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Research article

Hippocrates of Chios – His Elements and His Lunes A critique of circular reasoning

  • In memory of Johan Rosing (1907–1978) and J. A. Bundgaard (1898–1976) teachers who shaped me but left me free
  • Citation: Jens Høyrup. Hippocrates of Chios – His Elements and His Lunes A critique of circular reasoning[J]. AIMS Mathematics, 2020, 5(1): 158-184. doi: 10.3934/math.2020010

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  • Fractional differential equations with various types of fractional derivatives arise in modeling some dynamical processes (see, for example, [15] for the globally projective synchronization of Caputo fractional-order quaternion-valued neural networks with discrete and distributed delays, [18] for the quasi-uniform synchronization issue for fractional-order neural networks with leakage and discrete delays and [11] for Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field). In contrast to the classical derivative the fractional derivative is nonlocal and it depends significantly on its lower limit. As it is mentioned in [13], this leads to some obstacles for studying impulsive fractional differential equations.

    Since many phenomena are characterized by abrupt changes at certain moments it is important to consider differential equations with impulses. In the literature there are two main approaches used to introduce impulses to fractional equations:

    (i) With a fixed lower limit of the fractional derivative at the initial time- the fractional derivative of the unknown function has a lower limit equal to the initial time point over the whole interval of study;

    (ii) With a changeable lower limit of the fractional derivative at each time of impulse- the fractional derivative on each interval between two consecutive impulses is changed because the lower limit of the fractional derivative is equal to the time of impulse.

    Both interpretations of impulses are based on corresponding interpretations of impulses in ordinary differential equation, which coincide in the case of integer derivatives. However this is not the case for fractional derivatives. In the literature both types of interpretations are discussed and studied for Caputo fractional differential equations of order α(0,1). We refer the reader to the papers [6,7,12,13,16] as well as the monograph [3].

    We note in the case of the Caputo fractional derivative there is a similarity between both the initial conditions and the impulsive condition between fractional equations and ordinary equations (see, for example, [10] concerning the impulsive control law for the Caputo delay fractional-order neural network model). However for Riemann-Liouville fractional differential equations both the initial condition and impulsive conditions have to be appropriately given (which is different in the case of ordinary derivatives as well as the case of Caputo fractional derivatives). Riemann-Liouville fractional differential equations are considered, for example, in [1,2] for integral presentation of solutions in the case of the fractional order α(0,1), [5,8,17] for the case of the fractional order α(1,2).

    In [14] the authors studied the following coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives of the form

    {{Dαu(t)ϕ1(t,Iαu(t),Iβv(t))=0   for tI, tti,  i=1,2,,p,Δu(tj)Ej(u(tj))=0,      Δu(tj)Ej(u(tj))=0,  j=1,2,,p,ν1Dα2u(t)|t=0=u1,     μ1u(t)|t=T+ν2Iα1u(t)|t=T=u2,{Dβv(t)ϕ2(t,Iαu(t),Iβv(t))=0   for tI, ttk,  k=1,2,,q,Δv(tk)Ek(v(tk))=0,      Δv(tk)Ej(v(tk))=0,  k=1,2,,q,ν3Dβ2v(t)|t=0=v1,     μ2v(t)|t=T+ν4Iβ1v(t)|t=T=v2, (1.1)

    where α,β(1,2], I=[0,T], ϕ1,ϕ2:I×R×RR are continuous functions, Δu(tj)=u(t+j)u(tj),  Δu(tj)=u(t+j)u(tj), Δv(tk)=v(t+k)v(tk),  Δv(tj)=v(t+k)v(tk), where u(t+j),v(t+k) and v(tj),v(tk) are the right limits and left limits respectively, Ej,Ej,Ek,Ek:RR are continuous functions, and Dα,Iα are the α-order Riemann-Liouville fractional derivative and integral operators respectively and Dβ2=I2β.

    Since fractional integrals and derivatives have memories, and their lower limits are very important we will use the notations RLDαa,t and Iβa,t, respectively, instead of Dα and Iβ, i.e. the Riemann-Liouville fractional derivative is defined by (see, for example [9])

    RLDαa,tu(t)=1Γ(2α)(ddt)2tau(s)(ts)α1ds,   t>a,  α(1,2), (1.2)

    and the Riemann-Liouville fractional integral Iβa,t of order α>0 is defined by (see, for example, [9])

    Iβa,tu(t)=1Γ(β)ta(ts)β1u(s)ds,t>a, (1.3)

    where a0, β>0 are given numbers.

    Note there are some unclear parts in the statement of coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives (1.1), such as:

    a). The presence of two different integers p and q in (1.1) leads to different domains of both the unknown functions u and v. For example, in Corollary 1 [14] the solutions u(t) and v(t) are defined on [0,tp+1] and [0,tq+1] respectively, which causes some problems in the definitions of formulas (3.7) or (3.8) ([14]);

    b). The impulsive functions Ej,Ek, j=1,2,,p, k=1,2,,q are assumed different but they are not (it is clear for example, for j=k=1). The same is about the functions Ej,Ek, j=1,2,,p, k=1,2,,q.

    In this paper we sort out the above mentioned points by setting up the cleared statement of the boundary value problem with the Riemann-Liouville (RL) fractional integral for the impulsive Riemann-Liouville fractional differential equation studied in [14], and we prove a new the integral presentations of the solutions. To be more general, we study two different interpretations for the presence of impulses in fractional differential equations. The first one is the case of the fixed lower limit of the RL fractional derivative at the initial time 0 and the second one is the case of the changed lower limit of the fractional RL derivative at any point of impulse. In both cases the integral presentation of the solution is provided.

    Define two different sequences of points of impulses

    0=t0<t1<t2<<tp<tp+1=T  and   0=τ0<τ1<τ2<<τq<τq+1=T,

    where p,q are given natural numbers.

    We will consider the following nonlinear boundary value problem for the coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives with a lower limit at 0

    {{RLDα0,tu(t)ϕ1(t,Iα0,tu(t),Iβ0,tv(t))=0   for tI, tti,  i=1,2,,p,Δu(tj)Ej(u(tj))=0,      Δu(tj)Ej(u(tj))=0,  j=1,2,,p,I2α0,tu(t)|t=0=u1,     μ1u(t)|t=T+ν1Iα10,tu(t)|t=T=u2,{RLDβ0,tv(t)ϕ2(t,Iα0,tu(t),Iβ0,tv(t))=0   for tI, tτk,  k=1,2,,q,Δv(τk)Sk(v(τk))=0,      Δv(τk)Sk(v(τk))=0,  k=1,2,,q,I2β0,tv(t)|t=0=v1,     μ2v(t)|t=T+ν2Iβ10,tv(t)|t=T=v2, (2.1)

    where α,β(1,2], I=[0,T], ϕ1,ϕ2:I×R×RR are continuous functions, Δu(tj)=u(t+j)u(tj),  Δu(tj)=u(t+j)u(tj), Δv(τk)=v(τ+k)v(τk),  Δv(τk)=v(τ+k)v(τk), where u(t+j),v(τ+k),u(t+j),v(τ+k) and u(tj), v(tk), u(tj), v(tk) are the right limits and left limits respectively, Ej,Ej,Sk,Sk:RR are continuous functions, and RLDβ0,t,Iα0,t are the α-order Riemann-Liouville fractional derivative and integral operators, respectively, μi,νi,uk,vk, i=1,2, are given constants.

    In the statement of the problem (2.1) some parts of (1.1) are cleared: there are two different points of impulses; the lower limits of the fractional integrals and fractional integrals are written; different functions at different points of impulses are used.

    In our proofs we will use the following well known properties for fractional integrals (see, for example [9]).

    Iαa,tIβa,tu(t)=Iα+βa,tu(t),   α,β>0,Iαa,t(ta)q=Γ(q+1)Γ(q+α+1)(ta)q+α,  α>0, q>1.  t>a. (2.2)

    We will apply the following auxiliary result which is a generalization of the result in [4] for an arbitrary lower limit of the fractional derivative:

    Lemma 1. ([4]). The general solution of the Riemann-Liouville fractional differential equation

    RLDαa,tw(t)=g(t),   t(a,T],   α(1,2) (2.3)

    is given by

    w(t)=c1(ta)α1+c0(ta)α2+Iαa,t g(t)=c1(ta)α1+c0(ta)α2+1Γ(α)ta(ts)α1g(s)ds, t(a,T], (2.4)

    where c0,c1,a0 are arbitrary real constants.

    We will consider an appropriate boundary value problem for a scalar impulsive linear equation, we will prove a formula for its solution and later we will apply it to obatin the main result.

    Consider the following boundary value problem for the linear impulsive fractional differential equation with Riemann-Liouville derivatives of the form

    RLDα0,tu(t)=f(t),   t(0,T],  ttj, j=1,2,p,  α(1,2),Δu(tj)=Ej(u(tj)),      Δu(tj)=Ej(u(tj))  j=1,2,,p,I2α0,tu(t)|t=0=u1,     μ1u(t)|t=T+ν1Iα10,tu(t)|t=T=u2, (2.5)

    where f: [0,T]R is a continuous function, Ej,Ej:RR are continuous functions, u1,u2R.

    Lemma 2. The solution of (2.5) satisfies the integral equation

    u(t)={c0tα1+u1Γ(α1)tα2+1Γ(α)t0(ts)α1f(s)ds,t(0,t1],c0tα1+(jk=1[t2αkEk(u(tk))](α2)jk=1[t1αkEk(u(tk))])tα1     +(u1Γ(α1)+(α1)jk=1t2αkEk(u(tk))jk=1t3αkEk(u(tk)))tα2     +1Γ(α)t0(ts)α1f(s)ds,t(tj,tj+1],

    where

    c0=(p+1k=1[t2αkEk(u(tk))](α2)p+1k=1[t1αkEk(u(tk))])     (u1Γ(α1)+(α1)p+1k=1t2αkEk(u(tk))p+1k=1t3αkEk(u(tk)))T1     1Tα1Γ(α)T0(Ts)α1f(s)ds      ν1μ1Tα1Γ(α1)T0(Ts)α2u(s)ds+u2μ1. (2.6)

    Proof. We will use induction.

    For t(0,t1] we apply Lemma 1 with a=0 and we get

    u(t)=c0tα1+c1tα2+1Γ(α)t0(ts)α1f(s)ds (2.7)

    and

    u(t)=c0(α1)tα2+c1(α2)tα3+α1Γ(α)t0(ts)α2f(s)ds. (2.8)

    From the initial condition I2α0,tu(t)|t=0=u1 and equalities (2.2), (2.7) we get

    I2α0,tu(t)|t=0=c0Γ(α)Γ(1)t|t=0+c1Γ(α1)Γ(0)|t=0+I20,tf(t)|t=0=c1Γ(α1),

    i.e. c1=u1Γ(α1).

    For t(t1,t2] by Lemma 1 with a=0 we get

    u(t)=b0tα1+b1tα2+1Γ(α)t0(ts)α1f(s)ds (2.9)

    and

    u(t)=b0(α1)tα2+b1(α2)tα3+1Γ(α1)t0(ts)α2f(s)ds. (2.10)

    From the impulsive condition u(t1+0)u(t10)=E1(u(t1)) we obtain

    u(t+1)=b0tα11+b1tα21+1Γ(α)t10(t1s)α1f(s)dsc0tα11u1Γ(α1)tα211Γ(α)t10(t1s)α1f(s)ds=(b0c0)tα11+(b1u1Γ(α1))tα21=E1(u(t1)). (2.11)

    and from the impulsive condition u(t1+0)u(t1)=E1(u(t1)) we get

    u(t+1)=b0(α1)tα21+b1(α2)tα31+1Γ(α1)t10(t1s)α2f(s)dsc0(α1)tα21u1Γ(α1)(α2)tα311Γ(α1)t10(t1s)α2f(s)ds=(b0c0)(α1)tα21+(b1u1Γ(α1))(α2)tα31=E1(u(t1)). (2.12)

    Thus we get the linear system w.r.t. b0 and b1

    (b0c0)(α1)tα21+(b1u1Γ(α1))(α2)tα31=E1(u(t1))(b0c0)tα11+(b1u1Γ(α1))tα21=E1(u(t1))

    or

    b0=c0+t2α1E1(u(t1))(α2)t1α1E1(u(t1))
    b1=(α1)t2α1E1(u(t1))t3α1E1(u(t1))+u1Γ(α1).

    Therefore,

    u(t)=c0tα1+(t2α1E1(u(t1))(α2)t1α1E1(u(t1)))tα1     +((α1)t2α1E1(u(t1))t3α1E1(u(t1))+u1Γ(α1))tα2     +1Γ(α)t0(ts)α1f(s)ds,   t(t1,t2]. (2.13)

    Assume the integral presentation of u(t) is correct on (tj1,tj], i.e

    u(t)=c0tα1+(j1k=1t2αkEk(u(tk))(α2)j1k=1t1αkEk(u(tk)))tα1     +((α1)j1k=1t2αkEk(u(tk))j1k=1t3αkEk(u(tk))+u1Γ(α1))tα2     +1Γ(α)t0(ts)α1f(s)ds,   t(tj1,tj]. (2.14)

    Denote

    m0=c0+j1k=1t2αkEk(u(tk))(α2)j1k=1t1αkEk(u(tk))

    and

    m1=(α1)j1k=1t2αkEk(u(tk))j1k=1t3αkEk(u(tk))+u1Γ(α1).

    Let t(tj,tj+1], j=2,,p,. By Lemma 1 with a=0 we get

    u(t)=k0tα1+k1tα2+1Γ(α)t0(ts)α1f(s)ds, (2.15)

    and

    u(t)=k0(α1)tα2+k1(α2)tα3+1Γ(α1)t0(ts)α2f(s)ds. (2.16)

    From the impulsive conditions and the equality (2.14) we obtain the linear system w.r.t. k0 and k1

    (k0m0)(α1)tα2j+(k1m1)(α2)tα3j=E2(u(t2))(k0m0)tα1j+(k1m1)tα2j=Ej(u(tj))

    or

    k0=c0+jk=1[t2αkEk(u(tk))](α2)jk=1[t1αkEk(u(tk))]
    k1=u1Γ(α1)+(α1)jk=1t2αkEk(u(tk))jk=1t3αkEk(u(tk)).

    Therefore,

    u(t)=c0tα1+(jk=1[t2αkEk(u(tk))](α2)jk=1[t1αkEk(u(tk))])tα1     +(u1Γ(α1)+(α1)jk=1t2αkEk(u(tk))jk=1t3αkEk(u(tk)))tα2     +1Γ(α)t0(ts)α1f(s)ds,   t(tj,tj+1], j=1,2,,p. (2.17)

    From the boundary condition μ1u(t)|t=T+ν1Iα1u(t)|t=T=u2 we get

    Iα1u(t)|t=T=1Γ(α1)T0(Ts)α2u(s)ds

    and

    μ1u(t)|t=T+ν1Iα1u(t)|t=T=μ1c0Tα1+μ1(p+1k=1[t2αkEk(u(tk))](α2)p+1k=1[t1αkEk(u(tk))])Tα1     +μ1(u1Γ(α1)+(α1)p+1k=1t2αkEk(u(tk))p+1k=1t3αkEk(u(tk)))Tα2     +μ11Γ(α)T0(Ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds      +ν11Γ(α1)T0(Ts)α2u(s)ds=u2. (2.18)

    From (2.18) we have (2.6).

    We will give an example to illustrate the claim of Lemma 2.

    Example 1. Consider the following boundary value problem for the scalar RL fractional differential equation with an impulse at t=1

    RLD1.50,tu(t)=t,   t(0,1](1,2],Δu(1)=1,      Δu(1)=0,I0.50,tu(t)|t=0=0,     u(t)|t=2+I0.50,tu(t)|t=2=1. (2.19)

    The solution of (2.19) satisfies the integral equation

    u(t)={c0t0.5+t0.5Γ(0.5)+0.266667t2.5Γ(1.5),t(0,1]c0t0.5+0.5t0.5+(1Γ(0.5)+0.5)t0.5+0.266667t2.5Γ(1.5),t(1,2],

    where

    c0=0.251.5084920.5Γ(1.5)120.5Γ(0.5)20(2s)0.5u(s)ds. (2.20)

    Consider the boundary value problem for the nonlinear impulsive fractional integro-differential equations with Riemann-Liouville derivatives of the form

    RLDα0,tu(t)=ϕ1(t,Iα0,tu(t)),   t(0,T],  ttj, j=1,2,p,  α(1,2)Δu(tj)=Ej(u(tj)),      Δu(tj)=Ej(u(tj))  j=1,2,,p,I2α0,tu(t)|t=0=u1,     μ1u(t)|t=T+ν1Iα10,tu(t)|t=T=u2, (2.21)

    where ϕ1: [0,T]×RR is a continuous function, Ej,Ej:RR are continuous functions, u1,u2R.

    Corollary 1. The solution of (2.21) satisfies the integral equation

    u(t)={c0tα1+u1Γ(α1)tα2+1Γ(α)t0(ts)α1ϕ1(s,Iα0,su(s),)ds,  t(0,t1]c0tα1+(jk=1[t2αkEk(u(tk))](α2)jk=1[t1αkEk(u(tk))])tα1     +(u1Γ(α1)+(α1)jk=1t2αkEk(u(tk))jk=1t3αkEk(u(tk)))tα2     +1Γ(α)t0(ts)α1ϕ1(s,Iα0,su(s))ds,      t(tj,tj+1], (2.22)

    where

    c0=(p+1k=1[t2αkEk(u(tk))](α2)p+1k=1[t1αkEk(u(tk))])     (u1Γ(α1)+(α1)p+1k=1t2αkEk(u(tk))p+1k=1t3αkEk(u(tk)))T1     1Tα1Γ(α)T0(Ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds      ν1μ1Tα1Γ(α1)T0(Ts)α2u(s)ds+u2μ1. (2.23)

    The proof of Corollary 1 is similar to the one of Lemma 2 with f(t)=ϕ1(t,Iα0,tu(t)) and we omit it.

    Remark 1. Corollary 1 and the integral presentation (2.22) correct Theorem 3.1 and the formula (3.2) [14]. The main mistake in the proof of formula (3.2) [14] is the incorrect application of Lemma 1 with a=0 on (t1,t2] and taking the lower limit of the integral in (3.5) [14] incorrectly at t1(σ1) instead of 0. A similar comment applies to all the other intervals (tj,tj+1].

    Following the proof of Lemma 2 and the integral presentation (2.22) of problem (2.21), we have the following result:

    Theorem 1. The solution of (2.1) satisfies the integral equations

    u(t)={c0tα1+u1Γ(α1)tα2+1Γ(α)t0(ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds,  t(0,t1],c0tα1+(jk=1[t2αkEk(u(tk))](α2)jk=1[t1αkEk(u(tk))])tα1     +(u1Γ(α1)+(α1)jk=1t2αkEk(u(tk))jk=1t3αkEk(u(tk)))tα2     +1Γ(α)t0(ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds,    t(tj,tj+1], (2.24)

    and

    v(t)={b0tα1+v1Γ(α1)tα2+1Γ(α)t0(ts)α1ϕ2(s,Iα0,su(s),Iβ0,sv(s))ds,  t(0,t1]b0tα1+(jk=1[t2αkSk(v(tk))](α2)jk=1[t1αkSk(v(tk))])tα1     +(u1Γ(α1)+(α1)jk=1t2αkSk(v(tk))jk=1t3αkSk(v(tk)))tα2     +1Γ(α)t0(ts)α1ϕ2(s,Iα0,su(s),Iβ0,sv(s))ds,    t(tj,tj+1], (2.25)

    where

    c0=(p+1k=1[t2αkEk(u(tk))](α2)p+1k=1[t1αkEk(u(tk))])     (u1Γ(α1)+(α1)p+1k=1t2αkEk(u(tk))p+1k=1t3αkEk(u(tk)))T1     1Tα1Γ(α)T0(Ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds      ν1μ1Tα1Γ(α1)T0(Ts)α2u(s)ds+u2μ1b0=(p+1k=1[t2αkSk(v(tk))](α2)p+1k=1[t1αkSk(v(tk))])     (v1Γ(α1)+(α1)p+1k=1t2αkSk(v(tk))p+1k=1t3αkSk(v(tk)))T1     1Tα1Γ(α)T0(Ts)α1ϕ2(s,Iα0,su(s),Iβ0,sv(s))ds      ν2μ2Tα1Γ(α1)T0(Ts)α2v(s)ds+v2μ2. (2.26)

    The proof is similar to the one of Lemma 2 applied twice to each of the both components u and v of the coupled system (2.1) for impulsive points ti, i=1,2,,p and τi, i=1,2,,p and the functions f(t)=ϕ1(t,Iα0,tu(t),Iβ0,tv(t)) and f(t)=ϕ2(t,Iα0,tu(t),Iβ0,tv(t)) respectively.

    Remark 2. Note the integral presentation (2.24), (2.25) of the solutions of the coupled system is the correction of Corollary 1 and integral presentation (3.7), (3.8) in [14].

    Consider the following nonlinear boundary value problem for the coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives with lower limits at impulsive points ti, i=0,1,2,,p1 and τk, k=0,1,2,,q1, respectively,

    {{RLDαti,tu(t)ϕ1(t,Iα0,tu(t),Iβ0,tv(t))=0   for t(ti,ti+1]  i=0,1,2,,p,I2αtj,tu(t)|t=tj=Pju(tj)+Qj,  j=1,2,,p,I2α0,tu(t)|t=0=u1,     μ1u(t)|t=T+ν1Iα10,tu(t)|t=T=u2,{RLDβτk,tv(t)ϕ2(t,Iα0,tu(t),Iβ0,tv(t))=0   for t(τk,τk+1]  k=0,1,2,,q,I2ατk,tu(t)|t=τk=Pkv(τk)+Qjk,  k=1,2,,q,I2β0,tv(t)|t=0=v1,     μ2v(t)|t=T+ν2Iβ10,tv(t)|t=T=v2, (3.1)

    where α,β(1,2], ϕ1,ϕ2:I×R×RR are continuous functions, Pj,Qj, j=1,2,,p, and Pj,Qj, j=1,2,,q, are real numbers, RLDαti,t and RLDβτk,t are the α-order Riemann-Liouville fractional derivatives with lower limits at ti and τk, respectively, μk,νk,uk,vk, i=1,2, are given constants.

    Remark 3. Note problem (3.1) differs from problem (2.1):

    The lower limits of the RL fractional derivatives RLDαtj,t and RLDβτk,t in (3.1) are changed at any time of impulse tj and τk, respectively.

    The impulsive conditions are changed in (3.1). This is because the values of the unknown functions after the impulse, u(tj+0) and v(τk+0), respectively, are considered as initial values at that point. But the RL fractional derivative has a singularity at its lower limit. It requires the chang of the impulsive conditions for the unknown functions.

    Consider the following boundary value problem for the scalar linear impulsive fractional equation with Riemann-Liouville derivatives of the form

    RLDαti,tu(t)=f(t),   for t(ti,ti+1],  i=0,1,2,,p,  α(1,2)I2αtj,tu(t)|t=tj=Pju(tj)+Qj,  j=1,2,,p,I2α0,tu(t)|t=0=u1,     μ1u(t)|t=T+ν1Iα10,tu(t)|t=T=u2, (3.2)

    where the function f: [0,T]R is a continuous function, Pj,Qj, j=1,2,,p are real numbers, u1,u2R.

    Now we will provide an integral presentation of the solution of (3.2).

    Lemma 3. The solution of (3.2) satisfies the integral equation

    u(t)={c0tα1+u1Γ(α1)tα2+Iα0,tf(t),   t(0,t1](c0+u1t1Γ(α1))(ttm)α1mk=1Pk(tktk1)α1(α1)Γ(α1)    +Iαtm,tf(t)+(ttm)α1mk=1PkIαtk1,tf(t)|t=tk+Qk(α1)Γ(α1)mj=k+1Pj(tjtj1)α1(α1)Γ(α1),     for    t(tm,tm+1],m=1,2,,p,

    where

    c0=u2μ1(Ttp)1αMu1t1Γ(α1)(Ttp)1αMIαtp,tf(t)|t=TMpk=1PkIαtk1,tf(t)|t=tk+Qk(α1)Γ(α1)mj=k+1Pj(tjtj1)α1(α1)Γ(α1)      ν1μ1(Ttp)1αM1Γ(α1)T0(Ts)α2u(s)ds,M=(α1)pΓp(α1)pk=1Pk(tktk1)1α. (3.3)

    Proof. We will use induction.

    For t(0,t1] similar to Lemma 2 we get

    u(t)=c0tα1+c1tα2+1Γ(α)t0(ts)α1f(s)ds (3.4)

    where c1=u1Γ(α1).

    For t(t1,t2] by Lemma 1 with a=t1 we get

    u(t)=b0(tt1)α1+b1(tt1)α2+1Γ(α)tt1(ts)α1f(s)ds (3.5)

    and

    u(t)=b0(α1)(tt1)α2+b1(α2)(tt1)α3+1Γ(α1)tt1(ts)α2f(s)ds. (3.6)

    From the impulsive condition I2αt1,tu(t)|t=t1=P1u(t1)+Q1, equalities (2.2) and Iαt1,tf(t))|t=t1=0 we obtain

    I2αt1,tu(t)=b0(α1)I2αt1,t(tt1)α2+b1(α2)I2αt1,t(tt1)α3+I2αt1,tIα1t1,tf(t)=b0(α1)Γ(α1)+b1(α2)I2αt1,t(tt1)α3+I1t1,tf(t) (3.7)

    and

    I2αt1,tu(t)|t=t1=b0(α1)Γ(α1)+b1(α2)I2αt1,t(tt1)α3|t=t1=P1u(t1)+Q1<. (3.8)

    Since the integral I2αt1,t(tt1)α3 does not converge, it follows that b1=0 and

    b0=P1(α1)Γ(α1)(c0(t1t0)α1+u1t1Γ(α1)(t1t0)α1+Iαt0,tf(t)|t=t1)+Q1(α1)Γ(α1).

    Thus,

    u(t)=(c0+u1t1Γ(α1))P1(t1t0)α1(α1)Γ(α1)(tt1)α1   +P1(α1)Γ(α1)Iαt0,tf(t)|t=t1(tt1)α1   +Q1(α1)Γ(α1)(tt1)α1+Iαt1,tf(t). (3.9)

    Similarly, for t(tj,tj+1], j=1,2,,p, by Lemma 1 with a=tj we get

    u(t)=k0(ttj)α1+k1(ttj)α2+1Γ(α)ttj(ts)α1ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds (3.10)

    and

    u(t)=k0(α1)(ttj)α2+k1(α2)(ttj)α3+1Γ(α1)ttj(ts)α2ϕ1(s,Iα0,su(s),Iβ0,sv(s))ds. (3.11)

    From the impulsive conditions we obtain k1=0 and

    k0=Pj(α1)Γ(α1)u(tj)+Qj(α1)Γ(α1).

    From the boundary condition μ1u(t)|t=T+ν1Iα1u(t)|t=T=u2 we get

    Iα1u(t)|t=T=1Γ(α1)T0(Ts)α2u(s)ds

    and

    μ1u(t)|t=T+ν1Iα1u(t)|t=T=μ1(c0+u1t1Γ(α1))(Ttp)α1pk=1Pk(tktk1)α1(α1)Γ(α1)    +μ1Iαtp,tf(t)|t=T+μ1(Ttp)α1pk=1PkIαtk1,tf(t)|t=tk+Qk(α1)Γ(α1)mj=k+1Pj(tjtj1)α1(α1)Γ(α1)      +ν11Γ(α1)T0(Ts)α2u(s)ds=u2 (3.12)

    and we obtain (3.3).

    Example 2. Consider the equation

    RLD1.50,tu(t)=t,   t(0,1],       RLD1.51,tu(t)=t, t(1,2],I0.51,tu(t)|t=1=1,I0.50,tu(t)|t=0=0,     u(t)|t=2+I0.50,tu(t)|t=2=1. (3.13)

    Note the Eq (3.13) is similar to (2.19) but the lower limit of the fractional derivative is changed at the point of the impulse. The solution of (3.13) satisfies the integral equation

    u(t)={0.266667t2.5Γ(1.5),t(0,1],1Γ(1.5)(0.4(t1)2.5+2(t1)1.5t3)+(t1)0.5(0.5)Γ(0.5),t(1,2].

    It is clear the change of the lower limits of the fractional derivatives has a huge influence on the solution of the equation.

    Based on the integral presentation of the linear problem (3.2) and Lemma 3, we obtain the following result:

    Theorem 2. The solution of (3.1) satisfies the integral equations

    u(t)={c0tα1+u1Γ(α1)tα2+Iα0,tϕ1(t,Iα0,tu(t),Iβ0,tv(t)),   t(0,t1](c0+u1t1Γ(α1))(ttm)α1mk=1Pk(tktk1)α1(α1)Γ(α1)    +Iαtm,tϕ1(t,Iα0,tu(t),Iβ0,tv(t))    +(ttm)α1mk=1Pk Iαtk1,tϕ1(t,Iα0,tu(t),Iβ0,tv(t))|t=tk+Qk(α1)Γ(α1)mj=k+1Pj(tjtj1)α1(α1)Γ(α1),     for    t(tm,tm+1],m=1,2,,p,

    and

    v(t)={b0tα1+v1Γ(α1)tα2+Iα0,tϕ2(t,Iα0,tu(t),Iβ0,tv(t)),   t(0,τ1](b0+v1τ1Γ(α1))(tτm)α1mk=1Pk(τkτk1)α1(α1)Γ(α1)    +Iατm,tϕ2(t,Iα0,tu(t),Iβ0,tv(t))    +(tτm)α1mk=1Pk Iατk1,tϕ2(t,Iα0,tu(t),Iβ0,tv(t))|t=τk+Qk(α1)Γ(α1)mj=k+1Pj(τjτj1)α1(α1)Γ(α1),     for    t(τm,τm+1],m=1,2,,q,

    where

    c0=u2μ1(Ttp)1αMu1t1Γ(α1)   (Ttp)1αMIαtq,tϕ1(t,Iα0,tu(t),Iβ0,tv(t))|t=T   Mpk=1PkIαtk1,tϕ1(t,Iα0,tu(t),Iβ0,tv(t))|t=tk+Qk(α1)Γ(α1)pj=k+1Pj(tjtj1)α1(α1)Γ(α1)      ν1μ1(Ttp)1αM1Γ(α1)T0(Ts)α2u(s)ds,M=(α1)pΓp(α1)pk=1Pk(tktk1)1α,
    b0=v2μ2(Tτq)1αCv1τ1Γ(α1)   (Tτq)1αC Iατq,tϕ2(t,Iα0,tu(t),Iβ0,tv(t))t=T   Cqk=1Pk Iατk1,tϕ2(t,Iα0,tu(t),Iβ0,tv(t))|t=τk+Qk(α1)Γ(α1)qj=k+1Pj(τjτj1)α1(α1)Γ(α1)      ν2μ2(Ttq)1αC1Γ(α1)T0(Ts)α2u(s)ds,C=(α1)qΓq(α1)qk=1Pk(τkτk1)1α.

    The proof is similar to the one of Lemma 3 applied twice to each of the both components u and v of the coupled system (3.1) for impulsive points ti, i=1,2,,p and τi, i=1,2,,p and the functions f(t)=ϕ1(t,Iα0,tu(t),Iβ0,tv(t)) and f(t)=ϕ2(t,Iα0,tu(t),Iβ0,tv(t)) respectively.

    In this paper we set up and study a scalar nonlinear integro-differential equation with Riemann-Liouville fractional derivative and impulses. We consider a boundary value problem for the studied equation with Riemann-Liouville fractional derivative of order in (1,2). Note for Riemann-Liouville fractional differential equations both the initial condition and impulsive conditions have to be appropriately given (which is different in the case of ordinary derivatives as well as the case of Caputo fractional derivatives). We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We obtain integral presentations of the solutions in both cases. These presentations could be successfully used for furure studies of existence, stability and other qualitative properties of the solutions of the integro-differential equations with Riemann-Liouville fractional derivative of order in (1,2) and impulses.

    S. H. is partially supported by the Bulgarian National Science Fund under Project KP-06-N32/7.

    The authors declare that they have no competing interests.



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