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Research article

The existence of a compact global attractor for a class of competition model

  • Received: 10 July 2020 Accepted: 24 September 2020 Published: 09 October 2020
  • MSC : 35A01, 35B41, 35K57, 92D40

  • This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

    Citation: Yanxia Wu. The existence of a compact global attractor for a class of competition model[J]. AIMS Mathematics, 2021, 6(1): 210-222. doi: 10.3934/math.2021014

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  • This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.


    Consider the following second-order differential equation with Dirichlet boundary conditions,

    {x(t)+q(t)x(t)=0,t(a,b),x(a)=x(b)=0, (1.1)

    where q(t)C([a,b],R). If there exists a nontrivial solution x(t) of the boundary value problem (BVP for short) given in Eq (1.1), then the inequality,

    ba|q(s)|ds>4ba, (1.2)

    holds. The above inequality (1.2) is known as Lyapunov inequality, and it was first proved by Lyapunov [1]. The inequality given in Eq (1.2) and its generalizations have been used successfully in various branches of mathematics, such as stability problems, oscillation theory, and eigenvalue bounds for ordinary differential equations, see for instance [2,3] and the references cited therein.

    In recent years, with the successful development of fractional calculus theory, the Lyapunov inequalities have been generalized to fractional BVPs, see [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and the references cited therein. Especially, in 2013, Ferreira [4] firstly proved the following result.

    Theorem 1.1. If the fractional BVP

    {(aDαx)(t)+q(t)x(t)=0,t(a,b),1<α2,x(a)=x(b)=0,

    has a nontrivial solution, where aDα is the Riemann-Liouville fractional derivative of order α and q(t)C([a,b],R), then,

    ba|q(s)|ds>Γ(α)(4ba)α1. (1.3)

    Inequality expressed in Eq (1.3) is called Lyapunov-type inequality, and it is a generalization of the inequality given by Eq (1.2) above in the sense of fractional derivative. Since then, many scholars have been tremendously interested in developing Lyapunov-type inequalities, and based on different definitions of fractional calculus, the inequality (1.2) has been generalized to various forms. Examples include Lyapunov-type inequalities for BVPs involving Caputo fractional derivative, Hilfer fractional derivative, Caputo-Fabrizio fractional derivative, Hadamard fractional derivative, Katugampola fractional derivative, conformable fractional derivative, local fractional derivative, and so on. For more details, we refer the interested reader to the survey [15] for a review of recent developments in these problems.

    In recent years, several papers have been published on the study of Lyapunov-type inequalities for fractional differential equations with nonlocal boundary conditions, see for example [16,17,18,19,20,21,22,23,24]. However, only a few considered similar inequalities for fractional m-point BVPs, see [20,21,22,23,24]. In 2018, Wang et al. [20] derived a Lyapunov-type inequality for fractional differential equation involving Hilfer fractional derivative subject to m-point boundary conditions,

    {Dα,βa+x(t)+q(t)x(t)=0,t(a,b),1<α2,0β1,x(a)=0,x(b)=m2i=1βix(ξi), (1.4)

    where Dα,βa+ denotes the Hilfer fractional derivative of order α and type β; a<ξ1<ξ2<<ξm2<b, βi0(i=1,2,,m2), 0m2i=1βi(ξia)1(2α)(1β)<(ba)1(2α)(1β) and q(t)C([a,b],R). By converting the BVP (1.4) into the equivalent integral equation with corresponding Green's function and using norm estimation method, the authors reached the following conclusion.

    Theorem 1.2. If there exists a nontrivial continuous solution of the fractional BVP (1.4), then

    ba|q(s)|dsΓ(α)(ba)α1L11+m2i=1βiT(b),

    where

    L=(α1)α1(α1+2βαβ)α1+2βαβ(2α2+2βαβ)2α2+2βαβ,T(b)=(ba)1(2α)(1β)(ba)1(2α)(1β)m2i=1βi(ξia)1(2α)(1β).

    Later, Aouafi and Adjeroud [21], obtained Lyapunov-type inequality for the fractional differential equation of higher order under m-point boundary conditions

    {CaDαx(t)+q(t)x(t)=0,t(a,b),3<α4,x(a)=x(a)=x(a)=0,x(b)=m2i=1ηix(ξi), (1.5)

    where CaDα is the Caputo fractional derivative of order α; a<ηi,ξi<b, i=1,2,,m2, with a<ξ1<ξ2<<ξm2<b, 0<m2i=1ηi(ξia)2<2 and q(t)C([a,b],R). By converting the BVP (1.5) into the equivalent integral equation with corresponding Green's function and using norm estimation method, the authors obtained the following result.

    Theorem 1.3. If there exists a nontrivial continuous solution of the fractional BVP (1.5), then

    ba|q(s)|ds2Γ(α2)(ba)α(1+(ba)2m2i=1|ηi|2m2i=1ηi(ξia)2)1.

    More recently, in [24], the authors analyzed Lyapunov-type inequality for the fractional BVP involving Caputo-Hadamard fractional derivative supplemented with m-point boundary conditions

    {CHDαa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=m2i=1βix(ξi), (1.6)

    where CHDαa+ denotes the Caputo-Hadamard fractional derivative of order α; βi0, a<ξi<b, (i=1,2,,m2), with a<ξ1<ξ2<<ξm2<b, 0m2i=1βi<1 and q(t)C([a,b],R). By converting the BVP (1.6) into the equivalent integral equation with corresponding Green's function and using norm estimation method, the authors given the following result.

    Theorem 1.4. If there exists a nontrivial continuous solution of the Caputo-Hadamard fractional BVP (1.6), then

    ba|q(s)|dsaααΓ(α)[(α1)(lnblna)]α1lnbam2i=1βilnξialnba+m2i=1βilnbξi.

    Notice the diversity of definitions for fractional derivative, and thus it is challenging to know which definition is the most suitable to use in studying fractional differential equations. One way to overcome such a problem is to work with more general fractional operators, see for example [25,26]. In particular, Oliveira et al. [26] applying the idea of the fractional derivative in the Hilfer sense, proposed a new fractional derivative called Hilfer-Katugampola fractional derivative, which formulation interpolates the well-known fractional derivatives of Hilfer, Katugampola, Hilfer-Hadamard, Riemann-Liouville, Hadamard, Caputo, Caputo-Hadamard, Weyl. Recently, many scholars have been interested in Hilfer-Katugampola fractional derivative and have obtained many exciting and essential results of the existence, uniqueness, and stability of solutions for fractional differential equations using the Hilfer-Katugampola fractional derivative, such as [27,28].

    Motivated by the earlier papers, this study aims to establish new Lyapunov-type inequalities for fractional BVPs involving Hilfer-Katugampola fractional derivative subject to m-point boundary conditions. In precise terms, we consider here the following BVPs:

    {ρDα,βa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,ρ>0,x(a)=0,x(b)=m2i=1γix(ηi), (1.7)

    and

    {ρDα,βa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,ρ>0,x(a)=0,t1ρddtx(t)|t=b=m2i=1σix(ξi), (1.8)

    where ρDα,βa+ is Hilfer-Katugampola fractional derivative of order α and type β(0β1); q(t)C([a,b],R); γi,σi0, a<ηi,ξi<b,(i=1,2,,m2), with a<η1<η2<<ηm2<b, a<ξ1<ξ2<<ξm2<b and they are subject to the following conditions:

    (A1) m2i=1γi(ηρiaρ)1(2α)(1β)<(bρaρ)1(2α)(1β).

    (A2) m2i=1σi(ξρiaρ)1(2α)(1β)<[1(2α)(1β)]ρ(bρaρ)(2α)(1β).

    The main contributions to our results can be summarized as follows:

    We prove some new properties of Hilfer-Katugampola calculus and correct Lemma 2.11 in [28].

    We study the Lyapunov-type inequalities for the m-point fractional BVPs (1.7) and (1.8), which generalize and complement some previous results. Indeed, in the limit case β=0 or β=1 and ρ0+ or ρ1, the conclusions of this paper can be reduced to the results presented in [6,9,20,24].

    In the present work, we derive the Lyapunov-type inequalities for BVPs (1.7) and (1.8) by using Banach's contraction principle, which is quite different from the previous research work.

    To the best of our knowledge, the fractional BVPs involving Hilfer-Katugampola fractional derivatives are rarely studied for Lyapunov-type inequalities. So, the results obtained in this paper are new.

    The rest of the paper is organized as follows: In Section 2, we recall some definitions on the fractional integral and derivative, and related basic properties which will be used in the sequel. In Section 3, we prove some new properties of Hilfer-Katugampola fractional calculus. Our main results are given in Section 4. Finally, we summarize our results and specify new directions for the future works in Section 5.

    In this section, we recall some definitions and lemmas about fractional integral and fractional derivative which we used in this paper. For cR, p1, let Xpc(a,b) denote the space of all complex-valued Lebesgue measurable functions x on (a,b) with ||x||Xpc<, where the norm is defined by

    ||x||Xpc=(ba|tcx(t)|pdtt)1/p<.

    Definition 2.1. [29,30] The left-sided Katugampola fractional integral of order α>0 and ρ>0 of xXpc(a,b) for 0<a<t<b<, is defined by

    (ρIαa+x)(t)=ρ1αΓ(α)ta(tρsρ)α1sρ1x(s)ds,t[a,b]. (2.1)

    Definition 2.2. [29,30] Let α>0,n=[α]+1 and ρ>0. The left-side Katugampola fractional derivative, associated with the Katugampola fractional integral (2.1), is defined, for 0a<t<b, by

    (ρDαa+x)(t)=δnρ(ρInαa+x)(t)=ρ1n+αΓ(nα)(t1ρddt)ntasρ1x(s)(tρsρ)1n+αds,

    where δnρ=(t1ρd/dt)n.

    Definition 2.3. [26] Let α>0, n=[α]+1 and ρ>0. The left-side Hilfer-Katugampola fractional derivative of order α and type β(0β1) of a function x is defined by

    (ρDα,βa+x)(t)=(ρIβ(nα)a+(t1ρddt)nρI(1β)(nα)a+x)(t).

    Lemma 2.1. [29,30] Let α,β>0,1p,0<a<b< and ρ>0. Then, for xXpc(a,b) the semigroup property is valid. That is,

    (ρIαa+ρIβa+x)(t)=(ρIα+βa+x)(t).

    Lemma 2.2. [31] Let α>0, n=[α]+1, xXpc(a,b) and ρIαa+xACnδρ[a,b]. Then

    (ρIαa+ρDαa+x)(t)=x(t)nj=1(δnjρ(ρInαa+x))(a)Γ(αj+1)(tρaρρ)αj,

    where ACnδρ[a,b] is defined by

    ACnδρ[a,b]={x:[a,b]R|δn1ρxAC[a,b]},

    and AC[a,b] denote the space of all absolutely continuous real valued function on [a,b].

    Lemma 2.3. [26,31] Let α>0, n=[α]+1, ρ>0, a>0, ξ>0 and λ>α1. Then

    ρIαa+(tρaρρ)ξ1=Γ(ξ)Γ(α+ξ)(tρaρρ)α+ξ1, (2.2)
    ρDαa+(tρaρρ)λ=Γ(λ+1)Γ(λ+1α)(tρaρρ)λα, (2.3)
    ρDαa+(tρaρρ)αj=0,j=1,2,,n. (2.4)

    Lemma 2.4. [28] Let α>0, then the homogeneous differential equation with Hilfer-Katugampola fractional derivative

    ρDα,βa+x(t)=0,

    has a solution

    x(t)=c0(tρaρρ)γ1+c1(tρaρρ)γ+2β2++cn(tρaρρ)γ+n(2β)(n+1).

    Remark 2.1. By Lemma 2.4, let 0<α1, then the homogeneous differential equation with Hilfer-Katugampola fractional derivative

    ρDα,βa+x(t)=0,

    has a solution

    x(t)=c0(tρaρρ)γ1+c1(tρaρρ)γ+2β2.

    Remark 2.2. The conclusion of Remark 2.1 is incorrect, which means that Lemma 2.4 is not rigorous. Therefore, it is necessary to correct the conclusion of Lemma 2.4. To this end, we will give some new properties of Hilfer-Katugampola fractional calculus in the following section.

    In this section, we will present some new properties of Hilfer-Katugampola fractional calculus and give the modified results of Lemma 2.4.

    Lemma 3.1. Let α>0,n=[α]+1,ρ>0,cR and 1p<. If xXpc(a,b) and ρI(nα)(1β)a+xACnδρ[a,b], then

    (ρIαa+ρDα,βa+x)(t)=x(t)n1k=0(δkρ(ρI(nα)(1β)a+x))(a)Γ[k(nα)(1β)+1](tρaρρ)k(nα)(1β).

    Proof. According to the Definitions 2.2, 2.3, and Lemma 2.1, we have

    (ρIαa+ρDα,βa+x)(t)=(ρIαa+ρIβ(nα)a+δnρρI(1β)(nα)a+x)(t)=(ρIα+β(nα)a+ρDα+β(nα)a+x)(t).

    Let ν=α+β(nα), then n1<νn. An argument similar to the one used in Lemma 2.2 ([31], Theorem 2.7) shows that

    (ρIνa+ρDνa+x)(t)=δρ[ρI1a+x(t)nj=1(δnjρ(ρInνa+x))(a)Γ(α+2j)(tρaρρ)νj+1].

    Let k=nj, then we can rewrite

    (ρIνa+ρDνa+x)(t)=δρ[ρI1a+x(t)n1k=0(δkρ(ρInνa+x))(a)Γ(νn+k+2)(tρaρρ)νn+k+1].

    Therefore, by using the Lemma 2.2, we finally have

    (ρIνa+ρDνa+x)(t)=x(t)n1k=0(δkρ(ρInνa+x))(a)Γ(νn+k+1)(tρaρρ)νn+k,

    which completes the proof.

    As a direct consequence of Lemma 3.1, we will have

    Corollary 3.1. Let α>0,n=[α]+1,ρ>0,0β1, then the homogeneous fractional differential equation

    ρDα,βa+x(t)=0,

    has a general solution of the form

    x(t)=c0(tρaρρ)(nα)(1β)+c1(tρaρρ)1(nα)(1β)++cn1(tρaρρ)n1(nα)(1β),

    where cjR(j=0,1,,n1) are arbitrary constants.

    Lemma 3.2. Let α>0, n=[α]+1, 0β1, ρ>0, a>0 and λ>α1, then

    ρDα,βa+(tρaρρ)λ+β(nα)=Γ[λ+1+β(nα)]Γ[λ+1α+β(nα)](tρaρρ)β(nα)+λα, (3.1)

    in particular,

    ρDα,βa+(tρaρρ)λj+β(nα)=0,j=1,2,,n. (3.2)

    Proof. From the definition of Hilfer-Katugampola fractional derivative, we have

    ρDα,βa+(tρaρρ)λ+β(nα)=ρIβ(nα)a+δnρρI(1β)(nα)a+(tρaρρ)λ+β(nα)=ρIβ(nα)a+ρDα+β(nα)a+(tρaρρ)λ+β(nα), (3.3)

    and

    ρDα,βa+(tρaρρ)αj+β(nα)=ρIβ(nα)a+ρDα+β(nα)a+(tρaρρ)α+β(nα)j. (3.4)

    On the one hand, we obtain from the Eqs (2.2), (2.3) and (3.3) that

    ρDα,βa+(tρaρρ)λ+β(nα)=Γ[λ+1+β(nα)]Γ(λ+1α)ρIβ(nα)a+(tρaρρ)λα=Γ[λ+1+β(nα)]Γ[λ+1α+β(nα)](tρaρρ)λ+β(nα)α.

    On the other hand, by using the Eqs (2.4) and (3.4), we obtain (3.2) immediately. The proof is completed.

    In this subsection we discuss the Green's functions of problems (1.7) and (1.8), and present some of their properties.

    Lemma 4.1. Assume that (A1) holds. Then, for x(t)C[a,b] is a solution of the BVP (1.7) if and only if x(t) satisfies the integral equation

    x(t)=baG(t,s)q(s)x(s)ds+Q(t)m2i=1γibaG(ηi,s)q(s)x(s)ds,t[a,b], (4.1)

    where Q(t) is defined by

    Q(t)=(tρaρ)1(2α)(1β)(bρaρ)1(2α)(1β)m2i=1γi(ηρiaρ)1(2α)(1β),t[a,b],

    and G(t,s) is the Green's function given by

    G(t,s)=ρ1αsρ1Γ(α)(bρaρ)1(2α)(1β){h1(t,s),astb,h2(t,s),atsb,

    with

    h1(t,s)=(tρaρ)1(2α)(1β)(bρsρ)α1(bρaρ)1(2α)(1β)(tρsρ)α1,h2(t,s)=(tρaρ)1(2α)(1β)(bρsρ)α1.

    Proof. Using Lemma 3.1, the fractional differential equation in (1.7) can be transformed into an equivalent integral equation

    x(t)=ρIαa+q(t)x(t)+c0(tρaρρ)(2α)(1β)+c1(tρaρρ)1(2α)(1β),

    where c0,c1R. From the first boundary condition x(a)=0, we get c0=0, then

    x(t)=ρIαa+q(t)x(t)+c1(tρaρρ)1(2α)(1β). (4.2)

    The second boundary condition x(b)=m2i=1γix(ηi) yields

    x(b)=ρIαa+q(t)x(t)|t=b+c1(bρaρρ)1(2α)(1β)=m2i=1γix(ηi),

    from which we obtain

    c1=(bρaρρ)1+(2α)(1β)(m2i=1γix(ηi)+ρIαa+q(t)x(t)|t=b).

    Substituting the value of c1 into (4.2), we have

    x(t)=ρIαa+q(t)x(t)+(tρaρbρaρ)1(2α)(1β)(m2i=1γix(ηi)+ρIαa+q(t)x(t)|t=b)=baG(t,s)q(s)x(s)ds+m2i=1γix(ηi)(tρaρbρaρ)1(2α)(1β). (4.3)

    Then, we deduce

    m2i=1γix(ηi)=m2i=1γibaG(ηi,s)q(s)x(s)ds+m2i=1γix(ηi)m2i=1γi(ηρiaρbρaρ)1(2α)(1β),

    which gives

    m2i=1γix(ηi)=m2i=1γibaG(ηi,s)q(s)x(s)ds(bρaρ)1(2α)(1β)(bρaρ)1(2α)(1β)m2i=1γi(ηρiaρ)1(2α)(1β). (4.4)

    Using Eq (4.4) in Eq (4.3), we obtain the solution (4.1). The converse follows by direct computation. The proof is completed.

    Lemma 4.2. Assume that (A2) holds. Then, for x(t)C[a,b] is a solution of the BVP (1.8) if and only if x(t) satisfies the integral equation

    x(t)=baK(t,s)q(s)x(s)ds+R(t)m2i=1σibaK(ξi,s)q(s)x(s)ds,t[a,b], (4.5)

    where R(t) is defined by

    R(t)=(tρaρ)1(2α)(1β)[1(2α)(1β)]ρ(bρaρ)(2α)(1β)m2i=1σi(ξiρaρ)1(2α)(1β),t[a,b],

    and K(t,s) is the Green's function defined by

    K(t,s)=(bρsρ)α2ρ1αsρ1[1(2α)(1β)]Γ(α){k1(t,s),astb,k2(t,s),atsb,

    with

    k1(t,s)=(α1)(bρaρ)(2α)(1β)(tρaρ)1(2α)(1β)[1(2α)(1β)](tρsρ)α1(bρsρ)α2,k2(t,s)=(α1)(bρaρ)(2α)(1β)(tρaρ)1(2α)(1β).

    Proof. As argued in Lemma 4.1, the solutions of fractional differential equation in (1.8) can be written as

    x(t)=ρIαa+q(t)x(t)+c0(tρaρρ)(2α)(1β)+c1(tρaρρ)1(2α)(1β),

    where c0,c1R. Using the first boundary condition x(a)=0, we find that c0=0, which gives

    x(t)=ρIαa+q(t)x(t)+c1(tρaρρ)1(2α)(1β). (4.6)

    Differentiating the equality (4.6) with respect to t, and then multiplying the both sides of the equation by t1ρ, we get

    t1ρddtx(t)=ρIα1a+q(t)x(t)+c1[1(2α)(1β)](tρaρ)(2α)(1β)ρ(2α)(1β),

    which, together with the boundary condition t1ρddtx(t)|t=b=m2i=1σix(ξi), yields

    c1=ρ(2α)(1β)[1(2α)(1β)](bρaρ)(2α)(1β)[m2i=1σix(ξi)+ρIα1a+q(t)x(t)|t=b].

    Substituting the value of c1 into (4.6), we obtain the solution

    x(t)=ρIαa+q(t)x(t)+[m2i=1σix(ξi)+ρIα1a+q(t)x(t)|t=b](tρaρ)1(2α)(1β)[1(2α)(1β)]ρ(bρaρ)(2α)(1β)=baK(t,s)q(s)x(s)ds+(tρaρ)1(2α)(1β)[1(2α)(1β)]ρ(bρaρ)(2α)(1β)m2i=1σix(ξi), (4.7)

    it follows that

    m2i=1σix(ξi)=m2i=1σibaK(ξi,s)q(s)x(s)ds+m2i=1σi(ξρiaρ)1(2α)(1β)m2i=1σix(ξi)[1(2α)(1β)]ρ(bρaρ)(2α)(1β). (4.8)

    Solving Eq (4.8), we get

    m2i=1σix(ξi)=[1(2α)(1β)]ρ(bρaρ)(2α)(1β)m2i=1σibaK(ξi,s)q(s)x(s)ds[1(2α)(1β)]ρ(bρaρ)(2α)(1β)m2i=1σi(ξρiaρ)1(2α)(1β). (4.9)

    By substituting (4.9) into (4.7), we obtain (4.5). Conversely, by direct computation, it can be established that (4.5) satisfies the problem (1.8). This completes the proof.

    Lemma 4.3. [5] If 1<υ<2, then

    2υ(υ1)υ1υ2(υ1)υ1υυ.

    Lemma 4.4. The Green's functions G(t,s) and K(t,s) given by Lemmas 4.1 and 4.2, respectively, satisfy the following properties:

    (i)G(t,s) and K(t,s) are two continuous functions in [a,b]×[a,b];

    (ii)|G(t,s)|(α1)α1[α1+β(2α)]α1+β(2α)[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)ρ1αsρ1(bρaρ)α1 for any (t,s)[a,b]×[a,b];

    (iii) |K(t,s)|(bρsρ)α2ρ1αsρ1[1(2α)(1β)]Γ(α)(bρaρ)max{β(2α),α1} for any (t,s)[a,b]×[a,b].

    Proof. Clearly, (i) is true. Let's now prove that properties (ii) and (iii). Firstly, we show that (ii) holds. In fact, by the expression for the function h2(t,s), we can easily obtain that

    0h2(t,s)h2(s,s),(t,s)[a,b]×[a,b].

    Now, we turn our attention to the function h1(t,s). Differentiating h1(t,s) with respect to s for every fixed t[a,b], we get

    h1(t,s)s=(α1)ρsρ1(bρaρ)1(2α)(1β)(tρsρ)α2[1(tρsρbρsρ)2α(tρaρbρaρ)1(2α)(1β)]0.

    This means that h1(t,s) is increasing with respect to s[a,t] for any fixed t[a,b]. Hence, we have

    h1(t,a)h1(t,s)h1(t,t).

    Note that

    h1(t,a)=(tρaρ)1(2α)(1β)(bρaρ)α1(bρaρ)1(2α)(1β)(tρaρ)α1=(tρaρ)1(2α)(1β)(bρaρ)α1[1(bρaρtρaρ)2βαβ]0,

    we obtain

    |h1(t,s)|max{maxt[a,b]h1(t,t),maxt[a,b]h1(t,a)}.

    Denote

    g1(t)=h1(t,t)=(tρaρ)1(2α)(1β)(bρtρ)α1,t[a,b].

    Then, differentiating g1(t) on (a,b), we get

    g1(t)=ρtρ1(tρaρ)(2α)(1β)(bρtρ)α2×{[1(2α)(1β)](bρtρ)(α1)(tρaρ)}.

    Observe that g1(t)=0 has a unique zero on (a,b), attained at the point

    t=t={aρ+[β(2α)+(α1)](bρaρ)2(α1)+β(2α)}1ρ={bρ(α1)(bρaρ)2(α1)+β(2α)}1ρ.

    Since g1(a)=g1(b)=0 and g1(t)>0 on (a,b), we conclude that g1(t) reaches the maximum at t=t, that is,

    maxt[a,b]g1(t)=g1(t)={[β(2α)+(α1)](bρaρ)2(α1)+β(2α)}1(2α)(1β)[(α1)(bρaρ)2(α1)+β(2α)]α1=(2βαβ+α1)2βαβ+α1(bρaρ)2β+2ααβ2(α1)α1(2α2+2βαβ)2α2+2βαβ.

    Denote

    g2(t)=h1(t,a)=(bρaρ)1(2α)(1β)(tρaρ)α1(tρaρ)1(2α)(1β)(bρaρ)α1=(bρaρ)1(2α)(1β)(tρaρ)α1[1(tρaρbρaρ)β(2α)],t[a,b].

    We claim that

    maxt[a,b]g2(t)maxt[a,b]g1(t). (4.10)

    In fact, if β(2α)=0, then (4.10) holds obviously. If β(2α)0, differentiating g2(t) on (a,b), we have

    g2(t)=ρtρ1(bρaρ)α1(tρaρ)α2[(α1)(bρaρ)β(2α)(α+2βαβ1)(tρaρ)β(2α)].

    Observe that g2(t)=0 has a unique zero on (a,b), attained at the point

    t=˜t=[aρ+(α1α+2βαβ1)1β(2α)(bρaρ)]1ρ.

    It follows from g2(a)=g2(b)=0 and g2(t)>0 on (a,b) that g2(t) has maximum at point ˜t, that is,

    maxt[a,b]g2(t)=g2(˜t)=β(2α)α+2βαβ1(α1α+2βαβ1)α1β(2α)(bρaρ)α(2α)(1β).

    We now show that g2(˜t)g1(t). Let υ=2α2+2βαβα+2βαβ1, then by Lemma 4.3, we obtain

    g2(˜t)=β(2α)α+2βαβ1(α1α+2βαβ1)α1β(2α)(bρaρ)α(2α)(1β){(α1)α1[α1+β(2α)]α1+β(2α)[2(α1)+β(2α)]2(α1)+β(2α)}1α1+β(2α)(bρaρ)α(2α)(1β)(α1)α1[α1+β(2α)]α1+β(2α)[2(α1)+β(2α)]2(α1)+β(2α)(bρaρ)α(2α)(1β)=g1(t),

    which completes the proof of the claim. So we have

    |h1(t,s)|max{maxt[a,b]h1(t,t),maxt[a,b]h1(t,a)}=max{maxt[a,b]g1(t),maxt[a,b]g2(t)}=maxt[a,b]g1(t)=(α1)α1[α1+β(2α)]α1+β(2α)[2(α1)+β(2α)]2(α1)+β(2α)(bρaρ)α(2α)(1β).

    Thus we have derived that

    |G(t,s)|ρ1αsρ1Γ(α)(bρaρ)(2α)(1β)1maxt[a,b]h1(t,t)=(α1)α1[α1+β(2α)]α1+β(2α)[2(α1)+β(2α)]2(α1)+β(2α)ρ1αsρ1(bρaρ)α1Γ(α).

    Therefore, (ii) is true. Finally, we have to prove that (iii) is also holds. In fact, for any (t,s)[a,b]×[a,b], it is easy to see that

    0k2(t,s)k2(s,s)=k1(s,s).

    We now consider the function k1(t,s). Differentiating it with respect to t, we get

    k1(t,s)t=(α1)[1(2α)(1β)]ρtρ1(bρaρ)(2α)(1β)(tρaρ)(2α)(1β)(α1)[1(2α)(1β)]ρtρ1(tρsρ)α2(bρsρ)α2=(α1)[1(2α)(1β)]ρtρ1[(bρsρtρsρ)2α+(bρaρtρaρ)(2α)(1β)]0.

    This means that for fixed s[a,b], k1(t,s) is a decreasing function of t[s,b], it follows

    k1(b,s)k1(t,s)k1(s,s)=k2(s,s).

    Thus we have

    |k1(t,s)|max{maxs[a,b]|k1(b,s)|,maxs[a,b]|k1(s,s)|}. (4.11)

    Note that

    k1(s,s)=(α1)(bρaρ)(2α)(1β)(sρaρ)1(2α)(1β)k1(b,b)=(α1)(bρaρ).k1(b,s)=(α1)(bρaρ)[1(2α)(1β)](bρsρ). (4.12)

    It can easily be seen that k1(b,s) is an increasing function with respect to s[a,b]. Thus,

    k1(b,a)k1(b,s)k1(b,b).

    Since

    k1(b,a)=β(2α)(bρaρ)0,k1(b,b)=(α1)(bρaρ)>0,

    then we have

    |k1(b,s)|max{k1(b,b),k1(b,a)}=(bρaρ)max{β(2α),α1}. (4.13)

    Combined with (4.11)–(4.13), we get

    |k1(t,s)|(bρaρ)max{β(2α),α1}.

    Thus we are led to the conclusion that

    |K(t,s)|(bρsρ)α2ρ1αsρ1[1(2α)(1β)]Γ(α)(bρaρ)max{β(2α),α1}.

    The proof of the Lemma 4.4 is now completed.

    In this subsection we present the Lyapunov-type inequality for problem (1.7). To show this, we define X=C[a,b] be the Banach space endowed with norm x=maxt[a,b]|x(t)|.

    Theorem 4.1. Suppose that (A1) holds. If the BVP (1.7) has a nontrivial continuous solution x(t)X, where q(t) is a real and continuous function in [a,b], then

    ba|q(s)|ds[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)ρα1Δ1[1+Q(b)m2i=1γi]max{aρ1,bρ1}, (4.14)

    where

    Δ1:=(α1)α1[α1+β(2α)]α1+β(2α)(bρaρ)α1.

    Proof. By Lemma 4.1, we define the linear operator T:XX as follow:

    Tx(t)=baG(t,s)q(s)x(s)ds+Q(t)m2i=1γibaG(ηi,s)q(s)x(s)ds,x(t)C[a,b],t[a,b].

    Then x(t)X is a solution of BVP (1.7) if and only if x(t) is a fixed point of the operator T on X. Using Lemma 4.4 (ii), for any x1,x2X with t[a,b], we have

    |Tx1(t)Tx2(t)|ba|G(t,s)q(s)||x1(s)x2(s)|ds+Q(t)m2i=1γiba|G(ηi,s)q(s)||x1(s)x2(s)|ds[ba|G(t,s)q(s)|ds+Q(t)m2i=1γiba|G(ηi,s)q(s)|ds]||x1x2||Δ1ρ1αsρ1[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)[1+Q(b)m2i=1γi]ba|q(s)|ds||x1x2||Δ1ρ1αmax{aρ1,bρ1}[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)[1+Q(b)m2i=1γi]ba|q(s)|ds||x1x2||.

    Combining this with the Banach's contraction principle, it follows that x(t)X is a nontrivial solution of BVP (1.7) iff the inequality given in Eq (4.14) holds. Otherwise, (1.7) has a uniqueness solution x(t)0. Thus, Theorem 4.1 is proved.

    Notice that the fractional derivative ρDα,βa+ is an interpolator of the following fractional derivatives: Caputo-Hadamard (β=1,ρ0+), Katugampola (β=0), Hadamard (β=0,ρ0+), Hilfer (ρ1). As special cases of Theorem 4.1, we have the following corollaries:

    Corollary 4.1. Consider the following Caputo-Hadamard fractional m-point BVP

    {CHDαa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=m2i=1γix(ηi), (4.15)

    where q(t)C([a,b],R), CHDαa+ denotes the Caputo-Hadamard fractional derivative of order α, and γi, ηi are defined as (1.7). If (4.15) has a nontrivial continuous solution, then

    ba|q(s)|dsaααΓ(α)[(α1)(lnblna)]α1lnbam2i=1γilnηialnba+m2i=1γilnbηi. (4.16)

    Proof. If we put β=1 and let ρ0+ in the right-hand side of inequality (4.14), we have

    limβ=1,ρ0+[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)ρα1Δ1[1+Q(b)m2i=1γi]max{aρ1,bρ1}=aααΓ(α)(α1)α1limρ0+ρα1(bρaρ)α1limρ0+(bρaρ)m2i=1γi(ηρiaρ)(bρaρ)(1+m2i=1γi)m2i=1γi(ηρiaρ)L'Hospital'srule__aααΓ(α)[(α1)(lnblna)]α1lnbam2i=1γilnηialnba+m2i=1γilnbηi.

    Therefore, we obtain form (4.14) that (4.16) holds. Obviously, our results matches the results of Theorem 3.7 in [24].

    Corollary 4.2. Consider the following Katugampola fractional Dirichlet problem

    {ρDαa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=0, (4.17)

    where q(t)C([a,b],R), ρDαa+ denotes the Katugampola fractional derivative of order α. If (4.17) has a nontrivial continuous solution, then

    ba|q(s)|dsΓ(α)max{aρ1,bρ1}(4ρbρaρ)α1. (4.18)

    Proof. Apply Theorem 4.1 for β=0,γi=0, we derive (4.18) immediately. Clearly, our results matches the results of Theorem 5 in [6].

    Corollary 4.3. Consider the following Hadamard fractional Dirichlet problem

    {HDαa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=0, (4.19)

    where q(t)C([a,b],R), ρDαa+ denotes the Hadamard fractional derivative of order α. If (4.19) has a nontrivial continuous solution, then

    ba|q(s)|ds4(α1)aΓ(α)(lnba)1α. (4.20)

    Proof. If we take β=0, γi=0 and let ρ0+ in the right-hand side of inequality (4.14), we obtain

    limβ=0,ρ0+[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)ρα1Δ1[1+Q(b)m2i=1γi]max{aρ1,bρ1}=4(α1)aΓ(α)limρ0+ρα1(bρaρ)α1LHospitalsrule__(lnba)1α4(α1)aΓ(α).

    So we conclude from (4.14) that inequality (4.20) is valid. Evidently, our results matches the results of Theorem 2 in [9].

    Corollary 4.4. Consider the following Hilfer fractional m-point BVP

    {Dα,βa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=m2i=1γix(ηi), (4.21)

    where q(t)C([a,b],R), Dα,βa+ denotes the Hilfer fractional derivative of order α and type β (0β1), and γi, ηi are defined as (1.15). If (4.21) has a nontrivial continuous solution, then

    ba|q(s)|dsΓ(α)˜Δ111+m2i=1γi˜Q(b), (4.22)

    where

    ˜Δ1=limρ1[2(α1)+β(2α)]2(α1)+β(2α)Δ1=[2(α1)+β(2α)]2(α1)+β(2α)(α1)α1[α1+β(2α)]α1+β(2α)(ba)α1,˜Q(b)=limρ1Q(b)=(ba)1(2α)(1β)(ba)1(2α)(1β)m2i=1γi(ηia)1(2α)(1β).

    Proof. Taking ρ1 in the right-hand side of inequality (4.14), it follows

    limρ1[2(α1)+β(2α)]2(α1)+β(2α)Γ(α)ρα1Δ1[1+m2i=1γiQ(b)]max{aρ1,bρ1}=Γ(α)˜Δ111+m2i=1γi˜Q(b).

    Then, by Theorem 4.1, we derive (4.22) from (4.14). Apparently, for a>0, our results matches the results of Theorem 3.1 in [20].

    In this subsection we will prove a Lyapunov-type inequality for problem (1.8). To state our result, we set E=C[a,b] be the Banach space endowed with norm x=maxt[a,b]|x(t)|.

    Theorem 4.2. Suppose that (A2) holds. If the BVP (1.8) has a nontrivial continuous solution x(t)E, where q(t) is a real and continuous function in [a,b], then

    ba(bρsρ)α2|q(s)|ds[1(2α)(1β)]ρα1Γ(α)Δ2[1+R(b)m2i=1σi], (4.23)

    where

    Δ2:=(bρaρ)max{β(2α),α1}max{aρ1,bρ1}.

    Proof. By Lemma 4.2, we define the linear operator ˜T:EE as follow:

    ˜Tx(t)=baK(t,s)q(s)x(s)ds+R(t)m2i=1σibaK(ξi,s)q(s)x(s)ds,x(t)C[a,b],t[a,b],

    thus x(t)E is a solution of BVP (1.8) iff x(t) is a fixed point of the operator ˜T on E. Applying Lemma 4.4 (iii), for any x1,x2E with t[a,b], we get

    |˜Tx1(t)˜Tx2(t)|ba|K(t,s)q(s)||x1(s)x2(s)|ds+R(t)m2i=1σiba|K(ξi,s)q(s)||x1(s)x2(s)|ds[ba|K(t,s)q(s)|ds+R(t)m2i=1σiba|K(ξi,s)q(s)|ds]||x1x2||(bρaρ)max{β(2α),α1}ρ1αsρ1[1(2α)(1β)]Γ(α)[1+R(b)m2i=1σi]ba(bρsρ)α2|q(s)|ds||x1x2||Δ2ρ1α[1(2α)(1β)]Γ(α)[1+R(b)m2i=1σi]ba(bρsρ)α2|q(s)|ds||x1x2||.

    Thus, combined with the Banach's contraction principle, we deduce that x(t)E is a nontrivial solution of BVP (1.8) iff the inequality expressed in Eq (4.23) holds. Otherwise, (1.8) has a uniqueness solution x(t)0. Therefore, we finish the proof of Theorem 4.2.

    As special cases of Theorem 4.2, we have the following corollaries:

    Corollary 4.5. Consider the following Katugampola fractional m-point BVP

    {ρDαa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,t1ρddtx(t)|t=b=m2i=1σix(ξi), (4.24)

    where q(t)C([a,b],R), ρDαa+ denotes the Katugampola fractional derivative of order α, and σi, ξi are defined as (1.8). If (4.24) has a nontrivial continuous solution, then

    ba(bρsρ)α2|q(s)|dsρα1Γ(α)˜Δ2[1+˜R(b)m2i=1σi], (4.25)

    where

    ˜Δ2:=(bρaρ)max{aρ,bρ},˜R(b):=(bρaρ)α1(α1)ρ(bρaρ)α2m2i=1σi(ξρiaρ)α1.

    Proof. Substituting the value of β=0 in (4.23), we get (4.25) immediately.

    Corollary 4.6. Consider the following Hilfer fractional m-point BVP

    {Dα,βa+x(t)+q(t)x(t)=0,0<a<t<b,1<α<2,x(a)=0,x(b)=m2i=1σix(ξi), (4.26)

    where q(t)C([a,b],R), Dα,βa+ denotes the Hilfer fractional derivative of order α and type β (0β1), and σi, ξi are defined as (1.8). If (4.26) has a nontrivial continuous solution, then

    ba(bs)α2|q(s)|ds[1(2α)(1β)]Γ(α)ˆΔ2[1+ˆR(b)m2i=1σi], (4.27)

    where

    ˆΔ2=limρ1Δ2=(ba)max{β(2α),α1},ˆR(b)=limρ1R(b)=(ba)1(2α)(1β)[1(2α)(1β)](ba)(2α)(1β)m2i=1σi(ξia)1(2α)(1β).

    Proof. Taking ρ1 in the both sides of inequality (4.23) gives the desired result (4.27).

    In the last decades, the study of Lyapunov-type inequalities of fractional BVPs has received significant attention from researchers. This increasing interest is motivated by essential applications of the Lyapunov inequality and the development of the fractional calculus theory. In this study, we obtained Lyapunov-type inequalities for fractional m-point BVPs in the frame of Hilfer-Katugampola fractional derivative. In addition, we showed some new properties of the Hilfer-Katugampola fractional derivative, which play a crucial role in the study of BVPs (1.7) and (1.8). Differing from previous work, we established that new Lyapunov-type inequalities are based on a more general fractional derivative, especially in the limit case β=0 or β=1 and ρ0+ or ρ1, our results can be reduced to some known results in the literature. Finally, we point out that there is still some work to be done in the future, such as: discussing the Lyapunov-type inequalities for a nonlinear fractional hybrid boundary value problems involving Hilfer-Katugampola fractional derivative; studying Lyapunov-type inequalities for Hilfer-Katugampola fractional p-Laplacian equations, considering the Lyapunov-type inequalities for fractional Langevin equations, and so on.

    The authors wish to express their sincere appreciation to the editor and the anonymous referees for their valuable comments and suggestions. This research is supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0291) and the Natural Science Foundation of Anhui Province (2108085MA14).

    The authors declare that they have no competing interests.



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