Lyapunov-type inequalities for Hadamard type fractional boundary value problems

1.   Introduction

In the last few decades, fractional differential equations have gained more importance due to its applications in various sciences such as physics, mechanics, chemistry, engineering, etc. For a detailed introduction on this topic, we refer the monographs of Podlubny ^[23], Miller & Ross ^[16], Kibas et al. ^[12] and the references therein. Many mathematicians and scientists have contributed to the development of the theory of fractional differential equations. In this process, several types of fractional derivatives were introduced including the Hadamard fractional derivative, which is the focus of this article.

There has been a rigorous development in the theory and applications of fractional boundary value problems. However, most of the results are concerned with the Riemann-Liouville or the Caputo fractional derivatives. Recently, much attention has been paid to the study of two-point boundary value problems for fractional differential equations involving Hadamard fractional derivatives. In ^[2], Ahmad and Ntouyas studied a coupled system of Hadamard fractional differential equations together with integral boundary conditions. Also, they developed Hadamard fractional integro-differential boundary value problems in ^[3]. Wang et al. ^[26] investigated a non-local Hadamard fractional boundary value problem with Hadamard integral and discrete boundary condition on half line. Recently, Mao et al. ^[19] and Ardjouni ^[1] established sufficient conditions on positive solutions for Hadamard fractional boundary value problems. More recently, Wang et al. ^[27,28] analysed the stability properties of nonlinear Hadamard fractional differential system.

On the other hand, Lyapunov ^[14] proved a necessary condition for the existence of a nontrivial solution of Hill's equation associated with Dirichlet boundary conditions.

Theorem 1.1. ^[14] If the boundary value problem

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

The Lyapunov inequality (1.2) has several applications in various problems related to differential equations. Due to its importance, it has been generalized in many forms. Many researchers have derived Lyapunov-type inequalities for various classes of fractional boundary value problems in the recent years. For the first time, in 2013, Ferreira ^[8] generalized Theorem 1.1 to the case where the classical second-order derivative in (1.1) is replaced by the $\alpha^{\text{th}}$-order ($1 < \alpha \leq 2$) Riemann–Liouville fractional derivative. Further, in 2014, Ferreira ^[9] developed a Lyapunov-type inequality for the Caputo fractional derivative. In 2018, Ntouyas et al. ^[21] presented a survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions. For more details on Lyapunov-type inequalities and their applications, we refer ^{[4,6,10,11,22,24,25,29,30]} and the references therein.

In particular, Ma et al. ^[17] 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 non-trivial solution, where $q : [1, e]\rightarrow \mathbb{R}$ is a continuous function, then

Here $\lambda = \frac{2\alpha - 1 - \sqrt{(2\alpha - 2)^2 + 1}}{2}$ and $^{H}D^{\alpha}$ denotes the $\alpha^{\mathit{\text{th}}}$-order Hadamard fractional differential operator with $1 < \alpha \leq 2$.

Recently, Dhar ^[7] and Laadjal et al. ^[15] generalized the Lyapunov-type inequality in Theorem 1.2 by replacing the interval $[1, e]$ with $[a, b]$.

Theorem 1.3. If the Hadamard fractional boundary value problem

has a non-trivial solution and $y(t) \neq 0$ on $(a, b)$, where $q : [a, b]\rightarrow \mathbb{R}$ is a continuous function, then

Here $q_{+}(t) = \max \{q(t), 0\}$.

Theorem 1.4. If the Hadamard fractional boundary value problem

has a non-trivial solution, where $q : [a, b]\rightarrow \mathbb{R}$ is a continuous function, then

where

Motivated by the works in ^[7,15,17], in this article, we establish a few Lyapunov-type inequalities for Hadamard fractional boundary value problems associated with a variety of boundary conditions.

2.   Preliminaries

Throughout, we shall use the following notations, definitions and some lemmas from the theory of fractional calculus in the sense of Hadamard. These results can be found in the monographs ^[6,12]. Denote the set of all real numbers and complex numbers by $\mathbb{R}$ and $\mathbb{C}$, respectively.

Definition 2.1. ^[12] Let $\alpha > 0$ and $a \in \mathbb{R}$. The $\alpha^{\text{th}}$-order Hadamard fractional integral of a function $y : [a, b] \rightarrow \mathbb{R}$ is defined by

provided the right-hand side exists. Here $\Gamma(.)$ denotes the Euler's Gamma function.

Definition 2.2. ^[12] The $\alpha^{\text{th}}$-order Hadamard fractional derivative of a function $y : [a, b] \rightarrow \mathbb{R}$ is defined by

where $n = [\alpha] + 1$.

Definition 2.3. ^[12] $C[a, b]$ be the space of all continuous functions $y : [a, b] \rightarrow \mathbb{R}$ with the norm

Definition 2.4. ^[12] Let $0 \leq \gamma < 1$, $y : (a, b] \rightarrow \mathbb{R}$ and define

$C_{\gamma, log}[a, b]$ be the weighted space of functions $y$ such that $y_{\gamma, \log} \in C[a, b]$, and

Lemma 2.1. ^[12] If $\alpha > 0$, $\beta > 0$ and $0 < a < b < \infty$, then

Lemma 2.2. ^[6] Let $\alpha > 0$, $n = [\alpha] + 1$, and $0 < a < b < \infty$. Assume $y \in C(a, b)$. The equality

is valid if, and only if,

where $C_1$, $C_2$, $\cdots$, $C_n$ are arbitrary constants.

Lemma 2.3. ^[6] Let $\alpha > 0$, $n = [\alpha] + 1$, and $0 < a < b < \infty$. Assume $y \in C(a, b)$. Then,

and

where $C_1$, $C_2$, $\cdots$, $C_n$ are arbitrary constants.

3.   Main results

In this section, we obtain Lyapunov-type inequalities for Hadamard fractional boundary value problems associated with mixed, Sturm-Liouville, Robin and general boundary conditions, using the properties of the corresponding Green's functions.

Theorem 3.1. Let $1 < \alpha \leq 2$ and $h : [a, b] \rightarrow \mathbb{R}$. The fractional boundary value problem

has the unique solution

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

and

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

Proof. Applying ${}^{H}I^{\alpha}_{a}$ (the $\alpha^{\text{th}}$-order Hadamard fractional integral operator) on both sides of (3.1) and using Lemma 2.2, we have

for some $C_{1}$, $C_{2} \in \mathbb{R}$. Applying ${}^{H}I^{2 - \alpha}_{a}$ on both sides of (3.6) and using Lemmas 2.1, we get

Applying ${}^{H}D^{\alpha - 1}_{a}$ (the $(\alpha-1)^{\text{th}}$-order Hadamard fractional differential operator) on both sides of (3.6) and using Lemmas 2.1, we obtain

From the first boundary condition, we have

The second boundary condition yields

Solving (3.9) and (3.10) for $C_{1}$ and $C_{2}$, we have

and

Substituting $C_{1}$ and $C_{2}$ in (3.6), it follows that

The proof is complete.

Corollary 1. Let $1 < \alpha \leq 2$ and $h : [a, b] \rightarrow \mathbb{R}$. The fractional boundary value problem

has the unique solution

where $\bar{G}(t, s)$ is given by

and

Here $n \geq 0$, $p > 0$ and $\bar{A} = n{\left(\log{\frac{b}{a}}\right)}^{\alpha - 1} + p\Gamma(\alpha) > 0$.

Proof. The proof is similar to Theorem 3.1.

We define $H(t, s) = sG(t, s)$ and $\bar{H}(t, s) = s\bar{G}(t, s)$. Now, we prove that these Green's functions are positive and obtain upper bounds for both the Green's functions and their integrals.

Theorem 3.2. The Green's function $H(t, s)$ for (3.1) satisfies $H(t, s) > 0$ for $(t, s) \in (a, b] \times (a, b]$.

Proof. Clearly, for $a < t \leq s \leq b$,

Now, suppose $a < s \leq t \leq b$. Consider

Clearly, $A\Gamma(\alpha) > 0$. Consider

implying that

Since

we have

and

implying that

and

Clearly,

Using (3.17) - (3.20) in (3.16), we have $H(t, s) > 0$. The proof is complete.

Corollary 2. The Green's function $\bar{H}(t, s)$ for (3.11) satisfies $\bar{H}(t, s) \geq 0$ for $(t, s) \in [a, b] \times [a, b]$.

Proof. The proof is similar to Theorem 3.2.

Theorem 3.3. For the Green's function $H(t, s)$ defined in (3.3),

and

Proof. For the first part, we show that for any fixed $t \in (a, b]$, $H(t, s)$ increases with respect to $s$ from $a$ to $t$, and then decreases with respect $s$ from $t$ to $b$. Let $a < t \leq s \leq b$. Consider

implying that $H(t, s)$ is a decreasing function of $s$. Now, suppose $a < s \leq t \leq b$. Consider

Clearly, $\frac{(\alpha - 1)}{A\Gamma(\alpha)} > 0$. Since $a < s < t \leq b$, we have

implying that

Since

we have that

Clearly,

Using (3.22) - (3.24) in (3.21), we have $\frac{\partial}{\partial s}H(t, s) > 0$, implying that $H(t, s)$ is an increasing function of $s$. Then, it follows that

To prove the second part, for $t \in [a, b]$, consider

The proof is complete.

Corollary 3. For the Green's function $\bar{H}(t, s)$ defined in (3.13),

and

Proof. The first part of the proof is similar to the proof of Theorem 3.3. To prove the second part, for $t \in [a, b]$, consider

The proof is complete.

We are now able to formulate Lyapunov-type inequalities for the fractional boundary value problems (3.1) and (3.11).

Theorem 3.4. If the following fractional boundary value problem

has a nontrivial solution, then

Proof. Let $\mathfrak{B} = C_{\gamma, \log}[a, b]$ be the Banach space of functions $y$ endowed with norm

It follows from Theorem 3.1 that a solution to (3.25) satisfies the equation

Hence,

or, equivalently,

An application of Theorem 3.3 yields the result.

Corollary 4. If the following fractional boundary value problem

has a nontrivial solution, then

Proof. Let $\mathfrak{B} = C[a, b]$ be the Banach space of functions $y$ endowed with norm

It follows from Corollary 1 that a solution to (3.27) satisfies the equation

Hence,

or, equivalently,

An application of Corollary 3 yields the result.

Take $l = p = 0$ in Theorem 3.4. Then, we obtain the following Lyapunov-type inequality for the left-focal fractional boundary value problem.

Corollary 5. If the following fractional boundary value problem

has a nontrivial solution, then

Take $n = 0$ in Corollary 4. Then, we obtain the following Lyapunov-type inequality for the right-focal fractional boundary value problem.

Corollary 6. If the following fractional boundary value problem

has a nontrivial solution, then

Take $l = m = n = p = 1$ in Theorem 3.4. Then, we obtain the following Lyapunov-type inequality for the fractional boundary value problem with Robin boundary conditions.

Corollary 7. If the following fractional boundary value problem

has a nontrivial solution, then

Take $l > 0$ and $p = 0$ in Theorem 3.4. Then, we obtain the following Lyapunov-type inequality for the fractional boundary value problem with Sturm-Liouville boundary conditions.

Corollary 8. If the following fractional boundary value problem

has a nontrivial solution, then

4.   Anti-periodic boundary condition

In this section, we obtain a Lyapunov-type inequality for an anti-periodic fractional boundary value problem using the properties of the corresponding Green's function.

Theorem 4.1. Let $1 < \alpha \leq 2$ and $h : [1, T] \rightarrow \mathbb{R}$. The fractional boundary value problem

has the unique solution

where

Proof. Applying $^{H}I^{\alpha}$ on both sides of (4.1) and using Lemma 2.2, we have

for some $C_{1}$, $C_{2} \in \mathbb{R}$. Applying $^{H}I^{2-\alpha}$ on both sides of (4.4) and using Lemma (2.1), we get

Applying $^{H}D^{\alpha - 1}$ on both sides of (4.4) and using Lemma (2.1), we obtain

From the first boundary condition, we get

The second boundary condition yields

Solving (4.7) and (4.8) for $C_{1}$ and $C_{2}$, we have

and

Substituting $C_{1}$ and $C_{2}$ in (3.6), we obtain the unique solution of (4.1) as

The proof is complete.

Let us define $\tilde{H}(t, s) = s (\log t)^{2-\alpha} \tilde{G}(t, s)$ Now, we obtain an upper bound for the Green's function $\tilde{H}(t, s)$.

Theorem 4.2. For the Green's function $\tilde{H}(t, s)$ defined in (4.3), we observe that

Proof. Consider

Denote by

and

For a fixed $t \in [1, T]$,

for all $s\in [1, t]$. So, $\tilde{H_1}(t, s)$ is an increasing function of $s$. Thus,

We observe that $\tilde{H_1}(t, t)$ is an increasing function of $s$, since

Therefore, we have

Now, consider

implying that $H_1(t, 1)$ is a decreasing function of $t$. So, we have

Therefore,

For a fixed $s \in [1, T]$,

implying that $\tilde{H_2}(t, s)$ is an increasing function of $t$. So,

Since $\big{|}\tilde{H_2}(s, s)\big{|} = \big{|}\tilde{H_1}(s, s)\big{|}$ for $s \in [1, T]$, we only consider $\tilde{H_2}(1, s)$. Since

$\tilde{H_2}(1, s)$ is a decreasing function of $s$. Thus, we have

Hence, we have

The final result follows from (4.10) and (4.11).

We are now able to formulate a Lyapunov-type inequality for the Hadamard type fractional boundary value problem with anti-periodic boundary condition.

Theorem 4.3. If the following fractional boundary value problem

has a nontrivial solution, then

Proof. Let $\mathfrak{B} = C_{\gamma, \log}[1, T]$ be the Banach space of functions $y$ endowed with norm

It follows from Theorem 4.1 that a solution to (4.12) satisfies the equation

Hence,

or, equivalently,

An application of Theorem 4.2 yields the result.

5.   Applications

In this section, we discuss two applications of the results established in previous sections. First, we estimate lower bounds on the smallest eigenvalues of the eigenvalue problems corresponding to (3.25), (3.27) and (3.25).

Theorem 5.1. Assume $y$ is a nontrivial solution of the Hadamard fractional eigenvalue problem

where $y(t) \neq 0$ for each $t \in (a, b)$. Then,

Corollary 9. Assume $y$ is a nontrivial solution of the Hadamard fractional eigenvalue problem

where $y(t) \neq 0$ for each $t \in (a, b)$. Then,

Theorem 5.2. Assume $y$ is a nontrivial solution of the Hadamard fractional eigenvalue problem

where $y(t) \neq 0$ for each $t \in (1, T)$. Then,

Proof. From (4.13), we obtain

or, equivalently,

This proves the result (5.2).

The proof is complete. Now we will discuss the disconjugacy and disfocality for Hadamard fractional boundary value problems (1.5), (3.29) and (3.31).

Definition 5.1. The Hadamard fractional boundary value problem (1.5) is disconjugate on $[a, b]$ if and only if each nontrivial solution has less than $[\alpha] + 1$ zeros on $[a, b]$.

Definition 5.2. The Hadamard fractional boundary value problem (3.29) is left disfocal on $[a, b]$ if and only if each nontrivial solution has less than $[\alpha]$ zeros on $[a, b]$.

Definition 5.3. The Hadamard fractional boundary value problem (3.31) is right disfocal on $[a, b]$ if and only if each nontrivial solution has less than $[\alpha]$ zeros on $[a, b]$.

Using these definitions, we introduce non-existence criteria for non-trivial solutions as follows:

Theorem 5.3. The Hadamard fractional boundary value problem (1.5) is disconjugate if

Theorem 5.4. Assume that the assumptions of Theorem 5.3 are satisfied. Then, the Hadamard fractional boundary value problem (1.5) has no non-trivial solution on $[a, b]$.

Theorem 5.5. The Hadamard fractional boundary value problem (3.29) is left disfocal if

Theorem 5.6. Assume that the assumptions of Theorem 5.5 are satisfied. Then, the Hadamard fractional boundary value problem (3.29) has no non-trivial solution on $[a, b]$.

Theorem 5.7. The Hadamard fractional boundary value problem (3.31) is right disfocal if

Theorem 5.8. Assume that the assumptions of Theorem 5.7 are satisfied. Then, the Hadamard fractional boundary value problem (3.31) has no non-trivia l solution on $[a, b]$.

6.   Conclusions

In this article, we considered Hadamard fractional boundary value problems associated with two different types of boundary conditions-general and anti-periodic, and established Lyapunov-type inequalities for the same. In this process, we derived a few important properties of the corresponding Green's functions. In the later part of the article, we illustrated the applicability of established results through two applications. As the first application, we obtained lower bounds on the smallest eigenvalues for the corresponding Hadamard fractional eigenvalue problems. For the second application, we introduced the concepts of disconjugacy and disfocality and using which we obtained non-existence criteria for non-trivial solutions of Hadamard fractional boundary value problems.

Conflict of interest

All the authors declare no conflicts of interest.