# A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

Authors: Hasanen A. Hammad; Manuel De la Sen
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena typeΩ-contractions in the setting of extended b-metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).


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## Body

## 1. Introduction

The importance of fixed-point (FP) theory increased after Banach introduced his principle [1]. It became an essential tool in nonlinear analysis. In particular, there are many important applications in which the FP method is involved such as economics, microbiology, and physical applications. This technique permeates fractional analysis and becomes a basic method in finding solutions to some fractional systems. For more details, we suggest the reader to read the books of Kilbas et al. [2], Samko et al. [3], Wang et al. [4], and Atangana and Baleanu [5]. Furthermore, this methodology is widely used in finding solutions of integral equations, fractional differential equations, and boundary value problems, for example, see [6–11].Under Hausdorff metric space formalism, Nadler [12] introduced the notion of multivalued mapping as follows: Let V,τ be a metric space, VCB represents the set of nonempty closed bounded subsets of V, and let Ξ:VCB×VCB⟶ℝ+ be a function given by
(1)Ξz1,z2=maxsupξ∈z1τξ,z2,supζ∈z2τζ,z1,∀z1,z2∈VCB,where τξ,℧2=infτξ,ζ:ζ∈z2.The functionΞ is called also the Hausdorff-Pompeiu metric. Consider
(2)δz1,z2=supτξ,ζ:ξ∈z1,ζ∈z2,then, one can deduce that for all z1,z2,z3∈VCB,
(i)
δz1,z2=δz2,z1(ii)
δz1,z2=0⇔z1=z2=ξ(iii)
δz1,z3≤δz1,z2+δz2,z3(iv)
δz1,z1=diamz1In 2012, Wardowski [13] generalized the old principle of Banach to a broader kind. He called it Ω-contraction and presented it in the following definition:Definition 1.
A nonlinear mappingN:V⟶V described on the metric space V,τ is called a Ω-contraction if there exist F∈Σ and υ>0 so that
(3)τN℘,Nℏ>0⇒υ+ΩτN℘,Nℏ≤Ωτ℘,ℏ,∀℘,ℏ∈V,where Σ is the class of functions Ω:0,+∞⟶ℝ so that the assertions below hold:
(Ω1) for all z1,z1∈ℝ+, if z1<z2, then Ωz1<Ωz2
(Ω2) for all positive real sequence zr,zr⟶0 if and only if Ωzr⟶−∞ as r⟶∞
(Ω3) there exists ℓ∈0,1 so that lims⟶0+sℓΩs=0Many researchers have turned to generalize ordinary metric spaces to more comprehensive spaces such asb-metric, metric-like, and b-metric-like spaces. In the other word, Czerwik [14] was able to present b-metric spaces, Alghmandi et al. [15] introduced metric-like and b-metric-like spaces, and Hammad et al. [16] presented extended b-metric like spaces. It should be noted that the definitions of these spaces are mentioned in the next section.The concept of multivalued mapping andΩ-contraction mapping in abstract spaces are combined, and fruitful FP results in this line are obtained. For instance, see Cosentino and Vetro [17], Sgroi and Vetro [18], and Ali et al. [19].By the same approach, in this manuscript, some FP results under rational Gupta-Saxena typeΩ-contractions (for short, Φφ,ΩrGS-contractions) are obtained in the setting of extended b-metric-like space (for short, ¥b-spaces). Also, the results are supported by nontrivial examples. Finally, the theoretical results are applied to study the existence of the solution to the Fredholm integral equation which arises from the damped harmonic oscillator, to study the initial value problem (IVP) which arises from Newton’s law of cooling and to study infinite systems of fractional ODEs.


## 2. Preliminaries

This section discusses some extended spaces and their topological properties.Definition 2 [15].
LetV be a nonempty set. A function τ:V2⟶ℝ+ is called a metric-like on V, if for all z1,z2,z3∈V, the conditions below hold:
(a)
τz1,z2=0⇒z1=z2(b)
τz1,z2=τz2,z1(c)
τz1,z3≤τz1,z2+τz2,z3Here,V,τ is named as a metric-like space.Definition 3 [15].
Ab-metric-like on a nonempty set V is a function τ:V×V⟶ℝ+ so that for all z1,z2,z3∈V and a constant j≥1, the following three conditions hold:
(e)
τz1,z2=0⇒z1=z2(f)
τz1,z2=τz2,z1(g)
τz1,z3≤jτz1,z2+τz2,z3Here,V,τ is called a b-metric-like space with constant j.For more details about the conversion of metric-like andb-metric-like spaces, see [20–22].Very recently, Hammad et al. [16] merged the results of metric-like with b-metric-like spaces and formed a ¥b-space. Also, topological properties of this space were discussed, and some nice FP results of Wardowski and Suzuki contraction mappings were obtained. This space is introduced as follows:Definition 4 [16].
Assume thatV is a nonempty set and j:V×V⟶1,∞ is a given function. A function τj:V2⟶0,∞ is called a modified τj-metric-like if for all z1,z2,z3∈V:
(τj1) τjz1,z2=0⇒z1=z2
(τj2) τjz1,z2=τjz2,z1
(τj3) τjz1,z3≤jz1,z3τjz1,z2+τjz2,z3Then,V,τj is called a ¥b-space.Note that the class of¥b-spaces is larger than the class of b-metric-like spaces by replacing the constant j≥1 of Definition 3 by a nonconstant function j:V×V⟶1,∞ of Definition 4.For more examples and applications via the mentioned space, see [16].Definition 5 [16].
Letzι be a sequence in the ¥b-space V,τj.
(i)
Iflimi⟶∞τjzi,z=τjz,z, then zi is convergent to z(ii)
zi is called Cauchy if limi,k⟶∞τjzi,zk exists and is finite(iii)
If for each Cauchy sequencezi, there is z∈V, so that limi,k⟶∞τjzi,zk=τjz,z=limι⟶∞τjzi,z; then, V,τj is called completeLemma 6 [4].
Suppose thatV,τ is an extended b-metric space and that V1,V2∈VCB,μ∈V1 and λ>1, there exists ημ∈V2 so that ϖθμ,η≤λΞV1,V2.It should be noted that Lemma6 also satisfied in each of metric spaces [23] and b-metric spaces [24]. Hence, it can be easily achieved in ¥b-spaces.


## 3. Main Theorem

We begin this part with the following definition:Definition 7.
LetV,τj be a ¥b-space with a function j:V×V⟶1,∞. Assume that Φ,φ:V⟶VCB are two multivalued mappings. We say that the pair Φ,φ is a Φφ,ΩrGS-contraction if there exist υ>0 and Ω∈Σ so that for all z1,z2∈V, then
(4)υ+ΩλHΦz1,φz2≤Ωβz1,z2,where
(5)βz1,z2=maxτjz1,z2,1+τjz1,Φz1τjz2,φz21+τjz1,z2,τjz1,Φz1τjz2,φz21+τjz1,z2Theorem 8.
LetV,τj be a complete ¥b-space and let φ,Φ be a Φφ,ΩrGS-contraction so that for all z0∈V,limm.n⟶∞jzm,zn<1/λ. Then, φ and Φ have a unique common FP.Proof.
Letz0∈V and z1∈Φz0. Assume that z2=φz1, and by Lemma 6, there is z3∈Φz2 so that
(6)τjz3,z2≤λHΦz2,φz1.
Inductively, we letz2n=φz2n−1, and by Lemma 6, we select z2n+1∈Φz2n so that
(7)τjz2n−1,z2n≤λHΦz2n,φz2n−1.
Applying (4) and (7), we have
(8)υ+Ωτjz2n+1,z2n+2≤υ+ΩλHΦz2n,φz2n+1≤Ωmaxτjz2n,z2n+1,1+τjz2n,Φz2nτjz2n+1,φz2n+11+τjz2n,z2n+1,τjz2n,Φz2nτjz2n+1,φz2n+11+τjz2n,z2n+1=Ωmaxτjz2n,z2n+1,1+τjz2n,z2n+1τjz2n+1,z2n+21+τjz2n,z2n+1,τjz2n,z2n+1τjz2n+1,z2n+21+τjz2n,z2n+1≤Ωmaxτjz2n,z2n+1,τjz2n+1,z2n+2,τjz2n+1,z2n+2=Ωmaxτjz2n,z2n+1,τjz2n+1,z2n+2.
Ifmaxτjz2n,z2n+1,τjz2n+1,z2n+2=τjz2n+1,z2n+2, then
(9)υ+Ωτjz2n+1,z2n+2≤Ωτjz2n+1,z2n+2.
This is a contradiction according toΩ1. Therefore, maxτjz2n,z2n+1,τjz2n+1,z2n+2=τjz2n,z2n+1. Then, from (8),
(10)Ωτjz2n+1,z2n+2≤Ωτjz2n,z2n+1−υ,∀n≥0.
Similarly, we get(11)Ωτjz2n,z2n+1≤Ωτjz2n−1,z2n−υ.
Applying (10) and (11), we have
(12)Ωτjz2n+1,z2n+2≤Ωτjz2n−1,z2n−2υ.
Repeating consequently, one can obtain(13)Ωτjz2n+1,z2n+2≤Ωτjz0,z1−2n+1υ.
Similarly,(14)Ωτjz2n,z2n+1≤Ωτjz0,z1−2nυ.
By using (13) and (14), we have
(15)Ωτjzn,zn+1≤Ωτjz0,z1−nυ.
Taking the limit asn⟶∞ in both sides of (15), we get
(16)limn⟶∞Ωτjzn,zn+1=−∞.
It follows fromΩ2 that
(17)limn⟶∞τjzn,zn+1=0.
By (15), for n∈ℕ, we get
(18)τjzn,zn+1ℓΩτjzn,zn+1−Ωτjz0,z1≤−τjzn,zn+1ℓnυ≤0,for all ℓ∈ℕ∪0. Letting n⟶∞ in (18) and applying (16) and (17), one can write
(19)limn⟶∞nτjzn,zn+1ℓ=0.
From (19), there is n1∈ℕ so that nτjzn,zn+1ℓ≤1, for all n≥n1. It follows that
(20)τjzn,zn+1≤1n1/ℓ,∀n≥n1.
Using (20) for m>n>n1, we have
(21)τjzn,zm≤jzn,zmτjzn,zn+1+τjzn+1,zm≤jzn,zmτjzn,zn+1+jzn,zmjzn+1,zmτjzn+1,zn+2+τjzn+2,zm≤jzn,zmτjzn,zn+1+jzn,zmjzn+1,zmτjzn+1,zn+2+⋯+jzn,zmJzn+1,zmjzn+2,zm.⋯jzm−2,zmjzm−1,zmτjzm−1,zm≤jz1,zmjz2,zm⋯jzn,zmτjzn,zn+1+jz1,zmjz2,zm⋯jzn+1,zmτjzn+1,zn+1+⋯+jz1,zmjz2,zm⋯jzm−1,zmτjzm−1,zm.
It should be noted that the series(22)∑n=n1∞τjzn,zn+1∏i=1njzi,zm,converges because
(23)∑n=n1∞τjzn,zn+1∏i=1njzi,zm≤∑n=n1∞1n1/ℓ∏i=1njzi,zm≤1λ∑n=n1∞1n1/ℓ,which is convergent. Put E=∑n=n1∞τjzn,zn+1∏i=1njzi,zm and Ek=∑k=1∞τjzk,zk+1∏i=1njzi,zm. Thus, for m>n>n1, inequality (20) implies that τjzn,zm≤Em−1−En−1. Passing n⟶∞, we conclude that zn is a Cauchy sequence in V,τj but a ¥-space V,τj is complete so there exists z∈V such that limn⟶∞zn=z.
Now, we want to show thatz is a common FP of Φ and φ. It is clear that
(24)λτjz2n+2,φz≤λHΦz2n+1,φz≤λHΦz2n+1,φz,λ>1.
Hence,(25)λτjz2n+2,φz≤λHΦz2n+1,φz.
SinceΩ is strictly increasing, therefore
(26)Ωλτjz2n+2,φz≤ΩλHΦz2n+1,φz.
By addingυ to both sides and using (4), we have
(27)υ+Ωλτjz2n+2,φz≤υ+ΩλHΦz2n+1,φz≤Ωmaxτjz2n+1,z,1+τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z,τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z2.
Sinceυ>0, we have
(28)Ωλτjz2n+2,φz≤Ωmaxτjz2n+1,z,1+τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z,τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z2.
SinceΩ is strictly increasing, we get
(29)λτjz2n+2,φz≤maxτjz2n+1,z,1+τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z,τjz2n+1,Φz2n+1τjz,φz1+τjz2n+1,z2.
Lettingn⟶∞ in the above inequality, one can write
(30)λτjz,φz≤max0,τjz,φz,0=τjz,φz.
This implies thatλ−1τjz,φz≤0. So, we have τjz,φz⟶0. Thus, z∈φz. Similarly, one can prove that z∈Φz. Therefore, z is a common fixed point of Φ and φ. In order to show the uniqueness, let z∗ be another common FP of Φ and φ so that z∗≠z. Then, by ΦΩrGS-contraction (4), one can obtain
(31)υ+Ωλτjz,z∗≤υ+ΩλHΦz,φz∗≤Ωmaxτjz,z∗,1+τjz,Φzτjz∗,φz∗1+τjz,z∗,τjz,Φzτj+τjz,z∗,φ+τjz,z∗+τjz,z∗=Ωτjz,z∗.
Based on the properties ofΩ and since υ>0, we conclude that 1−λτjz,z∗≤0. This means τjz,z∗⟶0, i.e., z=z∗. This finishes the proof.☐Remark 9.
The contraction condition of Theorem8 is reduced to the following:
(i)
Φφ,ΩrGS-contraction in b-metric-like space, if jy,z=j≥1(ii)
Φφ,ΩrGS-contraction in metric-like space, if jy,z=1(iii)
ΦΩrGS-contraction in ¥-spaces, if Φ=φ(iv)
ΦΩrGS-contraction in b-metric-like space, if Φ=φ and jy,z=j≥1(v)
ΦΩrGS-contraction in b-metric-like space, if jy,z=1 and Φ=φCorollary 10.
Theorem8 is still valid if one of the following forms is chosen for βz1,z2 in Equation (5):
(1)
βz1,z2=τjz1,z2(2)
βz1,z2=1+τjz1,Φz1τjz2,φz2/1+τjz1,z2(3)
βz1,z2=1+τjz1,Φz1τjz2,φz2/1+τjz1,z2(4)
βz1,z2=maxτjz1,z2,1+τjz1,Φz1τjz2,φz2/1+τjz1,z2(5)
βz1,z2=maxτjz1,z2,τjz1,Φz1τjz2,φz2/τjz1,Φz1τjz2,φz2(6)
βz1,z2=max1+τjz1,Φz1τjz2,φz2/τjz1,Φz1τjz2,φz2,τjz1,Φz1τjz2,φz2/1+τjz1,z2


## 4. Illustrative Examples

In this section, we give some illustrative examples to support Theorem8 and its consequences.Example 1.
LetV=0,1. Define τj:V×V⟶ℝ+ by τjz1,z2=z1+z22 for all z1,z2∈V; then, the pair V,τj is ¥-space with jz1,z2=2+z12+z22. Define Φ,φ:V⟶VCB and Ω:ℝ+⟶ℝ+ by
(32)Φz1=0,z13,φz2=0,z23,forallz1,z2∈0,1,λ>1,and Ωσ=lnσ, for all σ>0, respectively. Now, we have
(33)3HΦz1,φz2=3H0,z13,0,z23≤3z13+z232=13z1+z22=13τjz1,z2≤13βz1,z2.
Taking log to the basee on both sides, we get
(34)ln3HΦz1,φz2≤ln13βz1,z2=ln13+lnβz1,z2.This implies that(35)ln3HΦz1,φz2−lnβz1,z2≤−ln3⇒ΩλHΦz1,φz2−Ωβz1,z2≤−υ.Hence, the contraction condition (7) is fulfilled with λ=3>1 and υ=ln3>0. Thus, all stipulations of Theorem 8 are fulfilled, and 0 is a unique common FP of Φ and φ.Example 2.
Assume thatV=0,1. Define a distance function τj over a set V by τjz1,z2=z1+z22 and j:V×V⟶1,∞ by jz1,z2=2+2z1+3z2. Then, V,τj forms a ¥-space. Define multivalued mappings Φ,φ:V⟶VCB by Φz=2/15z, for all z∈0,1 and
(36)φz=210z,ifz∈0,1,0,ifz=1.Define the functionΩ:ℝ+⟶ℝ+ by Ωσ=lnσ2+σ for all σ>0, υ>0, and λ=5/2.Now, to fulfill the conditions of Theorem8, we discuss the following cases:Case 1.
Ifz1∈0,1 and z2=1, consider
(37)ΩλHΦz1,φz2=Ω52H215z1,0=Ω13Hz1,0≤Ω13z1+02=lnz149+z123.Now, consider(38)βz1,z2=maxτjz1,1,1+τjz1,2/15z1τj1,01+τjz1,1,τjz1,2/15z1τj1,01+τjz1,1=maxz1+12,1+514/225z121+z1+12,289/225z121+z1+12=z1+12.Therefore,(39)Ωβz1,z2=Ωz1+12=lnz1+14+z1+12.Thus,(40)ΩλHΦz1,φz2−Ωβz1,z2≤lnz14/9+z12/3z1+14+z1+12=−υ,for any value of 0≤z1<1 and υ>0. Therefore,
(41)υ+ΩλHΦz1,φz2≤Ωβz1,z2.Case 2.
Ifz2∈0,1 and z1=1, then Φz1=Φ1=2/15 and φz2=1/5z2.
Consider(42)ΩλHΦz1,φz2=Ω52H215,210z2=ΩH13,12z2≤Ω13+z22=ln13+z24+13+z22.Now, consider(43)βz1,z2=maxτj1,z2,1+τj1,2/15τjz2,1/5z21+τj1,z2,τj1,2/15τjz2,1/5z21+τj1,z2=max1+z22,514/225.36/25z221+1+z22,289/225.36/25z221+1+z22=1+z22.Therefore,(44)Ωβz1,z2=Ω1+z22=ln1+z24+1+z22.Thus,(45)ΩλHΦz1,φz2−Ωβz1,z2≤ln1/3+z24+1/3+z221+z24+1+z22=−υ,for any value of 0≤z2<1 and υ>0. Therefore,
(46)υ+ΩλHΦz1,φz2≤Ωβz1,z2.Thus, all requirements of Theorem8 are fulfilled, and 0 is a unique common FP of Φ and φ.Example 3.
Assume thatV=0,1. Define a distance function τj over a set V by τjz1,z2=z1+z2+z12/4+z22/5 and j:V×V⟶1,∞ by jz1,z2=z1+z22. Then, V,τj forms a ¥-space. Define multivalued mappings Φ,φ:V⟶VCB by
(47)Φz=φz=z22,ifz∈0,12,log3z4,ifz∈12,1.Define the functionΩ:ℝ+⟶ℝ+ by Ωσ=logσ for all σ>0, υ>0, and λ=2>1.Case 1.
Ifz1,z2∈0,1/2, then φz1=z12/2 and φz2=z22/2. Consider
(48)ΩλHφz1,φz2=Ω2Hz122,z222≤Ωz12+z22+z144+z245=logz12+z22+z144+z245.Now, consider(49)βz1,z2=maxτjz1,z2,1+τjz1,z12/2τjz2,z22/21+τjz1,z2,τjz1,z12/2τjz2,z22/21+τjz1,z2=maxz1+z2+z124+z225,1+z1+z1/2+z12/4+z14/20z2+z2/2+z22/4+z24/201+z1+z2+z12/4+z22/5,z1+z1/2+z12/4+z14/20z2+z2/2+z22/4+z24/201+z1+z2+z12/4+z22/5=z1+z2+z124+z225.Therefore,(50)Ωβz1,z2=Ωz1+z2+z124+z225=logz1+z2+z124+z225.Thus,(51)ΩλHφz1,φz2−Ωβz1,z2≤logz12+z22+z14/4+z24/5z1+z2+z12/4+z22/5=−υ,forsomeυ>0.Hence,ΩλHφz1,φz2−Ωβz1,z2≤−υ, which implies
(52)υ+ΩλHφz1,φz2≤Ωβz1,z2.Case 2.
Ifz1,z2∈1/2,1, then φz1=log3z1/4 and φz2=log3z2/4. Consider
(53)ΩλHφz1,φz2=Ω2Hlog3z14,log3z24=ΩHlog3z12,log3z22≤Ωlog3z12+log3z22+log3z1216+log3z2220=loglog3z12+log3z22+log3z1216+log3z2220.Now, consider(54)βz1,z2=maxτjz1,z2,1+τjz1,log3z1/4τjz2,log3z2/41+τjz1,z2,τjz1,log3z1/4τjz2,log3z2/41+τjz1,z2=maxz1+z2+z124+z225,1+z1+log3z1/4+z12/4+log3z12/80z2+log3z2/4+z22/4+log3z22/801+z1+z2+z12/4+z22/5,z1+log3z1/4+z12/4+log3z12/80z1+log3z2/4+z22/4+log3z22/801+z1+z2+z12/4+z22/5=z1+z2+z124+z225.Therefore,(55)Ωβz1,z2=Ωz1+z2+z124+z225=logz1+z2+z124+z225.Thus,(56)ΩλHφz1,φz2−Ωβz1,z2≤loglog3z1/2+log3z2/2+log3z12/16+log3z22/20z1+z2+z12/4+z22/5=−υ,forsomeυ>0.Hence,ΩλHφz1,φz2−Ωβz1,z2≤−υ, which implies
(57)υ+ΩλHφz1,φz2≤Ωβz1,z2.Case 3.
Ifz1∈0,1/2 and z2∈1/2,1, then φz1=z12/2 and φz2=log3z2/4. Consider
(58)ΩλHφz1,φz2=Ω2Hz122,log3z24=ΩHz12,log3z22≤Ωz12+log3z22+z144+log3z2220=logz12+log3z22+z144+log3z2220.Now, consider(59)βz1,z2=maxτjz1,z2,1+τjz1,z12/2τjz2,log3z2/41+τjz1,z2,τjz1,z12/2τjz2,log3z2/41+τjz1,z2=maxz1+z2+z124+z225,1+z1+z12/2+z12/4+z14/20z2+log3z2/4+z22/4+log3z22/801+z1+z2+z12/4+z22/5,z1+z1+z12/2+z12/4+z14/20z2+log3z2/4+z22/4+log3z22/801+z1+z2+z12/4+z22/5=z1+z2+z124+z225.Therefore,(60)Ωβz1,z2=Ωz1+z2+z124+z225=logz1+z2+z124+z225.Thus,(61)ΩλHφz1,φz2−Ωβz1,z2≤logz12+log3z2/2+z14/4+log3z22/20z1+z2+z12/4+z22/5=−υ,forsomeυ>0.Hence,ΩλHφz1,φz2−Ωβz1,z2≤−υ, which implies
(62)υ+ΩλHφz1,φz2≤Ωβz1,z2.Case 4.
Ifz1∈1/2,1 and z2∈0,1/2, then φz1=log3z1/4 and φz2=z22/2. Consider
(63)ΩλHφz1,φz2=Ω2Hlog3z14,z222=ΩHlog3z12,z22≤Ωlog3z12+z22+log3z1220+z244=loglog3z12+z22+log3z1220+z244.Now, consider(64)βz1,z2=maxτjz1,z2,1+τjz1,log3z1/4τjz2,z22/21+τjz1,z2,τjz1,log3z1/4τjz2,z22/21+τjz1,z2=maxz1+z2+z124+z225,1+z1+log3z1/4+z12/4+log3z12/80z2+z22/2+z22/4+z24/201+z1+z2+z12/4+z22/5,1+z1+log3z1/4+z12/4+log3z12/80z2+z22/2+z22/4+z24/201+z1+z2+z12/4+z22/5=z1+z2+z124+z225.Therefore,(65)Ωβz1,z2=Ωz1+z2+z124+z225=logz1+z2+z124+z225.Thus,(66)ΩλHφz1,φz2−Ωβz1,z2≤loglog3z1/2+z22+log3z12/20+z24/4z1+z2+z12/4+z22/5=−υ,forsomeυ>0.Hence,ΩλHφz1,φz2−Ωβz1,z2≤−υ, which implies
(67)υ+ΩλHφz1,φz2≤Ωβz1,z2.From the four cases, we conclude the contraction condition (iii) of Remark9 is fulfilled and 0 is a unique FP of φ.Example 4.
LetV=It:t∈1,2,⋯,100 be a sequence defined by
(68)I1=1.2,I2=2.3,⋯,It=tt+1.Suppose thatτj:V×V⟶0,∞ and j:V×V⟶1,∞ are functions described by τjz1,z2=maxz1,z22 and jz1,z2=1+z1+z2, respectively, for all z1,z2∈V. Then, V,τj is ¥−space. Define the multivalued mappings Φ,φ:V⟶VCB and Ω:ℝ+⟶ℝ+by
(69)ΦI1=φI1=I1,ΦIt=φIt=It−1,and Ωσ=lnσ, for all σ>0, respectively. Now, we have the following calculations for two cases:Case 1.
For each∈ℕ,s>2, and t=1, we have
(70)HφIs,φI1βIs,I1=maxIs−1,I12maxτjIs,I1,1+τjIs,φIsτjI1,φI1/1+τjIs,I1,τjIs,φIsτjI1,φI1/1+τjIs,I1=maxIs−1,I12maxτjIs,I1,1+τjIs,Is−1τjI1,I1/1+τjIs,It,τjIs,Is−1τjI1,I1/1+τjIs,It=Is−12maxIs2,1+Is2I12/1+Is2,Is2.I12/1+Is2=Is−12maxIs2,I12,Is2.I12/1+Is2=Is−12Is2=s−12s2s2s+12=s−12s+12≤e−0.02.Case 2.
For eachs,t∈ℕ,s>t>1, we get
(71)HφIs,φItβIs,It=maxIs−1,It−12maxτjIs,It,1+τjIs,φIsτjIt,φIt/1+τjIs,It,τjIs,φIsτjIt,φIt/1+τjIs,It=maxIs−1,It−12maxτjIs,It,1+τjIs,Is−1τjIt,It−1/1+τjIs,It,τjIs,Is−1τjIt,It−1/1+τjIs,It=Is−12maxIs2,1+Is2It2,/1+Is2,Is2.It2/1+Is2=Is−12maxIs2,It2,Is2.It2/1+Is2=Is−12Is2=s−12s2s2s+12=s−12s+12≤e−0.02.Multiply1.02 on the both cases and taking ln, we have
(72)ln1.02HφIs,φIt−lnβIs,It≤ln1.02×e−0.02⇒ln1.02HφIs,φIt−lnβIs,It≤−0.00019737⇒lnλHφIs,φIt−lnβIs,It≤−υ⇒υ+ΩλHφIs,φIt≤ΩβIs,It.Therefore, the condition (iii) of Remark9 is fulfilled with υ=0.00019737 and λ=1.02. Hence, φ has a unique FP.


## 5. The Damped Harmonic Oscillator

Assume that we have a body of massμ moving back and forth on the x-axis around the equilibrium position x=0 and has a position xt at time t (see Figure 1). The force subjected to by the spring gives from
(73)Fs=−kx.Figure 1
Forced oscillations.In addition to the damping force that impedes the movement of the body, it gives from(74)Fd=bdxdt.From the second law of motion we find that(75)Fnet=md2xdt2,where ,b, and m are positive constants. Previously, this force is called the force damping oscillation.Now suppose that there is an additional time-dependent forceFt that affects the body. Based on Newton’s second law, we can write
(76)md2xdt2+bdxdt+kx=Ft,x0=x′0=0.By Green’s function, Problem (76) is equivalent to the following Fredholm integral equation:
(77)wt1=∫01Ot1,rBt1,r,wrdr,t1∈0,1,where B:0,1×0,1×ℝ⟶ℝ is a continuous function depends on F and the continuous Green’s function O for critically damping oscillation is described by
(78)Ot1,r=−reυt1−t,0≤r≤t1≤1,−teυt−t1,0≤t1≤r≤1,and υ can be found in terms of m,b, and k.Suppose thatV=ßC0,1,ℝ is the set of all bounded real continuous functions defined on 0,1, equipped with
(79)τjz1,z2=z1+z2∞2,∀z1,z2∈V,where z∞=supt∈0,1zte−υ. Define the function j:V×V⟶1,∞ by jz1,z2=1+z1+z2. Then, V,τj is a complete ¥-space.In order to study the existence of a solution to Problem (77), we shall consider the following stipulations:
(a)
There is a continuous functionB:0,1×0,1×ℝ⟶ℝ and for r,t1∈0,1 and λ>1 so that
(80)Bt1,r,z1r+Bt1,r,z1r2≤e−υλβz1r,z2r,for all z1,z2∈ℝ, where
(81)βz1,z2=maxz1+z22,1+z1+Φz12z2+Φz221+z1+z22,z1+Φz12z2+Φz221+z1+z22.and Φ:C0,1⟶ßC0,1 is a multivalued operator defined on the beginning proof.(b)
∫01Ot1,r≤1, for some t1>0Now, we present our main result in this part.Theorem 11.
Assume thatV,τj is a complete ¥-space. Then, Problem (77) has a unique solution provided that the stipulations (a) and (b) are fulfilled.Proof.
Define an operatorΦ:C0,1⟶ßC0,1 by
(82)Φwt1=∫01Ot1,rBt1,r,wrdr,t1∈0,1.
It should be noted that the unique FP of the mappingΦ is a solution of the nonlinear Problem (77). Hence, it is a solution to the differential equation of the second degree (76).
Forz1,z2∈V, then by the hypotheses (a) and (b), we can get
(83)Φz1t1+Φz2t12=∫01Ot1,rBt1,r,z1r+Bt1,r,z2rdr2≤∫01Ot1,r2Bt1,r,z1r+Bt1,r,z2r2dr≤∫01Bt1,r,z1r+Bt1,r,z2r2dr≤∫01e−υλβz1r,z2rdr=e−υλ∫01eυmaxz1+z22e−υ,1+z1+Φz12z2+Φz22e−2υ1+z1+z22e−υ,z1+Φz12z2+Φz22e−2υ1+z1+z22e−υdr=e−υλ∫01erυmaxτjz1,z2,1+τjz1,Φz1τjz2,φz21+τjz1,z2,τjz1,Φz1τjz2,φz21+τjz1,z2dr=e−υλβz1,z2∫01eυdr=e−υλβz1,z2×eυ.
Hence forz1,z2∈V,(84)λΦz1t1+Φz2t12×e−υ≤e−υβz1,z2,which yields
(85)λτjΦz1,Φz2≤e−υβz1,z2.
Taking thelog for the base e to both side, we have
(86)lnλτjΦz1,Φz2≤−υ+lnβz1,z2.
Describe the functionΩ:ℝ+⟶ℝ+ by Ωσ=lnσ,σ>0 in (86), so that Ω∈Σ, and we get
(87)υ+ΩλτjΦz1,Φz2≤Ωβz1,z2,where
(88)βz1,z2=maxτjz1,z2,1+τjz1,Φz1τjz2,Φz21+τjz1,z2,τjz1,Φz1τjz2,Φz21+τjz1,z2.
Therefore, all requirements of Theorem11 under a special case (iii) (ΦΩrGS-contraction) are fulfilled. Then, there is a unique FP of the mapping Φ, which is a unique solution to Problem (77) and then to (76).☐


## 6. Newton’s Law of Cooling

Newton’s law of cooling is a differential equation that foresees the cooling of somebody that placed in a colder environment; then, the fixed point method can be applied in finding the solution of this differential equation as follows. Consider the IVP in the form of(89)R′t=Ξt,Rt,Rt0=R0.We say thatRt is a solution of Problem (89) if and only if [25]
(90)Rt=R0+∫t0tΞℓ,Rℓdℓ,for all t,ℓ∈t0,t, where the functions R:t0,t⟶ℝ and Φ:t0,t×ℝ⟶ℝ are continuous.Define a distance normτj℘,ℏ=℘+ℏ∞2 on the set of all real continuous functions V=Ct0,t,ℝ defined on t0,t, for all ℘,ℏ∈V, where ℘∞=supt∈t0,t℘te−tυ. Then, the pair V,τj is a complete ¥-space with j℘,ℏ=1+℘+ℏ.Now, we present the main theorem in this section.Theorem 12.
Assume thatV,τj is a complete ¥-space. Assume also, for all t,ℓ∈t0,t, there is a constant υ∈ℝ+ so that
(91)Ξℓ,℘ℓ+Ξℓ,ℏℓ≤e−υ/2λt−t0℘ℓ+ℏℓ,where ℘,ℏ∈V,λ>1. Let ℘ù,ℏù∈V so that ℘ù+ℏù≤1, for all ù∈ℝ. Then, Equation (90) has a unique solution.Proof.
Define an operatorΦ:V⟶V by
(92)ΦRt=R0+∫t0tΞℓ,Rℓdℓ.
It is clear that, ifΦR=R, then R is a solution of the Equation (90) and then (89).
For℘,ℏ∈V, one can write
(93)Φ℘t+Φℏt2=℘0+∫t0tΞℓ,℘ℓdℓ+ℏ0+∫t0tΞℓ,ℏℓdℓ2≤∫t0tΞℓ,℘ℓ+Ξℓ,ℏℓdℓ2≤∫t0te−υ/2λt−t0℘ℓ+ℏℓdℓ2=∫t0te−υ/2λt−t0×e−2υt×e2υt×℘ℓ+ℏℓdℓ2=e−υλt−t02×τj℘,ℏ×e2υt∫t0tdℓ2=e−υλτj℘,ℏe2υt≤e−υλβ℘,ℏe2υt,where
(94)β℘,ℏ=maxτj℘,ℏ,1+τj℘,Φ℘τjℏ,Φℏ1+τj℘,ℏ,τj℘,Φ℘τjℏ,Φℏ1+τj℘,ℏ.
Hence, one can write(95)Φ℘t+Φℏt×e−υt2≤e−υλβ℘,ℏ,yields,
(96)Φ℘t+Φℏt∞2≤e−υλβ℘,ℏ.
It follows that(97)λτjΦ℘t,Φℏt≤e−υβ℘,ℏ.
ConsiderΩσ=lnσ for σ>0 and Ω∈Σ; then,
(98)lnλτjΦ℘t,Φℏt≤lne−υβ℘,ℏ,or
(99)υ+lnλτjΦ℘t,Φℏt≤lnβ℘,ℏ.
Equivalently,(100)υ+ΩλτjΦ℘t,Φℏt≤Ωβ℘,ℏ.
Based on Theorem8 under a special case (iii) of Remark 9 (ΦΩrGS-contraction), Φ possesses a unique FP, which is a unique solution to Equation (90) and then to (89).
Now, we describe Newton’s law of cooling as follows: Let(101)η′t=−ξηt−ηϰ,where ηt and ηù refer to the temperature of the object at the time t and the temperature of the environment, respectively, and ξ is proportionality constant. If we put ηt0=η0, then we obtain the IVP below:
(102)η′t=−ξηt−ηùηt0=η0.
Assume thatΞt,η=−ξηt−ηù. Then, we have
(103)Ξℓ,℘ℓ+Ξℓ,ℏℓ=−ξ℘t−℘ϰ−ξℏt−ℏϰ=ξ℘ℓ+ℏℓ−℘ù+ℏϰ≤ξ℘ℓ+ℏℓ+℘ϰ+ℏϰ≤ξ℘ℓ+ℏℓ=e−υ/2λt−t0℘ℓ+ℏℓ.
According to Theorem12, the initial value problem (102) has a unique solution. It should be noted that the exact solution of (102) is ηt=ηù+η0−ηùe−ξt−t0.☐


## 7. Infinite Systems of Fractional ODEs

In this part, we study the sufficient condition to find the solution to an infinite nonlinear fractional system from ODEs on the space of all real sequences whose limit is finite which denoted withc.Assume that the system is defined as follows:(104)ℸρi=e−υqtqitℸi+σiℸ,ℸ1,ℸ2,⋯,ρi∈0,1,ℸκj=e−υqtqjtℸj+σjℸ,ℸ1,ℸ2,⋯,κj∈0,1,ℸ0=ℸ0i,i,j=1,2,⋯,where t∈J,J is any fixed interval on the real line and υ is the positive real number.Suppose thatV=c equipped with τj℘,ℏ=sup℘i+ℏi2e−υ for all ℘,ℏ∈V, and j:V×V⟶1,∞ is a function described by j℘,ℏ=1+℘+ℏ. Then, V,τj is a complete ¥-space.System (104) will be studied under the following hypotheses:(hy1) ℸ0=ℸ0i∈c and σ=σ1,σ2,⋯:J×c⟶c(hy2) the function qit>0 is a continuous for all t∈J and i=1,2⋯(hy3) for all ℘,ℏ∈V, we have
(105)σit,℘+σit,ℏ≤e−υ2λβ℘,ℏ,λ>1,where ℘=℘1,℘2,⋯, ℘=ℏ1,ℏ2,⋯ and
(106)β℘,ℏ=maxτj℘,ℏ,1+τj℘,Φi℘τjℏ,φjℏ1+τj℘,ℏ,τj℘,Φi℘τjℏ,φjℏ1+τj℘,ℏ,(hy4) for all ℘,ℏ∈V and for t∈J,(107)qit℘+qjtℏ≤qt2λβ℘,ℏ.Now, we state and prove our main theorem.Theorem 13.
System (104) has a unique solution under hypotheses (hy1)-(hy4).Proof.
LetΦit,℘,φjt,ℏ:J×c⟶c be mappings described by
(108)Φit,℘=e−υqtqit℘i+σi℘,℘1,℘2,⋯,φjt,ℏ=e−υqtqjtℏj+σjℏ,ℏ1,ℏ2,⋯.
A sufficient condition for a solution to System (104) to exist is that there is a common FP of the mappings (108). To achieve this, consider
(109)Φit,℘+φjt,ℏ=e−υqtqit℘i+σi℘,℘1,℘2,⋯+e−υqtqjtℏj+σjℏ,ℏ1,ℏ2,⋯≤e−υqtqit℘i+e−υqtqjtℏj+σi℘,℘1,℘2,⋯+σjℏ,ℏ1,ℏ2,⋯≤e−υqt×qt2λβ℘,ℏ+e−υ2λβ℘,ℏ=e−υλβ℘,ℏ.
Squaring both sides, we get(110)Φit,℘+φjt,ℏ2≤e−2υλβ℘,ℏ,or
(111)λΦit,℘+φjt,ℏ2e−υ≤e−3υβ℘,ℏ≤e−υβ℘,ℏ.
It follows that after taking the supremum,(112)λτjΦi,φj≤e−υβ℘,ℏ.
By taking theln of both sides, we find that
(113)υ+lnλτjΦi,φj≤lnβ℘,ℏ.
LettingΩσ=lnσ,Ω∈Σ, one can write
(114)υ+ΩλτjΦi,φj≤Ωβ℘,ℏ,where
(115)β℘,ℏ=maxτj℘,ℏ,1+τj℘,Φi℘τjℏ,φjℏ1+τj℘,ℏ,τj℘,Φi℘τjℏ,φjℏ1+τj℘,ℏ.
Hence, we conclude by Theorem12 that System (104) has a unique solution.☐



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