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[1] | Lakhdar Ragoub, J. F. Gómez-Aguilar, Eduardo Pérez-Careta, Dumitru Baleanu . On a class of Lyapunov's inequality involving λ-Hilfer Hadamard fractional derivative. AIMS Mathematics, 2024, 9(2): 4907-4924. doi: 10.3934/math.2024239 |
[2] | Weerawat Sudsutad, Wicharn Lewkeeratiyutkul, Chatthai Thaiprayoon, Jutarat Kongson . Existence and stability results for impulsive (k,ψ)-Hilfer fractional double integro-differential equation with mixed nonlocal conditions. AIMS Mathematics, 2023, 8(9): 20437-20476. doi: 10.3934/math.20231042 |
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[7] | Saleh S. Redhwan, Sadikali L. Shaikh, Mohammed S. Abdo . Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type. AIMS Mathematics, 2020, 5(4): 3714-3730. doi: 10.3934/math.2020240 |
[8] | Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon . Nonlocal integro-multistrip-multipoint boundary value problems for ¯ψ∗-Hilfer proportional fractional differential equations and inclusions. AIMS Mathematics, 2023, 8(6): 14086-14110. doi: 10.3934/math.2023720 |
[9] | Muhammad Tariq, Sotiris K. Ntouyas, Hijaz Ahmad, Asif Ali Shaikh, Bandar Almohsen, Evren Hincal . A comprehensive review of Grüss-type fractional integral inequality. AIMS Mathematics, 2024, 9(1): 2244-2281. doi: 10.3934/math.2024112 |
[10] | Thabet Abdeljawad, Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Eman Al-Sarairah, Artion Kashuri, Kamsing Nonlaopon . Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application. AIMS Mathematics, 2023, 8(2): 3469-3483. doi: 10.3934/math.2023177 |
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>4b−a, | (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<α≤2,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)(4b−a)α−1. | (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<α≤2,0≤β≤1,x(a)=0,x(b)=m−2∑i=1βix(ξi), | (1.4) |
where Dα,βa+ denotes the Hilfer fractional derivative of order α and type β; a<ξ1<ξ2<⋯<ξm−2<b, βi≥0(i=1,2,⋯,m−2), 0≤∑m−2i=1βi(ξi−a)1−(2−α)(1−β)<(b−a)1−(2−α)(1−β) 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)|ds≥Γ(α)(b−a)α−1L⋅11+∑m−2i=1βiT(b), |
where
L=(α−1)α−1(α−1+2β−αβ)α−1+2β−αβ(2α−2+2β−αβ)2α−2+2β−αβ,T(b)=(b−a)1−(2−α)(1−β)(b−a)1−(2−α)(1−β)−∑m−2i=1βi(ξi−a)1−(2−α)(1−β). |
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<α≤4,x(a)=x′(a)=x‴(a)=0,x″(b)=m−2∑i=1ηix(ξi), | (1.5) |
where CaDα is the Caputo fractional derivative of order α; a<ηi,ξi<b, i=1,2,⋯,m−2, with a<ξ1<ξ2<⋯<ξm−2<b, 0<∑m−2i=1ηi(ξi−a)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)|ds≥2Γ(α−2)(b−a)α(1+(b−a)2∑m−2i=1|ηi|2−∑m−2i=1ηi(ξi−a)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)=m−2∑i=1βix(ξi), | (1.6) |
where CHDαa+ denotes the Caputo-Hadamard fractional derivative of order α; βi≥0, a<ξi<b, (i=1,2,⋯,m−2), with a<ξ1<ξ2<⋯<ξm−2<b, 0≤∑m−2i=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)|ds≥aααΓ(α)[(α−1)(lnb−lna)]α−1⋅lnba−∑m−2i=1βilnξialnba+∑m−2i=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)=m−2∑i=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,t1−ρddtx(t)|t=b=m−2∑i=1σix(ξi), | (1.8) |
where ρDα,βa+ is Hilfer-Katugampola fractional derivative of order α and type β(0≤β≤1); q(t)∈C([a,b],R); γi,σi≥0, a<ηi,ξi<b,(i=1,2,⋯,m−2), with a<η1<η2<⋯<ηm−2<b, a<ξ1<ξ2<⋯<ξm−2<b and they are subject to the following conditions:
(A1) m−2∑i=1γi(ηρi−aρ)1−(2−α)(1−β)<(bρ−aρ)1−(2−α)(1−β).
(A2) m−2∑i=1σi(ξρi−aρ)1−(2−α)(1−β)<[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β).
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 ρ→0+ or ρ→1, 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 c∈R, p≥1, 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 x∈Xpc(a,b) for 0<a<t<b<∞, is defined by
(ρIαa+x)(t)=ρ1−αΓ(α)∫ta(tρ−sρ)α−1sρ−1x(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 0≤a<t<b≤∞, by
(ρDαa+x)(t)=δnρ(ρIn−αa+x)(t)=ρ1−n+αΓ(n−α)(t1−ρddt)n∫tasρ−1x(s)(tρ−sρ)1−n+αds, |
where δnρ=(t1−ρd/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(0≤β≤1) of a function x is defined by
(ρDα,βa+x)(t)=(ρIβ(n−α)a+(t1−ρddt)nρI(1−β)(n−α)a+x)(t). |
Lemma 2.1. [29,30] Let α,β>0,1≤p≤∞,0<a<b<∞ and ρ>0. Then, for x∈Xpc(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, x∈Xpc(a,b) and ρIαa+x∈ACnδρ[a,b]. Then
(ρIαa+ρDαa+x)(t)=x(t)−n∑j=1(δn−jρ(ρIn−αa+x))(a)Γ(α−j+1)(tρ−aρρ)α−j, |
where ACnδρ[a,b] is defined by
ACnδρ[a,b]={x:[a,b]→R|δn−1ρx∈AC[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>α−1. Then
ρIαa+(tρ−aρρ)ξ−1=Γ(ξ)Γ(α+ξ)(tρ−aρρ)α+ξ−1, | (2.2) |
ρDαa+(tρ−aρρ)λ=Γ(λ+1)Γ(λ+1−α)(tρ−aρρ)λ−α, | (2.3) |
ρDαa+(tρ−aρρ)α−j=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(tρ−aρρ)γ−1+c1(tρ−aρρ)γ+2β−2+⋯+cn(tρ−aρρ)γ+n(2β)−(n+1). |
Remark 2.1. By Lemma 2.4, let 0<α≤1, then the homogeneous differential equation with Hilfer-Katugampola fractional derivative
ρDα,βa+x(t)=0, |
has a solution
x(t)=c0(tρ−aρρ)γ−1+c1(tρ−aρρ)γ+2β−2. |
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,c∈R and 1≤p<∞. If x∈Xpc(a,b) and ρI(n−α)(1−β)a+x∈ACnδρ[a,b], then
(ρIαa+ρDα,βa+x)(t)=x(t)−n−1∑k=0(δkρ(ρI(n−α)(1−β)a+x))(a)Γ[k−(n−α)(1−β)+1](tρ−aρρ)k−(n−α)(1−β). |
Proof. According to the Definitions 2.2, 2.3, and Lemma 2.1, we have
(ρIαa+ρDα,βa+x)(t)=(ρIαa+ρIβ(n−α)a+δnρρI(1−β)(n−α)a+x)(t)=(ρIα+β(n−α)a+ρDα+β(n−α)a+x)(t). |
Let ν=α+β(n−α), then n−1<ν≤n. 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)−n∑j=1(δn−jρ(ρIn−νa+x))(a)Γ(α+2−j)(tρ−aρρ)ν−j+1]. |
Let k=n−j, then we can rewrite
(ρIνa+ρDνa+x)(t)=δρ[ρI1a+x(t)−n−1∑k=0(δkρ(ρIn−νa+x))(a)Γ(ν−n+k+2)(tρ−aρρ)ν−n+k+1]. |
Therefore, by using the Lemma 2.2, we finally have
(ρIνa+ρDνa+x)(t)=x(t)−n−1∑k=0(δkρ(ρIn−νa+x))(a)Γ(ν−n+k+1)(tρ−aρρ)ν−n+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>0,0≤β≤1, then the homogeneous fractional differential equation
ρDα,βa+x(t)=0, |
has a general solution of the form
x(t)=c0(tρ−aρρ)−(n−α)(1−β)+c1(tρ−aρρ)1−(n−α)(1−β)+⋯+cn−1(tρ−aρρ)n−1−(n−α)(1−β), |
where cj∈R(j=0,1,⋯,n−1) are arbitrary constants.
Lemma 3.2. Let α>0, n=[α]+1, 0≤β≤1, ρ>0, a>0 and λ>α−1, then
ρDα,βa+(tρ−aρρ)λ+β(n−α)=Γ[λ+1+β(n−α)]Γ[λ+1−α+β(n−α)](tρ−aρρ)β(n−α)+λ−α, | (3.1) |
in particular,
ρDα,βa+(tρ−aρρ)λ−j+β(n−α)=0,j=1,2,⋯,n. | (3.2) |
Proof. From the definition of Hilfer-Katugampola fractional derivative, we have
ρDα,βa+(tρ−aρρ)λ+β(n−α)=ρIβ(n−α)a+δnρρI(1−β)(n−α)a+(tρ−aρρ)λ+β(n−α)=ρIβ(n−α)a+ρDα+β(n−α)a+(tρ−aρρ)λ+β(n−α), | (3.3) |
and
ρDα,βa+(tρ−aρρ)α−j+β(n−α)=ρIβ(n−α)a+ρDα+β(n−α)a+(tρ−aρρ)α+β(n−α)−j. | (3.4) |
On the one hand, we obtain from the Eqs (2.2), (2.3) and (3.3) that
ρDα,βa+(tρ−aρρ)λ+β(n−α)=Γ[λ+1+β(n−α)]Γ(λ+1−α)ρIβ(n−α)a+(tρ−aρρ)λ−α=Γ[λ+1+β(n−α)]Γ[λ+1−α+β(n−α)](tρ−aρρ)λ+β(n−α)−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)m−2∑i=1γi∫baG(ηi,s)q(s)x(s)ds,t∈[a,b], | (4.1) |
where Q(t) is defined by
Q(t)=(tρ−aρ)1−(2−α)(1−β)(bρ−aρ)1−(2−α)(1−β)−∑m−2i=1γi(ηρi−aρ)1−(2−α)(1−β),t∈[a,b], |
and G(t,s) is the Green's function given by
G(t,s)=ρ1−αsρ−1Γ(α)(bρ−aρ)1−(2−α)(1−β){h1(t,s),a≤s≤t≤b,h2(t,s),a≤t≤s≤b, |
with
h1(t,s)=(tρ−aρ)1−(2−α)(1−β)(bρ−sρ)α−1−(bρ−aρ)1−(2−α)(1−β)(tρ−sρ)α−1,h2(t,s)=(tρ−aρ)1−(2−α)(1−β)(bρ−sρ)α−1. |
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(tρ−aρρ)−(2−α)(1−β)+c1(tρ−aρρ)1−(2−α)(1−β), |
where c0,c1∈R. From the first boundary condition x(a)=0, we get c0=0, then
x(t)=−ρIαa+q(t)x(t)+c1(tρ−aρρ)1−(2−α)(1−β). | (4.2) |
The second boundary condition x(b)=m−2∑i=1γix(ηi) yields
x(b)=−ρIαa+q(t)x(t)|t=b+c1(bρ−aρρ)1−(2−α)(1−β)=m−2∑i=1γix(ηi), |
from which we obtain
c1=(bρ−aρρ)−1+(2−α)(1−β)(m−2∑i=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)+(tρ−aρbρ−aρ)1−(2−α)(1−β)(m−2∑i=1γix(ηi)+ρIαa+q(t)x(t)|t=b)=∫baG(t,s)q(s)x(s)ds+m−2∑i=1γix(ηi)(tρ−aρbρ−aρ)1−(2−α)(1−β). | (4.3) |
Then, we deduce
m−2∑i=1γix(ηi)=m−2∑i=1γi∫baG(ηi,s)q(s)x(s)ds+m−2∑i=1γix(ηi)m−2∑i=1γi(ηρi−aρbρ−aρ)1−(2−α)(1−β), |
which gives
m−2∑i=1γix(ηi)=∑m−2i=1γi∫baG(ηi,s)q(s)x(s)ds(bρ−aρ)1−(2−α)(1−β)(bρ−aρ)1−(2−α)(1−β)−∑m−2i=1γi(ηρi−aρ)1−(2−α)(1−β). | (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)m−2∑i=1σi∫baK(ξi,s)q(s)x(s)ds,t∈[a,b], | (4.5) |
where R(t) is defined by
R(t)=(tρ−aρ)1−(2−α)(1−β)[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β)−m−2∑i=1σi(ξiρ−aρ)1−(2−α)(1−β),t∈[a,b], |
and K(t,s) is the Green's function defined by
K(t,s)=(bρ−sρ)α−2ρ1−αsρ−1[1−(2−α)(1−β)]Γ(α){k1(t,s),a≤s≤t≤b,k2(t,s),a≤t≤s≤b, |
with
k1(t,s)=(α−1)(bρ−aρ)(2−α)(1−β)(tρ−aρ)1−(2−α)(1−β)−[1−(2−α)(1−β)](tρ−sρ)α−1(bρ−sρ)α−2,k2(t,s)=(α−1)(bρ−aρ)(2−α)(1−β)(tρ−aρ)1−(2−α)(1−β). |
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(tρ−aρρ)−(2−α)(1−β)+c1(tρ−aρρ)1−(2−α)(1−β), |
where c0,c1∈R. Using the first boundary condition x(a)=0, we find that c0=0, which gives
x(t)=−ρIαa+q(t)x(t)+c1(tρ−aρρ)1−(2−α)(1−β). | (4.6) |
Differentiating the equality (4.6) with respect to t, and then multiplying the both sides of the equation by t1−ρ, we get
t1−ρddtx(t)=−ρIα−1a+q(t)x(t)+c1[1−(2−α)(1−β)](tρ−aρ)−(2−α)(1−β)ρ−(2−α)(1−β), |
which, together with the boundary condition t1−ρddtx(t)|t=b=m−2∑i=1σix(ξi), yields
c1=ρ−(2−α)(1−β)[1−(2−α)(1−β)](bρ−aρ)−(2−α)(1−β)[m−2∑i=1σix(ξi)+ρIα−1a+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)+[∑m−2i=1σix(ξi)+ρIα−1a+q(t)x(t)|t=b](tρ−aρ)1−(2−α)(1−β)[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β)=∫baK(t,s)q(s)x(s)ds+(tρ−aρ)1−(2−α)(1−β)[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β)m−2∑i=1σix(ξi), | (4.7) |
it follows that
m−2∑i=1σix(ξi)=m−2∑i=1σi∫baK(ξi,s)q(s)x(s)ds+m−2∑i=1σi(ξρi−aρ)1−(2−α)(1−β)∑m−2i=1σix(ξi)[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β). | (4.8) |
Solving Eq (4.8), we get
m−2∑i=1σix(ξi)=[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β)∑m−2i=1σi∫baK(ξi,s)q(s)x(s)ds[1−(2−α)(1−β)]ρ(bρ−aρ)−(2−α)(1−β)−∑m−2i=1σi(ξρi−aρ)1−(2−α)(1−β). | (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
2−υ(υ−1)υ−1υ−2≤(υ−1)υ−1υυ. |
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)|≤(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)ρ1−αsρ−1(bρ−aρ)α−1 for any (t,s)∈[a,b]×[a,b];
(iii) |K(t,s)|≤(bρ−sρ)α−2ρ1−αsρ−1[1−(2−α)(1−β)]Γ(α)(bρ−aρ)max{β(2−α),α−1} 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
0≤h2(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=(α−1)ρsρ−1(bρ−aρ)1−(2−α)(1−β)(tρ−sρ)α−2[1−(tρ−sρbρ−sρ)2−α(tρ−aρbρ−aρ)1−(2−α)(1−β)]≥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)=(tρ−aρ)1−(2−α)(1−β)(bρ−aρ)α−1−(bρ−aρ)1−(2−α)(1−β)(tρ−aρ)α−1=(tρ−aρ)1−(2−α)(1−β)(bρ−aρ)α−1[1−(bρ−aρtρ−aρ)2β−αβ]≤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)=(tρ−aρ)1−(2−α)(1−β)(bρ−tρ)α−1,t∈[a,b]. |
Then, differentiating g1(t) on (a,b), we get
g′1(t)=ρtρ−1(tρ−aρ)−(2−α)(1−β)(bρ−tρ)α−2×{[1−(2−α)(1−β)](bρ−tρ)−(α−1)(tρ−aρ)}. |
Observe that g′1(t)=0 has a unique zero on (a,b), attained at the point
t=t∗={aρ+[β(2−α)+(α−1)](bρ−aρ)2(α−1)+β(2−α)}1ρ={bρ−(α−1)(bρ−aρ)2(α−1)+β(2−α)}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(2−α)+(α−1)](bρ−aρ)2(α−1)+β(2−α)}1−(2−α)(1−β)[(α−1)(bρ−aρ)2(α−1)+β(2−α)]α−1=(2β−αβ+α−1)2β−αβ+α−1(bρ−aρ)2β+2α−αβ−2(α−1)α−1(2α−2+2β−αβ)2α−2+2β−αβ. |
Denote
g2(t)=−h1(t,a)=(bρ−aρ)1−(2−α)(1−β)(tρ−aρ)α−1−(tρ−aρ)1−(2−α)(1−β)(bρ−aρ)α−1=(bρ−aρ)1−(2−α)(1−β)(tρ−aρ)α−1[1−(tρ−aρbρ−aρ)β(2−α)],t∈[a,b]. |
We claim that
maxt∈[a,b]g2(t)≤maxt∈[a,b]g1(t). | (4.10) |
In fact, if β(2−α)=0, then (4.10) holds obviously. If β(2−α)≠0, differentiating g2(t) on (a,b), we have
g′2(t)=ρtρ−1(bρ−aρ)α−1(tρ−aρ)α−2[(α−1)(bρ−aρ)β(2−α)−(α+2β−αβ−1)(tρ−aρ)β(2−α)]. |
Observe that g′2(t)=0 has a unique zero on (a,b), attained at the point
t=˜t=[aρ+(α−1α+2β−αβ−1)1β(2−α)(bρ−aρ)]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(2−α)α+2β−αβ−1(α−1α+2β−αβ−1)α−1β(2−α)(bρ−aρ)α−(2−α)(1−β). |
We now show that g2(˜t)≤g1(t∗). Let υ=2α−2+2β−αβα+2β−αβ−1, then by Lemma 4.3, we obtain
g2(˜t)=β(2−α)α+2β−αβ−1(α−1α+2β−αβ−1)α−1β(2−α)(bρ−aρ)α−(2−α)(1−β)≤{(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)[2(α−1)+β(2−α)]2(α−1)+β(2−α)}1α−1+β(2−α)(bρ−aρ)α−(2−α)(1−β)≤(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)[2(α−1)+β(2−α)]2(α−1)+β(2−α)(bρ−aρ)α−(2−α)(1−β)=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)=(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)[2(α−1)+β(2−α)]2(α−1)+β(2−α)(bρ−aρ)α−(2−α)(1−β). |
Thus we have derived that
|G(t,s)|≤ρ1−αsρ−1Γ(α)(bρ−aρ)(2−α)(1−β)−1maxt∈[a,b]h1(t,t)=(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)[2(α−1)+β(2−α)]2(α−1)+β(2−α)⋅ρ1−αsρ−1(bρ−aρ)α−1Γ(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
0≤k2(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=(α−1)[1−(2−α)(1−β)]ρtρ−1(bρ−aρ)(2−α)(1−β)(tρ−aρ)−(2−α)(1−β)−(α−1)[1−(2−α)(1−β)]ρtρ−1(tρ−sρ)α−2(bρ−sρ)α−2=(α−1)[1−(2−α)(1−β)]ρtρ−1[−(bρ−sρtρ−sρ)2−α+(bρ−aρtρ−aρ)(2−α)(1−β)]≤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)=(α−1)(bρ−aρ)(2−α)(1−β)(sρ−aρ)1−(2−α)(1−β)≤k1(b,b)=(α−1)(bρ−aρ).k1(b,s)=(α−1)(bρ−aρ)−[1−(2−α)(1−β)](bρ−sρ). | (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(2−α)(bρ−aρ)≤0,k1(b,b)=(α−1)(bρ−aρ)>0, |
then we have
|k1(b,s)|≤max{k1(b,b),−k1(b,a)}=(bρ−aρ)max{β(2−α),α−1}. | (4.13) |
Combined with (4.11)–(4.13), we get
|k1(t,s)|≤(bρ−aρ)max{β(2−α),α−1}. |
Thus we are led to the conclusion that
|K(t,s)|≤(bρ−sρ)α−2ρ1−αsρ−1[1−(2−α)(1−β)]Γ(α)(bρ−aρ)max{β(2−α),α−1}. |
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(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)ρα−1Δ1[1+Q(b)∑m−2i=1γi]max{aρ−1,bρ−1}, | (4.14) |
where
Δ1:=(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)(bρ−aρ)α−1. |
Proof. By Lemma 4.1, we define the linear operator T:X→X as follow:
Tx(t)=∫baG(t,s)q(s)x(s)ds+Q(t)m−2∑i=1γi∫baG(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,x2∈X with t∈[a,b], we have
|Tx1(t)−Tx2(t)|≤∫ba|G(t,s)q(s)||x1(s)−x2(s)|ds+Q(t)m−2∑i=1γi∫ba|G(ηi,s)q(s)||x1(s)−x2(s)|ds≤[∫ba|G(t,s)q(s)|ds+Q(t)m−2∑i=1γi∫ba|G(ηi,s)q(s)|ds]||x1−x2||∞≤Δ1ρ1−αsρ−1[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)[1+Q(b)m−2∑i=1γi]∫ba|q(s)|ds||x1−x2||∞≤Δ1ρ1−αmax{aρ−1,bρ−1}[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)[1+Q(b)m−2∑i=1γi]∫ba|q(s)|ds||x1−x2||∞. |
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,ρ→0+), Katugampola (β=0), Hadamard (β=0,ρ→0+), Hilfer (ρ→1). 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)=m−2∑i=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)|ds≥aααΓ(α)[(α−1)(lnb−lna)]α−1⋅lnba−∑m−2i=1γilnηialnba+∑m−2i=1γilnbηi. | (4.16) |
Proof. If we put β=1 and let ρ→0+ in the right-hand side of inequality (4.14), we have
limβ=1,ρ→0+[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)ρα−1Δ1[1+Q(b)∑m−2i=1γi]max{aρ−1,bρ−1}=aααΓ(α)(α−1)α−1limρ→0+ρα−1(bρ−aρ)α−1⋅limρ→0+(bρ−aρ)−∑m−2i=1γi(ηρi−aρ)(bρ−aρ)(1+∑m−2i=1γi)−∑m−2i=1γi(ηρi−aρ)L'Hospital'srule__aααΓ(α)[(α−1)(lnb−lna)]α−1⋅lnba−∑m−2i=1γilnηialnba+∑m−2i=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)|ds≥Γ(α)max{aρ−1,bρ−1}(4ρbρ−aρ)α−1. | (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)|ds≥4(α−1)aΓ(α)(lnba)1−α. | (4.20) |
Proof. If we take β=0, γi=0 and let ρ→0+ in the right-hand side of inequality (4.14), we obtain
limβ=0,ρ→0+[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)ρα−1Δ1[1+Q(b)∑m−2i=1γi]max{aρ−1,bρ−1}=4(α−1)aΓ(α)limρ→0+ρα−1(bρ−aρ)α−1L′Hospital′srule__(lnba)1−α4(α−1)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)=m−2∑i=1γix(ηi), | (4.21) |
where q(t)∈C([a,b],R), Dα,βa+ denotes the Hilfer fractional derivative of order α and type β (0≤β≤1), and γi, ηi are defined as (1.15). If (4.21) has a nontrivial continuous solution, then
∫ba|q(s)|ds≥Γ(α)˜Δ1⋅11+∑m−2i=1γi˜Q(b), | (4.22) |
where
˜Δ1=limρ→1[2(α−1)+β(2−α)]2(α−1)+β(2−α)Δ1=[2(α−1)+β(2−α)]2(α−1)+β(2−α)(α−1)α−1[α−1+β(2−α)]α−1+β(2−α)(b−a)α−1,˜Q(b)=limρ→1Q(b)=(b−a)1−(2−α)(1−β)(b−a)1−(2−α)(1−β)−∑m−2i=1γi(ηi−a)1−(2−α)(1−β). |
Proof. Taking ρ→1 in the right-hand side of inequality (4.14), it follows
limρ→1[2(α−1)+β(2−α)]2(α−1)+β(2−α)Γ(α)ρα−1Δ1[1+∑m−2i=1γiQ(b)]max{aρ−1,bρ−1}=Γ(α)˜Δ1⋅11+∑m−2i=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(bρ−sρ)α−2|q(s)|ds≥[1−(2−α)(1−β)]ρα−1Γ(α)Δ2[1+R(b)∑m−2i=1σi], | (4.23) |
where
Δ2:=(bρ−aρ)max{β(2−α),α−1}max{aρ−1,bρ−1}. |
Proof. By Lemma 4.2, we define the linear operator ˜T:E→E as follow:
˜Tx(t)=∫baK(t,s)q(s)x(s)ds+R(t)m−2∑i=1σi∫baK(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,x2∈E with t∈[a,b], we get
|˜Tx1(t)−˜Tx2(t)|≤∫ba|K(t,s)q(s)||x1(s)−x2(s)|ds+R(t)m−2∑i=1σi∫ba|K(ξi,s)q(s)||x1(s)−x2(s)|ds≤[∫ba|K(t,s)q(s)|ds+R(t)m−2∑i=1σi∫ba|K(ξi,s)q(s)|ds]||x1−x2||∞≤(bρ−aρ)max{β(2−α),α−1}ρ1−αsρ−1[1−(2−α)(1−β)]Γ(α)[1+R(b)∑m−2i=1σi]∫ba(bρ−sρ)α−2|q(s)|ds||x1−x2||∞≤Δ2ρ1−α[1−(2−α)(1−β)]Γ(α)[1+R(b)∑m−2i=1σi]∫ba(bρ−sρ)α−2|q(s)|ds||x1−x2||∞. |
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,t1−ρddtx(t)|t=b=m−2∑i=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(bρ−sρ)α−2|q(s)|ds≥ρα−1Γ(α)˜Δ2[1+˜R(b)∑m−2i=1σi], | (4.25) |
where
˜Δ2:=(bρ−aρ)max{aρ,bρ},˜R(b):=(bρ−aρ)α−1(α−1)ρ(bρ−aρ)α−2−∑m−2i=1σi(ξρi−aρ)α−1. |
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)=m−2∑i=1σix(ξi), | (4.26) |
where q(t)∈C([a,b],R), Dα,βa+ denotes the Hilfer fractional derivative of order α and type β (0≤β≤1), and σi, ξi are defined as (1.8). If (4.26) has a nontrivial continuous solution, then
∫ba(b−s)α−2|q(s)|ds≥[1−(2−α)(1−β)]Γ(α)ˆΔ2[1+ˆR(b)∑m−2i=1σi], | (4.27) |
where
ˆΔ2=limρ→1Δ2=(b−a)max{β(2−α),α−1},ˆR(b)=limρ→1R(b)=(b−a)1−(2−α)(1−β)[1−(2−α)(1−β)](b−a)−(2−α)(1−β)−∑m−2i=1σi(ξi−a)1−(2−α)(1−β). |
Proof. Taking ρ→1 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 ρ→0+ or ρ→1, 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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