Lyapunov-type inequalities for Hadamard fractional differential equation under Sturm-Liouville boundary conditions

1.   Introduction

For the second order Hill's equation,

where $r(t)$ is a positive continuous function on $[a, b]$, Lyapunov ^[1] obtain the interesting following result.

Theorem 1.1. If Eq (1.1) has a nontrivial solution u(t) satisfying $u(a) = u(b) = 0$, then

Because of the importance of Lyapunov inequality (1.2) in application, Lyapunov inequality has been extended in many directions. In the last few decades, with the increasing enthusiasm for the study of fractional differential equations, a large number of Lyapunov inequalities for fractional differential equations have appeared. For example, Ferreira ^[2] first obtained a Lyapunov-type inequality for Riemann-Liouville fractional differential equation, in 2014, Ferreira ^[3] developed a Lyapunov-type inequality for the Caputo fractional differential equation. For more details on Lyapunov-type inequalities and their applications, we refer ^{[4,5,6,7,8,9,10,11,12,13,14,15]} and the references therein.

Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard ^[16]. Hadamard-type integrals arise in the formulation of many problems in mechanics such as in fracture analysis. For details and applications of Hadamard fractional derivative and integral, we refer the reader to the works in ^{[17,18,19,20,21,22]}. It must be noted that there are few papers studied Lyapunov-type inequality for Hadamard fractional differential equation. For instance, Ma et al. ^[23] developed a Lyapunov-type inequality for the Hadamard fractional boundary value problem in 2017.

Theorem 1.2. If the Hadamard fractional boundary value problem

has a nontrivial solution, where $q:[1, e]\to \mathbb{R}$ is a continuous function, then

where $\lambda = \frac{2\alpha-1-\sqrt{(2\alpha-2)^2+1}}{2}$ and $^HD_{a+}^{\alpha}$ denotes the Hadamard fractional derivative of order $\alpha$.

Recently, Dhar ^[24] and Laadjal et al. ^[25] generalized the Lyapunov-type inequality in Theorem 1.2 by replacing the interval $[1, e]$ with a general interval $[a, b](1\leq a < b),$.

Theorem 1.3. If the Hadamard fractional boundary value problem

has a nontrivial solution, where $q:[a, b]\to \mathbb{R}$ is a continuous function, then

Theorem 1.4. If the Hadamard fractional boundary value problem

has a nontrivial solution, where $q:[a, b]\to \mathbb{R}$ is a continuous function, then

where

Very recently, J. Jonnalagadda and B. Debananda ^[26] obtained Lyapunov-type inequalities for Hadamard fractional boundary value problems associated with different sets of boundary conditions, the main results are as follows.

Theorem 1.5. If the following fractional boundary value problem

has a nontrivial solution, then

where $l, p\geq 0; m, n > 0$ and

Theorem 1.6. If the following fractional boundary value problem

has a nontrivial solution, then

Inspired by papers ^[24,25,26], in this paper, we establish a few Lyapunov-type inequalities for Hadamard fractional differential equations

with the following Sturm-Liouville multi-point and integral boundary conditions,

where $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, $a < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < b$, $\beta_i\geq 0 (i = 1, 2, \cdots, m-2)$ and $h: [a, b]\to [0, \infty)$ with $h\in L^1(a, b)$.

The main difficulty of this paper is to express the solution for boundary value problems (1.20) and (1.21) with Green's function. We solve this problem by properly decomposing coefficients. For convenience, we shall adopt the following notations and assumptions

2.   Preliminaries

In this paper, we shall use the following notations, definitions and some lemmas from the theory of fractional calculus in the sense of Hadamard. For more details, we refer to ^[27].

Definition 2.1. ^[27] The Hadamard fractional integral of order $\alpha\in \mathbb{R}_+$ for a continuous function $f:[a, b]\to \mathbb{R}$ is defined by

Definition 2.2. ^[27] The Hadamard fractional derivative of order $\alpha\in \mathbb{R}_+$ for a continuous function $f:[a, b]\to \mathbb{R}$ is defined by

where $n-1 < \alpha < n, \quad n = [\alpha]+1$.

Lemma 2.3. ^[27] Let $\alpha > 0, n = [\alpha]+1$ and $0 < a < b < \infty.$ if $u\in L^{1}(a, b)$ and $(^{H}I_{a+}^{n-\alpha}u)(t)\in AC_{\delta}^{n}[a, b],$ then

where $AC_{\delta}^{n}[a, b] = \bigg\{\varphi:[a, b]\rightarrow \mathbb{C}:\delta^{(n-1)}\varphi\in AC[a, b], \delta = t\frac{d}{dt}\bigg\}\;$.

Lemma 2.4. Assume that $r\in C[a, b]$. The Sturm-Liouville Hadamard fractional boundary value problem

has the unique solution

where $G(t, s)$ is given by

Proof. Using Lemma 2.3, we have

for some $c_1, c_2\in\mathbb{R}$. Applying the boundary condition $u(a) = 0$, we have $c_2 = 0$, hence,

it is easy to obtain

the boundary condition $\gamma u(b)+\delta u'(b) = \sum_{i = 1}^{m-2}\beta_i u(\xi_i)$ imply that,

we obtain

by the relation

we have

therefore,

The proof is complete.

Lemma 2.5. Assume that $g\in C[a, b]$. The Sturm-Liouville Hadamard fractional boundary value problem

has the unique solution

where $h: [a, b]\to [0, \infty)$ with $h\in L^1(a, b),$ $G(t, s)$ is defined in Lemma 2.4.

Proof. Using Lemma 2.3, we have

for some $c_1, c_2\in\mathbb{R}$. Using the boundary condition $u(a) = 0$, we have $c_2 = 0$, therefore,

and

the boundary condition $\gamma u(b)+\delta u'(b) = \lambda\int_a^bh(t)u(t){\rm{d}}t$ imply that,

we obtain

therefore, the solution of the boundary value problem is

Multiplying both side of above equation by $h(t)$ and integrating from $a$ to $b$, we obtain

and

thus

which concludes the proof.

Lemma 2.6. The function $G$ defined in Lemma 2.4 satisfies the following properties:

Proof. (1). Firstly, we define two functions as follows

Obviously, $g_2(t, s)\geq 0$. Next, we consider function $g_1(t, s)$, differentiating $g_1(t, s)$ with respect to $t$, we get

since $a\leq s\leq t\leq b$, we have

which imply that

The inequality

implying that

so we have $\frac{\partial g_1(t, s)}{\partial t}\leq 0$, this means that $g_1(b, s)\leq g_1(t, s)\leq g_1(s, s)$. On the other hand,

hence, $g_1(t, s)\geq g_1(b, s)\geq 0$ and $G(t, s)\geq 0$.

(2). By the above discussion, for function $g_1(t, s)$ satisfy $0\leq g_1(t, s)\leq g_1(s, s) = G(s, s)$. It is easy to see that $0\leq g_2(t, s)\leq g_2(s, s) = g_1(s, s) = G(s, s)$ for $a\leq t\leq s\leq b$.

Lemma 2.7. Assume $0 < a\leq s\leq b$ and $1 < \alpha < 2$, then

Proof. Let

Clearly $f(a) = f(b) = 0$ and $f(s) > 0$ on $(a, b)$. By Rolle's Theorem, there exists $s^*\in(a, b)$ such that $f(s^*) = \max f(s)$ on $(a, b)$, i.e., $f'(s^*) = 0$. Note that

denote $x = \ln\frac{b}{s}, \ y = \ln\frac{s}{a}$, let $f'(s) = 0$, we obtain $(\alpha-1)(x-y) = xy$ and $x+y = \ln\frac{b}{a}$, by these two equalities, we get

note the fact $xy > 0$, we have

and

so,

it is easy to show that $s^*\in(a, b)$, thus

which concludes the proof.

3.   Main results

We now present Lyapunov-type inequalities for the Sturm-Liouville Hadamard fractional boundary value problems (1.20) and (1.21).

Theorem 3.1. If a nontrivial continuous solution of the Sturm-Liouville Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, and $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, $a < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < b$, $\beta_i\geq 0 (i = 1, 2, \cdots, m-2)$, then we have

Proof. Let $\mathbf{B} = C[a, b]$ be the Banach space endowed with norm $\|x\| = \sup_{t\in[a, b]}|x(t)|$. From Lemma 2.4, for all $t\in[a, b]$, we have

Now, an application of Lemma 2.6 yields

Since $u$ is non trivial, then $\|u\|\neq 0$, so

from which inequality in (3.1) follows.

Let $\beta_i = 0 (i = 1, 2, \cdots, m-2)$ in Theorem 3.1, we have

Corollary 3.2. If a nontrivial continuous solution of the Sturm-Liouville Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, and $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, then we have

Let $\gamma = 1, \delta = 0$ or $\gamma = 0, \delta = 1$ in Corollary 3.2, we can obtain the following Lyapunov-type inequalities.

Corollary 3.3. If a nontrivial continuous solution of the Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, then we have

This is Theorem 1 in ^[25].

Corollary 3.4. If a nontrivial continuous solution of the Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, then we have

Let $\gamma = 1, \delta = 0$ in Theorem 3.1, we have the following Lyapunov-type inequality.

Corollary 3.5. If a nontrivial continuous solution of the Hadamard fractional multi-point boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, $a < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < b$, $\beta_i\geq 0 (i = 1, 2, \cdots, m-2)$ with $0\leq \sum_{i = 1}^{m-2}\beta_i < 1$, then we have

Let $\gamma = 0, \delta = 1$ in Theorem 3.1, we have the following Lyapunov-type inequality.

Corollary 3.6. If a nontrivial continuous solution of the Hadamard fractional multi-point boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, $a < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < b$, $\beta_i\geq 0 (i = 1, 2, \cdots, m-2)$ with $0\leq \sum_{i = 1}^{m-2}\beta_i < 1$, then we have

Let $\beta = \beta_1\geq 0, \beta_2 = \beta_3 = \cdots = \beta_{m-2} = 0, \xi = \xi_1$ in Corollary 3.5, we obtain three-point Lyapunov-type inequality.

Corollary 3.7. If a nontrivial continuous solution of the Hadamard fractional three-point boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, $a < \xi < b$, $0\leq\beta < 1$, then we have

By the relation

and applying Theorem 3.1 and Lemma 2.7, we easily get the following results.

Theorem 3.8. If a nontrivial continuous solution of the Sturm-Liouville Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function, and $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, $a < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < b$, $\beta_i\geq 0 (i = 1, 2, \cdots, m-2)$, then we have

A similar discussion can be made for Theorem 3.8. We omit the details here.

Next we give Lyapunov-type inequalities for Sturm-Liouville Hadamard fractional boundary value problems (1.20)–(1.22). The proof is essentially the same as that of Theorem 3.1 and Theorem 3.8. Therefore, we omit the proof.

Theorem 3.9. If a nontrivial continuous solution of the Sturm-Liouville Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function and $h: [a, b]\to [0, \infty)$ with $h\in L^1(a, b),$ $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, then we have

Theorem 3.10. If a nontrivial continuous solution of the Sturm-Liouville Hadamard fractional boundary value problem

exists, where $q: [a, b]\to \mathbb{R}$ is a continuous function and $h: [a, b]\to [0, \infty)$ with $h\in L^1(a, b),$ $\gamma\geq 0, \delta\geq 0, \gamma\delta > 0$, then we have

Remark 3.11. The similar discussion can be made for Theorem 3.9 and Theorem 3.10. We omit the details here.

Remark 3.12. Comparisons with previous literatures, the conclusion of this paper has the following characteristics. Firstly, our results are new and include all the conclusions of ^[23,24,25]. Secondly, the result of Lemma 2.7 is better than that of Lemma 5 in ^[25].

Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions. This work is supported by the Tianjin Natural Science Foundation (grant no. 20JCYBJC00210).

Conflict of interest

The authors declare no conflict of interest.