# Existence Theorems on Advanced Contractions with Applications

Authors: Sang Og Kim; Muhammad Nazam
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this research article, by introducing a mappingφ defined on 0,∞4, with some axioms, we define two generalized contractions called FH+φ-contractions and φHP+-contractions. We investigate their mutual relation and establish an existence theorem addressing FH+φ-contractions with some applications.


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## Body

## 1. Introduction

Frechet gave an abstraction to the notion of distance in Euclidean spaces by introducing metric spaces. Partial metrics (denoted byP) were introduced in [1] as a generalization of the notion of metric to allow nonzero self-distance for the purpose of modeling partial objects in reasoning about data flow networks. The self-distance Pα,α is to be understood as a quantification of the extent to which α is unknown. Matthews [1] proved an analogue of Banach’s fixed point theorem in partial metric spaces. This remarkable fixed point theorem led many researchers to investigate fixed points of self-mappings in partial metric spaces (see [2–7]).The investigation of fixed points of multivalued or set-valued mappings was started by Nadler [8]. For this purpose, Nadler introduced a metric function H to measure distance between two nonempty closed and bounded sets. This metric function is also known as the Hausdorff metric in literature. Aydi et al. [5] generalized the Hausdorff metric to the partial Hausdorff metric and hence generalized the Nadler fixed point theorem. Nazam et al. [7] established various fixed point results using the partial Hausdorff metric. Recently, Pathak et al. [9] introduced another metric function H+ to measure the distance between two nonempty closed and bounded sets and hence proved some fixed point results. Nashine et al. [10] also proved some fixed points theorems on H+-multivalued contractions and their application to homotopy theory. For recent research in this direction, see [11–13].In 1922, Banach introduced the Banach Contraction Principle in his PhD thesis. Since then, there has been a trend to generalize and apply it to show the existence of the solutions to various mathematical models (both linear and nonlinear). A large number of research articles contain many useful generalizations of Banach Contraction Principle. In one such attempt, Wardowski [14] introduced F-contractions, where F represents the class of nonlinear real-valued functions satisfying three axioms (F1,F2,F3). The concept of F-contractions proved to be a useful addition in fixed point theory (see for instant [15–18] and references therein). The advancement in the study of F-contraction is in progress, and in this direction recently, Abbas et al. [19] introduced the Presic-type F-contraction and established a fixed point theorem for such kind of mappings. Tomar et al. [20] provided an existence theorem for six self-mappings under the notion of F-contraction. Durmaz et al. [17] studied F-contraction under the effect of a partial order. Sgroi et al. [21] extended the notion of F-contraction to multivalued F-contraction by combining the ideas of Wardowski and Nadler. Durmaz et al. [22] generalized the results given in [16, 17, 21] by introducing α,F-contraction. Similarly, Piri et al. [23] proved some theorems on the F-Suzuki type inequalities under some weaker conditions, and Shukla et al. [24] established a common fixed point theorem for weak F-contraction under 0-complete partial metric spaces. Recently, Karapinar et al. [25] presented a survey paper which encompasses almost all the results addressing F-contractions.The motivation to write this article is the contents of the article [26]. In [26], authors introduced a function (called auxiliary function) defined on 0,∞4 satisfying some axioms and used it to establish a fixed point theorem. It was then shown that the Banach fixed point theorem, Kannan fixed point theorem, Chatterjea fixed point theorem, Reich fixed point theorem, Hardy and Rogers fixed point theorem, and Círíc type fixed point theorems are particular cases of this fixed point theorem. Since all the above mentioned fixed point theorems have been generalized using the notion of F-contraction both in metric spaces and partial metric spaces (see [25]), we develop a general fixed point theorem, representing them all, in the Hausdorff partial metric spaces.This article is organized as follows. In Section2, some basic notions are given. In Section 3, we give highlights of Hausdorff p-ms. In Section 4, we introduce the FH+φ-contractions and φHP+-contractions and investigate the relations between them. We also study the existence theorem and its consequences. And in Section 5, we derive two results regarding applications by applying the existence theorem given in Section 4. The presented existence theorem generalizes, improves, and extends the results established by Pathak et al. [9].


## 2. Basic Notions

Let partial metric spaces be denoted by p-m-s.Matthews [1], while working on networking topologies, noticed the nonzero self-distance (loop is the best example to understand his point). The self-distance played a key role in introduction of p-m-s. Matthews [1] defined the p-m-s as follows: let I be a nonempty set, and the function P:I×I→0,∞ is said to be a partial metric (p-m) on I if for all α,β,γ∈I, the axioms (p1)-(p4) are satisfied.(p1) α=β⇔Pα,α=Pα,β=Pβ,β(p2) Pα,α≤Pα,β(p3) Pα,β=Pβ,α(p4) Pα,γ≤Pα,β+Pβ,γ−Pβ,β.Some examples ofI,P are as follows. The function P:I2→0,∞ defined by
(1)
Pα,β=∣α−β∣+C;C≥0 for all α,β∈I is a I,P(2)
Pα,β=maxα,β, is a I,P(3)
Pα,β=e∣α−β∣+maxα,β, is a I,P.It is noted thatPα,β=0 implies α=β. The p-m function P is continuous. If P is a p-m then the function dP:I×I→0,∞ defined by
(1)dPα,β=2Pα,β−Pα,α+Pβ,βforallα,β∈Idefines a metric on I. A T0 topology can be defined on I,P with P-open balls being its elements. The P-open ball centered at σ0 having radius ε is defined by OPσ0,ε=σ∈I:Pσ0,σ<Pσ0,σ0+ε. A set G is said to be bounded in I,P if there exist σ0∈I and Δ≥0 such that Pσ0,η<Pη,η+Δ for all η∈G. Also it is easy to write η∈G¯ (closure of G) ⇔Pη,G=Pη,η and G is closed in I,P if and only if G=G¯. If Pσ,σ=limn→∞Pσ,σn; then we say that σn converges to σ and conversely. If limn,m→∞Pσn,σm is finite, then the sequence σn is said to be Cauchy, and in particular, if this Cauchy sequence converges in I,P, then we say that the p-m-s I,P is complete. Lemma 1 provides fundamental rules to work in the p-m-s.Lemma 1 [1].
(1)
If the sequenceσn is Cauchy sequence in I,P, then it is Cauchy sequence in the metric space I,dP and conversely(2)
The completeness ofI,P implies the completeness of I,dP and conversely(3)
limn→∞dPσ,σn=0⇔limn→∞Pσ,σn=Pσ,σ=limn,m→∞Pσn,σm, provided I,P is complete.Remark 1.
There are sequences which converge in p-m-s but not in metric spaces. Indeed, for the sequence1/n:n∈ℕ in I=0,1 and p-m P defined by Pρ,ς=∣ρ−ς∣+CC≥0∀ρ,ς∈I, it is easy to check that the sequence 1/n converges to 0 with respect to P but does not converge to 0 with respect to metric d defined by dρ,ς=Pρ,ς if ρ≠ς and 0 otherwise.


## 3. Hausdorff Partial Metric

Let the set of nonempty closed and bounded subsets ofI,P be denoted by CBPI. Let Pσ,A=infPσ,a:a∈A, A∈CBPI. Let ΔP:CBPI×CBPI→0,∞ be defined by ΔPX,Y=supPa,Y:a∈X. Let HP:CBPI×CBPI→0,∞ be defined by
(2)HPX,Y=maxΔPX,Y,ΔPY,X.LetHP+:CBPI×CBPI→0,∞ be defined by
(3)HP+X,Y=12ΔPX,Y+ΔPY,X.Sincemaxσ,ς≥1/2σ+ς, HPX,Y≥HP+X,Y for all X,Y∈CBPI. A comprehensive study of the distance H+X,Y with reference to metric d was presented by Pathak et al. in [9]. We claim that
(a)
HP+X,Y and HPX,Y are topological equivalent(b)
the mappingHP+:CBPI×CBPI→0,∞ defines a p-m on CBPI(c)
if the p-m-sI,P is complete then CBPI,HP+ is also complete and vice versa(d)
the mappingHP+:CBPI×CBPI→0,∞ is continuous.Proposition 1 [7].
LetI,P be p-m-s. For any J,K,L∈CBPI, we have the following:
(1)
ΔPJ,J=supPu,v:u,v∈J(2)
ΔPJ,K=ΔPK,J(3)
ΔPJ,K=0⇒J⊆K(4)
ΔPJ,L≤ΔPJ,K+ΔPK,L−infk∈KPk,k.Proposition 2.
LetI,P be p-m-s. For any J,K,L∈CBPI, we have the following:
(1)
HP+J,K=0 implies J=K(2)
HP+J,J≤HP+J,K(3)
HP+J,K=HP+K,J(4)
HP+J,L≤HP+J,K+HP+K,L−infk∈KPk,k.Proof.
Following the arguments given in ([5], Proposition 2.2 and Proposition 2.3), we get the result. We omit its details.
HP-contraction: Let I,P be a p-m-s, the mapping T:I→CBPI is called an HP-contraction, if there exists k<1 such that HPTσ\σ,Tς\ς≤kPσ,ς for all σ,ς∈I (see [7]).
HP+-contraction: Let I,P be a p-m-s; the mapping T:I→CBPI is called an HP+-contraction, if (1) there exists k<1 such that HP+Tσ\σ,Tς\ς≤kPσ,ς for all σ,ς∈I; (2) for all σ∈I, ς∈Tσ, ε>0 there exists ξ∈Tς such that Pς,ξ≤HP+Tσ,Tς+ε (see [27]).
SinceHPX,Y≥HP+X,Y for all X,Y∈CBPI, HP-contraction implies HP+-contraction but not conversely (see Example 1).Example 1.
LetI=0,1/7,1. Define the functionP:I×I→0,∞by(4)Pσ,ς=maxσ,ςforallσ,ς∈I.ThenI,P is a p-m-s. Let T:I→CBPI be defined by
(5)Tσ=0ifσ=00,17ifσ=170,1ifσ=1.We have three cases (Case1: σ=0,ς=1/7, Case 2: σ=0,ς=1, and Case 3: σ=1/7,ς=1).Case 1.
Ifσ=0,ς=1/7, then Pσ,ς=1/7, HPT0,R1/7=1/7, and HP+T0,R1/7=1/14. This clearly shows that
(6)HP+T0,R17≤LP0,17holdsforallL≥12,whereas
(7)HPT0,R17>LP0,17foranyL<1.Case 2.
Ifσ=0,ς=1, then Pσ,ς=1, HPT0,R1=1, and HP+T0,R1=1/2. This clearly shows that
(8)HP+T0,R1≤LP0,1holdsforallL≥12,whereas
(9)HPT0,R1>LP0,1foranyL<1.Case 3.
Ifσ=1/7,ς=1, then Pσ,ς=1, HPR1/7,R1=1, and HP+R1/7,R1=4/7. This clearly shows that
(10)HP+R17,R1≤LP17,1holdsforallL≥12,whereas
(11)HPR17,R1>LP17,1foranyL<1.Note: the inequalityPς,ξ≤HP+Tσ,Tς+ε also holds for each case, and for all σ∈I, ς∈Tσ, ξ∈Tς.


## 4. Fixed Points ofFH+φ-Contraction

LetT:I→I be a self-mapping defined on nonempty set I. The problem “to find σ∗∈I such that σ∗=Tσ∗” is called fixed point problem. If T:I→CBI, then the fixed point problem turns into the form “to find σ∗∈I such that σ∗∈Tσ∗.” For the solution of fixed point problem, generally, a Picard iterative sequence σnsuchthatσn+1=Tσn is proved to be a Cauchy sequence subject to contractive condition and completeness of the underlying abstract metric space leads to such σ∗. In this section, at first, we introduce and compare FH+φ-contraction and φHP+-contraction, and secondly, we obtain a theorem assuring unique fixed point of FH+φ-contraction. We proceed with definitions of functions F and φ associated with some axioms.Wardowski [14] considered a nonlinear function F:0,∞→ℝ with the following axioms: (F1): F is strictly increasing. (F2): For each sequence σn of positive numbers, limn→∞σn=0 if and only if limn→∞Fσn=−∞. (F3): For each sequence σn of positive numbers limn→∞σn=0, there exists θ∈0,1 such that limσn→0+σnθFσn=0. Let F=F:0,∞→ℝFsatisfiesF1−F3.The collectionF is nonempty: fσ=lnσ, gσ=σ+lnσ, hσ=lnσ2+σ, and kσ=−1/σ are members of this collection.Let us consider the functionφ:0,∞4→0,∞ satisfying the following axioms:(C1) φ is continuous and non-decreasing in each coordinate(C2) if there exist σ,ς∈0,∞ such that σ<ς then φς,ς,σ,ς≤ς(C3) if there exists σ∈0,∞ such that σ≤φ0,0,σ,σ/2 then φ0,0,σ,σ/2=σ.LetCφ=φ:0,∞4→0,∞φsatisfiesC1−C3. The following examples show that the set Cφ is nonempty:
(1)
φaσ,ς,ω,θ=maxσ,ς,ω,θ(2)
φbσ,ς,ω,θ=θ(3)
φcσ,ς,ω,θ=maxσ,ς,ω(4)
φdσ,ς,ω,θ=maxς,ω(5)
φeσ,ς,ω,θ=σ(6)
φfσ,ς,ω,θ=1/2ς+ω(7)
φgσ,ς,ω,θ=maxσ,ς+ω/2,θ(8)
φZσ,ς,ω,θ=aσ+bς+ω+2cθ,a+2b+2c=1(9)
φiσ,ς,ω,θ=aσ+bς+cω,a+b+c=1.Definition 1.
LetT:I→PIandα:I×I→0,∞ be two functions. A mapping T is said to be strictly α-admissible if for each σ∈Iandς∈Tσ with ασ,ς>1, there exists ω∈Tς such that ας,ω>1.Definition 2.
LetI,P be a p-m-s and let α:I×I→0,∞ be a function. The space I,P is said to be strictly α-regular if for any sequence σn⊂I such that ασn,σn+1>1 for all n∈ℕ and σn→σasn→∞, we have ασn,σ>1 for all n∈ℕ.Definition 3.
LetI,P be a p-m-s. A mapping T:I→CBPI is said to be a φHP+-contraction if there exist k∈0,1 and φ∈Cφ such that
(12)ασ,ςHP+Tσ\σ,Tς\ς≤kφPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2,for all σ,ς∈I.LetA∗=σ,ς∈I2∣ασ,ς≥1andH+Tσ\σ,Tς\ς>0.Definition 4.
LetI,P be a p-m-s. A mapping T:I→CBPI is said to be an FH+φ-contraction if
(a)
there existφ∈Cφ, τ>0 and F∈F such that(13)τ+Fασ,ςHP+Tσ\σ,Tς\ς≤FφPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2,for all σ,ς∈A∗.
(b)
For everyε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that(14)Pς,ξ≤HP+Tσ,Tς+ε.Remark 2.
In particular ifPσ,σ=0, then for φe∈Cφ, the inequality (13) turns into H+-contraction [9] for Fσ=lnσ.Proposition 3.
EveryφHP+-contraction is an FH+φ-contraction, but the converse may not be true.Proof.
LetT:I→CBPI be a φHP+-contraction defined on I,P; then for all σ,ς∈I there exist k∈0,1 and φ∈Cφ such that
(15)ασ,ςHP+Tσ\σ,Tς\ς≤kφPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2.
This can be written as(16)ln1k+lnασ,ςHP+Tσ\σ,Tς\ς≤lnPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2.
LetF∈F be defined by Fσ=lnσ for all σ>0 and put τ=ln1/k. The inequality (16) leads to
(17)τ+Fασ,ςHP+Tσ\σ,Tς\ς≤FφPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2.The following example (Example2) shows that an FH+φ-contraction needs not to be a φHP+-contraction.Example 2.
Letφa∈Cφτ=1 and F∈F defined by Fσ=lnσ+σ where φa=φau,r,s,t:u=Pσ,ς,r=Pσ,Tσ\σ,s=Pς,Tς\ς,t=Pς,Tσ\σ+Pσ,Tς\ς/4. Let I=0,1,2,⋯ equipped with p-m P:I×I→0,∞ defined by
(18)Pσ,ς=σ+ςforallσ≤ς.Then,I,P is a p-m-s. Define the mapping T:I→2I by
(19)Rσ=0ifσ∈0,1;0,σ−1ifσ≥2,ασ,ς=0ifσ,ς∈−∞,0;ePσ,ςifσ,ς∈0,1,2,⋯.The mappingT is α-admissible, closed, and bounded. We show that this mapping satisfies inequality (13) for all σ,ς∈I. We observe that HP+Tσ,Tς>0 if and only if σ≥2 and ς>0. Also for all σ,ς∈I with ς∈Tσ and taking ζ=0∈Tς, we have
(20)ασ,ςHP+Tσ,Tς=ασ,ςPς,ζ=ασ,ςς<ασ,ςσ+ς=ασ,ςPσ,ς,and thus
(21)ασ,ςHP+Tσ,Tς−φau,r,s,t≤ασ,ςHP+Tσ,Tς−Pσ,ς≤−2.Consequently,(22)ασ,ςHP+Tσ,Tςφau,r,s,teασ,ςH+Tσ,Tς−φau,r,s,t≤e−1.Hence,(23)1+Fας,ωHP+Tσ\σ,Tς\ς≤FφPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2.Similarly, for every member ofCφ, the mapping T is FH+φ-contraction. However, the mapping T is not φHP+-contraction: for φe∈Cφ and σ≠ς=0, we have
(24)ασ,0HP+Tσ,T0≤kφeu,r,s,t⇒eσσ−1≤kσ,which then gives eσσ−1/σ≤k, and limσ→∞eσσ−1/σ≤k implies k≥∞, a contradiction. Hence, T is not φHP+-contraction for this particular member of Cφ. Similarly, for φbu,r,s,t=t∈Cφ and σ≠ς=1, we have
(25)ασ,1HP+Tσ,T1≤kφbu,r,s,tdoesnotexist.Hence,T is not φHP+-contraction for this member of Cφ. The mapping T has similar nature for other members of Cφ.The following theorem (Theorem1) gives the proof of all particular problems corresponding to members of Cφ in one attempt.Theorem 1.
LetI,P be a complete p-m-s and T:I→BCPI be an FH+φ-contraction such that
(1)
T is a strictly α-admissible mapping(2)
∃σ0 and σ1∈Tσ0 in I such that ασ0,σ1>1(3)
I is a strictly α-regular space(4)
F is continuous.Then, there existsx∗∈I such that x∗∈Tx∗.Proof.
By assumption (2), there existσ0 and σ1∈Tσ0 in A such that ασ0,σ1>1. Note that if σ0∈Tσ0, then σ0 is a fixed point of T, and if σ1∈Tσ1, then σ1 is a fixed point of T as required. We proceed by assuming σ0∉Tσ0 and σ1∉Tσ1; thus, σ0,σ1∈A∗. Given ασ0,σ1>1 and Tσ0,Tσ1 are nonempty, closed, and bounded sets, so, by Definition 4(b), there exists σ2∈Tσ1 such that
(26)Pσ1,σ2≤HP+Tσ0,Tσ1+ε.
Lettingε=ασ0,σ1−1HP+Tσ0,Tσ1, we have
(27)Pσ1,σ2≤HP+Tσ0,Tσ1+ασ0,σ1−1HP+Tσ0,Tσ1=ασ0,σ1HP+Tσ0,Tσ1.
ByF1, (13) and C1, we have
(28)FPσ1,σ2≤Fασ0,σ1H+Tσ0,Tσ1≤FφPσ0,σ1,Pσ0,Tσ0,Pσ1,Tσ1,Pσ1,Tσ0+Pσ0,Tσ12−τ,≤FφPσ0,σ1,Pσ0,σ1,Pσ1,σ2,Pσ1,σ1+Pσ0,σ22−τ.
By the triangular inequality, we have(29)Pσ0,σ2+Pσ1,σ1≤Pσ0,σ1+Pσ1,σ2.
We claim thatPσ1,σ2<Pσ0,σ1. On the contrary, if Pσ1,σ2≥Pσ0,σ1, then due to (29), we get Pσ0,σ2≤2Pσ1,σ2. The inequality (28) implies
(30)FPσ1,σ2<FφPσ1,σ2,Pσ0,σ1,Pσ1,σ2,Pσ1,σ2.
ByC2, we have φPσ1,σ2,Pσ0,σ1,Pσ1,σ2,Pσ1,σ2≤Pσ1,σ2, and by axiom F1, the inequality (30) reduces to
(31)FPσ1,σ2<FPσ1,σ2.
This is an absurdity. This indicates that our claim is valid. Thus,Pσ1,σ2<Pσ0,σ1. Let Pn=Pσn,σn+1 for all positive integers n, and by inequality (28) we obtain
(32)FP1≤FφP0,P0,P1,P0−τ.
ApplyingC2 and F1 consecutively, we have
(33)FP1≤FP0−τ.
SinceT is a strictly α-admissible mapping, ασ0,σ1>1 implies ασ1,σ2>1; thus, σ1,σ2∈A∗ (assume σ2∉Tσ2). Since, Tσ1,Tσ2 are nonempty, closed, and bounded sets. By Definition 4(b), there exists σ3∈Tσ2 such that
(34)Pσ2,σ3≤HP+Tσ1,Tσ2+ε.
Lettingε=ασ1,σ2−1HP+Tσ1,Tσ2, we have
(35)Pσ2,σ3≤HP+Tσ1,Tσ2+ασ1,σ2−1HP+Tσ1,Tσ2=ασ1,σ2HP+Tσ1,Tσ2.
ByF1, (13) and C1, we have
(36)FPσ2,σ3≤Fασ1,σ2HP+Tσ1,Tσ2≤FφPσ1,σ2,Pσ1,Tσ1,Pσ2,Tσ2,Pσ2,Tσ1+Pσ1,Tσ22−τ,≤FφPσ1,σ2,Pσ1,σ2,Pσ2,σ3,Pσ2,σ2+Pσ1,σ32−τ.
By the triangular inequality, we have(37)Pσ1,σ3+Pσ2,σ2≤Pσ1,σ2+Pσ2,σ3.
We claim thatPσ2,σ3<Pσ1,σ2. On the contrary, if Pσ2,σ3≥Pσ1,σ2, then by (37), we get Pσ1,σ3≤2Pσ2,σ3. The inequality (36) implies
(38)FPσ2,σ3<FφPσ2,σ3,Pσ1,σ2,Pσ2,σ3,Pσ2,σ3−τ.
ByC2, φPσ2,σ3,Pσ1,σ2,Pσ2,σ3,Pσ2,σ3≤Pσ2,σ3. By F1 and (38), we have
(39)FPσ2,σ3<FPσ2,σ3.
This is an absurdity. Thus,Pσ2,σ3<Pσ1,σ2. By (36), we obtain
(40)FP2≤FφP1,P1,P2,P1−τ.
Again applying the conditionC2 followed by F1, we have
(41)FP2≤FP1−τ≤FP0−2τ.
Similarly, there existsσ4∈Tσ3σ3∉Tσ3, such that
(42)FP3≤FP2−τ≤FP0−3τ.
Thus, we are able to construct an iterative sequenceσn⊂X such that
(43)σn∈Tσn−1,σn−1∉Tσn−1,ασn−1,σn>1,Pn<Pn−1foralln∈ℕand(44)FPn≤FP0−nτ.
By (44), we obtain limn→∞FPn=−∞, by F2 we have limn→∞Pn=0, and by F3, there exists κ∈0,1 such that
(45)limn→∞PnκFPn=0.
Following (44), for all n∈ℕ, we obtain
(46)PnκFPn−FP0≤−Pnκnτ≤0.
Lettingn→∞, in (46), we have limn→∞nPnκ=0; thus, there exists n1∈ℕ, such that nPnκ≤1 for all n≥n1, that is Pn≤1/n1/κforalln≥n1.
Form>n≥n1,
(47)Pσn,σm≤Pσn,σn+1+Pσn+1,σn+2+Pσn+2,σn+3+⋯+Pσm−1,σm≤∑i=nm−1Pσi,σi+1≤∑i=n∞Pσi,σi+1≤∑i=n∞1i1/k.
Since the series∑i=n∞1/i1/κ on the right hand side is convergent and by definition of metric dP defined on I, we know that dPσn,σm≤2Pσn,σm; thus, limn,m→∞dPσn,σm=0. This implies σn is a Cauchy sequence in I,dP. Since I,P is complete, so by Lemma 1(2), the metric space I,dP is complete. Thus, there exists x∗∈I such that σn→x∗ as n→∞ with respect to metric dP. Then Lemma 1(3) implies
(48)limn→∞Px∗,σn=Px∗,x∗=limn,m→∞Pσn,σm.
This shows thatσn is a Cauchy sequence in I,P. Now, we show that x∗∈Tx∗, and to do so, we claim that Px∗,Tx∗=0. If on the other hand Px∗,Tx∗>0, then there exists n1∈ℕ such that Pσn,Tx∗>0 for each n≥n1. By assumption (3), ασn,x∗>1. By (13),
(49)FPσn+1,Tx∗≤Fασn,x∗H+Tσn,Tx∗≤FφPσn,x∗,Pσn,Tσn,Px∗,Tx∗,Px∗,Tσn+Pσn,Tx∗2−τ≤FφPσn,x∗,Pσn,σn+1,Px∗,Tx∗,Px∗,σn+1+Pσn,Tx∗2−τ.
Thus,(50)FPσn+1,Tx∗<FφPσn,x∗,Pσn,σn+1,Px∗,Tx∗,Px∗,σn+1+Pσn,Tx∗2.
Sinceφ is a coordinate-wise continuous function, letting n→∞ in the above inequality, we obtain
(51)FPx∗,Tx∗<Fφ0,0,Px∗,Tx∗,Px∗,Tx∗2.
ByC3, we have
(52)FPx∗,Tx∗<FPx∗,Tx∗.
This is an absurdity and consequentlyPx∗,Tx∗=0; thus, we have Px∗,Tx∗=Px∗,x∗ which implies that x∗∈Tx∗¯=Tx∗. Hence, x∗ is a fixed point of T.The following example explains Theorem1.Example 3.
Consistent with ([28], Example 3.3), let φa∈Cφ where
(53)φa=φau,r,s,t:u=Pσ,ς,r=Pσ,Tσ,s=Pς,Tς,t=Pς,Tσ+Pσ,Tς2,
τ=1 and F∈F defined by Fσ=lnσ+σ. Let I=0,1,2,⋯ equipped with p-m P:I×I→0,∞ defined by
(54)Pσ,ς=σ+ςforallσ≠ς.Then,I,P is a complete p-m-s. Define the mapping T:I→2I by
(55)Rσ=0ifσ∈0,1;0,σ−1ifσ≥2,ασ,ς=0ifσ,ς∈−∞,0;ePσ,ςifσ,ς∈0,1,2,⋯.The mappingT is strict α-admissible, closed, and bounded. We show that T is FH+φ-contraction. We observe that H+Tσ,Tς>0 if and only if σ≥2 and ς>0. Also for all σ,ς∈I with ς∈Tσ and taking ζ=0∈Tς, we have
(56)ασ,ςH+Tσ,Tς=ePσ,ςPς,ζ=ePσ,ςς<ePσ,ςσ+ς=ePσ,ςPσ,ς,andthus,ePσ,ςH+Tσ,Tς−φau,r,s,t≤ePσ,ςH+Tσ,Tς−Pσ,ς≤−2.Consequently,(57)ePσ,ςH+Tσ,Tςφau,r,s,teePσ,ςH+Tσ,Tς−φau,r,s,t≤e−1.Hence,(58)1+Fασ,ςH+Tσ,Tς≤FφPσ,ς,Pσ,Tσ,Pς,Tς,Pς,Tσ+Pσ,Tς2.Similarly, for every member ofCφ, the mapping T satisfies all assumptions in Theorem 1. As it is clear from Proposition 3 that FH+φ-contraction needs not to be φHP+-contraction, and hence, it is not HP+-contraction. Consequently, FH+φ-contraction needs not to be H+-contraction. Thus, the results in [9, 10, 27] are not applicable in this case.Remark 3.
In the following section, we obtain the corollaries of Theorem1. To simplify the expression of the corollaries, we consider the three conditions below.
Let
(A1) there existσ0 in I such that ασ0,Tσ0>1
(A2)I be a strictly α-regular space
(A3)F be continuous.Corollary 1.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admisible mapping. Assume that
(59)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FPσ,ς,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(60)Pς,ξ≤HP+Tσ,Tς+ε.Then, the mappingT has a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(61)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=Pσ,ςand following the proof of Theorem 1, we obtain the result.Corollary 2.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admisible mapping. Assume that
(62)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FmaxPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, ∃ξ∈Tς such that
(63)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(64)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=maxPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς
and following the proof of Theorem1, we obtain the result.Corollary 3.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admissible mapping. Assume that
(65)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FmaxPσ,Tσ\σ,Pς,Tς\ς,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(66)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(67)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=maxPσ,Tσ\σ,Pς,Tς\ς
and following the steps given in the proof of Theorem1, we obtain the result.Corollary 4.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admissible mapping. Assume that
(68)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FmaxPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(69)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(70)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=maxPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2,
and following the proof of Theorem1, we obtain the result.Corollary 5.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admissible mapping. Assume that
(71)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FPς,Tσ\σ+Pσ,Tς\ς2,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(72)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(73)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=Pς,Tσ\σ+Pσ,Tς\ς2
in the proof of Theorem1, we get the result.Corollary 6.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admissible mapping. Assume that
(74)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FPσ,Tσ\σ+Pς,Tς\ς2,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(75)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(76)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=Pσ,Tσ\σ+Pς,Tς\ς2
in the proof of Theorem1, we get the result.Corollary 7.
LetI,P be a complete p-m-s and T:I→BCPI be strictly α-admissible mapping. Assume that
(77)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FmaxPσ,ς,Pσ,Tσ\σ+Pς,Tς\ς2,Pς,Tσ\σ+Pσ,Tς\ς2,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(78)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(79)φPσ,ς,Pσ,Tσ,Pς,Tς,Pς,Tσ+Pσ,Tς2=maxPσ,ς,Pσ,Tσ+Pς,Tς2,Pς,Tσ+Pσ,Tς2
and following the proof of Theorem1, we obtain the result.Corollary 8.
LetI,P be a complete p-m-s and T:I→BCPI be strictly α-admisible mapping. Assume that there exist a≥0,b≥0,c≥0 satisfying a+2b+2c=1, such that
(80)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FaPσ,ς+bPσ,Tσ\σ+Pς,Tς\ς+cPς,Tσ\σ+Pσ,Tς\ς,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(81)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
Definingφ:0,∞4→0,∞ by
(82)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=aPσ,ς+bPσ,Tσ\σ+Pς,Tς\ς+2cPς,Tσσ+Pσ,Tςς2,
in the proof of Theorem1, we obtain the result.Corollary 9.
LetI,P be a complete p-m-s and T:I→BCPI be a strictly α-admissible mapping. Assume that there exist a≥0,b≥0,c≥0 satisfying a+b+c=1, such that
(83)τ+Fασ,ςH+Tσ\σ,Tς\ς≤FaPσ,ς+bPσ,Tσ\σ+cPς,Tς\ς,
for allσ,ς∈A∗, and for every ε>0, σ∈I, and ς∈Tσ, there exists ξ∈Tς such that
(84)Pς,ξ≤HP+Tσ,Tς+ε.Then,T admits a fixed point provided (A1)-(A3) hold.Proof.
If we defineφ:0,∞4→0,∞ by
(85)φPσ,ς,Pσ,Tσ\σ,Pς,Tς\ς,Pς,Tσ\σ+Pσ,Tς\ς2=aPσ,ς+bPσ,Tσ\σ+cPς,Tς\ς,
in the proof of Theorem1, then the result follows.
Let(86)I∗=σ,ς∈I2∣ασ,ς>1andPTσ,Tς>0.For a single-valued self-mapping, Theorem1 can be stated as follows:Theorem 2.
LetI,P be a complete p-m-s and T:I→I be a φF-contraction, that is, there exist φ∈Cφ and F∈F such that
(87)τ+Fασ,ςPTσ,Tς≤FφPσ,ς,Pσ,Tσ,Pς,Tς,Pς,Tσ+Pσ,Tς2,
for allσ,ς∈I∗ and
(1)
T is a strictly α-admissible mapping(2)
there existsσ0 in I such that ασ0,Tσ0>1(3)
I is a strictly α-regular space(4)
F is continuous.Then,T admits a fixed point.We omit its proof as it is a mere repetition of the proof of Theorem1 with some minor modifications.


## 5. Applications of Theorem2

### 5.1. Applications to Fractional Differential Equations

Lacroix (1819) introduced and investigated several applicable properties of fractional differentials. Recently, various new models involving Caputo-Fabrizio derivative (CFD) were discovered and analyzed in [29–31]. We investigate one of these models in p-m-s. We introduce some notations as follows:LetC0,1=f∣f:0,1→ℝandfiscontinuous. Define the metric function d:C0,1×C0,1→0,∞ by
(88)df,g=f−g∞=maxν∈0,1fν−gν,forallf,g∈C0,1.Then, the spaceC0,1,d is a complete metric space. The function α:C0,1×C0,1→1,∞ by
(89)αr,t=er+t∞forallr,t∈C0,1.LetK1:0,1×ℝ→ℝ be a continuous function. We shall investigate the following CFDE:
(90)CDβfν=K1ν,fν,with boundary conditions
(91)σ0=0,Iσ1=σ′0.Here,CDβ denotes CFD of order β defined by
(92)CDβK1ν=1Γn−β∫0νν−ηn−β−1K1nηdη,where
(93)n−1<β<nandn=β+1,and IβK1 is given by
(94)IβK1ν=1Γβ∫0νν−ηβ−1K1ηdη,withβ>0.Then, the equation (90) can be modified to
(95)fν=1Γβ∫0νν−ηβ−1K1η,fηdη+2νΓβ∫01∫0ηη−uβ−1K1u,fududη,Theorem 3.
Equation (90) admits a solution inC0,1provided(I)
there existsτ>0 such that for all σ,ς∈C0,1, we have(96)K1η,ση−K1η,ςη≤e−τΓβ+14ασ,ςση−ςη(II)
there existsσ0∈C0,1 such that for all ν∈0,1, we have(97)σ0ν≤1Γβ∫0νν−ηβ−1K1η,σ0ηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σ0ududη.Proof.
Consistent with the notations introduced above and defining the mappingR:C0,1→C0,1 by
(98)Rσν=1Γβ∫0νν−ηβ−1K1η,σηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σududη.By (II), there existsσ0∈C0,1 such that σn=tnσ0. The continuity of function K1 leads to the continuity of mapping t on C0,1. It is easy to verify the assumptions (1)-(4) in Theorem 2. In the following, we verify the contractive condition (87) of Theorem 2.
(99)Rσν−Rςν=1Γβ∫0νν−ηβ−1K1η,σηdη−1Γβ∫0νν−ηβ−1K1η,ςηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σududη−2νΓβ∫01∫0ηη−uβ−1K1u,ςududηimpliesRσν−Rςν≤∫0ν1Γβν−ηβ−1K1η,ση−1Γβν−ηβ−1K1η,ςηdη+∫01∫0η2Γβη−uβ−1K1η,ση−2Γβη−uβ−1K1u,ςududη≤1Γβe−τΓβ+14ασ,ς·∫0νν−ηβ−1ση−ςηdη+2Γβe−τΓβ+14ασ,ς·∫01∫0ηη−uβ−1ςu−σududη≤1Γβe−τΓβ+14ασ,ς·dσ,ς·∫0νν−ηβ−1dη+2Γβe−τΓβ·Γβ+14ασ,ςΓα·Γβ+1·dσ,ς·∫01∫0ηη−uβ−1dudη≤e−τΓβ·Γβ+14ασ,ςΓβ·Γβ+1·dσ,ς+2e−τBβ+1,1Γβ·Γβ+14ασ,ςΓβ·Γβ+1·dσ,ς≤e−τ4ασ,ςdσ,ς+e−τ2ασ,ςdσ,ς,where B is the beta function. The last inequality can be written by that
(100)ασ,ςdRσ,Rς≤e−τdσ,ς.Let us define the metricd on C0,1 by
(101)dσ,ς=Pσ,ς=σ−ς∞+ll≥0ifσ≠ς0ifσ=ς.Thus, (100) can be written as
(102)ασ,ςPRσ,Rς≤e−τPσ,ς.Define the functionsφ∈Cφ and F by
(103)φaPσ,ς,Pσ,tσ,Pς,tς,Pς,tσ+Pσ,tς2=maxPσ,ς,Pσ,tσ,Pς,tς,Pς,tσ+Pσ,tς2,Fσν=lnσν for all σ,ς∈C0,1. Under these definitions, the inequality (102) gets the form
(104)τ+Fασ,ςPRσ,Rς≤FφPσ,ς,Pσ,Tσ,Pς,Tς,Pς,Tσ+Pσ,Tς2.Hence, by Theorem2, the self-mapping t admits a fixed point, and hence, the equation (90) has a solution.


### 5.2. Applications to the Matrix Equations

In this section, by Theorem2, we shall investigate study the existence of the solutions to
(105)X=D+1m+θ∑i=1mWi∗XWi+∑i=1mGi∗XGi,whereθ∈0,1, D∈Pm (set of m×m positive definite matrices), and Wi,Gi are arbitrary m×m matrices for each i and are entries of block matrices given by
(106)W=W1W2W3⋮WmG=G1G2G3⋮Gm.LetWZ∈Zm (set of m×m Hermitian matrices) be an arbitrary matrix; then, its eigenvalues e1,e2,e3,⋯em are real. Moreover, if WZ∈Z+m, then the eigenvalues are nonnegative. Let the functional .tr:Zm→ℝ be defined by
(107)WZtr=∑i=1mei.LetX∈Pm be arbitrary and define WZtr,X=X1/2WZX1/2tr. By ([32], Theorem IX.2.2), Zm,.tr,X is a Banach space (see also [33–35]). Hence, Zm,d is a complete metric space. The induced metric d:Zm×Zm→ℝ is defined by
(108)dWZ,GZ=WZ−GZtr,XforallWZ,GZ∈Zm.To establish the existence result we need the following lemma.Lemma 2 [35].
IfWZ,GZ∈Z+m, then
(109)0≤TrWZGZ≤WZTrGZ.Define the operatorE:Zm→Zm by
(110)EU=G+1m+θ∑i=1mWi∗UWi+∑i=1mGi∗UGi,forallU∈Zm.Remark 4.
SinceEU−G∈Pm for all U∈Zm, in particular, we have EG−G∈Pm. The operator E is continuous on Zm.The solution of the matrix equation (105) is the fixed point of the operator E.Theorem 4.
LetX and Y be two positive definite matrices such that ∑i=1mWi∗XWi≺1/2X and ∑i=1mGi∗XGi≺1/2X. Then, the operator E has a fixed point in Zm.Proof.
LetU and V be any matrices in Pm. We observe that the operator E and the space Zm,.tr,X fulfill the assumptions (1)-(4) in Theorem 2. To prove that E is an φF-contraction, we proceed with
(111)EV−EUtr,X=trX1/2EV−EUX1/2=tr1m+θ∑i=1mX1/2Wi∗V−UWi+Gi∗V−UGiX1/2=tr1m+θ∑i=1mX1/2Wi∗V−UWiX1/2+1m+θ∑i=1mX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+X1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+trX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+1m+θ∑i=1mtrX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrWiXWi∗V−U+1m+θ∑i=1mtrGiXGi∗V−U=1m+θ∑i=1mtrWiXWi∗X−1/2X1/2V−UX1/2X−1/2+1m+θ∑i=1mtrGiXGi∗X−1/2X1/2V−UX1/2X−1/2=1m+θ∑i=1mtrX−1/2WiXWi∗X−1/2X1/2V−UX1/2+1m+θ∑i=1mtrX−1/2GiXGi∗X−1/2X1/2V−UX1/2=1m+θtr∑i=1mX−1/2WiXWi∗X−1/2X1/2V−UX1/2+1m+θtr∑i=1mX−1/2GiXGi∗X−1/2X1/2V−UX1/2≤1m+θ∑i=1mX−1/2WiXWi∗X−1/2V−Utr,X+1m+θ∑i=1mX−1/2GiXGi∗X−1/2V−Utr,XbyLemma2=1m+θ∑i=1mX−1/2WiXWi∗X−1/2+∑i=1mX−1/2GiXGi∗X−1/2V−Utr,X.
Given∑i=1mWi∗XWi≺1/2X, ∑i=1mGi∗XGi≺1/2X, and letting K be a number such that
(112)K=∑i=1mX−1/2WiXWi∗X−1/2+∑i=1mX−1/2GiXGi∗X−1/2<1,wehaveEV−EUtr,X≤Km+θV−Utr,X.
Thus,(113)m+θKdEV,EU≤KdV,U.
We defineα:Zm×Zm→1,∞ by
(114)αU,V=m+θforallU,V∈Zmandθ∈0,1,and the metric d on Zm by
(115)dWi,Gi=PWi,GiifWi≠Gi;∑i=1mWi∗XWi≺12Xand∑i=1mGi∗XGi≺12X0ifWi=Gi,
In view of the metric defined above, the inequality (113) can be written as
(116)m+θPEV,EUK≤PV,UlnαU,VPEV,EUK≤lnPV,U.
Define the functionsφ∈Cφ and F by
(117)φaPU,V,PU,EU,PV,EV,PV,EU+PU,EV2=maxPU,V,PU,EU,PV,EV,PV,EU+PU,TV2,
Fσ=lnσ for all σ∈0,∞, respectively. Under these definitions, we have
(118)τ+FαU,VPEV,EU≤FPV,Uputτ=lnK−1≤FmaxPU,V,PU,EU,PV,EV,PV,EU+PU,EV2.
By Theorem2, the operator E has a fixed point, and hence, the matrix equation (105) has a solution.Remark 5.
The numerical explanation of the conditions∑i=1mWi∗XWi≺1/2X and ∑i=1mGi∗XGi≺1/2X imposed in Theorem 4 for i=2 and taking 4×4 matrices is as follows:
(119)letW1=0.10.050.050.050.050.10.050.050.050.050.10.050.050.050.050.1W2=0.5−0.02−0.02−0.02−0.020.5−0.02−0.02−0.02−0.020.5−0.02−0.02−0.02−0.020.5.Then, for a matrix(120)X=10.20.20.20.210.20.20.20.210.20.20.20.21,we have
(121)∑i=12Wi∗XWi=0.26620.04790.04790.04790.04790.26620.04790.04790.04790.04790.26620.04790.04790.04790.04790.2662≺12X.Similarly,letG1=0.010.0010.010.010.0010.010.010.0010.010.0010.0010.010.0010.010.0010.001G2=0.14130.0082940.14130.14130.0082940.09970.0082940.14130.14130.0082940.14130.09970.11090.14130.0082940.0997.Then, for a matrix(122)X=10.20.20.20.210.20.20.20.210.20.20.20.21,we have
(123)∑i=12Gi∗XGi=0.07440.03590.05700.0.07600.03590.03760.01910.04910.05700.01910.05020.05790.07600.04910.05790.0.0946≺12X.


## 5.1. Applications to Fractional Differential Equations

Lacroix (1819) introduced and investigated several applicable properties of fractional differentials. Recently, various new models involving Caputo-Fabrizio derivative (CFD) were discovered and analyzed in [29–31]. We investigate one of these models in p-m-s. We introduce some notations as follows:LetC0,1=f∣f:0,1→ℝandfiscontinuous. Define the metric function d:C0,1×C0,1→0,∞ by
(88)df,g=f−g∞=maxν∈0,1fν−gν,forallf,g∈C0,1.Then, the spaceC0,1,d is a complete metric space. The function α:C0,1×C0,1→1,∞ by
(89)αr,t=er+t∞forallr,t∈C0,1.LetK1:0,1×ℝ→ℝ be a continuous function. We shall investigate the following CFDE:
(90)CDβfν=K1ν,fν,with boundary conditions
(91)σ0=0,Iσ1=σ′0.Here,CDβ denotes CFD of order β defined by
(92)CDβK1ν=1Γn−β∫0νν−ηn−β−1K1nηdη,where
(93)n−1<β<nandn=β+1,and IβK1 is given by
(94)IβK1ν=1Γβ∫0νν−ηβ−1K1ηdη,withβ>0.Then, the equation (90) can be modified to
(95)fν=1Γβ∫0νν−ηβ−1K1η,fηdη+2νΓβ∫01∫0ηη−uβ−1K1u,fududη,Theorem 3.
Equation (90) admits a solution inC0,1provided(I)
there existsτ>0 such that for all σ,ς∈C0,1, we have(96)K1η,ση−K1η,ςη≤e−τΓβ+14ασ,ςση−ςη(II)
there existsσ0∈C0,1 such that for all ν∈0,1, we have(97)σ0ν≤1Γβ∫0νν−ηβ−1K1η,σ0ηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σ0ududη.Proof.
Consistent with the notations introduced above and defining the mappingR:C0,1→C0,1 by
(98)Rσν=1Γβ∫0νν−ηβ−1K1η,σηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σududη.By (II), there existsσ0∈C0,1 such that σn=tnσ0. The continuity of function K1 leads to the continuity of mapping t on C0,1. It is easy to verify the assumptions (1)-(4) in Theorem 2. In the following, we verify the contractive condition (87) of Theorem 2.
(99)Rσν−Rςν=1Γβ∫0νν−ηβ−1K1η,σηdη−1Γβ∫0νν−ηβ−1K1η,ςηdη+2νΓβ∫01∫0ηη−uβ−1K1u,σududη−2νΓβ∫01∫0ηη−uβ−1K1u,ςududηimpliesRσν−Rςν≤∫0ν1Γβν−ηβ−1K1η,ση−1Γβν−ηβ−1K1η,ςηdη+∫01∫0η2Γβη−uβ−1K1η,ση−2Γβη−uβ−1K1u,ςududη≤1Γβe−τΓβ+14ασ,ς·∫0νν−ηβ−1ση−ςηdη+2Γβe−τΓβ+14ασ,ς·∫01∫0ηη−uβ−1ςu−σududη≤1Γβe−τΓβ+14ασ,ς·dσ,ς·∫0νν−ηβ−1dη+2Γβe−τΓβ·Γβ+14ασ,ςΓα·Γβ+1·dσ,ς·∫01∫0ηη−uβ−1dudη≤e−τΓβ·Γβ+14ασ,ςΓβ·Γβ+1·dσ,ς+2e−τBβ+1,1Γβ·Γβ+14ασ,ςΓβ·Γβ+1·dσ,ς≤e−τ4ασ,ςdσ,ς+e−τ2ασ,ςdσ,ς,where B is the beta function. The last inequality can be written by that
(100)ασ,ςdRσ,Rς≤e−τdσ,ς.Let us define the metricd on C0,1 by
(101)dσ,ς=Pσ,ς=σ−ς∞+ll≥0ifσ≠ς0ifσ=ς.Thus, (100) can be written as
(102)ασ,ςPRσ,Rς≤e−τPσ,ς.Define the functionsφ∈Cφ and F by
(103)φaPσ,ς,Pσ,tσ,Pς,tς,Pς,tσ+Pσ,tς2=maxPσ,ς,Pσ,tσ,Pς,tς,Pς,tσ+Pσ,tς2,Fσν=lnσν for all σ,ς∈C0,1. Under these definitions, the inequality (102) gets the form
(104)τ+Fασ,ςPRσ,Rς≤FφPσ,ς,Pσ,Tσ,Pς,Tς,Pς,Tσ+Pσ,Tς2.Hence, by Theorem2, the self-mapping t admits a fixed point, and hence, the equation (90) has a solution.


## 5.2. Applications to the Matrix Equations

In this section, by Theorem2, we shall investigate study the existence of the solutions to
(105)X=D+1m+θ∑i=1mWi∗XWi+∑i=1mGi∗XGi,whereθ∈0,1, D∈Pm (set of m×m positive definite matrices), and Wi,Gi are arbitrary m×m matrices for each i and are entries of block matrices given by
(106)W=W1W2W3⋮WmG=G1G2G3⋮Gm.LetWZ∈Zm (set of m×m Hermitian matrices) be an arbitrary matrix; then, its eigenvalues e1,e2,e3,⋯em are real. Moreover, if WZ∈Z+m, then the eigenvalues are nonnegative. Let the functional .tr:Zm→ℝ be defined by
(107)WZtr=∑i=1mei.LetX∈Pm be arbitrary and define WZtr,X=X1/2WZX1/2tr. By ([32], Theorem IX.2.2), Zm,.tr,X is a Banach space (see also [33–35]). Hence, Zm,d is a complete metric space. The induced metric d:Zm×Zm→ℝ is defined by
(108)dWZ,GZ=WZ−GZtr,XforallWZ,GZ∈Zm.To establish the existence result we need the following lemma.Lemma 2 [35].
IfWZ,GZ∈Z+m, then
(109)0≤TrWZGZ≤WZTrGZ.Define the operatorE:Zm→Zm by
(110)EU=G+1m+θ∑i=1mWi∗UWi+∑i=1mGi∗UGi,forallU∈Zm.Remark 4.
SinceEU−G∈Pm for all U∈Zm, in particular, we have EG−G∈Pm. The operator E is continuous on Zm.The solution of the matrix equation (105) is the fixed point of the operator E.Theorem 4.
LetX and Y be two positive definite matrices such that ∑i=1mWi∗XWi≺1/2X and ∑i=1mGi∗XGi≺1/2X. Then, the operator E has a fixed point in Zm.Proof.
LetU and V be any matrices in Pm. We observe that the operator E and the space Zm,.tr,X fulfill the assumptions (1)-(4) in Theorem 2. To prove that E is an φF-contraction, we proceed with
(111)EV−EUtr,X=trX1/2EV−EUX1/2=tr1m+θ∑i=1mX1/2Wi∗V−UWi+Gi∗V−UGiX1/2=tr1m+θ∑i=1mX1/2Wi∗V−UWiX1/2+1m+θ∑i=1mX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+X1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+trX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrX1/2Wi∗V−UWiX1/2+1m+θ∑i=1mtrX1/2Gi∗V−UGiX1/2=1m+θ∑i=1mtrWiXWi∗V−U+1m+θ∑i=1mtrGiXGi∗V−U=1m+θ∑i=1mtrWiXWi∗X−1/2X1/2V−UX1/2X−1/2+1m+θ∑i=1mtrGiXGi∗X−1/2X1/2V−UX1/2X−1/2=1m+θ∑i=1mtrX−1/2WiXWi∗X−1/2X1/2V−UX1/2+1m+θ∑i=1mtrX−1/2GiXGi∗X−1/2X1/2V−UX1/2=1m+θtr∑i=1mX−1/2WiXWi∗X−1/2X1/2V−UX1/2+1m+θtr∑i=1mX−1/2GiXGi∗X−1/2X1/2V−UX1/2≤1m+θ∑i=1mX−1/2WiXWi∗X−1/2V−Utr,X+1m+θ∑i=1mX−1/2GiXGi∗X−1/2V−Utr,XbyLemma2=1m+θ∑i=1mX−1/2WiXWi∗X−1/2+∑i=1mX−1/2GiXGi∗X−1/2V−Utr,X.
Given∑i=1mWi∗XWi≺1/2X, ∑i=1mGi∗XGi≺1/2X, and letting K be a number such that
(112)K=∑i=1mX−1/2WiXWi∗X−1/2+∑i=1mX−1/2GiXGi∗X−1/2<1,wehaveEV−EUtr,X≤Km+θV−Utr,X.
Thus,(113)m+θKdEV,EU≤KdV,U.
We defineα:Zm×Zm→1,∞ by
(114)αU,V=m+θforallU,V∈Zmandθ∈0,1,and the metric d on Zm by
(115)dWi,Gi=PWi,GiifWi≠Gi;∑i=1mWi∗XWi≺12Xand∑i=1mGi∗XGi≺12X0ifWi=Gi,
In view of the metric defined above, the inequality (113) can be written as
(116)m+θPEV,EUK≤PV,UlnαU,VPEV,EUK≤lnPV,U.
Define the functionsφ∈Cφ and F by
(117)φaPU,V,PU,EU,PV,EV,PV,EU+PU,EV2=maxPU,V,PU,EU,PV,EV,PV,EU+PU,TV2,
Fσ=lnσ for all σ∈0,∞, respectively. Under these definitions, we have
(118)τ+FαU,VPEV,EU≤FPV,Uputτ=lnK−1≤FmaxPU,V,PU,EU,PV,EV,PV,EU+PU,EV2.
By Theorem2, the operator E has a fixed point, and hence, the matrix equation (105) has a solution.Remark 5.
The numerical explanation of the conditions∑i=1mWi∗XWi≺1/2X and ∑i=1mGi∗XGi≺1/2X imposed in Theorem 4 for i=2 and taking 4×4 matrices is as follows:
(119)letW1=0.10.050.050.050.050.10.050.050.050.050.10.050.050.050.050.1W2=0.5−0.02−0.02−0.02−0.020.5−0.02−0.02−0.02−0.020.5−0.02−0.02−0.02−0.020.5.Then, for a matrix(120)X=10.20.20.20.210.20.20.20.210.20.20.20.21,we have
(121)∑i=12Wi∗XWi=0.26620.04790.04790.04790.04790.26620.04790.04790.04790.04790.26620.04790.04790.04790.04790.2662≺12X.Similarly,letG1=0.010.0010.010.010.0010.010.010.0010.010.0010.0010.010.0010.010.0010.001G2=0.14130.0082940.14130.14130.0082940.09970.0082940.14130.14130.0082940.14130.09970.11090.14130.0082940.0997.Then, for a matrix(122)X=10.20.20.20.210.20.20.20.210.20.20.20.21,we have
(123)∑i=12Gi∗XGi=0.07440.03590.05700.0.07600.03590.03760.01910.04910.05700.01910.05020.05790.07600.04910.05790.0.0946≺12X.


## 6. Conclusion

The introduced contractions encompass theF-contractions and multivalued contractions and hence the Banach contractions, Kannan contractions, Chatterjea contractions, Reich contractions, Hardy-Rogers contractions, and Ciric-type contractions (both metric and p-m versions). It is a real generalization of Matthews contractions and F-contractions. The theorems give general criteria for the existence of the uniqueness of the fixed point.



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