# Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Authors: Manzoor Ahmad; Jiqiang Jiang; Akbar Zada; Syed Omar Shah; Jiafa Xu
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.


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## Body

## 1. Introduction

From the last few years, fractional differential equations (FDES) theory has gained significant attraction and importance. It arises naturally in various models in areas such as control theory, biology, nonlinear waves of earthquake, mechanics, signal processing, modeling the seepage flow in porous media, and in fluid dynamics, memory mechanism and hereditary properties of materials. Some recent existence and uniqueness (EU) results of solutions for FDES with initial as well as boundary conditions can be found in [1–8]. In fact, FDES are the effective tools in real-world problems that motivate many researchers to work in this field, see [6, 9–20] and references cited therein.Another important aspect in the qualitative theory of differential equations (DES), which is exclusively studied for integer-order DES, is Hyers–Ulam (HU) stability and its various types. This stability concept was originated in 1940 from the question of Ulam [21], which was answered by Hyers [22]. Many researchers extended and generalized Hyers’s results in which the work of Rassias [23] is considered to be the first notable contribution. Many researchers studied HU and HU–Rassias stability of various functional equations, see [24–44] and references cited therein. This field got notorious attention when mathematicians started studying the HU stability for the solution of differential equations, initiated by Obloza [45, 46]. Motivated by the work of Obloza, various classes of integer-order ordinary differential equations were investigated [39, 47]. The idea was then extended for nonintegral-order differential equations; for some recent work, we refer to [48, 49]. As far as we know, only few researchers studied the different kinds of Ulam’s type stabilities for the coupled system of FDES. For details, see [38, 50–52].Nowadays, both Riemann–Liouville-type (ℝL) and Caputo-type derivatives are introduced generally, and the impact of applying it in mathematical physics and equations associated with probability is exposed. The fractional integral that generalizes both ℝL- and Hadamard-type integrals into a single form was initiated by Katugampola [53]. Later on, in [54], new fractional derivative that generalizes the two derivatives was introduced by Katugampola.Motivated by the work [54, 55], in this paper, we study the EU and HU stability of the following implicit switched coupled system of FDE involving the Katugampola–Caputo (Kℂ) fractional derivative:(1)Dcσuασ−fς,uσ,Dcσuασ=ψσ,uσ,vσ,σ∈J=0,T,T>0,0<α<1,Dcσvασ−gς,vσ,Dcσvασ=φσ,uσ,vσ,u0+hu=u0,v0+hv=v0,where σ is a positive real number and Dcσuασ and Dcσvασ are the Katugampola–Caputo fractional derivatives of uσ and vσ, respectively. The functions f,g:J×X×X×⟶X are closed and bounded. Also, ψandφ are nonlocal continuous functions.


## 2. Preliminaries

In this portion, we introduce some notions and preliminaries. SupposeℂJ,X denotes the Banach space (BS) of continuous functions from J=0,T into X, defined by ℂJ,X=u:J⟶X,σ∈J, endowed with norms as u=maxσ∈Juσ,v=maxσ∈Jvσ; indeed, these are BSs’ under these norms, and hence, their product is also BS with u,v=u+v, where uandv are in ℂJ,X.Definition 1.
(see [5]). Let α>0, and the arbitrary order integral in the ℝL sense for a function p:J⟶ℝ is(2)Iαpσ=1Γα∫0σσ−υα−1pυdυ,where the integral on the right-hand side (RHS) is pointwise defined on ℝ+.Definition 2.
(see [54]). Kℂ left-sided noninteger-order integral Iρfa+α of the function f on a closed interval a,b of order α is defined as(3)Iρfασ=ρ1−αΓα∫aσσρ−υρα−1υρ−1dυ,whereα∈C,withℜα>0.
The correspondingKℂ fractional derivative to the above integral is given by(4)Dcρfa+ασ=ρα−n+1Γn−1σ1−ρddtn∫aσσρ−υρn−α−1fυdυ.Definition 3.
(see [5]). The noninteger-order derivative in the Caputo sense of p on closed interval a,b is(5)dαdσαpσ=∫aσσ−υn−α−1Γn−αdndυnpυdυ,α∈n−1,n,where n−1=α. In particular,(6)dαdσαpσ=1Γ1−α∫aσp′υσ−υαdυ,whereφ′υ=dφυdυ,α∈0,1.Moreover, the integral on the RHS is pointwise defined on ℝ+.Lemma 1 (see [56]).
Letα∈n−1,n, for p∈ℂa,b, and the only one solution of dα/dσαpσ=0 has the formula pσ=∑k=0αckσk, where ck∈ℝ, k=1,2,…,α, α=n−1.Lemma 2 (see [56]).
Letα∈n−1,n, for p∈ℂa,b and Iαdα/dσαpσ=pσ+∑k=0αakσk, for some ak∈ℝ, k=1,2,…,α, α=n−1.Definition 4.
(see [57]). Consider BSX. Let Ω1,Ω2:X×X⟶X be two operators, and the operator system(7)pσ=Ω1p,qσqσ=Ω2p,qσis called HU stable if we can find constants Cii=1,2,3,4>0 such that, for each ϱjj=1,2>0 and for each solution p^,q^∈X×X of the inequalities(8)p^−ψp^,q^≤ϱ1q^−φp^,q^≤ϱ2hold, then there exists a solution p˜,q˜∈X of system (7), which satisfies the inequalities(9)p^−p˜≤C1ϱ1+C2ϱ2q^−q˜≤C3ϱ1+C4ϱ2.Lemma 3 (see [57], Theorem 4).
ConsiderBSX with operators Ω1,Ω2:X×X⟶X such that(10)Ω1p,q−Ω1p^,q^≤Λ1p−p^+Λ2q−q^,Ω2p,q−Ω2p^,q^≤Λ3p−p^+Λ4q−q^.
If the spectral radius of(11)ℍ=Λ1Λ2Λ3Λ4,is less than one, then the fixed points corresponding to operational system (10) are HU stable.


## 3. Existence and Uniqueness of the Solution

In this section, we proveEU of system (1). We consider the following assumptions:ℍ
1: suppose ψ,φ:J×X×X⟶X are continuous, and for all μ,ν,μ¯,ν¯∈X×X with σ∈J, there exist Mψ,Mφ,Mψ′,Mφ′>0 such that(12)ψσ,μσ,νσ−ψσ,μ¯σ,ν¯σ≤Mψμσ−μ¯σ+Mψ′νσ−ν¯σ,φσ,μσ,νσ−φσ,μ¯σ,ν¯σ≤Mφμσ−μ¯σ+Mφ′νσ−ν¯σ.ℍ
2: there exist Lf,Lg,Lf′,Lg′>0 such that(13)fσ,μσ,νσ−fσ,μ¯σ,ν¯σ≤Lfμ−μ¯+Lf′ν−ν¯,gσ,μσ,νσ−gσ,μ¯σ,ν¯σ≤Lgμ−μ¯+Lg′ν−ν¯,for all σ∈J,μ,ν,μ¯,ν¯∈X×X.ℍ
3: suppose h:CJ,X⟶ℝ is continuous, and for all μ,v,μ¯,ν¯∈X, there exist a,b>0 such that(14)hμ−hμ¯≤aμ−μ¯andhν−hν¯≤bν−ν¯.ℍ
4: let lf,lg,p,q,p∗,q∗∈CJ,X with lgh=suplfσ,lgσ<1 such that(15)fσ,μσ,νσ≤lfσ+pσμPC+qσν,gσ,μσ,νσ≤lgσ+p∗σμPC+q∗σν,where σ∈J.ℍ
5: let ψ,φ:J×X×X⟶X be continuous, and for all μ,ν∈X with ς∈J, there exist lψ,lψ1,lψ2,lφ,lφ1,lφ2∈CJ,X such that(16)ψσ,μσ,νσ≤lψσ+lψ1σμσ+lψ2νσ,φσ,μσ,νσ≤lφσ+lφ1σμσ+lφ2νσ.Theorem 1.
Letθ1,θ2∈CJ,X and f,g:J×X×X⟶X be continuous, and the solution of(17)Dcσμασ−fς,μσ,Dcσμασ=θ1σ,Dcσvασ−fς,νσ,Dcσvασ=θ2σ,μ0+hμ=μ0,ν0+hν=ν0,is equivalent to(18)μσ=μ0−hμ+σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1θ1υdυ,νσ=ν0−hν+σ1−αΓα∫0σσσ−υσα−1υσ−1gυ,νυ,Dcσναυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1θ2υdυ.Proof.
Consider(19)Dcσμασ−fσ,μσ,Dcσμασ=θ1σ,0<α<1,σ∈J.
Applying integralIρα, we get(20)IσαDcσμασ−Iραfσ,μσ,Dcσμασ=ρIαθ1σ−c0μσ=σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1θ1υdυ−c0.
Usingμ0+hμ=μ0, we have(21)μσ=μ0−hμ+σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1θ1υdυ.
Similarly,(22)νσ=ν0−hν+σ1−αΓα∫0σσσ−υσα−1υσ−1gυ,νυ,Dcσναυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1θ2υdυ.
For the upcoming result, here, we define an operator.
ℍ
6: choose Br=μ,ν∈X×X:μ,ν≤r,u≤r/2,ν≤r/2⊂X, with r≥μ1+μ0+ν0+H+G/1−μ2, p=suppσ:σ∈J, q=supqσ:σ∈J, H=maxhμ,μ∈X, G=maxhν,ν∈X, lψ=suplψσ:σ∈J, lψ1=suplψ1σ:σ∈J, and lψ2=suplψ2σ:σ∈J.(23)μ1=Tσαlgh2−q+q∗+1−q1−q∗lψ+lφσα1−q1−q∗Γα+1,μ2=Tσα1−q∗p+1−qp∗+1−q1−q∗lψ1+lφ1+lψ2+lφ22σα1−q1−q∗Γα+1,maxμ1,μ2<1.
ConstructF=F1,F2andG=G1,G2 on Br as(24)F1μσ=μ0−hμ+σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ,F2νσ=ν0−hν+σ1−αΓα∫0σσσ−υσα−1υσ−1gυ,μυ,Dcσμαυdυ,G1μσ=σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μυ,νυdυ,G2νσ=σ1−αΓα∫0σσσ−υσα−1υσ−1φυ,μυ,νυdυ.Theorem 2.
Let the conditions fromℍ1 to ℍ6 hold; then, system (1) has only one solution.Proof.
For anyμ,ν∈Br,(25)Fμ,ν+Gμ,ν≤Fμ,ν+Gμ,ν=F1μ+F2ν+G1μ,ν+G2μ,ν.
From (24), we have(26)F1μσ=μ0−hμ+σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ≤μ0−hμ+σ1−αΓα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ≤μ0+hμ+σ1−αΓα∫0ςσσ−υσα−1υσ−1Kμυdυ,whereKμσ=fς,μσ,Dcσμασ=fσ,μ0−hμ+IcσKαμσ,Kμσ≤lgh+pσμ+qσKμσ≤lgh+pμ+qKμσ,orKμσ≤lgh+pμ1−q.
Therefore,(27)F1μ≤μ0+H+2lgh+rpTσα21−qσαΓα+1,F2μ≤ν0+G+2lgh+rp∗Tσα21−q∗σαΓα+1.
Furthermore, (24) gives(28)G1μ,νσ=σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μυ,νυdυ≤σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μυ,νυdυ≤σ1−αΓα∫0σσσ−υσα−1υσ−1lψσ+lψ1σμσ+lψ2σνσdυ≤Tσαlψσ+lψ1σμσ+lψ2σνσσαΓα+1≤Tσαlψ+lψ1μ+lψ2νσαΓα+1≤Tσα2lψ+rlψ1+lψ22σαΓα+1.
In a similar way, we get(29)G2μ,ν≤Tσα2lφ+rlφ1+lψ22σαΓα+1.
Therefore, (25) gives(30)Fμ,ν+Gμ,ν≤r.
So,Fμ,ν+Gμ,ν∈Br.
Now, forμ,ν,μ¯,ν¯∈Br and ς∈J, we have(31)Fμ,ν−Fμ¯,ν¯≤F1μ−F1μ¯+F2ν−F1ν¯≤hμ−hμ¯+σα−1Γα∫0σσσ−υσα−1υσ−1Kμυ−Kμ¯υdυ+hν−hν¯+σα−1Γα∫0σσσ−υσα−1υσ−1Kμυ−Kμ¯υdυ,where(32)Kμυ=fυ,μυ,Dcσμαυ,Kμ¯υ=fυ,μ¯υ,Dcσμ¯αυ,Kνυ=gυ,μυ,Dcσμαυ,Kν¯υ=gυ,μ¯υ,Dcσμαυ.
SinceKμυ=fυ,μυ,Dcσμαυ, therefore,(33)Kμσ−Kμ¯σ≤fυ,μσ,Kμσ−fυ,μ¯σ,Kμ¯σ≤Lfμσ−μ¯σ+Lf′Kμσ−Kμ¯σKμσ−Kμ¯σ≤Lf‖μ−μ¯‖1−Lf′,
Similarly, we get(34)Kνσ−Kν¯σ≤Lgν−ν¯1−Lg′.
Using (31), we get(35)Fμ,ν−Fμ¯,ν¯≤aμ−μ¯+LfTσαμ−μ¯σα1−Lf′Γα+1+bν−ν¯+TσαLgν−ν¯σα1−Lg′Γα+1≤a+LfTσασα1−Lf′Γα+1μ−μ¯+b+TσαLgσα1−Lg′Γα+1ν−ν¯≤γ1μ−μ¯+γ2ν−ν¯≤γμ−μ¯+ν−ν¯.Here, γ=maxγ1,γ2, with(36)γ1=a+LfTσασα1−Lf′Γα+1,γ2=b+TσαLgσα1−Lg′Γα+1.
Thus,F is a contraction. For the continuity and compactness of G, take a sequence Sn=μn,νn in Br with μn,νn approaching to μ,ν as n approaches to ∞ in Br. So,(37)Gμn,νnσ−Gμ,νσ=G1μn,νnσ−G1μ,νσ+G2μn,νnσ−G2μ,νσ≤σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μnυ,νnυ−ψυ,μυ,νυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1φυ,μnυ,νnυ−φυ,μυ,νυdυ≤TσαMψμn−μ+Mψ′νn−νσαΓα+1+TσαMφμn−μ+Mφ′νn−νσαΓα+1implies Gμn,νnσ−Gμ,νσ⟶0, as n⟶∞. That is why G is continuous on Br. Now, for the uniform boundedness of G on Br, consider Gμ,ν, and by using (31), we have(38)Gμ,ν≤G1μ,ν+G2μ,ν≤Tσα2lψ+rlψ1+lψ22σαΓα+1+Tσα2lφ+rlφ1+lφ22σαΓα+1=Tσαlψ+lφrσαΓα+1+Tσαlψ1+lψ2+lφ1+lφ22σαΓα+1r.
Thus,G is uniformly bounded on Br. Now, for the equicontinuity of the operator G, take σ1,σ2 from J with σ1≥σ2 and μ,ν∈Br. Since ψσ,μσ,νσ is bounded on Br, we can take supψσ,μσ,νσ=M0. Thus,(39)G1μ,νσ1−G1μ,νσ2≤σ1−αΓα∫0σ1σ1σ−υσα−1υσ−1ψυ,μυ,νυ−σ1−αΓα∫0σ2σ2σ−υσα−1υσ−1ψυ,μυ,νυdυ≤σ1−αΓα∫0σ1σ1σ−υσα−1−σ2σ−υσα−1υσ−1ψυ,μυ,νυdυ+σ1−αΓα∫σ1σ2σ2σ−υσα−1υσ−1ψυ,μυ,νυdυ≤M0σ1−αΓα∫0σ1σ1σ−υσα−1−σ2σ−υσα−1υσ−1dυ+∫σ1σ2σ2σ−υσα−1υσ−1dυ=M0σ1−αΓασ2σ−σ1σασα+σ2σ−σ1σασα=2M0σαΓα+1σ2σ−σ1σα,givesG1μ,νσ1−G1μ,νσ2⟶0 as σ2⟶σ1. Using the same approach with supφσ,μσ,νσ=M0′, we have(40)G2μ,νσ1−G2μ,νσ2=2M0′σαΓα+1σ2σ−σ1σα⟶0,σ1⟶σ2.
Combining these inequalities, we getGμ,νσ1−Gμ,νσ2⟶0 as ς2⟶ς1; hence, G is relatively compact on Br. Thus, by Arzela–Ascolli AA theorem, G is compact and continuous, so (1) has a unique solution.Theorem 3.
Under conditionsℍ1 to ℍ5 with ζfg<1, system (1) has a unique solution.Proof.
We define the operatorΨ=Ψ1,Ψ2:X×X⟶X by(41)Ψ=Ψ1μ,νσ,Ψ2μ,νσ,σ∈J,where(42)Ψ1μ,νσ=μ0−hμ+σα−1Γα∫0σσσ−υσα−1υσ−1fυ,μυ,Dcσμαυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μυ,νυdυ,Ψ2μ,νσ=ν0−hν+σα−1Γα∫0σσσ−υσα−1υσ−1gυ,νυ,Dcσναυdυ+σ1−αΓα∫0σσσ−υσα−1υσ−1ψυ,μυ,νυdυ.
Forμ,ν,μ¯,ν¯∈X×X and ς∈J, we get(43)Ψμ,νσ−Ψμ¯,ν¯σ≤aμ−μ¯+TσαLfμ−μ¯σα1−Lf′Γα+1+TσαMψμ−μ¯+Mψ′ν−ν¯σαΓα+1+bν−ν¯+TσαLgν−ν¯σα1−Lg′Γα+1+TσαMφμ−μ¯+Mψ′ν−ν¯σαΓα+1=a+TσασαΓα+1Lf1−Lf′+Mψ+Mφ‖μ−μ¯‖+b+TσασαΓα+1Lg1−Lg′+Mψ′+Mψ′ν−ν¯≤ζfgμ−μ¯,ν−ν¯,withζf=a+TσασαΓα+1Lf1−Lf′+Mψ+Mφ,ζg=b+TσασαΓα+1Lg1−Lg′+Mψ′+Mψ′,ζfg=maxζf,ζg.Hence, Ψ is a contraction, and by the assumption that ζfg<1, (1) has only one solution.


## 4.HU Stability

Now, we are analyzing different kinds of stabilities such asHU stability of the proposed system given in (1).Theorem 4.
Let the hypothesis fromℍ1 to ℍ5 hold true with the conditions ζfg<1, and if the matrix ℚ⟶0, then (1) is HU stable.Proof.
Proceeding from Theorem3, for any μ,ν,μ∗,ν∗∈X and ς∈J, we have(44)Ψ1μ,νσ−Ψ1μ∗,ν∗σ≤aμ−μ∗+TσαLfμ−μ∗σα1−Lf′Γα+1+TσαMψμ−μ∗+Mψ′ν−ν¯σαΓα+1≤a+TσαLfσα1−Lf′Γα+1+MψTσασαΓα+1μ−μ∗+TσαMψ′σαΓα+1ν−ν¯=ℂ1μ−μ∗+ℂ2ν−ν∗.
Similarly,(45)Ψ2μ,νσ−Ψ2μ∗,ν∗σ≤MφTσασαΓα+1μ−μ∗+b+TσαLgσα1−Lg′Γα+1+MφTσασαΓα+1ν−ν∗=ℂ3μ−μ∗+ℂ4ν−ν∗,where(46)ℂ1=a+TσαLfσα1−Lf′Γα+1+MψTσασαΓα+1,ℂ2=TσαMψ′σαΓα+1,ℂ3=MφTσασαΓα+1,ℂ4=b+TσαLgσα1−Lg′Γα+1+MφTσασαΓα+1.
Writing together the above inequalities, we have(47)Ψ1μ,νσ−Ψ1μ∗,ν∗σ≤ℂ1μ−μ∗+ℂ2ν−ν∗,Ψ2μ,νσ−Ψ2μ∗,ν∗σ≤ℂ3μ−μ∗+ℂ4ν−ν∗.
From this, we get(48)Ψμ,ν−Ψμ∗,ν∗≤ℚμ−μ∗,ν−ν∗,where ℚ=ℂ1ℂ2ℂ3ℂ4. As given, ℚ converges to 0; therefore, (1) is HU stable.Example 1.
Consider(49)Dcσνσα=14eσ+15eσμσ+Dcσνσα+t+sinμσ+cosμσ50,σ∈J,Dcσvσα=110e2t+120eσvσ+Dcσvσα+cosμσ+vσt+100.μ0+∑k=1nakμσk=0,ν0+∑k=1nbkνσk=0,where σ=0.4,α∈0,1,ak,bk>0,k=0,1,2,…,n. Set functions as(50)fσ,μσ,νσ=14eσ+15eσμσ+Dcσvσα,gσ,μσ,νσ=110e2t+120eσvσ+Dcσνσα,hμ=∑k=1nbkνσk,hν=∑k=1nbkνσk.
Takingμ,ν,μ¯,ν¯ from X and σ∈0,1, we have(51)fσ,μσ,νσ−fσ,μ¯σ,ν¯σ≤15μ−μ¯+15ν−ν¯,gσ,μσ,νσ−gσ,μ¯σ,ν¯σ≤120μ−μ¯+120ν−ν¯,ψσ,μσ,νσ−ψσ,μ¯σ,ν¯σ≤150μ−μ¯+150ν−ν¯,φσ,μσ,νσ−φσ,μ¯σ,ν¯σ≤1100μ−μ¯+1100ν−ν¯,hμ−hμ¯≤∑k=1makμσk−μ¯σk,hν−hν¯≤∑k=1mbkνσk−ν¯σk.
In particular, let we takeT=1,α=1/2>0,σ=0.4, and from the inequalities in (51), Lf=Lf′=1/5,Lg=Lg′=1/20,Mψ=Mψ′=1/50,Mφ=Mφ′=1/100,∑ak=∑bk=1/3.
Thus, we haveζfg=0.8328<1. Therefore, by Theorem 3, (5.1) has a unique solution. After calculations, we get(52)ℚ=0.81510.03570.01780.4450.
Upon calculations, we obtained the eigenvalues 0.81675 and 0.4433, which show thatℚ is converging to zero. Using Theorem 4, the solution of (5.1) is HU stable.


## 5. Conclusion

In this manuscript, we usedAA theorem and Banach contraction principle to achieve the sufficient conditions for EU of solutions to a nonlocal implicit switched system. With the help of assumptions, we proved the HU stability result for the couple system given in (1).



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