen Aydi; Fahim Uddin; Muhammad Arshad
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we initiate the concept of orthogonal partialb-metric spaces. We ensure the existence of a unique fixed point for some orthogonal contractive type mappings. Some useful examples are given, and an application is also provided in support of the obtained results.


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## Body

## 1. Introduction

The notion of a metric space plays a vital role in metric fixed point theory. The Banach contraction principle [1] is a very famous result in the literature. Fixed point hypotheses are significant apparatuses for demonstrating the presence and uniqueness of solutions for different numerical models. Given a nonempty set U˜ and a map Ψ from Ψ into itself, the problem of finding a point ϰ∈U˜ such that Ψϰ=ϰ is considered as a fixed point problem, and the point ϰ∈U˜ is called a fixed point of Ψ.A natural question is that, under what conditions onϰ and U˜ does a fixed point exist? Theorems which establish the existence (and uniqueness) of such points are called fixed point theorems. These results allow us to find the existence of solutions that satisfy certain conditions for operator equations.There exist many generalizations of the concept of a metric space in the literature. In [2], Matthews introduced the notion of a partial metric space as part of the study of denotational semantics of dataflow networks. In this setting, the self-distance of any point may not be zero. A lot of fixed point theorems have been investigated in partial spaces (see [3–7] and references therein). Another important generalization of a metric space introduced by Czerwik [8] is a b-metric space, where the triangular inequality was replaced by sdx,z+dz,y with s≥1. Since then, many articles dealing with fixed point theory and variation principle in the setting of b-metric spaces for single- and multivalued operators have been appeared (see [9–20]).Recently, Eshaghi Gordji et al. [21] initiated the concept of orthogonal sets and gave an extension of the Banach contraction principle. Furthermore, they presented applications for their results to ensure the existence and uniqueness of solutions for first-order differential equations.The purpose of this paper is to improve and generalize the concept of an orthogonal contraction in the sense of metric spaces due to Gordji et al. [22]. Namely, we introduce the concept of an orthogonal partial b-metric and establish some fixed point theorems for related contractions. We also enrich this paper with a nontrivial example involving an orthogonal partial b-metric, which is not a partial b-metric. We set up some hypotheses for the proposed construction, and additionally, we present a potential application for the arrangement of Volterra integral equation to guarantee the legitimacy of the outcomes.


## 2. Preliminaries

In 1993, Czerwik [8] presented the idea of b-metric spaces (in short, b-M.Ss).Definition 1.
(see [8]). Let U˜ be a nonempty universal set and ρ:U˜×U˜⟶ℝ+ be a mapping so that for all ϰ1, ϰ2, ϰ3∈U˜ and s≥1:(1)bm1ρϰ1,ϰ2=0,if and only ifϰ1=ϰ2,ρϰ1,ϰ2=ρϰ2,ϰ1,bm3ρϰ1,ϰ2≤sρϰ1,ϰ3+ρϰ3,ϰ2.
Then,ρ is said as a b-metric on U˜ and U˜,ρ is said as a b-M.S.The notion of a partial metric space (in short,ρ-M.S) has been initiated by Matthews [2].Definition 2.
(see [2]). Let U˜≠∅ be a universal set and ζ∗:U˜×U˜⟶ℝ+ be a mapping so that for all ϰ1, ϰ2, ϰ3∈U˜:(2)pm1ϰ1=ϰ2,if and only ifζ∗ϰ1,ϰ1=ζ∗ϰ1,ϰ2=ζ∗ϰ2,ϰ2,pm2ζ∗ϰ1,ϰ1≤ζ∗ϰ1,ϰ2,pm3ζ∗ϰ1,ϰ2=ζ∗ϰ2,ϰ1,pm4ζ∗ϰ1,ϰ2≤ζ∗ϰ1,ϰ3+ζ∗ϰ3,ϰ2−ζ∗ϰ3,ϰ3.
Then,ζ∗ is said as a ρ-metric on U˜ and U˜,ζ∗ is said as a ρ-M.S.In 2014, Shukla [23] introduced the following concept of partial b-metric spaces (in short, ρb-M.Ss) and proved some fixed point results.Definition 3.
(see [23]). Let U˜ be a nonempty universal set and σ:U˜×U˜⟶ℝ+ be a mapping so that for all ϰ1, ϰ2, ϰ3∈U˜ and s≥1:(3)σ1ϰ1=ϰ2,if and only ifσϰ1,ϰ1=σϰ1,ϰ2=σϰ2,ϰ2,σ2σϰ1,ϰ1≤σϰ1,ϰ2,σ3σϰ1,ϰ2=σϰ2,ϰ1,σ4σϰ1,ϰ2≤sσϰ1,ϰ3+σϰ3,ϰ2−σϰ3,ϰ3.
Then,σ is said as a ρb-metric on U˜ and U˜,σ is said as a ρb-M.S with coefficient s≥1.Remark 1.
(see [23]). If ϰ1 and ϰ2 are in a ρb-M.S U˜,σ so that σϰ1,ϰ2=0, then ϰ1=ϰ2. However, the converse may not be true.Example 1.
(see [23]). Let U˜=ℝ+, ϰ>1 be a constant, and σ:U˜×U˜⟶ℝ+ be defined by(4)σϰ1,ϰ2=maxϰ1,ϰ22+ϰ1−ϰ22,for allϰ1,ϰ2∈U˜.
Then,U˜,σ is a ρb-M.S with coefficient s=2ϰ>1. However, it is neither a b-M.S nor a ρ -M.S.In 2017, the authors of [22] introduced the notion of orthogonal sets and gave a real generalization of Banach fixed point theorem.Definition 4.
(see [22]). Let U˜≠∅ and ⊥ be a binary relation defined on U˜×U˜. Then, U˜,⊥ is called an orthogonal set OS if there is ϰ1∈U˜ so that(5)∀ϰ2∈U˜,ϰ1⊥ϰ2or∀ϰ2∈U˜,ϰ2⊥ϰ1.
The elementϰ1 is said to be an O-element.Definition 5.
(see [22]). Let U˜,⊥ be an OS. A sequence ϰη is said to be an orthogonal sequence O−seq if(6)∀η∈ℕ,ϰη⊥ϰη+1or∀η∈ℕ,ϰη+1⊥ϰη.Definition 6.
(see [21]). A mapping Ψ:U˜⟶U˜ is orthogonal continuous OC in ϰ∈U˜ if for each O-sequence ϰηη∈ℕ in U˜ such that ϰη⟶ϰ, then Ψϰη⟶Ψϰ. Also, Ψ is said to be OC on U˜ if Ψ is OC at each ϰ∈U˜.Definition 7.
(see [22]). Let U˜,⊥,σ⊥ be an orthogonal M.S. Then, U˜ is called O-complete if every Cauchy O-seq is convergent.Definition 8.
(see [21]). Let U˜,⊥,σ⊥ be an orthogonal M.S and 0<z<1. A mapping Ψ:U˜⟶U˜ is called an orthogonal contraction O−contraction with the Lipschitz constant z if(7)σ⊥Ψϰ1,Ψϰ2≤zσ⊥ϰ1,ϰ2,for all ϰ1, ϰ2∈U˜ with ϰ1⊥ϰ2.Definition 9.
(see [22]). Let U˜,⊥ be an OS. A mapping Ψ:U˜⟶U˜ is said to be O-preserving OP if Ψϰ1⊥Ψϰ2, whenever ϰ1⊥ϰ2.


## 3. Main Results

Throughout the paper, O-comp designs orthogonal complete. First, we introduce the concept of an orthogonal partialb-metric space (in short, orthogonal ρb-M.S).Definition 10.
A mapσ⊥:U˜×U˜⟶ℝ+ is called an orthogonal ρb-M.S on the OS U˜,⊥ if the following axioms are satisfied:(8)σ⊥1ϰ1=ϰ2,if and only ifσ⊥ϰ1,ϰ1=σ⊥ϰ1,ϰ2=σ⊥ϰ2,ϰ2,σ⊥2σ⊥ϰ1,ϰ1≤σ⊥ϰ1,ϰ2,σ⊥3σ⊥ϰ1,ϰ2=σ⊥ϰ2,ϰ1,σ⊥4σ⊥ϰ1,ϰ2≤sσ⊥ϰ1,ϰ3+σ⊥ϰ3,ϰ2−σ⊥ϰ3,ϰ3,for all ϰ1, ϰ2, ϰ3∈U˜ with ϰ1⊥ϰ3⊥ϰ2.
The pairU˜,σ⊥ is said as an orthogonal ρb-M.S with a coefficient s≥1.Remark 2.
Every orthogonal partialb-metric space is a partial b-metric, but the converse is not true in general. Example 2 describes an orthogonal partial b-metric, which is not a partial b-metric. In [22], Eshaghi Gordji and Habibi considered orthogonal metric spaces to prove some fixed point theorems, while in Theorem 3.2 and Theorem 3.3 of the paper [21], the authors considered generalized orthogonal metric spaces to prove some related fixed point theorems. In this paper, we consider orthogonal partial b metric spaces to prove some fixed point results. Note that an orthogonal partial b-metric is more generalized than an orthogonal metric, an orthogonal b-metric, and an orthogonal partial metric. That is, our results are generalizations and extensions of the results given in [21, 22].Example 2.
LetU˜=−1,−2,−3,1,2,3 and let the binary relation be defined by ϰ1⊥ϰ2 iff ϰ1=ϰ2 or ϰ1, ϰ2>0. It is easy to prove that σ⊥ϰ1,ϰ2=maxϰ1,ϰ2 is an orthogonal ρb-metric on U˜ (with a coefficient s≥1). However, σ⊥ is not a ρb-metric on U˜ (with a coefficient s≥1). Indeed, for ϰ1=−3 and ϰ2=3, we have σ⊥ϰ1,ϰ1=σ⊥ϰ1,ϰ2=σ⊥ϰ2,ϰ2=3.As related topological notions for this new setting, we state the following definitions.Definition 11.
LetU˜,σ⊥ be an orthogonal ρb-M.S with s≥1. Then, an O-seq ϰη is called(i)
Convergent iff there existsϰ∈U˜ such that σ⊥ϰη,ϰ⟶0 as η⟶∞(ii)
Cauchy iffσ⊥ϰη,ϰm⟶0 as η, m⟶∞Definition 12.
LetU˜,σ⊥ be an orthogonal ρb-M.S. Then, Ψ:U˜⟶U˜ is called O-continuous OC at ϰ∈U˜ if for each O-seq ϰη in U˜ with σ⊥ϰη,ϰ⟶0, we have σ⊥Ψϰη,Ψϰ⟶0. Also, Ψ is said to be OC on U˜ if Ψ is OC at each ϰ∈U˜.Definition 13.
LetU˜,σ⊥ be an orthogonal ρb-M.S with s≥1. Then, U˜ is called O-complete if every Cauchy O-seq is convergent in U˜.Our first essential main result is as follows.Theorem 1.
LetU˜,σ⊥ be an O-comp ρb-M.S with s≥1 and Ψ:U˜⟶U˜ be an OP and OC mapping so that(9)σ⊥Ψϰ1,Ψϰ2≤zσ⊥ϰ1,ϰ2,for allϰ1,ϰ2∈U˜withϰ1⊥ϰ2,where z∈0,1/s. Then, Ψ has a unique fixed point ϰ∗∈U˜ and σ⊥ϰ∗,ϰ∗=0.Proof.
By the definition of orthogonality, there isϰ1∈U˜ such that for all ϰ2∈U˜, ϰ1⊥ϰ2 or for all ϰ2∈U˜, ϰ2⊥ϰ1. It follows that ϰ1⊥Ψϰ1 or Ψϰ1⊥ϰ1. Let ϰ1=Ψϰ0, ϰ2=Ψϰ1, ϰ3=Ψϰ2, ϰ4=Ψϰ3, …, ϰη+1=Ψϰη, for all η∈ℕ. Since Ψ is OP, ϰη is an O-seq. Then, by (9), we obtain(10)σ⊥ϰη,ϰη+1=σ⊥Ψϰη−1,Ψϰη≤zησ⊥ϰ0,ϰ1,for all η∈ℕ. For all m≥1 and p≥1, it follows that(11)σ⊥ϰm+p,ϰm≤sσ⊥ϰm+p,ϰm+p−1+σ⊥ϰm+p−1,ϰm−σ⊥ϰm+p−1,ϰm+p−1≤sσ⊥ϰm+p,ϰm+p−1+s2σ⊥ϰm+p−1,ϰm+p−2+σ⊥ϰm+p−2,ϰm−sσ⊥ϰm+p−2,ϰm+p−2≤sσ⊥ϰm+p,ϰm+p−1+s2σ⊥ϰm+p−1,ϰm+p−2+s2σ⊥ϰm+p−2,ϰm≤sσ⊥ϰm+p,ϰm+p−1+s2σ⊥ϰm+p−1,ϰm+p−2+s3σ⊥ϰm+p−2,ϰm+p−3+⋯+sp−1σ⊥ϰm+2,ϰm+1+sp−1σ⊥ϰm+1,ϰm≤szm+p−1σ⊥ϰ1,ϰ0+s2zm+p−2σ⊥ϰ1,ϰ0+s3zm+p−3σ⊥ϰ1,ϰ0+⋯+sp−1zm+1σ⊥ϰ1,ϰ0+sp−1zmσ⊥ϰ1,ϰ0≤szm+p−1+s2zm+p−2+s3zm+p−3+⋯+sp−1zm+1+sp−1zmσ⊥ϰ1,ϰ0≤spzm1−szσ⊥ϰ1,ϰ0.
Taking limit asm⟶∞, we have(12)limη⟶∞σ⊥ϰm+p,ϰm=0.
Therefore,ϰη is a Cauchy O-seq. Since U˜ is O-comp, there is ϰ1∗∈U˜ so that ϰη⟶ϰ∗ as η⟶∞. Since Ψ is OC, we obtain(13)Ψϰ∗=Ψlimη⟶∞ϰη=limη⟶∞Ψϰη=limη⟶∞ϰη+1=ϰ∗.
Hence,ϰ∗ is a fixed point of Ψ. Next, we demonstrate its uniqueness. Let ϰ2∗∈U˜ be a fixed point of Ψ. So, we obtain Ψηϰ∗=ϰ∗ and Ψηϰ2∗=ϰ2∗ for all η∈ℕ. By the definition of orthogonality, there is ϰ1∈U˜ so that(14)ϰ1⊥ϰ∗andϰ1⊥ϰ2∗orϰ∗⊥ϰ1andϰ2∗⊥ϰ1.
SinceΨ is OP, one writes(15)Ψηϰ1⊥Ψηϰ∗andΨηϰ1∗⊥Ψηϰ2∗,or(16)Ψηϰ∗⊥Ψηϰ1andΨηϰ2∗⊥Ψηϰ1,for all η∈ℕ. Therefore, by the triangle inequality, we obtain(17)σ⊥ϰ1∗,ϰ2∗=σ⊥Ψηϰ1∗,Ψηϰ2∗=sσ⊥Ψηϰ1∗,Ψηϰ1+σ⊥Ψηϰ1,Ψηϰ2∗−σ⊥Ψηϰ1,Ψηϰ1≤szησ⊥ϰ1∗,ϰ1+szησ⊥ϰ1,ϰ2∗.
Taking limit asη⟶∞, we obtain(18)σ⊥ϰ1∗,ϰ2∗=0,and so ϰ1∗=ϰ2∗.Example 3.
ConsiderU˜=ℝ. Given σ⊥:U˜×U˜⟶ℝ+ as(19)σ⊥ϰ1,ϰ2=ϰ1−ϰ22,ifϰ1,ϰ2≥0,0,otherwise.
Define⊥ on U˜ by ϰ1⊥ϰ2 iff ϰ1=ϰ2 or ϰ1, ϰ2≥0. Then, U˜,σ⊥ is an O-comp ρb-M.S with coefficient s=4. Define the mapping Ψ:U˜⟶U˜ by(20)Ψϰ=ϰ4,ϰ≥0,0,otherwise.
We have the following cases:(a)
Ifϰ1=ϰ2, then Ψϰ1=Ψϰ2. If ϰ1, ϰ2≥0, then Ψϰ1, Ψϰ2≥0. Thus, Ψ is OP.(b)
If any O-seqϰη in U˜ with limn⟶∞σ⊥ϰη,ϰ=σ⊥ϰ,ϰ, ϰη⊥ϰ for some η≥k and limn⟶∞σ⊥ϰη,Ψϰ=σ⊥ϰ,ϰ, then we obtain ϰ⊥Ψϰ.(c)
Ifϰ0≥0 any real number, then Ψϰ0≥0. Thus, ϰ0 and Ψϰ0≥0, that is, ϰ0⊥Ψϰ0.(d)
Letϰ1, ϰ2∈U˜ with ϰ1⊥ϰ2.
Ifϰ1=ϰ2, then the following result holds:(21)σ⊥Ψϰ1,Ψϰ2=0=116σ⊥ϰ1,ϰ2.
Ifϰ1, ϰ2≥0, then the following result holds:(22)σ⊥Ψϰ1,Ψϰ2=ϰ14−ϰ242=116ϰ1−ϰ22=116σ⊥ϰ1,ϰ2.
By Theorem1, Ψ has a fixed point.Our second result as a generalization of Theorem1 is as follows.Theorem 2.
LetU˜,σ⊥ be an O-comp ρb-M.S with s≥1 and Ψ:U˜⟶U˜ be an OP and OC mapping so that(23)σ⊥Ψϰ1,Ψϰ2≤zmaxσ⊥ϰ1,ϰ2,σ⊥ϰ1,Ψϰ1,σ⊥ϰ2,Ψϰ2,for all ϰ1, ϰ2∈U˜, where z∈0,1/s. Then, Ψ has a unique fixed point ϰ∗∈U˜ and σ⊥ϰ∗,ϰ∗=0.Proof.
By the definition of orthogonality, there isϰ1∈U˜ so that for all ϰ2∈U˜, ϰ1⊥ϰ2 or for all ϰ2∈U˜, ϰ2⊥ϰ1. It follows that ϰ1⊥Ψϰ1 or Ψϰ1⊥ϰ1. Let ϰ1=Ψϰ0, ϰ2=Ψϰ1, …, ϰη+1=Ψϰη for all η∈ℕ. Since Ψ is OP, ϰη is an O-seq. Then, by (23), we have(24)σ⊥ϰη+1,ϰη=σ⊥Ψϰη,Ψϰη−1≤zmaxσ⊥ϰη,ϰη−1,σ⊥ϰη,Ψϰη,σ⊥ϰη−1,Ψϰη−1=zmaxσ⊥ϰη,ϰη−1,σ⊥ϰη,ϰη+1,σ⊥ϰη−1,ϰη=zmaxσ⊥ϰη,ϰη−1,σ⊥ϰη,ϰη+1.
If for someη, maxσ⊥ϰη,ϰη−1,σ⊥ϰη,ϰη+1=σ⊥ϰη+1,ϰη, we obtain that σ⊥ϰη+1,ϰη≤zσ⊥ϰη+1,ϰη<σ⊥ϰη+1,ϰη, which is a contradiction. Thus,(25)maxσ⊥ϰη,ϰη−1,σ⊥ϰη,ϰη+1=σ⊥ϰη,ϰη−1,for allη≥1.
Again, we have(26)σ⊥ϰη+1,ϰη≤zσ⊥ϰη,ϰη−1.
Repeating this cycle, we obtain(27)σ⊥ϰη+1,ϰη≤zησ⊥ϰ1,ϰ0,for all η≥0. For m, η∈ℕ with m>η, we obtain(28)σ⊥ϰη,ϰm≤sσ⊥ϰη,ϰη+1+σ⊥ϰη+1,ϰm−σ⊥ϰη+1,ϰη+1≤sσ⊥ϰη,ϰη+1+s2σ⊥ϰη+1,ϰη+2+σ⊥ϰη+2,ϰm−sσ⊥ϰη+2,ϰη+2≤sσ⊥ϰη,ϰη+1+s2σ⊥ϰη+1,ϰη+2+s3σ⊥ϰη+2,ϰη+3+⋯+sm−ησ⊥ϰm−1,ϰm.
Using (27) in the above inequality,(29)σ⊥ϰη,ϰm≤szησ⊥ϰ1,ϰ0+s2zη+1σ⊥ϰ1,ϰ0+s3zη+2σ⊥ϰ1,ϰ0+⋯+sm−ηzm−1σ⊥ϰ1,ϰ0≤szη1+sz+sz2+⋯σ⊥ϰ1,ϰ0=szη1−szσ⊥ϰ1,ϰ0.
Asz∈0,1/s and s′≥1, it follows from the above inequality that(30)limη,m⟶∞σ⊥ϰη,ϰm=0.
Therefore,ϰη is a Cauchy O-seq. Since U˜ is O-comp, there is ϰ∗∈U˜ so that ϰη⟶ϰ∗ as η⟶∞. As Ψ is OC, one writes(31)Ψϰ∗=Ψlimη⟶∞ϰη=limη⟶∞Ψϰη=limη⟶∞ϰη+1=ϰ∗.
Therefore,ϰ∗ is a fixed point. To show that it is unique, consider ϰ2∗≠ϰ1∗∈U˜ as a fixed point of Ψ. So, we obtain Ψηϰ1∗=ϰ1∗ and Ψηϰ2∗=ϰ2∗ for all η∈ℕ′. By the definition of orthogonality, there exists ϰ1∈U˜ so that(32)ϰ1⊥ϰ1∗andϰ1⊥ϰ2∗orϰ1∗⊥ϰ1andϰ2∗⊥ϰ1.
AsΨ is OP, we obtain(33)Ψηϰ1⊥Ψηϰ1∗andΨηϰ1⊥Ψηϰ2∗,or(34)Ψηϰ1∗⊥Ψηϰ1andΨηϰ2∗⊥Ψηϰ1,for all η∈ℕ. Then, we obtain(35)σ⊥ϰ1∗,ϰ2∗=σ⊥Ψηϰ1∗,Ψηϰ2∗≤zmaxσ⊥ϰ1∗,ϰ2∗,σ⊥ϰ1∗,Ψϰ1∗,σ⊥ϰ2∗,Ψϰ2∗=zmaxσ⊥ϰ1∗,ϰ2∗,σ⊥ϰ1∗,ϰ1∗,σ⊥ϰ2∗,ϰ2∗=zσ⊥ϰ1∗,ϰ2∗<σ⊥ϰ1∗,ϰ2∗.
It is a contradiction. So, we need to haveσ⊥ϰ1∗,ϰ2∗=0, that is, ϰ1∗=ϰ2∗. Thus, if fixed point of Ψ exists, then it is unique.The following corollary is the analog of Kannan fixed point theorem [24] in orthogonal partial b-metric spaces.Corollary 1.
LetU˜,σ⊥ be an O-comp ρb-M.S with s≥1 and Ψ:U˜⟶U˜ be an OP and OC mapping so that(36)σ⊥Ψϰ1,Ψϰ2≤zσ⊥ϰ1,Ψϰ1+σ⊥ϰ2,Ψϰ2,for all ϰ1, ϰ2∈U˜, where z∈0,1/s. Then, Ψ has a unique fixed point ϰ∗∈U˜ and σ⊥ϰ∗,ϰ∗=0.The following corollary is the analog of Bianchini fixed point theorem [25] in orthogonal partial b-metric spaces.Corollary 2.
LetU˜,σ⊥ be an O-comp ρb-M.S with s≥1 and Ψ:U˜⟶U˜ be an OP and OC mapping so that(37)σ⊥Ψϰ1,Ψϰ2≤zmaxσ⊥ϰ1,Ψϰ1,σ⊥ϰ2,Ψϰ2,for all ϰ1, ϰ2∈U˜, where 0≤z<1. Then, Ψ has a unique fixed point ϰ∗∈U˜ and σ⊥ϰ∗,ϰ∗=0.


## 4. Application

In this section, we consider the Volterra integral type equation:(38)ϰ1e=he+k∫01ue,sfs,ϰ1sds,e∈I=0,1,k≥0.Take the spaceU˜=CI of continuous functions defined on I endowed with a metric given by(39)pϰ1,ϰ2=supe∈Iϰ1e−ϰ2e,for all ϰ1, ϰ2∈CI.Letγ denote the class of function γ:ℝ+⟶ℝ+ so that γμq≤γμq for each q≥1 and μ≥0.We consider the following assumptions:(i)
f:I×ℝ⟶ℝ is nondecreasing with respect to its second variable and continuous so that there is 0≤L≤1:(40)fe,v1−fe,v2≤Lγv1−v2,for allv1, v2∈ℝ with v1≥v2.(ii)
h:I⟶ℝ is continuous on I.(iii)
u:I×I⟶ℝ is continuous with respect to its first variable and measurable with respect to its second variable such that for each e∈I,(41)∫01ue,sds≤K.(iv)
LqkqKq≤1/24q−4.We consider onU˜ the following: ϰ1, ϰ2∈CI and ϰ1⊥ϰ2⟺ϰ1≤ϰ2.Now, forq≥1, we define(42)σ⊥ϰ1,ϰ2=pϰ1,ϰ2q=supe∈I|ϰ1e−ϰ2e|q=supe∈Iϰ1e−ϰ2eq,for ϰ1, ϰ2∈CI.We conclude thatU˜,ρb is a O-comp ρb M.S with s=2q−1.Now, we formulate the main result of this section.Theorem 3.
Under the assumptions (i)-(iv), equation (38) has a unique solution in CI.Proof.
We consider the operatorΨ:U˜⟶U˜ defined by(43)Ψϰe=he+k∫01ue,sfs,ϰsds,for e∈I and k≥0.
By virtue of our assumptions,Ψ is well defined ifϰ∈U˜,thenΨϰ∈U˜.
Forϰ1≤ϰ2 and e∈I, we have(44)Ψϰ1e−Ψϰ2e=he+k∫01ue,sfs,ϰ1sds−he−k∫01ue,sfs,ϰ2sds=k∫01ue,sfs,ϰ1s−fs,ϰ2sds≤0.
Therefore,Ψ has the monotone nondecreasing property. Also, for ϰ1⊥ϰ2, we have(45)Ψϰ1e−Ψϰ2e=he+k∫01ue,sfs,ϰ1sds−he−k∫01ue,sfs,ϰ2sds≤k∫01ue,sfs,ϰ1s−fs,ϰ2sds≤k∫01ue,sLγϰ1−ϰ2.
Sinceϰ1⊥ϰ2, we have(46)γϰ2s−ϰ1s≤γsupe∈Iϰ1s−ϰ2s=γpϰ1,ϰ2,hence(47)Ψϰ1e−Ψϰ2e≤k∫01ue,sLγpϰ1,ϰ2ds≤kKLγpϰ1,ϰ2.
Then, we obtain(48)σ⊥Ψϰ1,Ψϰ2=supe∈IΨϰ1e−Ψϰ2eq≤kKLγpϰ1,ϰ2q=kqKqLqγpϰ1,ϰ2q=kqKqLqγσ⊥ϰ1,ϰ2≤124q−4σ⊥ϰ1,ϰ2.
This proves that the operatorΨ satisfies the contractive condition (9) appearing in Theorem 1. So, (38) has a solution and the proof is complete.


## 5. Conclusion

The study of fixed points of mappings satisfying orthogonal sets has been focused vigorously on different research activities in the recent decade. As a consequence, many mathematicians obtained more results in this direction. In this paper, the concept of generalized orthogonal contractive conditions in partialb-metric spaces was introduced. Based on this notion, fixed point results have been discussed. Some illustrative examples are furnished, which demonstrate the validity of the hypotheses and degree of utility of the proposed results. It would be interesting to consider more generalized orthogonal contractions in this setting.



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*Source: 6692063-2021-01-12.xml*