# Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives

Authors: Usman Riaz; Akbar Zada; Zeeshan Ali; Manzoor Ahmad; Jiafa Xu; Zhengqing Fu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This work is committed to establishing the assumptions essential for at least one and unique solution of a switched coupled system of impulsive fractional differential equations having derivative of Hadamard type. Using Krasnoselskii’s fixed point theorem, the existence, as well as uniqueness results, is obtained. Along with this, different kinds of Hyers–Ulam stability are discussed. For supporting the theory, example is provided.


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## Body

## 1. Introduction

Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals and derivatives of arbitrary order. Fractional differential equations (FDEs) have played a significant role in many engineering and scientific disciplines, e.g., as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, capacitor theory, electrical circuits, electron analytical chemistry, biology, control theory, and fitting of experimental data [1–3]. FDEs also serve as an excellent tool for the description of hereditary properties of various materials and processes [4]. The theory of FDEs, involving different kinds of boundary conditions, has been a field of interest in pure and applied sciences. Nonlocal conditions are used to describe certain features of applied mathematics and physics such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, and population dynamics [5–16].In the classical text [17], it has been mentioned that Hadamard in 1892 [18] suggested a concept of fractional integro-differentiation in terms of the fractional power of the type (x(d/dx))p in contrast to its Riemann–Liouville counterpart of the form (d/dx)p. The kind of derivative introduced by Hadamard contains the logarithmic function of the arbitrary exponent in the kernel of the integral appearing in its definition. Hadamard construction is invariant in relation to dilation and is well suited to the problems containing half-axes. Coupled systems of FDEs have also been investigated by many authors. Such systems appear naturally in many real-world situations. Some recent results on the topic can be found in a series of papers [19–40].Another aspect of FDEs which has very recently got attention of the researchers is concerning the Ulam-type stability analysis of the aforesaid equations. The mentioned stability was first pointed out by Ulam [41] in 1940, which was further explained by Hyers [42], over Banach space. Later on, many researchers have done valuable work on the same task and interesting results were formed for linear and nonlinear integral and differential equations; for details see [43, 44]. This stability analysis is very useful in many applications, such as numerical analysis and optimization, where finding the exact solution is quite difficult. For detailed study of Ulam-type stability with different approaches, we recommend papers [44–52].Existence and uniqueness of Cauchy problems for fractional differential equations involving the Hadamard derivatives have been discussed by Kilbas et al. [53]. Using the contraction principle, existence and uniqueness of the solution of sequential fractional differential equations with Hadamard derivative have been explored by Klimek [54]. Recently, Wang et al. [55] discussed the existence, blowing-up solutions, and Ulam–Hyers stability of fractional differential equations with Hadamard derivative by using some classical methods. Further, Ahmad and Ntouyas [20] and Ma et al. [56] studied two-dimensional fractional differential systems with Hadamard derivative. Wang et al. [57] studied the fractional impulsive Cauchy problem of the form (1)HD1+αut-ft,ut=0,α∈0,1,t∈1,e-t1,t2,…,tm,Δuti=HI+11-αuti+-IH+11-αuti-=pi,pi∈R,i=1,2,…,m,HI+11-αu1+=u0,u0∈R,where HD1+α denotes left-sided Hadamard fractional derivative of order α with the lower limit 1 and HI+11-α denotes left-sided Hadamard fractional integral of order 1-α. In [58], Wang et al. studied the existence and Hyers–Ulam stability of switched coupled problem: (2)cDσpt-ftpt=Θt,pt,qt,t∈J,t≠tk,cDδqt-gtqt=Ψt,pt,qt,t∈J,t≠tk,Δptt=tk=Ikpk,Δqtt=tk=Ikqk,ptt=0+ϕp=p0,qtt=0+φq=q0, where cD denotes the Caputo derivative of order σ,δ∈(0,1].Motivated by the work in [58], we consider the following switched coupled impulsive FDEs involving Hadamard derivatives:(3)HDαpt-ft,pt,qt=0,t∈J,t≠ti,i=1,2,…,m,HDβqt-gt,pt,qt=0,t∈J,t≠tj,j=1,2,…,n,Δpti=Iipti,Δp′ti=I~ipti,i=1,2,…,m,Δqtj=Ijqtj,Δq′tj=I~jqtj,j=1,2,…,n,μln2p2+νln2p′2=ϕp,μpT+νp′T=φp,μln2q2+νln2q′2=ϕq,μqT+νq′T=φq,where 1<α,β≤2, f,g:J×R2→R, and ϕ,φ:C(J,R)→R are continuous functions defined as (4)ϕp=∑i=1a^ħipζi,φp=∑i=1a^℘ipηi,ϕq=∑j=1b^ħjqζj,φq=∑j=1b^℘jqηj,ζi, ηi, ζj, ηj∈(0,1) for i=1,2,…,a^, j=1,2,…,b^ and (5)Δpti=pti+-pti-,Δp′ti=p′ti+-p′ti-,Δqtj=qtj+-qtj-,Δq′tj=q′tj+-q′tj-. The notations p(ti+), q(tj+) are right limits and p(ti-), q(tj-) are left limits; Ii, I~i, Ij, I~j:R→R are continuous functions; HDα, HDβ are the Hadamard derivative operators of order α and β, respectively. For some other recent results on Hadamard fractional differential equations, we refer the reader to [59–65].For system (3), we discuss necessary and sufficient conditions for the existence and uniqueness of a positive solution by using the Krasnoselskii’s fixed point and Banach contraction theorems. Further, we investigate various kinds of Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stabilities.This paper is organized as follows. Section2 contains basic definitions, auxiliary lemmas, and theorems regarding problem (3). Existence, uniqueness, and at least one solution of the problem (3) are presented in Section 3. Section 4 contains Hyers–Ulam types stability results and Section 5 contains an example, which shows the applicability of our main results.


## 2. Preliminaries

In this fragment, we are introducing some fundamental descriptions and lemmas, which are used throughout the paper.Let the norms bep=max{pt,t∈J}, q=max{qt,t∈J} in PC(J,R+), which is Banach space under these norms, and hence their product is also Banach space with norm (p,q)=p+q.LetE1 and E2 denote spaces of the piecewise continuous functions defined as (6)E1=PC2-α,ln⁡J,R+=p:J→R+,pti+,pti-andp′ti+,p′ti-exist  fori=1,2,…,m,E2=PC2-β,ln⁡J,R+=q:J→R+,qtj+,qtj-andq′tj+,q′tj-exist  forj=1,2,…,n,with norms (7)pE1=supptlnt2-α,t∈J,qE2=supqtln⁡t2-β,t∈J,respectively. Their product E=E1×E2 is also a Banach space with norm p,qE=pE1+qE2.We recall the following definitions from [57].Definition 1.
The Hadamard fractional derivative of orderα∈[a-1,a),a∈Z+ of function p(t) is defined by(8)HDαpt=1Γa-αtddta∫1tlntsa-α+1psdss,1<t≤T,where Γ(·) is the Gamma function.Definition 2.
The Hadamard fractional integral of orderα∈R+ of function p(t) is defined by(9)HJαpt=1Γα∫1tln⁡tsα-1psdss,1<t≤T,where Γ(·) is the well-known Gamma function.Lemma 3 (see [66]).
Letα>0 and p be any functions; then the homogenous differential equation along with Hadamard fractional order HDαp(t)=0 has solution (10)pt=b1lntα-1+b2lntα-2+b3lntα-3+⋯+bnlntα-a, and the following formula holds: (11)HJαHDαpt=pt+b1lntα-1+b2lntα-2+b3lntα-3+⋯+bnlntα-a, where bj∈R, j=1,2,…,a, and a-1<α<a.Lemma 4 (see [53]).
Let0<α<1 and f:J×R→R. A function p∈C1-α,ln[a,b] is a solution of the fractional differential equation (12)HDαpt=ft,pt,t∈1,T,HJ1-αp1+=p0,p0∈R,if and only if p satisfies the following fractional integral equation: (13)pt=p0Γαlntα-1+1Γα∫1tlntsα-1fs,psdss.Theorem 5 (Altman [67]).
LetS≠∅ be a convex and closed subset of Banach space E. Consider two operators F, G such that(i)
F(p,q)+G(p,q)∈S, where (p,q)∈S.(ii)
F is contractive operator.(iii)
G is completely continuous operator.
Then the operator systemF(p,q)+G(p,q)=(p,q)∈E has a solution (p,q)∈S.Definition 6 (see [45]).
The switched coupled impulsive FDE (3) is said to be Hyers–Ulam stable if there exist Kα,β=max{Kα,Kβ}>0 such that, for ϱ=max{ϱα,ϱβ}>0 and for every solution (p,q)∈E of the inequality (14)HDαpt-ft,pt,qt≤ϱα,t∈J,Δpti-Iipti≤ϱα,i=1,2,…,m,Δp′^ti-I~ipti≤ϱα,i=1,2,…,m,HDβqt-gt,pt,qt≤ϱβ,t∈J,Δqtj-Ijqtj≤ϱβ,j=1,2,…,n,Δq′^tj-I~jqtj≤ϱβ,j=1,2,…,n,there exists a unique solution (p^,q^)∈E with (15)p,q-p^,q^E≤Kα,βϱ,t∈J.Definition 7 (see [45]).
The switched coupled impulsive FDE (3) is said to be generalized Hyers–Ulam stable if there exist Φ∈C(R+,R+) with Φ(0)=0 such that, for any approximate solution (p,q)∈E of inequality (14), there exists a unique solution (p^,q^)∈E of (3) satisfying (16)p,q-p^,q^E≤Φϱ,t∈J.DenoteΨα,β=max{Ψα,Ψβ}∈C(J,R) and KΨα,Ψβ=max{KΨα,KΨβ}>0.Definition 8 (see [45]).
The switched coupled impulsive FDE (3) is said to be Hyers–Ulam–Rassias stable with respect to Ψα,β if there exists a constant KΨα,Ψβ such that, for some ϱ>0 and for any approximate solution (p,q)∈E of the inequality(17)HDαpt-ft,pt,qt≤Ψαtϱα,t∈J,HDβqt-gt,pt,qt≤Ψβtϱβ,t∈J,there exists a unique solution (p^,q^)∈E with(18)p,q-p^,q^E≤KΨα,ΨβΨα,βϱ,t∈J.Definition 9 (see [45]).
The switched coupled impulsive FDE (3) is said to be generalized Hyers–Ulam–Rassias stable with respect to Ψα,β if there exists a constant KΨα,Ψβ such that, for any approximate solution (p,q)∈E of inequality (17), there exists a unique solution (p^,q^)∈E of (3) satisfying(19)p,q-p^,q^E≤KΨα,ΨβΨα,βt,t∈J.Remark 10.
We say that(p,q)∈E is a solution of the system of inequalities (14) if there exist functions Υf, Υg∈C(J,R) depending upon p,q, respectively, such that
( I )   Υ f t ≤ ϱ α, Υgt≤ϱβ, t∈J;
(II)(20)HDαpt=ft,pt,qt+Υft,Δpti=Iipti+Υfi,Δp′ti=I~ipti+Υfi,HDβqt=gt,pt,qt+Υgt.Δqtj=Ijqtj+Υgj,Δq′tj=I~jqtj+Υgj.


## 3. Existence Results

In this fragment, we present our main results.Theorem 11.
The solution(p,q)∈E of coupled system(21)HDαpt=ft,t∈J,t≠ti,i=1,2,…,m,HDβqt=gt,t∈J,t≠tj,j=1,2,…,n,Δpti=Iipti,Δp′ti=I~ipti,i=1,2,…,m,Δqtj=Ijqtj,Δq′tj=I~jqtj,j=1,2,…,n,μln2p2+νln2p′2=ϕp,μpT+νp′T=φp,μln2q2+νln2q′2=ϕq,μqT+νq′T=φq,is given by the integral equations(22)pt=lntα-2φpαΩ~tαΩ+lntα-2ϕpαA1t+∑i=1klntα-2αA2itΓα∫ti-1tilntisα-1fsdss+∑i=1klntα-2αA2itIipti+∑i=1klntα-2αA3itI~ipti+∑i=1klntα-2αA3itΓα-1∫ti-1tilntisα-2fsdss-μlntα-2αΩ~tαΩΓα∫tkTlnTsα-1fsdss-νlntα-2αΩ~tαΩΓα-1∫tkTlnTsα-2fsdss+1Γα∫tktlntsα-1fsdss,k=1,2,…,m,qt=lntβ-2φqβΩ~tβΩ+lntβ-2ϕqβA1t+∑j=1klntβ-2βA2jtΓβ∫tj-1tjlntjsβ-1gsdss+∑j=1klntβ-2βA2jtIjqtj+∑j=1klntβ-2βA3jtI~jqtj+∑j=1klntβ-2βA3jtΓβ-1∫tj-1tjlntjsβ-2gsdss-μlntβ-2βΩ~tβΩΓβ∫tkTlnTsβ-1gsdss-νlntβ-2βΩ~tβΩΓβ-1∫tkTlnTsβ-2gsdss+1Γβ∫tktlntsβ-1gsdss,k=1,2,…,n,where (23)αΩ=lnTα-3lnTμ+να-2T-1-αΩ~TlnTμ+να-1T≠0,αΩ~t=1-log2tln2μ+να-2/2ln2μ+να-1/2,αA1t=ln22-ααΩ-αΩ~tlnTα-2μ+log2Tνα-1/ln2αΩln2μ+να-1/2,αA2it=ln22-ααΩlogtitiα-1/tα-2-lnTα-2αΩ~tlogtitiα-1/Tα-2+να-1α-2logTe-logtie/TαΩ,αA3it=tilnti2-ααΩlnt/ti-αΩ~tlnTα-2lnT/tiμ+logTTα-1/tiα-2ν/TαΩ.Proof.
Consider(24)HDαpt=ft,t∈J,t≠ti.For t∈[2,t1], applying Jα on (24), we get(25)pt=c1lntα-1+c2lntα-2+1Γα∫2tlntsα-1fsdss.This implies (26)p′t=c1α-1tlntα-2+c2α-2tlntα-3+1Γα-1∫2tlntsα-2fsdss. Applying the initial condition, we obtain c1=(ϕ(p)(ln2)2-α-c2ln2μ+να-2/2)/ln2ln2μ+ν(α-1/2). Therefore (25) becomes(27)pt=ϕpln22-αlntα-1ln2ln2μ+να-1/2+c2lntα-2-lntα-1ln2μ+να-2/2ln2ln2μ+να-1/2+1Γα∫2tlntsα-1fsdss.Now for t∈(t1,t2], applying Jα on (24), we have(28)pt=b1lntα-1+b2lntα-2+1Γα∫t1tlntsα-1fsdss.This implies (29)p′t=b1α-1tlntα-2+b2α-2tlntα-3+1Γα-1∫t1tlntsα-2fsdss. Using initial impulses (30)pt1-=ϕpln22-αlnt1α-1ln2ln2μ+να-1/2+c2lnt1α-2-lnt1α-1ln2μ+να-2/2ln2ln2μ+να-1/2+1Γα∫2t1lnt1sα-1fsdss and p(t1+)=b1(lnt1)α-1+b2(lnt1)α-2.
Using impulsive conditions, we get(31)b1=ϕpln22-αln2ln2μ+να-1/2-c2ln2μ+να-2/2ln2ln2μ+να-1/2-α-2lnt11-αI1pt1+t1lnt12-αI~1pt1-α-2lnt11-αΓα∫2t1lnt1sα-1fsdss+t1lnt12-αΓα-1∫2t1lnt1sα-2fsdss,b2=c2+α-1lnt12-αI1pt1-t1lnt13-αI~1pt1+lnt12-αΓα-1∫2t1lnt1sα-1fsdss-t1lnt13-αΓα-1∫2t1lnt1sα-2fsdss.Substituting the values of b1, b2 in (28), we have (32)pt=ϕplog2tα-2ln2μ+να-1/2+c2lntα-21-log2tln2μ+να-2/2ln2μ+να-1/2+logt1tα-2logt1t1α-1tα-2I1pt1+t1logt1tα-2lntt1I~1pt1+logt1tα-2logt1t1α-1/tα-2Γα∫2t1lnt1sα-1fsdss+t1logt1tα-2ln⁡t/t1Γα-1∫2t1lnt1sα-2fsdss+1Γα∫t1tlntsα-1fsdss. Similarly for t∈(tk,T), we get(33)pt=ϕplog2tα-2ln2μ+να-1/2+c2lntα-21-log2tln2μ+να-2/2ln2μ+να-1/2+∑i=1klogtitα-2logtitiα-1tα-2Iipti+∑i=1ktilogtitα-2lnttiI~ipti+∑i=1klogtitα-2logtitiα-1/tα-2Γα∫ti-1tilntisα-1fsdss+∑i=1ktilogtitα-2ln⁡t/tiΓα-1∫ti-1tilntisα-2fsdss+1Γα∫tktlntsα-1fsdss.
This implies(34)p′t=ϕpα-1log2tα-1ln2ln2μ+να-1/2+∑i=1kα-1α-2logte-logtielogtitα-2tIipti+c2α-2-log2tα-1ln2μ+να-2/2/ln2μ+να-1/2lntα-3t+∑i=1ktilogttα-1/tiα-2logtitα-2tI~ipti+∑i=1kα-1α-2logte-logtielogtitα-2tΓα∫ti-1tilntisα-1fsdss+∑i=1ktilogttα-1/tiα-2logtitα-2tΓα-1∫ti-1tilntisα-2fsdss+1Γα-1∫tktlntsα-2fsdss.
Utilizingμp(T)+νp′(T)=φ(p), we obtain (35)c2=φpαΩ-ϕplog2Tα-1logT2μ+log2eνα-1αΩln2μ+να-1/2-∑i=1kI~iptitilogtiTα-2lnT/tiμ+logTTα-1/tiα-2ν/TαΩ-∑i=1kIiptilogtiTα-2logtitiα-1/Tα-2μ+να-1α-2logTe-logtie/TαΩ-∑i=1klogtiTα-2logtitiα-1/Tα-2μ+να-1α-2logTe-logtie/TαΩΓα∫ti-1tilntisα-1fsdss-∑i=1ktilogtiTα-2lnT/tiμ+logTTα-1/tiα-2ν/TαΩΓα-1∫ti-1tilntisα-2fsdss-μαΩΓα∫tkTlnTsα-1fsdss-ναΩΓα-1∫tkTlnTsα-2fsdss,k=1,2,…,m.
Substituting the value ofc2 in (33), we achieve p(t) of (22). In the same way, we can obtain q(t) of (22).Corollary 12.
In view of Theorem11, our coupled system (3) has the following solution:(36)pt=lntα-2αΩ~tφpαΩ+lntα-2αA1tϕp+∑i=1klntα-2αA2itIipti+∑i=1klntα-2αA3itI~ipti+∑i=1klntα-2αA2itΓα∫ti-1tilntisα-1fs,ps,qsdss+∑i=1klntα-2αA3itΓα-1∫ti-1tilntisα-2fs,ps,qsdss-νlntα-2αΩ~tαΩΓα-1×∫tkTlnTsα-2fs,ps,qsdss-μlntα-2αΩ~tαΩΓα∫tkTlnTsα-1fs,ps,qsdss+1Γα∫tktlntsα-1fs,ps,qsdss,k=1,2,…,m,qt=lntβ-2βΩ~tφqβΩ+lntβ-2βA1tϕq+∑j=1klntβ-2βA2jtIjqtj+∑j=1klntβ-2βA3jtI~jqtj+∑j=1klntβ-2βA2jtΓβ∫tj-1tjlntjsβ-1gs,ps,qsdss+∑j=1klntβ-2Aβ3jtΓβ-1∫tj-1tjlntjsβ-2gs,ps,qsdss-νlntβ-2βΩ~tβΩΓβ-1×∫tkTlnTsβ-2gs,ps,qsdss-μlntβ-2βΩ~tβΩΓβ∫tkTlnTsβ-1gs,ps,qsdss+1Γβ∫tktlntsβ-1gs,ps,qsdss,k=1,2,…,n.To convert the considered problem into a fixed point problem, we define the operatorsF:E→E by F=(F1,F2) and G:E→E by G=(G1,G2) such that(37a)F=F1pt=lntα-2αΩ~tφpαΩ+lntα-2αA1tϕp+∑i=1klntα-2αA2itIipti+∑i=1klntα-2αA3itI~ipti,k=1,2,…,m,(37b)F=F2qt=lntβ-2βΩ~tφqβΩ+lntβ-2βA1tϕq+∑j=1klntβ-2βA2jtIjqtj+∑j=1klntβ-2A3jtI~jqtj,k=1,2,…,n,(38a)G=G1pt,qt=∑i=1klntα-2αA2itΓα∫ti-1tilntisα-1fs,ps,qsdss+∑i=1klntα-2αA3itΓα-1×∫ti-1tilntisα-2fs,ps,qsdss-μαΩ~tlntα-2αΩΓα∫tkTlnTsα-1×fs,ps,qsdss-ναΩ~tlntα-2αΩΓα-1∫tkTlnTsα-2fs,ps,qsdss+1Γα∫tktlntsα-1fs,ps,qsdss,k=1,2,…,m,(38b)G=G2pt,qt=∑j=1klntβ-2βA2jtΓβ∫tj-1tjlntjsβ-1gs,ps,qsdss+∑j=1klntβ-2βA3jtΓβ-1×∫tj-1tjlntjsβ-2gs,ps,qsdss-μβΩ~tlntβ-2βΩΓβ∫tkTlnTsβ-1×gs,ps,qsdss-νβΩ~tlntβ-2βΩΓβ-1∫tkTlnTsβ-2gs,ps,qsdss+1Γβ∫tktlntsβ-1gs,ps,qsdss,k=1,2,…,n,respectively.The following assumptions will be helpful for our results.(H1)
f,g:J×R×R→R+ are continuous; for all (p,q),(p~,q~)∈E, and t∈J, there exist Lf,Lg>0 such that(39)ft,pt,qt-ft,p~t,q~t≤Lfp-p~,q-q~,gt,pt,qt-gt,p~t,q~t≤Lgp-p~,q-q~.(H2)
Ii,I~i,Ij,I~j:R→R are continuous and there exist LI,LI~,LI′,LI~′>0 such that for any p,q,p~,q~∈E(40)Iipti-Iip~ti≤LIp-p~,I~ipti-I~ip~ti≤LI~p-p~,k=1,2,…,m,Ijqtj-Ijq~tj≤LI′q-q~,I~jqtj-I~jq~tj≤LI~′q-q~,k=1,2,…,n.(H3)
ϕ,φ:R→R are continuous, and there exist Lϕ,Lφ,Lϕ′,Lφ′,Mϕ,Mφ,Mϕ′,Mφ′>0, for any (p,q),(p~,q~)∈E such that(41)ϕp-ϕp~≤Lϕp-p~,φp-φp~≤Lφp-p~,ϕq-ϕq~≤Lϕ′q-q~,φq-φq~≤Lφ′q-q~,ϕp≤Mϕ,φp≤Mφ,ϕq≤Mϕ′,φq≤Mφ′.(H4)
f,g:J×R×R→R+ are continuous; for all (p,q)∈E and t∈J, there exist Mf,Mg>0 such that(42)ft,pt,qt≤Mfp+q,gt,pt,qt≤Mgp+q.(H5)
Ii,I~i,Ij,I~j:R→R are continuous and there exist NI,MI,NI~,MI~,NI′,MI′,NI~′,MI~′>0 such that for any (p,q)∈E,(43)Iipti≤NIp+MI,I~ipti≤NI~p+MI~,k=1,2,…,m,Ijqtj≤NI′q+MI′,I~jqtj≤NI~′q+MI~′,k=1,2,…,n.Theorem 13.
Under the hypothesis(H1)-(H3), and(44)Λf+Λg<1L, (3) has unique solution.Proof.
Define an operatorΦ=(Φ1,Φ2):E→E, i.e., Φ(p,q)(t)=(Φ1(p,q),Φ2(p,q))(t), where (45)Φ1p,qt=φplntα-2αΩ~tαΩ+ϕplntα-2αA1t+∑i=1klntα-2αA2itIipti+∑i=1klntα-2αA3itI~ipti+∑i=1klntα-2αA2itΓα∫ti-1tilntisα-1fs,ps,qsdss-μαΩ~tlntα-2αΩΓα∫tkTlnTsα-1fs,ps,qsdss+∑i=1klntα-2αA3itΓα-1∫ti-1tilntisα-2×fs,ps,qsdss-ναΩ~tlntα-2αΩΓα-1∫tkTlnTsα-2fs,ps,qsdss+1Γα∫tktlntsα-1fs,ps,qsdss,Φ2p,qt=φqlntβ-2βΩ~tβΩ+ϕqlntβ-2βA1t+∑j=1klntβ-2βA2jtIjqtj+∑j=1klntβ-2βA3jtI~jqtj+∑j=1klntβ-2βA2jtΓβ∫tj-1tjlntjsβ-1gs,ps,qsdss-μβΩ~tlntβ-2βΩΓβ∫tkTlnTsβ-1gs,ps,qsdss+∑j=1klntβ-2βA3jtΓβ-1∫tj-1tjlntjsβ-2×gs,ps,qsdss-νβΩ~tlntβ-2βΩΓβ-1∫tkTlnTsβ-2gs,ps,qsdss+1Γβ∫tktlntsβ-1gs,ps,qsdss. In view of Theorem 13, we have (46)Φ1p,q-Φ1p~,q~lnt2-α≤LαΩ~tαΩ+αA1t+∑i=1kαA2it+∑i=1kA3itp-p~+L∑i=1kαA2itln⁡ti/ti-1ααΩ+lnt2-αln⁡t/tkααΩ-μln⁡T/tkααΩ~tαΩΓα+1+∑i=1kA3itln⁡ti/ti-1α-1αΩ-νln⁡T/tkα-1αΩ~tαΩΓαp-p~,q-q~.Taking sup, we achieve(47)Φ1p,q-Φ1p~,q~E1≤LαΩ~αΩ+αA1+mαA2+mαA3+mαA2ln⁡tm/tm-1αΩ+lnt2-αln⁡t/tmααΩ-μln⁡T/tmααΩ~αΩΓα+1+mαA3ln⁡tm/tm-1α-1αΩ-νln⁡T/tmα-1αΩ~αΩΓαp-p~+LmαA2ln⁡tm/tm-1ααΩ+lnt2-αln⁡t/tmααΩ-μln⁡T/tmααΩ~αΩΓα+1+αA3ln⁡tm/tm-1α-1αΩ-νln⁡T/tmα-1αΩ~αΩΓαq-q~≤LΛfp,q-p~,q~,where Λf=max{Λ1,Λ2}, such that (48)Λ1=αΩ~αΩ+αA1+mαA3ln⁡tm/tm-1α-1αΩ-νln⁡T/tmα-1αΩ~αΩΓα+mαA2ln⁡tm/tm-1αΩ+lnt2-αln⁡t/tmααΩ-μln⁡T/tmααΩ~αΩΓα+1+mαA2+mαA3Λ2=mαA2ln⁡tm/tm-1ααΩ+lnt2-αln⁡t/tmααΩ-μln⁡T/tmααΩ~αΩΓα+1+αA3ln⁡tm/tm-1α-1αΩ-νln⁡T/tmα-1αΩ~αΩΓα.Similarly(49)Φ2p,q-Φ2p~,q~E2≤LΛgp,q-p~,q~,where Λg=max{Λ3,Λ4}, (50)Λ3=βΩ~βΩ+βA1+nβA3ln⁡tn/tn-1β-1βΩ-νln⁡T/tnβ-1βΩ~βΩΓβ+nβA2lntn/tn-1ββΩ+lnt2-βln⁡t/tnββΩ-μln⁡T/tnββΩ~βΩΓβ+1+nβA2+nβA3Λ4=nβA2ln⁡tn/tn-1ββΩ+lnt2-βln⁡t/tnββΩ-μln⁡T/tnββΩ~βΩΓβ+1+βA3ln⁡tn/tn-1β-1βΩ-νln⁡T/tnβ-1βΩ~βΩΓβ.From (47) and (49), we have (51)Φp,q-Φp~,q~E≤LΛf+Λgp,q-p~,q~, where L=max{Lϕ,Lϕ′,Lφ,Lφ′,LI,LI′,LI~′,LI~}, which implies that the operator Φ is contraction due to (44). Therefore, (3) has a unique solution.Here we use the Krasnoselskii’s fixed point theorem to show that the operatorF+G has at least one fixed point. Therefore, we choose a closed ball(52)Er=p,q∈E,p,q≤r,p≤r2andq≤r2⊂E, where(53)r≥MαΩ~/αΩ+βΩ~/βΩ+αA1+βA1+mαA2+nβA2+mαA3+nβA31-NmαA2+nβA2+mαA3+nβA3-MCα+Cβ.Theorem 14.
If the assumptions(H1)-(H5) are true and αAl=sup1≤i,j≤kAαli,j, then (3) has at least one solution.Proof.
For any(p,q)∈Er, we have(54)Fp,q+Gp,qE≤F1pE1+F2qE2+G1p,qE1+G2p,qE2. From (37a) and (37b), we get (55)F1ptlnt2-α≤φpαΩ~tαΩ+ϕpαA1t+∑i=1kαA2itIipti+∑i=1kαA3itI~ipti,k=1,2,…,m,≤MφαΩ~tαΩ+MϕαA1t+mMI+NIpαA2mt+mMI~+NI~pαA3mt.Taking supt∈J, we get(56)F1pE1≤MφαΩ~αΩ+MϕαA1+mMI+NIpαA2+mMI~+NI~pαA3. Similarly, we can obtain(57)F2qE2≤Mφ′βΩ~βΩ+Mϕ′βA1+nMI′+NI′qβA2+nMI~′+NI~′qβA3.Also, we have(58)G1p,qtlnt2-α≤∑i=1kαA2itΓα∫ti-1tilntisα-1fs,ps,qsdss+∑i=1kαA3itΓα-1∫ti-1tilntisα-1×fs,ps,qsdss-μαΩ~tαΩΓα∫tkTlnTsα-1fs,ps,qsdss-ναΩ~tαΩΓα-1×∫tkTlnTsα-2fs,ps,qsdss+lnt2-αΓα∫tktlntsα-1fs,ps,qsdss≤∑i=1kMfln⁡ti/ti-1ααA2itΓα+1p,q+∑i=1kMfln⁡ti/ti-1ααA3itΓαp,q-μMfln⁡T/tkααΩ~tαΩΓα+1p,q-νMfln⁡T/tkα-1αΩ~tαΩΓαp,q+Mflnt2-αln⁡t/tkαΓα+1p,q,k=1,2,…,m.
Takingsupt∈J, we obtain(59)G1p,qE1≤Mflnt2-αln⁡t/tmααΩ+mln⁡tm/tm-1ααA2αΩ-μln⁡T/tmααΩ~αΩΓα+1+mln⁡tm/tm-1ααA3αΩ-νln⁡T/tmα-1αΩ~αΩΓαp,q≤MfCαp,q,where (60)Cα=lnt2-αln⁡t/tmααΩ+mln⁡tm/tm-1ααA2αΩ-μln⁡T/tmααΩ~αΩΓα+1+mln⁡tm/tm-1ααA3αΩ-νln⁡T/tmα-1αΩ~αΩΓα.Also(61)G2p,qtE2≤MgCβp,q,where (62)Cβ=lnt2-βln⁡t/tnββΩ+nln⁡tn/tn-1ββA2βΩ-μln⁡T/tnββΩ~βΩΓβ+1+mln⁡tn/tn-1ββA3βΩ-νln⁡T/tnβ-1βΩ~βΩΓβ.Substituting all inequalities from (59) to (61) in (54), we get (63)Fp,q+Gp,qE≤MαΩ~αΩ+βΩ~βΩ+αA1+βA1+mαA2+nβA2+mαA3+nβA3+NmαA2+nβA2+mαA3+nβA3r+MCα+Cβr≤r,
whereM=max{Mϕ,Mϕ′,Mφ,Mφ′,MI,MI′,MI~,MI~′,Mf, Mg} and N = max{NI,NI′,NI~,NI~′}. Hence, F(p,q)+G(p,q)∈Er.
Next, for anyt∈J, (p,q),(p~,q~)∈E(64)Fp,q-Fp~,q~E1≤F1p-F1p~E1+F2q-F2q~E1≤LϕαΩ~αΩ+LφαA1+mLIαA2+mLI~αA3p-p~+Lϕ′βΩ~βΩ+Lφ′βA1+nLI′βA2+nLI~′βA3q-q~≤LαΩ~αΩ+βΩ~βΩ+αA1+βA1+mαA2+nβA2+mαA3+nβA3p-p~,q-q~≤Lξ∗p-p~,q-q~,where(65)ξ∗=αΩ~αΩ+βΩ~βΩ+αA1+βA1+mαA2+nβA2+mαA3+nβA3. Therefore, F is contraction mapping.
Now, we are proving the continuity and compactness ofG and, for this reason, construct a sequence Ts=(ps,qs) in Er such that (ps,qs)→(p,q) for s→∞ in Er. Thus, we have (66)Gps,qs-Gp,qE≤G1ps,qs-G1p,qE1+G2ps,qs-G2p,qE2≤Lfμln⁡T/tmααΩ~αΩΓα+1+νln⁡T/tmα-1αΩ~αΩΓα-mαA2ln⁡tm/tm-1αΓα+1-mαA3ln⁡tm/tm-1α-1Γα+lnt2-αln⁡t/tmαΓα+1ps-p,qs-q+Lgμln⁡T/tnββΩ~βΩΓβ+1+νln⁡T/tnβ-1βΩ~βΩΓβ-nβA2ln⁡tn/tn-1βΓβ+1-nβA3ln⁡tn/tn-1β-1Γβ+lnt2-βln⁡t/tnβΓβ+1ps-p,qs-q.This implies Gpn,qn-Gp,qE→0 as n→∞; therefore G is continuous.
Next, we show thatG is uniformly bounded on Er. From (59) and (61), we have (67)Gp,qtE≤G1p,qtE1+G2p,qtE2≤MfC∗+MgC∗∗r. Thus, G is uniformly bounded on Er.
For equi-continuity, takeτ1,τ2∈J with τ1<τ2 and for any (p,q)∈Er⊂E, where Er is clearly bounded, we have (68)G1p,qτ1-G1p,qτ2lnt2-α≤MfΩ~τ1-Ω~τ2μln⁡T/tmαΩΓα+1+νln⁡T/tmα-1ΩΓα-∑k=1mA2kτ1-A2kτ2ln⁡tk/tk-1αΓα+1-∑k=1mA3kτ1-A3kτ2ln⁡tk/tk-1α-1Γαp,q+1Γα∫tmτ1lnτ1sα-1fs,ps,qsdss-∫tmτ2lnτ2sα-1fs,ps,qsdss. This implies that G1p,qτ1-G1p,qτ2E1→0 as τ1→τ2. In the same way, we have G2p,qτ1-G2p,qτ2E2→0 as τ1→τ2. Hence Gp,qτ1-Gp,qτ2E→0 as τ1→τ2. Therefore, G is relatively compact on Er. ByArzelä-Ascoli theorem, G is compact and hence completely continuous operator, so (3) has at least one solution.


## 4. Ulam Stability Analysis

In this portion, we analyze different kinds of stability like Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability of the proposed system.Theorem 15.
If assumptions(H1)-(H3) and inequality (44) are satisfied and (69)ϝ=1-L2Λ2Λ4lntα-2-LΛ1lntβ-2-LΛ3>0, then the unique solution of the coupled system (3) is Hyers–Ulam stable and consequently generalized Hyers–Ulam stable.Proof.
Let(p,q)∈E be an approximate solution of inequality (14) and let (p^,q^)∈E be the unique solution of the coupled system given by(70)HDαp^t=ft,p^t,q^t,t∈J,t≠ti,i=1,2,…,m,HDβq^t=gt,p^t,q^t,t∈J,t≠tj,j=1,2,…,n,Δp^ti=Iip^ti,Δp^′ti=I~ip^ti,i=1,2,…,m,Δq^tj=Ijq^tj,Δq^′tj=I~jq^tj,j=1,2,…,n,μln2p^2+νln2p^′2=ϕp^,μp^T+νp^′T=φp^,μln2q^2+νln2q^′2=ϕq^,μq^T+νq^′T=φq^.By Remark 10 we have(71)HDαpt=ft,pt,qt+Υft,t∈J,t≠ti,i=1,2,…,m,Δpti=Iipti+Υfi,Δp′ti=I~ipti+Υfi,i=1,2,…,m,HDβqt=gt,pt,qt+Υgt,t∈J,t≠tj,j=1,2,…,n,Δqtj=Ijqtj+Υgj,Δq′tj=I~jqtj+Υgj,j=1,2,…,n.Therefore, the solution of problem (71) is(72)pt=lntα-2αΩ~tφpαΩ+∑i=1klntα-2αA3itI~ipti+Υfi+∑i=1klntα-2αA2itIipti+Υfi+lntα-2αA1tϕp+∑i=1klntα-2αA2itΓα∫ti-1tilntisα-1fs,ps,qs+Υfsdss+∑i=1klntα-2αA3itΓα-1∫ti-1tilntisα-2fs,ps,qs+Υfsdss-νlntα-2αΩ~tαΩΓα-1∫tkTlnTsα-2fs,ps,qs+Υfsdss-μlntα-2αΩ~tαΩΓα∫tkTlnTsα-1fs,ps,qs+Υfsdss+1Γα∫tktlntsα-1fs,ps,qs+Υfsdss,k=1,2,…,m,qt=lntβ-2βΩ~tφqβΩ+∑j=1klntβ-2βA3jtI~jqtj+Υgj+∑j=1klntβ-2βA2jtIjqtj+Υgj+lntβ-2βA1tϕq+∑j=1klntβ-2βA2jtΓβ∫tj-1tjlntjsβ-1gs,ps,qs+Υgsdss+∑j=1klntβ-2βA3jtΓβ-1∫tj-1tjlntjsβ-2gs,ps,qs+Υgsdss-νlntβ-2βΩ~tβΩΓβ-1∫tkTlnTsβ-2gs,ps,qs+Υgsdss-μlntβ-2βΩ~tβΩΓβ∫tkTlnTsβ-1gs,ps,qs+Υgsdss+1Γβ∫tktlntsβ-1gs,ps,qs+Υgsdss,k=1,2,…,n.We consider (73)pt-p^tlntα-2≤αΩ~tφp-φp^αΩ+∑i=1kαA3itI~ipti-I~ip^ti+∑i=1kαA2itIipti-Iip^ti+αA1tϕp-ϕp^+∑i=1kαA2itΓα∫ti-1tilntisα-1fs,ps,qs-fs,p^s,q^sdss+∑i=1kαA3itΓα-1∫ti-1tilntisα-2fs,ps,qs-fs,p^s,q^sdss-ναΩ~tαΩΓα-1∫tkTlnTsα-2fs,ps,qs-fs,p^s,q^s-μαΩ~tαΩΓα∫tkTlnTsα-1fs,ps,qs-fs,p^s,q^s+lnt2-αΓα∫tktlntsα-1fs,ps,qs-fs,p^s,q^sdss+∑i=1kαA3itΥfi+∑i=1kαA2itΥfi+∑i=1kαA2itΓα∫ti-1tilntisα-1Υfsdss+∑i=1kαA3itΓα-1∫ti-1tilntisα-2Υfsdss-ναΩ~tαΩΓα-1∫tkTlnTsα-2Υfsdss-μαΩ~tαΩΓα∫tkTlnTsα-1Υfsdss+lnt2-αΓα∫tktlntsα-1Υfsdss.As in Theorem 13, we get(74)p-p^E1≤LΛ1lnt2-αp-p^E1+LΛ2lnt2-αq-q^E1+Λ2+mαA2+mαA3ϱα,(75)q-q^E2≤LΛ3lnt2-βp-p^E2+LΛ4lnt2-βq-q^E2+Λ4+nβA2+nβA3ϱβ.From (74) and (75) we have (76)p-p^E1-LΛ2lntα-2-LΛ1q-q^E1≤Λ2+mαA2+mαA31-LΛ1lnt2-αϱα,q-q^E2-LΛ4lntβ-2-LΛ3p-p^E2≤Λ4+nβA2+nβA31-LΛ3lnt2-βϱβ,respectively. Let Gα=(Λ2+mαA2+mαA3)/(1-LΛ1(lnt)2-α) and Gβ=(Λ4+nβA2+nβA3)/(1-LΛ3(lnt)2-β). Then the last two inequalities can be written in matrix form as (77)1-LΛ2lntα-2-LΛ1-LΛ4lntβ-2-LΛ31p-p^E1q-q^E2≤GαϱαGβϱβ,p-p^E1q-q^E2≤1ϝLΛ2lntα-2-LΛ1ϝLΛ4lntβ-2-LΛ3ϝ1ϝGαϱαGβϱβ,where (78)ϝ=1-L2Λ2Λ4lntα-2-LΛ1lntβ-2-LΛ3>0. From system (77) we have (79)p-p^E1≤Gαϱαϝ+LΛ2Gβϱβlntα-2-LΛ1ϝ,q-q^E2≤Gβϱβϝ+LΛ4Gαϱαlntβ-2-LΛ3ϝ, which implies that (80)p-p^E1+q-q^E2≤Gαϱαϝ+Gβϱβϝ+LΛ2Gβϱβlntα-2-LΛ1ϝ+LΛ4Gαϱαlntβ-2-LΛ3ϝ. If max{ϱα,ϱβ}=ϱ and Gα/ϝ+Gβ/ϝ+LΛ2Gβ/((lnt)α-2-LΛ1)ϝ+LΛ4Gα/((lnt)β-2-LΛ3)ϝ=Gα,β, then (81)p,q-p^,q^E≤Gα,βϱ. This shows that system (3) is Hyers–Ulam stable. Also, if (82)p,q-p^,q^E≤Gα,βΦϱ with Φ(0)=0, then the solution of system (3) is generalized Hyers–Ulam stable.For the upcoming result, we suppose the following:(H6)
There exist two nondecreasing functionsw¯α,w¯β∈C(J,R+) such that (83)HIαw¯αt≤Lαw¯αt,HIβw¯βt≤Lβw¯βt,whereLα,Lβ>0.Theorem 16.
If assumptions(H1)-(H3) and (H6) and inequality (44) are satisfied and (84)ϝ=1-L2Λ2Λ4lntα-2-LΛ1lntβ-2-LΛ3>0, then the unique solution of the coupled system (3) is Hyers–Ulam–Rassias stable and consequently generalized Hyers–Ulam–Rassias stable.Proof.
By using Definitions9 and 8, we can gain our result to perform the same steps as in Theorem 15.


## 5. Example

To testify our results established in the previous section, we provide an adequate problem.Example 1.
Consider(85)HD3/2pt=t2+sinpt+cosqt50,t∈J,t≠52,HD3/2qt=qt+cospt70+t2,t∈J,t≠73,ln2p2-ln2p′2=∑i=110ħipζi,ln2q2-ln2q′2=∑j=110ħjqηj,2<ηi,ζi<3,ħi>0pe-p′e=∑i=110℘ipζi,qe-q′e=∑j=110℘jqηj,2<ηj,ζj<3℘j>0,Δp52=I1p52=p5/275+p5/2,Δp′52=I~1p52=p5/225+p5/2,Δq73=I1q73=q7/375+q7/3,Δq′73=I~1q73=q7/325+q7/3.In system (85), we see that α=β=3/2 and ϕ(p)=∑i=110‍ħipζi, φ(p)=∑i=110‍℘ipζi, ϕ(q)=∑j=110‍ħjqηj, and φ(q)=∑j=110‍℘jqηj, where ∑i=110‍ħi<1/25 and ∑i=110‍℘i<1/75.
Fort∈J=(2,e], and (p,q),(p~,q~)∈E, we gain(86)ft,pt,qt-ft,p~t,q~t≤150p,q-p~,q~,gt,pt,qt-gt,p~t,q~t≤170p,q-p~,q~. From this we get Lf=1/50 and Lg=1/75. Also,(87)Ipt-Ip~t≤175p-p~,I~pt-I~p~t≤125p-p~,ϕpt-ϕp~t≤125p-p~,φpt-φp~t≤175p-p~.From this we get that LI=LI′=1/75, LI~=LI~′=1/25, Lϕ=1/25, Lφ=1/75, Lϕ′=1/25, Lφ′=1/75, and m=n=1. Finding Λ1=0.51282, Λ2=0.18059, Λ3=0.12899, and Λ4=0.46122, it is clear that Λf=0.51282 and Λg=0.46122. By the help of Theorem 13, the following inequality is true (88)Λf+Λg<1L, and hence (85) has a unique solution. Also, (89)ϝ=1-L2Λ2Λ4lntα-2-LΛ1lntβ-2-LΛ3≈0.02280>0, and hence by Theorem 15 the coupled system (85) is Hyers–Ulam stable and thus generalized Hyers–Ulam stable. Similarly, we can verify the condition of Theorems 16 and 14.


## 6. Conclusion

In this manuscript, we used theArzelä-Ascoli theorem, Banach contraction principle, and Krasnoselskii’s fixed point theorem to achieve the necessary criteria for the existence and uniqueness of the solution of considered switched coupled impulsive fractional differential systems given in (3). Similarly, under particular assumptions and conditions, we have established the Hyers–Ulam stability results of different kinds for the solution of the considered problem in (3). From the obtained results, we conclude that such a method is very powerful, effectual, and suitable for the solution of nonlinear fractional differential equations.



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*Source: 5093572-2019-06-11.xml*