# Generalized Interpolative Contractions and an Application

Authors: Muhammad Nazam; Hassen Aydi; Aftab Hussain
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we introduce a property (P) and the generalized interpolative contractions of types I, II, III, and IV. We investigate certain conditions for the existence of fixed points of generalized interpolative contractions. We derive several new results from the main theorems. As an application, we resolve the Urysohn integral equation.


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## Body

## 1. Introduction

Fixed-point theory is an outstanding example of a central principle with multiple implementations. In diverse areas, such as differential equations and artificial intelligence, it has always been a significant theoretical method. Furthermore, the development of accurate and efficient techniques for computing fixed points has significantly increased the concept’s utility for applications, making fixed-point methods a major tool in the arsenal of the applied mathematician. The key element in the metric fixed-point theory is the Banach contraction principle (BCP). It states thatevery contraction, in the complete metric space, admits a unique fixed point. This principle has been generalized by many ways (see [1]). Recently, Gordji et al. [2] presented a new generalization of the BCP by defining the notion of orthogonal sets and hence orthogonal metric spaces. They presented an example supporting the fact that their main theorem is a real generalization of the BCP. Baghani et al. [3] extended the work of [2] to F-contractions. Chandok et al. [4] extended the results given in [3] to multivalued F-contractions.On the contrary, Karapinar [5] introduced interpolative contractions and presented a method to obtain fixed points of such contractions. Karapinar et al. [6–9], in subsequent papers, investigated Rus–Reich–Ćirić-type interpolative contractions, Hardy–Rogers-type interpolative contractions, Rus–Reich–Ćirić-type ω-interpolative contractions, and Boyd–Wong- and Matkowski-type interpolative contractions to ensure the existence of fixed points in variant (generalized) metric spaces. Gautam et al. [10] presented some fixed-point results for Chatterjea and cyclic Chatterjea interpolative contractions in complete quasi-partial b-metric spaces. Debnath et al. [11] proved some fixed-point theorems for Rus–Reich–Ćirić- and Hardy–Rogers-type interpolative contractions in b-metric spaces.Boyd–Wong [12] generalized the well-known Banach contraction principle (BCP) [13] by introducing a control function Ψ:0,∞⟶0,∞, verifying the below conditions for each J>0:(1)
ΨJ<J(2)
limℓ⟶J+Ψℓ<JThe related result of Boyd–Wong [12] is as follows.Theorem 1.
LetS:X⟶X be a self-mapping on a complete metric space X,d so that(1)dSℓ,SJ≤Ψdℓ,J,for allℓ,J∈X,where Ψ:0,∞⟶0,∞ verifies (1)-(2). Then, S has a unique fixed point in X (say, ρ) and the sequence Snℓ is convergent to ρ, for each ℓ∈X.It is noted that Theorem1 is an improvement of main results of Rakotch [14] and Browder [15]. The Boyd–Wong idea has been generalized by Matkowski [16], Samet et al. [17], Karapinar et al. [18], Pasicki [19], and Proinov [20], respectively. Recently, Nazam et al. [21] introduced several conditions on the newly introduced functions Ψ,ϕ:0,∞⟶ℝ to generalize and improve the results in [12, 16–20].The Banach contraction principle (BCP) and its generalization (GBCP) have been extensively applied to show the existence of solutions to various mathematical models. For instant, in [22–27], authors have applied GBCP to show the existence of solution to a matrix equation:(2)X=D+∑i=1mWi∗XWi+∑i=1mGi∗XGi,where D∈Pm (set of m×m positive definite matrices) and WiandGi are arbitrary m×m matrices for each i and are entries of block matrices given by(3)W=W1W2W3⋮Wm,G=G1G2G3⋮Gm.Consider the system of fractional differential equations:(4)DCβfν=K1ν,fν,DCβgν=K2ν,gν,under boundary conditions,(5)f0=0,If1=f′0,g0=0,Ig1=g′0,where DCβ denotes CFD of order β defined by(6)DCβfν=1Γn−β∫0νν−ηn−β−1fnηdη,DCβgν=1Γn−β∫0νν−ηn−β−1gnηdη.The existence of solutions of the above system has been shown in [21] by using GBCP. In [28], authors have employed the GBCP for the existence of solutions to a system of integral equations:(7)ut=ft+∫atKt,x,Sutdx,wt=ft+∫atJt,x,Twtdx,for all u,w∈Ca,b, x,t∈a,b, and a>0, where f:M⟶ℝ is a continuous function, K,J:a,b×a,b×M⟶ℝ are lower semicontinuous operators, and S,T:Ca,b⟶Ca,b.In this paper, motivated by the interpolation notion of contractions and the applications of GBCP, we investigate different conditions on the functionsΨ,Φ to show the existence of fixed points of generalized interpolative contractions (a new GBCP) of type I, II, III, and IV and hence, we apply GBCP of type I to resolve the Urysohn integral equation.


## 2. Preliminaries

Before stating our main results, we need to define some basic notions for better understanding of readers.Definition 1 (see [2]).
Let⊥ be a binary relation defined on a nonempty set A (i.e., ⊥⊂A×A) verifying the property (O). Then, A,⊥ is called an orthogonal set (in short, O-set):(8)O: there isa∈Asuch that eithera⊥JorJ⊥a,∀J∈A.Example 1.
LetA be the set of integers. Consider a⊥θ if and only if a≡1modθ. Then, A,⊥ is an O-set. Indeed, 1⊥θ for each θ.Definition 2 (see [2]).
A sequenceℏn:nis a positive integer is said to be an O-sequence if either ℏn⊥ℏn+1 or ℏn+1⊥ℏn, for all n.Definition 3 (see [2]).
The O-setA,⊥ endowed with a metric d is called an O-metric space (in short, OMS) denoted by A,⊥,d.Definition 4 (see [2]).
The O-sequenceℏn⊂A is said to be O-Cauchy if limn,m⟶∞dℏn,ℏm=0. If each O-Cauchy sequence converges in A, then A is called O-complete.Remark 1.
Each complete metric space is O-complete, but the converse is not true in general (see [2], for details).Lemma 1.
LetX,⊥,d be an OMS and ın⊂X be an O-sequence, verifying limn⟶∞dın,ın+1=0. If the sequence ın is not Cauchy, then there are ınk, ımk, and ξ>0 such that(9)limk⟶∞dlnk+1,lmk+1=ξ+,(10)limk⟶∞dlnk,lmk=dlnk+1,lmk=dlnk,lmk+1=ξ.The proof of this lemma has the same arguments that are given in [20]. We omit details.Definition 5.
LetT:A⟶A be a self-mapping. An element υ∈A is said to be a fixed point of T if υ=Tυ.Definition 6 (see [3]).
LetA,⊥,d be an OMS and ⊥⊂A×A be a binary relation. A is called ⊥-regular if, for each sequence ın⊂A so that ın⊥ın+1 for each n≥0 and ın⟶ı as n⟶∞, we have either ın⊥ı, or ı⊥ın, for all n≥0.Definition 7 (see [2]).
A mappingT:A⟶A is said to be asymptotically regular at a point υ of X if(11)limı⟶∞dTıυ,Tı+1υ=0.
IfT is asymptotically regular at each point in A, then it is named as an asymptotically regular mapping.


## 3.Ψ,Φ⊥-Interpolative Contractions and Related Fixed-Point Results

In this section, we initiate the notion ofΨ,Φ⊥-interpolative contractions. We consider various conditions on control functions Ψ,Φ to ensure the existence of fixed points of Ψ,Φ⊥-interpolative contractions. In the following, we develop the strategy towards main results.LetΛ=a,υ∈A×A:a⊥υ.Definition 8.
A mappingf:A×A⟶1,∞ is said to be strictly ⊥-admissible if fa,θ>1, for all a,θ∈A, with a⊥θ and fa,θ=1 otherwise.Example 2.
LetA=0,1, and we define the relation ⊥⊂A×A by(12)a⊥θifaθ∈a,θ⊂A.
Then,A is O-set. Define f:A×A⟶1,∞ by(13)fa,θ=a+21+θ,ifa⊥θ,1,otherwise.
Then,f is ⊥-admissible.Definition 9.
LetT:A⟶A and ⊥⊂A×A be a binary relation. Such T is called ⊥-preserving if, for each q∈A and p=Tq such that q⊥p or p⊥q, there is ω=Tp such that p⊥ω or ω⊥p.Example 3.
LetA=0,1, and we define the relation ⊥⊂A×A by(14)a⊥θifaθ∈a,θ⊂A.
Then,A is an O-set. We define S:A⟶A by(15)Sa=a+17,ifa∈ℚ∩A,0,ifa∈ℚc∩A.
Then,S is ⊥-preserving. Indeed, for a=0, there is θ=S0=1/7 such that either a⊥θ or θ⊥a, and then, there is ℓ=Sθ such that either ℓ⊥θ or θ⊥ℓ.
LetA,d be a metric space. For a mapping S:A⟶A and positive real numbers a,b,c, we define the mappings Fˇ1,Fˇ2,Fˇ3,Fˇ4:A×A⟶0,∞ by(16)Fˇ1ℓ,J=dℓ,Jdℓ,Sℓ1/a−ba−cdJ,SJ1/a−ba−cdℓ,Sℓ+dJ,SJ1/b−ab−cdℓ,SJ+dJ,Sℓ1/c−ac−b,Fˇ2ℓ,J=dℓ,Jdℓ,Sℓa/a−ba−cdJ,SJa/a−ba−cdℓ,Sℓ+dJ,SJb/b−ab−cdℓ,SJ+dJ,Sℓc/c−ac−b,Fˇ3ℓ,J=maxdℓ,J,dℓ,Sℓa2/a−ba−cdJ,SJa2/a−ba−cdℓ,Sℓ+dJ,SJb2/b−ab−cdℓ,SJ+dJ,Sℓc2/c−ac−b,Fˇ4ℓ,J=dℓ,Ja3/a−ba−cdJ,SJa3/a−ba−cdℓ,Sℓ+dJ,SJb3/b−ab−cdℓ,SJ+dJ,Sℓc3/c−ac−b.
It is important to note that, despitea,b,c>0, some exponents are negative; for example, if a>b, a>c, and b>c, then 1/b−ab−c<0. If any one of a,b,c goes to ∞, then Fˇ1ℓ,J=dℓ,J. Moreover, we have the following interesting facts about the exponents that can be proved by using basic algebraic tools:(17)1a−ba−c+1b−ab−c+1c−ac−b=0,aa−ba−c+bb−ab−c+cc−ac−b=0,a2a−ba−c+b2b−ab−c+c2c−ac−b=1,a3a−ba−c+b3b−ab−c+c3c−ac−b=a+b+c.
The following observations are essential for the proofs of main theorems.Observation 1.
The following inequality holds for alla,b≥2 and r≥1:(18)a+br≤abr.Proof.
We note that the equality holds fora=b=2. We can assume that a≥b; then, a=ηb, η≥1. Let b=t so that a=ηt, t≥2. Define the function f:2,∞⟶−∞,∞ by(19)ft=ηt2r−ηt+tr,∀t∈2,∞.
This implies that(20)f′t=ddtft=rtr−1η+1r2trηη+1r−1.
Since2trη/η+1r>1 (otherwise t<1), we have f′t>0. This implies that ft≥0; hence, ηt2r−ηt+tr≥0, that is, a+br≤abr.Observation 2.
LetK≥2. For any nonempty set A, we define the mapping d:A×A⟶0,∞ by(21)du,v=K,ifu≠v,0,ifu=v.
Then, the pairA,d is a metric space.Definition 10.
LetA,d be a metric space. A mapping T:A⟶A is said to have property P if, for any real number r, it satisfies the following inequality:(22)dℓ,Tℓ+dTℓ,T2ℓr≤dℓ,TℓdTℓ,T2ℓr,∀ℓ∈A.Example 4.
LetA=1,∞ and consider the metric d defined by du,v=u−v for all u,v∈A. The mapping T:A⟶A defined by Tℓ=Kℓ, for all ℓ∈A and K≥5/2, satisfies the property P. Indeed,(23)dℓ,Tℓ+dTℓ,T2ℓr=K−1ℓ+K−1Kℓr≤K−1K+1ℓr≤K−12Kℓ2r=dℓ,TℓdTℓ,T2ℓr.Example 5.
Every identity mapping satisfies the propertyP. The constant mapping does not satisfy the property P. The mapping T:A⟶A is defined by Tℓ=0 for all ℓ∈A which satisfies the property P only for ℓ=0.Example 6.
LetA=−∞,∞. The mapping T:A⟶A defined by Tℓ=2−3ℓ for all ℓ∈A satisfies the property P. In fact, the mapping T:A⟶A defined by Tℓ=a−bℓ, for all ℓ∈A, for b>a, satisfies the property P.Example 7.
LetA=2.5,∞. The mapping T:A⟶A defined by Tℓ=2−3ℓ for all ℓ∈A satisfies the property P.Example 8.
LetA=1,∞. The mapping T:A⟶A defined by Tℓ=1/ℓ2 for all ℓ∈A satisfies the property P.Remark 2.
The proof of Theorem2 depends largely on the use of either “Observations 1 and 2” or “Property P.”
We proceed with the propertyP.Definition 11.
LetA,⊥,d be an OMS. A mapping S:A⟶A is said to be a Ψ,Φ⊥-interpolative fractional contraction of types I, II, III, and IV, for i=1,2,3,4, respectively, if there exist a strictly ⊥-admissible mapping f and a,b,c∈0,∞, for i=1, and a,b,c∈0,∞, for i=2,3,4, such that(24)Ψfℓ,JdSℓ,SJ≤ΦFˇiℓ,J,for all ℓ,J∈Λ and dSℓ,SJ>0.
If eithera=∞ or b=∞ or c=∞ in Ψ,Φ⊥-interpolative fractional contraction of type I, we receive the recently announced ψ,ϕ-contraction by Proinov [20] which provided ℓ,J∉Λ.
We also note that, forΦℓ=Ψℓ−τ and Ψℓ=lnℓ, for all ℓ∈0,∞, τ>0, contraction (24) (i=1) can be written as follows:(25)τ+lnfℓ,JdSℓ,SJ≤lndℓ,J+1a−ba−clndℓ,Sℓ+1a−ba−clndJ,SJ+1a−ba−clndℓ,Sℓ+dJ,SJ+1a−ba−clndℓ,SJ+dJ,Sℓ,and then, we have(26)τ+Ψfℓ,JdSℓ,SJ≤Ψdℓ,J+1a−ba−cΨdℓ,Sℓ+1a−ba−cΨdJ,SJ+1a−ba−cΨdℓ,Sℓ+dJ,SJ+1a−ba−cΨdℓ,SJ+dJ,Sℓ.
This represents a general version of the contraction introduced by Wardowski [29], and if either a=∞ or b=∞ or c=∞ and ℓ,J∉Λ, then type I represents an F-contraction [29].Remark 3.
It is very important to note that the set of self-mappings satisfying propertyP and contraction (24) is not empty. For example, the mappings Sℓ=2−3ℓ, for all ℓ∈∞,∞, and Sℓ=2ℓ−1, for all ℓ∈2.5,∞, satisfy both the property P and contraction (24) with Φℓ=Ψℓ−τ and Ψℓ=lnℓ, for all ℓ∈0,∞, where τ>0.
In the next result, we give a set of conditions that guarantee the existence of a fixed point of a self-mappingS.Theorem 2.
LetA,⊥,d be an ⊥-regular O-complete metric space (in short, OCMS). Let S:A⟶A be an ⊥-preserving mapping verifying (24) for i=1 and property P. Suppose the relation ⊥ is transitive and the functions Ψ,Φ:0,∞⟶−∞,∞ are so that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ1⊥ℏ0 or ℏ0⊥ℏ1(ii)
Ψ,Φ are nondecreasing and ΦJ<ΨJ, for all J>0(iii)
limsupJ⟶δ+ΦJ<Ψδ+, for all δ>0(iv)
limsupa⟶0Φa≤liminfa⟶ξ+Ψa
Then,S admits a fixed point in A.Proof.
Step 1: simplification ofFˇ1ℏn−1,ℏn:(27)Fˇ1ℏn−1,ℏn=dℏn−1,ℏndℏn−1,Sℏn−11/a−ba−cdℏn,Sℏn1/a−ba−cdℏn−1,Sℏn−1+dℏn,Sℏn1/b−ab−cdℏn−1,Sℏn+dℏn,Sℏn−11/c−ac−b≤dℏn−1,ℏndℏn−1,ℏn1/a−ba−cdℏn,ℏn+11/a−ba−cdℏn−1,ℏn+dℏn,ℏn+11/b−ab−cdℏn−1,ℏn+1+dℏn,ℏn1/c−ac−b≤dℏn−1,ℏndℏn−1,ℏn1/a−ba−cdℏn,ℏn+11/a−ba−cdℏn−1,ℏn+dℏn,ℏn+11/b−ab−cdℏn−1,ℏn+dℏn,ℏn+11/c−ac−b=dℏn−1,ℏndℏn−1,ℏn1/a−ba−cdℏn,ℏn+11/a−ba−cdℏn−1,ℏn+dℏn,ℏn+11/b−ab−c+1/c−ac−b≤dℏn−1,ℏndℏn−1,ℏn1/a−ba−cdℏn,ℏn+11/a−ba−cdℏn−1,ℏndℏn,ℏn+11/b−ab−c+1/c−ac−bby propertyP=dℏn−1,ℏn1+1/a−ba−c+1/b−ab−c+1/c−ac−bdℏn,ℏn+11/a−ba−c+1/b−ab−c+1/c−ac−b=dℏn−1,ℏn,
Step 2: by (i), for an arbitraryℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0. It is assumed that S is an ⊥-preserving mapping, so there is ℏ2=Sℏ1 such that ℏ1⊥ℏ2 or ℏ2⊥ℏ1, and then, there is ℏ3=Sℏ2 such that ℏ2⊥ℏ3 or ℏ3⊥ℏ2. In general, there is ℏn+1=Sℏn such that ℏn⊥ℏn+1 or ℏn+1⊥ℏn for all. Hence, fℏn,ℏn+1>1, for all n≥0. Note that if ℏn=Sℏn, then ℏn is a fixed point of S, for all n≥0. We assume that ℏn≠Sℏn, for all n≥0. Thus, dSℏn−1,Sℏn>0, for each n≠1 (otherwise, ℏn=Sℏn, for some n). Let hn=dℏn,ℏn+1, for all n≥0. By the first part of (ii) and (24) ( i = 1), we have(28)Ψhn<Ψfℏn−1,ℏndSℏn−1,Sℏn≤ΦFˇ1ℏn−1,ℏn≤Φhn−1.
In view of second part of (ii), we write(29)Ψhn≤Φhn−1<Ψhn−1.
SinceΨ is nondecreasing, one gets hn<hn−1, for each n≥1. This shows that the sequence hn is decreasing, so there is L≥0 such that limn⟶∞hn=L+. If L>0, by (29), one obtains(30)ΨL+=limn⟶∞Ψhn≤limn⟶∞supΦhn−1≤lima⟶L+supΦa.
This contradicts (iii), soL=0, i.e., S is an asymptotically regular mapping.
Step 3: we claim thatℏn is a Cauchy sequence. If not, then, by Lemma 1, there are ℏnk and ℏmk of ℏn and ξ>0 such that (9) and (10) hold. By (9), we infer that dℏnk+1,ℏmk+1>ξ. Since ℏn⊥ℏn+1, for all n≥0, by transitivity of ⊥, we have ℏnk⊥ℏmk and hence, fℏnk,ℏmk>1 for all k≥1. Letting ℓ=ℏnk and J=ℏmk in (24)  (i=1), we have, for each k≥1,(31)Ψdℏnk+1,ℏmk+1≤Ψfℏnk,ℏmkdSℏnk,Sℏmk≤ΦFˇ1ℏnk,ℏmk.
We note that(32)Fˇ1ℏnk,ℏmk=dℏnk,ℏmkdℏnk,Sℏnk1/a−ba−cdℏmk,Sℏmk1/a−ba−cdℏnk,Sℏnk+dℏmk,Sℏmk1/b−ab−cdℏnk,Sℏmk+dℏmk,Sℏnk1/c−ac−b≤dℏnk,ℏmkdℏnk,ℏnk+11/a−ba−cdℏmk,ℏmk+11/a−ba−cdℏnk,ℏnk+1+dℏmk,ℏmk+11/b−ab−cdℏnk,ℏmk+1+dℏmk,ℏnk+11/c−ac−b=Bk.
Ifℏk=dℏnk+1,ℏmk+1, we have(33)Ψℏk≤ΦBk,for allk≥1.
By (9), we have limk⟶∞ℏk=ξ+, and (33) implies(34)liminfa⟶ξ+Ψa≤liminfk⟶∞Ψℏk≤limsupk⟶∞ΦBk≤limsupa⟶0Φa.
It is a contradiction to (iv), soℏn is a Cauchy sequence in the OCMS A,⊥,d; hence, there is a∗∈A so that ℏn⟶a∗ as n⟶∞, and the ⊥-regularity of A,⊥,d yields that ℏn⊥a∗ or a∗⊥ℏn. Thus, fℏn,a∗>1. We claim that da∗,Sa∗=0. Assume that dℏn+1,Sa∗>0 for infinitely many values of n. By (24)  (i=1),(35)Ψdℏn+1,Sa∗≤Ψfℏn,a∗dSℏn,Sa∗≤ΦFˇ1ℏn,a∗.
By the first part of (ii), we getdℏn+1,Sa∗<Fˇ1ℏn,a∗. Applying limit n⟶∞, we obtain da∗,Sa∗≤0. This implies that da∗,Sa∗=0; hence, a∗=Sa∗.Next result gives an idea on conditions ensuring the existence of fixed points ofS verifying (24) (i=1).Theorem 3.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping verifying (24) (i=1) and property P. Assume the relation ⊥ is transitive and the functions Ψ,Φ:0,∞⟶−∞,∞ are such that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0(ii)
ΦJ<ΨJ, for all J>0(iii)
infa>ξ>0Ψa>−∞(iv)
IfΨℏn and Φℏn are converging to the same limit and Ψℏn is strictly decreasing, then limn⟶∞ℏn=0(v)
limsupa⟶0Φa<liminfa⟶ξ+Ψa, for all ξ>0(vi)
limsupa⟶0Φa<liminfa⟶ξΨa, for all ξ>0
Then,S possesses a fixed point in A.Proof.
Note that we need (i)-(iv) to show thatS is an asymptotically regular. Condition (v) is needed to establish that ℏn is Cauchy and (vi) is useful to ensure that the mapping S has a fixed point.
By (i), for an arbitraryℏ0∈A, there is ℏ1=Sℏ0 so that ℏ0⊥ℏ1 or ℏ1⊥ℏ0. Since S is ⊥-preserving, there is ℏ2=Sℏ1 so that ℏ1⊥ℏ2 or ℏ2⊥ℏ1, and then, ℏ3=Sℏ2 so that ℏ2⊥ℏ3 or ℏ3⊥ℏ2. In general, there is ℏn+1=Sℏn in order that ℏn⊥ℏn+1 or ℏn+1⊥ℏn, for all n≥0. Hence, fℏn,ℏn+1>1. Note that if ℏn=Sℏn, then ℏn is a fixed point of S. Suppose that ℏn≠Sℏn, for all n≥0. Thus, dSℏn−1,Sℏn>0 (otherwise ℏn=Sℏn). Since fℏn,ℏn+1>1, by (ii) and (24) (i=1), we write(36)Ψdℏn,ℏn+1≤Ψfℏn−1,ℏndSℏn−1,Sℏn≤ΦFˇ1ℏn−1,ℏn≤Ψdℏn−1,ℏn.
Inequality (36) shows that Ψdℏn−1,ℏn is strictly decreasing. If it is not bounded below, in view of (iii), we get infdℏn−1,ℏn>ξΨdℏn−1,ℏn>−∞. This implies that(37)liminfdℏn−1,ℏn⟶ξ+Ψdℏn−1,ℏn>−∞.
Thus,limn⟶∞dℏn−1,ℏn=0; otherwise, we have(38)liminfdℏn−1,ℏn⟶ξ+Ψdℏn−1,ℏn=−∞,(i.e., a contradiction to (iii)). If it is bounded below, then Ψdℏn−1,ℏn is a convergent sequence, and by (36), Φdℏn−1,ℏn also converges and both have the same limit. Thus, by (iv), one gets limn⟶∞dℏn−1,ℏn=0. Hence, S is asymptotically regular.
Now, we claim thatℏn is a Cauchy sequence. If ℏn is not a Cauchy sequence, so, by Lemma 1, there exist ℏnk and ℏmk and ξ>0 such that (9) and (10) hold. By (9), we infer that dℏnk+1,ℏmk+1>ξ. Since ℏn⊥ℏn+1, for all n≥0, so, by transitivity of ⊥, we have ℏnk⊥ℏmk, and hence, fℏnk,ℏmk>1 for all k≥1. Letting x=ℏnk and y=ℏmk in (24), one writes, for all k≥1,(39)Ψdℏnk+1,ℏmk+1≤Ψfℏnk,ℏmkdSℏnk,Sℏmk≤ΦFˇ1ℏnk,ℏmk.
We note that(40)Fˇ1ℏnk,ℏmk=dℏnk,ℏmkdℏnk,Sℏnk1/a−ba−cdℏmk,Sℏmk1/a−ba−cdℏnk,Sℏnk+dℏmk,Sℏmk1/b−ab−cdℏnk,Sℏmk+dℏmk,Sℏnk1/c−ac−b≤dℏnk,ℏmkdℏnk,ℏnk+11/a−ba−cdℏmk,ℏmk+11/a−ba−cdℏnk,ℏnk+1+dℏmk,ℏmk+11/b−ab−cdℏnk,ℏmk+1+dℏmk,ℏnk+11/c−ac−b=Bk.
Ifℏk=dℏnk+1,ℏmk+1, we have(41)Ψℏk≤ΦBk,for allk≥1.
By (9), we have limk⟶∞ℏk=ξ+ and (41) implies(42)lim infa⟶ξ+Ψa≤liminfk⟶∞Ψℏk≤limsupk⟶∞ΦBk≤limsupa⟶0Φa.
It contradicts (v), soℏn is a Cauchy sequence in the OCMS A. Hence, there is a∗∈A in order that ℏn⟶a∗ as n⟶∞.
To show thatSa∗=a∗, we have two cases:Case 1: ifdℏn+1,Sa∗=0, for some n≥0, then, since(43)da∗,Sa∗≤da∗,ℏn+1+dℏn+1,Sa∗=da∗,ℏn+1,taking limitn⟶∞ on both sides, we have da∗,Sa∗≤0. This implies da∗,Sa∗=0; thus, a∗=Sa∗.Case 2: if, for alln≥0, dℏn+1,Sa∗>0, then by ⊥-regularity of A, we find ℏn⊥a∗ or a∗⊥ℏn, so fℏn,a∗>1. By (24) (i=1), one writes(44)Ψdℏn+1,Sa∗≤Ψfℏn,a∗dSℏn,Sa∗≤ΦFˇ1ℏn,a∗for alln≥0.
By takingHn=dℏn+1,Sa∗ and bn=Fˇ1ℏn,a∗, one writes(45)ΨHn≤Φbnfor alln≥0.
Takeξ=da∗,Sa∗. Note that Hn⟶ξ and bn⟶0 as n⟶∞. Applying limits on (45), we have(46)liminfa⟶ξΨa≤liminfn⟶∞ΨHn≤limsupn⟶∞Φbn≤liminfa⟶0Φa.
This contradicts (vi) ifξ>0. Thus, we have da∗,Sa∗=0, i.e., a∗=Sa∗, that is, a∗ is a fixed point of S.Remark 4.
Observe that(47)Fˇ2ℏn−1,ℏn=dℏn−1,ℏndℏn−1,Sℏn−1a/a−ba−cdℏn,Sℏna/a−ba−cdℏn−1,Sℏn−1+dℏn,Sℏnb/b−ab−cdℏn−1,Sℏn+dℏn,Sℏn−1c/c−ac−b≤dℏn−1,ℏndℏn−1,ℏna/a−ba−cdℏn,ℏn+1a/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b/b−ab−cdℏn−1,ℏn+1+dℏn,ℏnc/c−ac−b≤dℏn−1,ℏndℏn−1,ℏna/a−ba−cdℏn,ℏn+1a/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b/b−ab−cdℏn−1,ℏn+dℏn,ℏn+1c/c−ac−b=dℏn−1,ℏndℏn−1,ℏna/a−ba−cdℏn,ℏn+1a/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b/b−ab−c+c/c−ac−b≤dℏn−1,ℏndℏn−1,ℏna/a−ba−cdℏn,ℏn+1a/a−ba−cdℏn−1,ℏndℏn,ℏn+1b/b−ab−c+c/c−ac−bby propertyP=dℏn−1,ℏn1+a/a−ba−c+b/b−ab−c+c/c−ac−bdℏn,ℏn+1a/a−ba−c+b/b−ab−c+c/c−ac−b=dℏn−1,ℏn,Fˇ3ℏn−1,ℏn=maxdℏn−1,ℏn,dℏn−1,Sℏn−1a2/a−ba−cdℏn,Sℏna2/a−ba−cdℏn−1,Sℏn−1+dℏn,Sℏnb2/b−ab−cdℏn−1,Sℏn+dℏn,Sℏn−1c2/c−ac−b=maxdℏn−1,ℏn,dℏn−1,ℏna2/a−ba−cdℏn,ℏn+1a2/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b2/b−ab−cdℏn−1,ℏn+1+dℏn,ℏnc2/c−ac−b≤maxdℏn−1,ℏn,dℏn−1,ℏna2/a−ba−cdℏn,ℏn+1a2/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b2/b−ab−cdℏn−1,ℏn+dℏn,ℏn+1c2/c−ac−b≤maxdℏn−1,ℏn,dℏn−1,ℏna2/a−ba−cdℏn,ℏn+1a2/a−ba−cdℏn−1,ℏndℏn,ℏn+1b2/b−ab−cdℏn−1,ℏndℏn,ℏn+1c2/c−ac−bby propertyP=maxdℏn−1,ℏn,dℏn−1,ℏna2/a−ba−c+b2/b−ab−c+c2/c−ac−bdℏn,ℏn+1a2/a−ba−c+b2/b−ab−c+c2/c−ac−b=maxdℏn−1,ℏn,dℏn,ℏn+1,Fˇ4ℏn−1,ℏn=dℏn−1,ℏna3/a−ba−cdℏn,Sℏna3/a−ba−cdℏn−1,Sℏn−1+dℏn,Sℏnb3/b−ab−cdℏn−1,Sℏn+dℏn,Sℏn−1c3/c−ac−b=dℏn−1,ℏna3/a−ba−cdℏn,ℏn+1a3/a−ba−cdℏn−1,ℏn+dℏn,ℏn+1b3/b−ab−cdℏn−1,ℏn+1+dℏn,ℏnc3/c−ac−b≤dℏn−1,ℏna3/a−ba−cdℏn,ℏn+1a3/a−ba−cdℏn−1,ℏndℏn,ℏn+1b3/b−ab−cdℏn−1,ℏndℏn,ℏn+1c3/c−ac−b=dℏn−1,ℏndℏn,ℏn+1a3/a−ba−c+b3/b−ab−c+c3/c−ac−b=dℏn−1,ℏndℏn,ℏn+1a+b+c≤maxdℏn−1,ℏn,dℏn,ℏn+1.
The next two results address theΨ,Φ⊥-interpolative fractional contractions of types II and III.Theorem 4.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping verifying (24) for i=2,3 and property P. Suppose the relation ⊥ is transitive, and the functions Ψ,Φ:0,∞⟶−∞,∞ are so that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ1⊥ℏ0 or ℏ0⊥ℏ1(ii)
Ψ,Φ are nondecreasing and ΦJ<ΨJ, for all J>0(iii)
limsupJ⟶δ+ΦJ<Ψδ+, for all δ>0(iv)
limsupa⟶0Φa≤liminfa⟶ξ+Ψa
Then,S has a fixed point in A.Proof.
Keeping in view the simplifications forFˇ2ℏn−1,ℏn and Fˇ3ℏn−1,ℏn given in Remark 4 with the fact that dℏn−1,ℏn>dℏn,ℏn+1 and following the proof of Theorem 2, we assert that S admits a fixed point in A. If dℏn−1,ℏn<dℏn,ℏn+1, then we have a contradiction to the definition of function Ψ.Theorem 5.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping verifying (24) (i=2,3) and property P. Assume the relation ⊥ is transitive, and the functions Ψ,Φ:0,∞⟶−∞,∞ are so that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0(ii)
ΦJ<ΨJ, for all J>0(iii)
infa>ξ>0Ψa>−∞(iv)
IfΨℏn and Φℏn are converging to the same limit and Ψℏn is strictly decreasing, then limn⟶∞ℏn=0(v)
limsupa⟶0Φa<liminfa⟶ξ+Ψa, for all ξ>0(vi)
limsupa⟶0Φa<liminfa⟶ξΨa, for all ξ>0
Then,S possesses a fixed point in A.Proof.
Keeping in view the simplifications forFˇ2ℏn−1,ℏn and Fˇ3ℏn−1,ℏn given in Remark 4 with the fact that dℏn−1,ℏn>dℏn,ℏn+1 and following the proof of Theorem 2, we assert that S admits a fixed point in A. If dℏn−1,ℏn<dℏn,ℏn+1, then we have a contradiction to the definition of function Ψ.The next two results address theΨ,Φ⊥-interpolative fractional contraction of type IV.Theorem 6.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping verifying (24) for i=4 with a+b+c<0.5 and property P. Suppose the relation ⊥ is transitive and the functions Ψ,Φ:0,∞⟶−∞,∞ are so that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ1⊥ℏ0 or ℏ0⊥ℏ1(ii)
Ψ,Φ are nondecreasing and ΦJ<ΨJ, for all J>0(iii)
limsupJ⟶δ+ΦJ<Ψδ+, for all δ>0(iv)
limsupa⟶0Φa≤liminfa⟶ξ+Ψa
Then,S has a fixed point in A.Proof.
Keeping in view the simplifications forFˇ4ℏn−1,ℏn given in Remark 4 and following the proof of Theorem 4, we assert that S admits a fixed point in A.Theorem 7.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping verifying (24) (i=4) with a+b+c<0.5 and property P. Assume the relation ⊥ is transitive and the functions Ψ,Φ:0,∞⟶−∞,∞ are so that(i)
For eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0(ii)
ΦJ<ΨJ, for all J>0(iii)
infa>ξ>0Ψa>−∞(iv)
IfΨℏn and Φℏn are converging to the same limit and Ψℏn is strictly decreasing, then limn⟶∞ℏn=0(v)
limsupa⟶0Φa<liminfa⟶ξ+Ψa, for all ξ>0(vi)
limsupa⟶0Φa<liminfa⟶ξΨa, for all ξ>0
Then,S possesses a fixed point in A.Proof.
Keeping in view the simplifications forFˇ4ℏn−1,ℏn given in Remark 4 and following the proof of Theorem 5, we assert that S admits a fixed point in A.


## 4. The Generality of the Main Results

Let us defineΨJ=J, for all J>0, in any one of Theorems 2 and 3, we receive a general version of the interpolative Boyd–Wong fixed-point theorem proved in [9], and defining ΦJ=ΨJ−τ in Theorem 2, we receive the following result (interpolative fractional version of Wardowski fixed-point theorem with only monotonicity condition on Ψ).Corollary 1.
LetA,d be a complete metric space. Let S:A⟶A be a mapping so that(48)ΨdSℓ,SJ≤ΨFˇiℓ,J−τ∀ℓ,J∈A,i=1,2,3,4provideddSℓ,SJ>0,where Ψ:0,∞⟶ℝ is nondecreasing and τ>0. Then, there is a fixed point of S in A.If we defineΦJ=ΨJ−τJ in Theorem 2, we get an interpolative fractional version of fixed-point theorem presented in [4].Corollary 2.
LetA,d be a complete metric space. Let S:A⟶A be a mapping so that(49)τdℓ,J+ΨdSℓ,SJ≤ΨFˇiℓ,J∀ℓ,J∈A,i=1,2,3,4provideddSℓ,SJ>0,where Ψ:0,∞⟶ℝ is nondecreasing and liminfa⟶t+τa>0,∀t≥0. Then, S has a fixed point in A.We receive the following interpolative fractional version of Moradi theorem [30] if we take ΦJ=hΨJ in Theorem 2.Corollary 3.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping so that(50)Ψfℓ,JdSℓ,SJ≤hΨFˇiℓ,J∀ℓ,J∈Λ,i=1,2,3,4provideddSℓ,SJ>0,where(i)
h:I⟶0,∞ is an upper semicontinuous function with hJ<J, for all J∈I⊂ℝ(ii)
Ψ:0,∞⟶I is nondecreasingAssume that, for eachℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0. Then, S has a unique fixed point in A.DefininghJ=Jδ and δ∈0,1 in Corollary 3, we have the next result.Corollary 4.
LetA,⊥,d be an ⊥-regular and OCMS. Let S:A⟶A be an ⊥-preserving mapping so that(51)Ψfℓ,JdSℓ,SJ≤ΨFˇiℓ,Jr∀ℓ,J∈Λ,i=1,2,3,4provideddSℓ,SJ>0,where Ψ:0,∞⟶0,1 is nondecreasing. Assume that, for each ℏ0∈A, there is ℏ1=Sℏ0 such that ℏ0⊥ℏ1 or ℏ1⊥ℏ0. Then, S has a fixed point in A.Observe that Corollary4 is an improvement of Jleli–Samet fixed-point theorem [31] and the results of Li and Jiang [32] and Ahmad et al. [33].An improvement of Skof fixed-point theorem [34] may be stated by putting ΦJ=λΨJ in Theorem 2, for i=1, with either a=∞ or b=∞ or c=∞.Corollary 5.
LetA,⊥,d be an ⊥-regular OCMS. Let S:A⟶A be an ⊥-preserving mapping so that(52)Ψfℓ,JdSℓ,SJ≤λΨFˇ1ℓ,J∀ℓ,J∈Λ,provideddSℓ,SJ>0,where Ψ:0,∞⟶0,∞ is nondecreasing and λ∈0,1. Assume that, for each ℏ0∈A, there is ℏ1=Sℏ0 so that ℏ0⊥ℏ1 or ℏ1⊥ℏ0. Then, S has a unique fixed point in A.


## 5. The Existence of the Solution to Urysohn Integral Equation (UIE)

In this section, we will apply Theorem2 for the existence of the unique solution to UIE:(53)ℓℏ=fℏ+∫IRK1ℏ,s,ℓsds.This integral equation encapsulates both Volterra integral equation (VIE) and Fredholm integral equation (FIE), depending on the region of integration (IR). If IR=a,x, where a is fixed, then UIE is VIE, and for IR =a,b, where a,b are fixed, UIE is FIE. In the literature, one can find many approaches to find a unique solution to UIE (see [35–39] and references therein). We are interested to use a fixed-point technique for this purpose. The fixed-point technique is simple and elegant to show the existence of a unique solution to further mathematical models.Let IR be a set of finite measure andℒIR2=ℓ|∫IRℓs2ds<∞.Define the norm.:ℒIR2⟶0,∞ by(54)ℓ2=∫IRℓs2ds,for allℓ,J∈ℒIR2.An equivalent norm can be defined as follows:(55)ℓ2,ν=supe−ν∫IRαsds∫IRℓs2ds,for allℓ∈ℒIR2,ν>1.Then,ℒIR2,.2,ν is a Banach space. Let A=ℓ∈ℒIR2:ℓs>0for almost everys. The metric dν associated to norm .2,ν is given by dνℓ,J=ℓ−J2,ν, for all ℓ,J∈A. Define an orthogonal relation ⊥ on A by(56)a⊥υif and only ifasυs≥υs,for alla,υ∈A.Then,A,⊥,d is an OCMS (see Theorem 4.1 in [3]). Let L:A×A⟶1,∞ be defined by(57)Lδ,t=eδ+tℒ2for allδ,t∈Awithδ⊥t.Then,L is a strictly ⊥-admissible mapping. Put M=infLδ,t,∀δ,t∈Awithδ⊥t. Let(A1) The kernelK1:IR×IR×ℝ⟶ℝ satisfies Carathéodory conditions and(58)K1ℏ,s,ℓs≤wℏ,s+eℏ,sℓs;w,e∈ℒ2IR×IR,eℏ,s>0.(A2) The functionf:IR⟶1,∞ is continuous and bounded on IR.(A3) There exists a positive constantC such that(59)supℏ∈IR∫IRK1ℏ,sds≤C.(A4) For anyℓ0∈ℒIR2, there is ℓ1=Rℓ0 such that ℓ1⊥ℓ0 or ℓ0⊥ℓ1.(A5) There exists a nonnegative and measurable functionq:IR×IR⟶ℝ such that(60)αℏ≔∫IRq2ℏ,sds≤1νM2and integrable over IR with(61)K1ℏ,s,ℓs−K1ℏ,s,Js≤qℏ,sℓs−Js,for allℏ,s∈IR and ℓ,J∈A with ℓ⊥J.Theorem 8.
Suppose that the mappingsf and K1 mentioned above satisfy conditions (A1)–(A5); then, the UIE (53) has a unique solution.Proof.
Define the mappingR:A⟶A, in accordance with the abovementioned notations, by(62)Rℓℏ=fℏ+∫IRK1ℏ,s,ℓsds.
The operatorR is ⊥-preserving: let ℓ,J∈A with ℓ⊥J; then, ℓsJs≥Js. Since, for almost every ℏ∈ IR,(63)Rℓℏ=fℏ+∫IRK1ℏ,s,ℓsds≥1,this implies that RℓℏRJℏ≥RJℏ. Thus, Rℓ⊥RJ.
Self-operator: conditions (A1) and (A3) imply thatR is continuous and compact mapping from A to A (see Lemma 3 in [35]).
By (A4), for anyℓ0∈A, there is ℓ1=Rℓ0 such that ℓ1⊥ℓ0 or ℓ0⊥ℓ1, and using the fact that R is ⊥-preserving, we have ℓn=Rnℓ0 with ℓn⊥ℓn+1 or ℓn+1⊥ℓn, for all n≥0. We will check the contractive condition (24) of Theorem 2 in the next lines. By (A5) and Holder inequality, we have(64)Rℓℏ−RJℏ2=∫IRK1ℏ,s,ℓsds−∫IRK1ℏ,s,Jsds2≤∫IRK1ℏ,s,ℓs−K1ℏ,s,Jsds2≤∫IRqℏ,sℓs−Jsds2≤∫IRq2ℏ,sds⋅∫IRℓs−Js2ds=αℏ∫IRℓs−Js2ds.
This implies, by integrating with respect toℏ,(65)∫IRRℓℏ−RJℏ2dℏ≤∫IRαℏ∫IRℓs−Js2dsdℏ=∫IRαℏeν∫IRαsds⋅e−ν∫IRαsds∫IRℓs−Js2dsdℏ≤ℓ−J2,ν2∫IRαℏeν∫IRαsdsdℏ≤1νM2ℓ−J2,ν2eν∫IRαsds.
Thus, we have(66)M2e−ν∫IRαsds∫IRRℓℏ−RJℏ2dℏ≤1νℓ−J2,ν2.
This implies that(67)M2Rℓ−RJ2,ν2≤1νℓ−J2,ν2.
That is,(68)Lℓ,JdνRℓ,RJ≤1νdνℓ,J.
Takingln on both sides and defining Ψt=lnt with Φt=Ψt−τ, τ>0, we have(69)ΨLℓ,JdνRℓ,RJ≤ΦFˇ1ℓ,J,τ=−ln1ν,a=∞.
The definedΨ and Φ satisfy remaining conditions of Theorem 2. Hence, by Theorem 2, the operator R has a unique fixed point. This means that the UIE (53) has a unique solution.


## 6. Conclusion

TheΨ,Φ⊥ interpolative contractions are broad enough to include well-known contractions. The presented theorems provide a general criterion for the existence of a unique fixed point of Ψ,Φ⊥ interpolative contraction mappings. Fixed-point methodology is used to investigate the presence of a solution to a UIE.



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