# Existence and Uniqueness for a System of Caputo-Hadamard Fractional Differential Equations with Multipoint Boundary Conditions

Authors: S. Nageswara Rao; Ahmed Hussein Msmali; Manoj Singh; Abdullah Ali H. Ahmadini
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we study existence and uniqueness of solutions for a system of Caputo-Hadamard fractional differential equations supplemented with multi-point boundary conditions. Our results are based on some classical fixed point theorems such as Banach contraction mapping principle, Leray-Schauder fixed point theorems. At last, we have presented two examples for the illustration of main results.


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## Body

## 1. Introduction

In recent years, fractional differential equations (FDE) gain enormous attention among scientists due to the applications which were not possible with ordinary or partial differential equations of integer order. FDEs becomes a very successful tool in modeling anomalous diffusion and fractal-like nature. Agrawal discusses diffusion and heat equations of fractional order in [1–3]. Agrawal et al., Baleanu, and others investigated the boundary value problems for fractional differential equations [4]. Fractional dynamic models, fractional control systems, fractional population dynamics models, and fractional fluid dynamics all involve at least one ordinary or partial fractional derivative.Fractional differential equations have several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, and Grunwald-Letnikov fractional derivative. Another kind of fractional derivative is Hadamard type which was introduced in 1892 [5]. This derivative differs from various derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [6]. The readers who are interested in the subject of fractional calculus is referred to the books by Kilbas et al. [7], Podlubny [8], Miller and Ross [9], Samko et al. [10], Diethelm [11], and Zhou [12] and the references therein.Coupled systems of fractional differential equations play a key role in developing differential models such as the synchronization of chaotic systems [13–15], anomalous diffusion [16, 17], disease models [18, 19], ecological models [20], Lorenz system [21], and nonlocal thermoelectricity systems [22, 23]. For recent theoretical results on the topic, we refer the reader to a series of papers [24–37] and the references cited therein. Ahmad and Ntouyas [32, 33] discussed some fractional integral boundary value problems involving Hadamard fractional differential equations/systems and obtained the existence and uniqueness of solutions by applying the Banach fixed point theorem and Leray–Schauder alternative, respectively.In [35], the authors investigated the existence and uniqueness of solutions for the coupled system of nonlinear fractional differential equations with three-point boundary conditions
(1)Dαut=ft,vt,Dpvt,t∈0,1,Dβvt=gt,vt,Dqut,t∈0,1,u0=0,u1=γuη,v0=0,v1=γvη,where 1<α,β<2,p,q,γ>0,0<η<1,α−q≥1,β−p≥1,γηα−1<1,γηβ−1<1 and Dα,Dβ are the standard Riemann-Liouville fractional derivative and f,g:0,1×R×R→R are given continuous functions.Recently, Alsulami et al. [36] established the existence and uniqueness results for a nonlinear coupled system of Caputo type fractional differential equations supplemented with nonseparated coupled boundary conditions.
(2)cDαxt=ft,xt,yt,t∈0,T,1<α≤2cDβyt=gt,xt,yt,t∈0,T,1<β≤2,x0=λ1yT,x′0=λ2y′T,y0=μ1yT,y′0=μ2x′T,where cDα,cDβ denote the Caputo fractional derivatives of order α and β, respectively, f,g:0,T×R×R→R are appropriately chosen functions, and λi,μi,i=1,2, are real constants with λiμi≠1,i=1,2.Motivated by the research going on in this direction, in this paper, we study existence and uniqueness of solutions for a coupled system of Caputo-Hadamard fractional differential equations.(3)cDa+αut=ft,ut,vt,t∈a,b,cDa+βvt=gt,ut,vt,t∈a,b,with multipoint boundary conditions
(4)ua=λ1vb,λ2CDa+γ1ub=μ2∑i=1NCDa+δ1vηi,va=μ1ub,λ3CDa+γ2vb=μ3∑i=1MCDa+δ2vξi,where α,β∈1,2,γi,δi∈0,1,i=1,2, ηi∈R, for i=1,2,⋯NN∈N,a<η1<η2<⋯<b,ξi∈R, for i=1,2,⋯MM∈N,a<ξ1<ξ2<⋯<b,λi,μi,i=1,2,3 are real positive constants Da+κ denotes the Caputo-Hadamard fractional derivatives of order κ for κ=α,β,γi,δi,fori=1,2, f,g∈a,b×R×R→R are appropriately chosen functions.The paper is organized as follows. In Sect.2, we present some preliminary concepts of fractional calculus. Sect. 3 contains main results concerning the existence and uniqueness of solutions for the given problem (3), (4). The Leray-Schauder alternative theorem is applied to prove existence, while the uniqueness result was obtained via the Banach contraction mapping principle. Finally, we also discuss some examples for illustration of the existence-uniqueness results.


## 2. Preliminaries

For the convenience of the reader, we present some concepts of Hadamard type fractional calculus to facilitate the analysis of system (3), (4).Definition 1 [7].
The Hadamard fractional integral of orderq>0 of a function xt for all t>a>0 is defined by
(5)HIa+qxt=1Γq∫atlntsq−1xsdss,where Γq=∫0∞tq−1e−tdt is the gamma function,s provided the right side is pointwise defined on R+.Definition 2 [7].
The Hadamard fractional derivative of orderq>0 of a function xt for all t>a>0 is defined by
(6)HDa+qxt=1Γn−qtddtn∫atlntsn−q−1xsdss,where n=q+1 with q denotes the integral part of the real number q and ln·=lne·.Definition 3 [38].
Letq≥0 and n=q+1. If yx∈ACδna,b, where 0<a<b<∞ and
(7)ACδna,b=g:a,b→C:δn−1gx∈ACa,b,δ=xddx.
The Caputo type modification of the Hadamard fractional derivative of orderq is defined by
(8)CDa+qyx=Da+qyt−∑k=0n−1δkyak!lntakx.Theorem 4 [38].
Letq≥0, and n=q+1. If yt∈ACδna,b, where 0<a<b<∞. Then CDa+qft exist everywhere on a,b and
(i)
ifq∉N0, CDa+qft can be represented by
(9)CDa+qyt=Ia+n−qδnyt=1Γn−q∫atlntsn−q−1δnysdss,(ii)
ifq∈N0, then CDa+qyt=δnytRemark 5.
Ifa,α,β>0 then
(10)HDaαlntaβ−1x=ΓβΓβ−αlnxaβ−α−1.Lemma 6 [38].
Letq≥0 and n=q+1. If xt∈ACδna,b, then the Caputo-Hadamard fractional differential equation CDa+qxt=0 has a solution:
(11)xt=∑k=0n−1cklogtak,and the following formula holds:
(12)Ia+qCDa+qxt=xt+∑k=0n−1cklntak,where ck∈R,k=1,2,⋯,n−1.
Now, we present an auxiliary lemma for boundary value problem of linear fractional differential equation with Caputo-Hadamard derivative.Lemma 7.
LetΔ=λ2λ3/Γ2−γ1Γ2−γ2lnb/a2−γ1−γ2−μ2μ3/Γ2−δ1Γ2−δ2∑i=1Nlnηi/a1−δ1∑i=1Mlnξi/a1−δ2≠0 and μ1λ1≠1. Let x,y∈ACnδa,b. Then, the solution of the linear Caputo-Hadamard fractional differential system
(13)cDa+αut=xt,t∈a,b,1<α≤2,cDa+βvt=yt,t∈a,b,1<β≤2,ua=λ1vb,λ2cDa+γ1ub=μ2∑i=1NcDa+δ1vηi,va=μ1ub,λ3cDa+γ2vb=μ3∑i=1McDa+δ2vξi,is equivalent to the system of integral equations
(14)ut=μ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnt/a∑i=1Nlnηi/a1−δ1Γ2−δ1B3−λ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnt/a∑i=1Nlnηi/a1−δ1Γ2−δ1A3+μ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnt/alnb/a1−γ2Γ2−γ2A2−λ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnt/alnb/a1−γ2Γ2−γ2B2+λ11−μ1λ1μ1B1+A1+∫atlnt/sα−1Γαxsdss,(15)vt=μ3Δμ1λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ2lnt/alnb/a1−γ1Γ2−γ1B3−λ3Δμ1λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ2lnt/alnb/a1−γ1Γ2−γ1A3+μ2Δμ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+μ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+μ3lnt/a∑i=1Mlnξi/a1−δ2Γ2−δ2A2−λ2Δμ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+μ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+μ3lnt/a∑i=1Mlnξi/a1−δ2Γ2−δ2B2+μ11−μ1λ1λ1A1+B1+∫atlnt/sβ−1Γβysdss,where
(16)B1=∫ablnb/sα−1Γαxsdss,A1=∫ablnb/sβ−1Γβysdss,B2=∫ablnb/sα−γ1−1Γα−γ1xsdss,A2=∑i=1N∫aηilnηi/sβ−δ1−1Γβ−δ1ysdss,B3=∑i=1M∫aξilnξi/sα−δ2−1Γα−δ2xsdss,A3=∫ablnb/sβ−γ2−1Γβ−γ2ysdss.Proof.
We apply Lemma [6] that the general solution of the Caputo-Hadamard fractional differential equation in (13) can be written as:
(17)ut=c0+c1lnta+∫atlnt/sα−1Γαxsdss,(18)vt=d0+d1lnta+∫atlnt/sβ−1Γβysdss,where ci,di,i=0,1, are arbitrary real constants. From (17) and (18) we have
(19)cDa+γ1ut=c11Γ2−γ1lnta1−γ1+∫atlnt/sα−γ1−1Γα−γ1xsdss,(20)cDa+γ2vt=d11Γ2−γ2lnta1−γ2+∫atlnt/sβ−γ2−1Γβ−γ2ysdss,(21)cDa+δ1vt=d11Γ2−δ1lnta1−δ1+∫atlnt/sβ−δ1−1Γβ−δ1ysdss,(22)cDa+δ2ut=c11Γ2−δ2lnta1−δ2+∫atlnt/sα−δ2−1Γα−δ2xsdss.
Using the boundary conditionsua=λ1vb and va=μ1ub from (17) and (18), we have
(23)⇒c0=λ1d0+d1lnba+A1,(24)⇒d0=μ1c0+c1lnba+B1.
Using the boundary conditionsλ2CDa+γ1ub=μ2∑i=1NDCa+δ1vηi and λ3CDa+γ2vb=μ3∑i=1MDCa+δ2uξi from (19) to (22), we have
(25)⇒c1λ2Γ2−γ1lnba1−γ1−d1μ2Γ2−δ1∑i=1Nlnηia1−δ1=A2μ2−λ2B2,⇒c1−μ3Γ2−δ2∑i=1Mlnξia1−δ2+d1λ3Γ2−γ2lnba1−γ2=B3μ3−λ3A3.
Solving the resulting equations forc1 and d1, we find that
(26)c1=1Δλ3A2μ2−λ2B2Γ2−γ2lnba1−γ2+μ2B3μ3−λ3A3Γ2−δ1∑i=1Nlnηia1−δ1,d1=1Δλ2B3μ3−λ3A3Γ2−γ1lnba1−γ1+μ3A2μ2−λ2B2Γ2−δ2∑i=1Mlnξia1−δ2,substituting c1 and d1 in (23) and (24), we have
(27)c0=λ11−μ1λ1μ1μ2μ3ΔΓ2−δ1∑i=1Nlnηia1−δ1lnba+λ2μ3ΔΓ2−γ1lnba2−γ1B3−μ1μ2λ3ΔΓ2−δ1∑i=1Nlnηia1−δ1lnba+λ2λ3ΔΓ2−γ1lnba2−γ1A3+μ1μ2λ3ΔΓ2−γ2lnba2−γ2+μ2μ3ΔΓ2−δ2∑i=1Mlnξia1−δ2lnbaA2−μ1λ2λ3ΔΓ2−γ2lnba2−γ2+λ2μ3ΔΓ2−δ2∑i=1Mlnξia1−δ2lnbaB2+μ1B1+A1,and
(28)d0=λ11−μ1λ1λ1λ2μ3ΔΓ2−γ1lnba2−γ1+μ2μ3ΔΓ2−δ1∑i=1Nlnηia1−δ1lnbaB3−λ1λ2λ3ΔΓ2−γ1lnba2−γ1+μ2λ3ΔΓ2−δ1∑i=1Nlnηia1−δ1lnbaA3+λ1μ2μ3ΔΓ2−δ2∑i=1Mlnξia1−δ2lnba+μ2λ3ΔΓ2−γ2lnba2−γ2A2−λ1λ2μ3ΔΓ2−δ2∑i=1Mlnξia1−δ2lnba+λ2λ3ΔΓ2−γ2lnba2−γ2B2+λ1A1+B1.
Inserting the values ofci,di,i=0,1 in (17) and (18), which leads to the solution system (14), (15). The converse follows by direct computation. The proof is completed.


## 3. Existence and Uniqueness Results

This section is concerned with the main results of the paper. First of all, we fix our terminology. LetC=Ca,b,R,a>0 be the Banach space of all continuous functions from a,b to R. Space X=ut:ut∈C2a,b,R endowed with the norm u=sup∣ut∣,t∈a,b is a Banach space. In addition, let Y=vt:vt∈C2a,b,R with the norm v=sup∣vt∣,t∈a,b. It is obvious that product space X×Y,u,v is a Banach space with the norm u,v=u+v.In view of Lemma7, we introduce an operator T:X×Y→X×Y as follows:
(29)Tu,vt=T1u,vt,T2u,vt,where
(30)T1u,vt=μ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnt/a∑i=1Nlnηi/a1−δ1Γ2−δ1B3f−λ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnt/a∑i=1Nlnηi/a1−δ1Γ2−δ1A3g+μ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnt/alnb/a1−γ2Γ2−γ2A2g−λ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnt/alnb/a1−γ2Γ2−γ2B2f+λ11−μ1λ1μ1B1f+A1g+∫atlnt/sα−1Γαfs,us,vsdss,and(31)T2u,vt=μ3Δμ1λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ2lnt/alnb/a1−γ1Γ2−γ1B3f−λ3Δμ1λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ2lnt/alnb/a1−γ1Γ2−γ1A3g+μ2Δμ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+μ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+μ3lnt/a∑i=1Mlnξi/a1−δ2Γ2−δ2A2g−λ2Δμ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+μ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+μ3lnt/a∑i=1Mlnξi/a1−δ2Γ2−δ2B2f+μ11−μ1λ1λ1A1g+B1f+∫atlnt/sβ−1Γβgs,us,vsdss.Here,(32)B1f=∫ablnb/sα−1Γαfs,us,vsdss,A1g=∫ablnb/sβ−1Γβgs,us,vsdss,B2f=∫ablnb/sα−γ1−1Γα−γ1fs,us,vsdss,A2g=∑i=1N∫aηilnηi/sβ−δ1−1Γβ−δ1gs,us,vsdss,B3f=∑i=1M∫aξilnξi/sα−δ2−1Γα−δ2fs,us,vsdss,A3g=∫ablnb/sβ−γ2−1Γβ−γ2gs,us,vsdss.For computational convenience, we set(33)K1=μ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×∑i=1Mlnξi/aα−δ2Γα−δ2+1+λ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnb/a2−γ2Γ2−γ2×lnb/aα−γ1Γα−γ1+1+μ1λ11−μ1λ1+1lnb/aαΓα+1,K2=∣λ3∣∣Δ∣μ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+∣μ2∣lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×lnb/aβ−γ2Γβ−γ2+1+μ2∣Δ∣μ1λ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+∣λ3∣lnb/a2−γ2Γ2−γ2×∑i=1Nlnηi/aβ−δ1Γβ−δ1+1+μ1∣1−μ1λ1∣lnb/aβΓβ+1,K3=∣μ3∣∣Δ∣μ1λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+∣λ2∣lnb/a2−γ1Γ2−γ1∑i=1Mlnξi/aα−δ2Γα−δ2+1+∣λ2∣∣Δ∣μ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+μ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2×lnb/aα−γ1Γα−γ1+1+∣μ1∣∣1−λ1μ1∣lnb/aαΓα+1,K4=∣λ3∣∣Δ∣μ1λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+μ1μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ2lnb/a2−γ1Γ2−γ1lnb/aβ−γ2Γβ−γ2+1+∣μ2∣∣Δ∣μ1μ3λ1lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+μ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2×∑i=1Nlnηi/aβ−δ1Γβ−δ1+1+λ1μ11−λ1μ1+1lnb/aβΓβ+1.Now, we are in a position to present our main results. The methods used to prove the existence and uniqueness solutions of boundary value problem (3), (4) via Banach’s contraction principle.Theorem 8.
Suppose thatf,g:a,b×R×R→R are continuous functions. In addition, we assume that:
(H1) there exist constantsmi and ni,i=1,2, such that for all t∈a,b and ui,vi∈R,i=1,2, we have
(34)ft,u1,v1−ft,u2,v2≤m1u1−u2+m2v1−v2,gt,u1,v1−gt,u2,v2≤n1u1−u2+n2v1−v2.
Then, the system (3), (4) has a unique solution on a,b, if
(35)K1+K3m1+m2+K2+K4n1+n2<1.Proof.
Definesupt∈a,bft,0,0=σ1<∞ and supt∈a,bgt,0,0=σ2<∞ and r>0 such that
(36)r>K1+K3σ1+K2+K4σ21−K1+K3m1+m2+K2+K4n1+n2.
Now, we show thatTBr⊂Br, where Br=u,v∈X×Y:∥u,v∥≤r.
By assumptionH1, for u,v∈Br,t∈a,b, we have that
(37)∣ft,ut,vt∣≤∣ft,ut,vt−ft,0,0∣+∣ft,0,0∣≤m1∣ut∣+m2∣vt∣+σ1≤m1∥u∥+m2∥v∥+σ1∣gt,ut,vt∣≤n1∥u∥+n2∥v∥+σ2,which leads to
(38)T1u,vt≤μ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×∑i=1Mlnξi/aα−δ2Γα−δ2+1m1u+m2v+σ1+λ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+∣μ2∣lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×lnb/aβ−γ2Γβ−γ2+1n1u+n2v+σ2+∣μ2∣∣Δ∣μ1λ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+∣λ3∣lnb/a2−γ2Γ2−γ2×∑i=1Nlnηi/aβ−δ1Γβ−δ1+1n1u+n2v+σ2+∣λ2∣∣Δ∣μ1λ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+∣λ3∣lnb/a2−γ2Γ2−γ2×lnb/aα−γ1Γα−γ1+1m1u+m2v+σ1+∣λ1∣∣1−μ1λ1∣∣μ1∣lnb/aαΓα+1m1u+m2v+σ1+lnb/aβΓβ+1n1u+n2v+σ2+lnb/aαΓα+1m1u+m2v+σ1=K1m1u+m2v+σ1+K2n1u+n2v+σ2=K1m1+K2n1u+K1m2+K2n2v+K1σ1+K2σ2≤K1m1+m2+K2n1+n2r+K1σ1+k2σ2.
Hence,(39)∥T1u,v∥≤K1m1+m2+K2n1+n2r+K1σ1+K2σ2.
In the same way, we can obtain that(40)∥T2u,v∥≤K3m1+m2+K4n1+n2r+K3σ1+K4σ2.
Consequently, it follows that(41)∥Tu,v∥≤K1m1+m2+K2n1+n2r+K1σ1+K2σ2+K3m1+m2+K4n1+n2r+K3σ1+K4σ2≤r,which implies TBr⊂Br. Next, we show that operator T is contraction mapping.
For anyu1,v1,u2,v2∈X×Y and for any t∈a,b, we obtain
(42)∣T1u1,v1t−T1u2,v2t∣≤∣μ3∣∣Δ∣μ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+∣μ2∣lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×∑i=1Mlnξi/aα−δ2Γα−δ2+1m1u1−u2+m2v1−v2+∣λ3∣∣Δ∣μ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1∣1−μ1λ1∣+λ1λ2lnb/a2−γ1Γ2−γ1∣1−μ1λ1∣+∣μ2∣lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×lnb/aβ−γ2Γβ−γ2+1n1∥u1−u2∥+n2∥v1−v2∥+∣μ2∣∣Δ∣μ1λ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+∣λ3∣lnb/a2−γ2Γ2−γ2×∑i=1Nlnηi/aβ−δ1Γβ−δ1+1n1∥u1−u2∥+n2∥v1−v2∥+∣ë2∣∣Δ∣μ1λ1λ3lnb/a2−γ2Γ2−γ2∣1−μ1λ1∣+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ2∣1−μ1λ1∣+λ3lnb/a2−γ2Γ2−γ2×lnb/aα−γ1Γα−γ1+1m1∥u1−u2∥+m2∥v1−v2∥+λ11−μ1λ1μ1lnb/aαΓα+1m1∥u1−u2∥+m2∥v1−v2∥+lnb/aβΓβ+1n1∥u1−u2∥+n2∥v1−v2∥+lnb/aαΓα+1m1∥u1−u2∥+m2∥v1−v2∥=K1m1∥u1−u2∥+m2∥v1−v2∥+K2n1∥u1−u2∥+n2∥v1−v2∥=K1m1+K2n1∥u1−u2∥+K1m2+K2n2∥v1−v2∥.
Therefore, we get the following inequality(43)∥T1u1,v1t−T1u2,v2t∥≤K1m1+m2+K2n1+n2∥u1−u2∥+∥v1−v2∥.
Similarly,(44)∥T2u1,v1t−T2u2,v2t∥≤K3m1+m2+K4n1+n2∥u1−u2∥+∥v1−v2∥.
From inequalities (43) and (44), it yields
(45)∥Tu1,v1t−Tu2,v2∥≤K1+K3m1+m2+K2+K4n1+n2∥u1−u2∥+∥v1−v2∥.
SinceK1+K3m1+m2+K2+K4n1+n2<1, therefore, T is a contraction operator. So, by applying Banach’s fixed point theorem, the operator T has a unique fixed point in Br. Hence, there exists a unique solution of problem (3), (4) on a,b.
Now, we prove our second existence result via the Leray-Schauder alternative.Lemma 9 (Leray-Schauder alternative [39]).
LetF:E→E be a completely continuous operator (i.e., a map restricted to any bounded set in E is compact). Let
(46)εF=x∈E:x=λFxforsome0<λ<1.
Then, either the setεF is unbounded or F has at least one fixed point.Theorem 10.
Assume that:
(H2)f,g:a,b×R×R→R are continuous functions and there exist real constants ai,bi≥0i=0,1,2 and a0>0,b0>0 such that ∀xi∈Ri=1,2, we have
(47)ft,x1,x2≤a0+a1x1+a2x2,∣gt,x1,x2∣≤b0+b1∣x1∣+b2x2.
IfK1+K3a1+K2+K4b1<1 and K1+K3a2+K2+K4b2<1 then system (3), (4) has at least one solution on a,b.Proof.
By the continuity of functionsf,g on a,b×R×R, the operator T is continuous. Now, we show that the operator T:X×Y→X×Y is completely continuous. Let Ω⊂X×Y be bounded. Then, there exist two positive constants, M1 and M2, such that
(48)∣ft,ut,vt∣≤M1,∣gt,ut,vt∣≤M2∀u,v∈Ω.
Then, for anyu,v∈Ω, we have
(49)T1u,vt≤μ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×∑i=1Mlnξi/aα−δ2Γα−δ2+1M1+λ3Δμ1μ2λ1lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ11−μ1λ1+λ1λ2lnb/a2−γ1Γ2−γ11−μ1λ1+μ2lnb/a∑i=1Nlnηi/a1−δ1Γ2−δ1×lnb/aβ−γ2Γβ−γ2+1M2+μ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnb/a2−γ2Γ2−γ2×∑i=1Nlnηi/aβ−δ1Γβ−δ1+1M2+λ2Δμ1λ1λ3lnb/a2−γ2Γ2−γ21−μ1λ1+λ1μ3lnb/a∑i=1Mlnξi/a1−δ2Γ2−δ21−μ1λ1+λ3lnb/a2−γ2Γ2−γ2×lnb/aα−γ1Γα−γ1+1M1+λ11−μ1λ1μ1lnb/aαΓα+1M1+lnb/aβΓβ+1M2+lnb/aαΓα+1M1,which yields,
(50)∥T1u,v∥≤K1M1+K2M2.
In the same way, we can obtain that∥T2u,v∥≤K3M1+K4M2. Hence, from the above inequalities, we get that the operator T is uniformly bounded, since ∥Tu,v∥≤K1+K3M1+K2+K4M2.
Next, we show thatT is equicontinuous. For any u,v∈Ω, and τ1,τ2∈a,b with τ1<τ2. Then, we have
(51)∣T1u,vτ2−T1u,vτ1∣≤M1μ2μ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1∑i=1Mlnξi/aα−δ2Γα−δ2+1lnτ2a−lnτ1a+M2μ2λ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1lnb/aβ−γ2Γβ−γ2+1lnτ2a−lnτ1a+M2μ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2∑i=1Nlnηi/aβ−δ1Γβ−δ1+1lnτ2a−lnτ1a+M1λ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2lnb/aα−γ1Γα−γ1+1lnτ2a−lnτ1a+M11Γα∫aτ2lnτ2sα−1dss−1Γα∫aτ1lnτ1sα−1dss≤M1μ2μ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1∑i=1Mlnξi/aα−δ2Γα−δ2+1lnτ2a−lnτ1a+M2μ2μ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1lnb/aβ−γ2Γβ−γ2+1lnτ2a−lnτ1a+M2μ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2∑i=1Nlnηi/aβ−δ1Γβ−δ1+1lnτ2a−lnτ1a+M1λ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2lnb/aα−γ1Γα−γ1+1lnτ2a−lnτ1a+M11Γα∫aτ1lnτ2sα−1−lnτ1sα−1dss+M11Γα∫τ1τ2lnτ2sα−1dss=M1μ2μ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1∑i=1Mlnξi/aα−δ2Γα−δ2+1lnτ2a−lnτ1a+M2μ2λ3∑i=1Nlnηi/a1−δ1∣Δ∣Γ2−δ1lnb/aβ−γ2Γβ−γ2+1lnτ2a−lnτ1a+M2μ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2∑i=1Nlnηi/aβ−δ1Γβ−δ1+1lnτ2a−lnτ1a+M1λ2λ3lnb/a1−γ2∣Δ∣Γ2−γ2lnb/aα−γ1Γα−γ1+1lnτ2a−lnτ1a+M1Γα+12lnτ2τ1α+lnτ2aα−lnτ1aα.
Therefore, we obtain(52)∣T1u,vτ2−T1u,vτ1∣⟶0,asτ1⟶τ2.
Analogously, we can get the following inequality:(53)∣T2u,vτ2−T2u,vτ1∣⟶0,asτ1⟶τ2.
Then we can easily show that the operatorTu,v is equicontinuous. As a consequence of steps together with the Arzela′-Ascoli theorem, we get that the operator Tu,v is completely continuous.
Finally, it will be verified that the setε=u,v∈X×Y:u,v=λTu,v,0≤λ≤1 is bounded. Let u,v∈ε, with u,v=λTu,v. For any t∈a,b, we have
(54)ut=λT1u,vt,vt=λT2u,vt.
Then, we have(55)∥ut∥≤K1a0+a1∥u∥+a2∥v∥+K2b0+b1∥u∥+b2∥v∥=K1a0+K2b0+K1a1+K2b1∥u∥+K1a2+K2b2∥v∥,∥vt∥≤K3a0+a1∥u∥+a2∥v∥+K4b0+b1∥u∥+b2∥v∥=K3a0+K4b0+K3a1+K4b1∥u∥+K3a2+K4b2∥v∥,which implies that
(56)∥u∥+∥v∥≤K1+K3a0+K2+K4b0+K1+K3a1+K2+K4b1∥u∥+K1+K3a2+K2+K4b2∥v∥.
Consequently,(57)∥u,v∥≤K1+K3a0+K2+K4b0K0,where
(58)K0=min1−K1+K3a1+K2+K4b1,1−K1+K3a2+K2+K4b2,which proves that ε is bounded. Therefore, by applying Lemma 9, the operator T has at least one fixed point in Ω. Therefore, we deduce that the boundary value problem (3), (4) has at least one solution on a,b.


## 4. Some Examples

In this section, we give an example to illustrate our main results.Example 11.
Consider the following system of Caputo-Hadamard boundary value problem:(59)cD1+3/2ut=ft,ut,vt,t∈1,e,cD1+3/2vt=gft,ut,vt,t∈1,e,u1=ve,1/2cD1+1/2ue=1/3cD1+1/3v3/2+1/3cD1+1/3v4/3,v1=2ue,1/4cD1+1/4ve=1/5cD1+1/5u5/3+1/5cD1+1/5u5/4.
Here,α=β=3/2,a=1,b=e,γ1=1/2,γ2=1/4,δ1=1/3,δ2=1/5,N=M=2,η1=3/2,η2=4/3,ξ1=5/3,ξ2=5/4,λ1=1,λ2=1/2,λ3=1/4,μ1=2,μ2=1/3,μ3=1/5. By simple calculation, we found that Δ=0.078172,K1=10.36402,K2=8.38734,K3=11.58173,K4=11.18721.
(i)
Let two nonlinear functionsf,g:1,e×R×R⟶R be given by(60)ft,x,y=11524+t2∣x∣1+∣xt∣+sinyt64+t2+12,(61)gt,x,y=sin∣x∣124+t2+tan−1y120t2+2+23.
Note that(62)ft,x1,x2−ft,y1,y2≤175x1−x2+165y1−y2,∣gt,x1,x2−gt,y1,y2∣≤1125∣x1−x2∣+1122y1−y2,we obtain K1+K31/75+1/65+K2+K41/125+1/122≈0.9552435311<1. Thus, all the conditions of Theorem 8 are satisfied. Problem (59) with (60) and (61)has a unique solution on 1,e.
(ii)
Let two nonlinear functionsf,g:1,e×R×R→R be given by(63)ft,x,y=e−2t4+x2cos2t391+∣x∣+y4sin2t451+y3,(64)gt,x,y=2t2+2+sinx12t+4+tan−1y143+t2.
Note that(65)ft,x,y≤14+139x+145y,gt,x,y≤23+160x+156y.
We geta1=1/39,a2=1/45,b1=1/60,b2=1/56. By simple calculation, we have K1+K3a1+K2+K4b1≈0.8960930769<1 and K1+K3a2+K2+K4b2≈0.8434206667<1. By Theorem 10, the coupled boundary value problem (59) with (63) and (64) has at least one positive solution on [1,e].


## 5. Conclusions

In this paper, we studied existence and uniqueness of solutions for the system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions. The existence theory of solutions of a Caputo-Hadamard system using a variety of fixed point theorems. The Leray-Schauder alternative was applied to prove existence, while the uniqueness result was obtained via the Banach contradiction mapping principle. Finally, we have given two examples to demonstrate our result.



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