1 Introduction

Let u be a nontrivial solution to the second order differential equation

$$ u''(t)+q(t)u(t)=0,\quad a< t< b $$
(1.1)

with the Dirichlet boundary condition

$$ u(a)=u(b)=0, $$
(1.2)

where \(q:[a,b]\to\mathbb{R}\) is continuous. Then the so-called Lyapunov inequality [1]

$$ (b-a)\int_{a}^{b}\bigl\vert q(s) \bigr\vert \,ds>4 $$
(1.3)

holds, and constant 4 in (1.3) cannot be replaced by a larger number. The above inequality has several applications to various problems related to differential equations.

There are several generalizations and extensions of Lyapunov’s result. Hartman and Wintner [2] proved that if u is a nontrivial solution to (1.1)-(1.2), then

$$\int_{a}^{b} (b-s) (s-a)q^{+}(s)\,ds>b-a, $$

where \(q^{+}(s)\) is the positive part of q, defined as

$$q^{+}(s)=\max\bigl\{ q(s),0\bigr\} . $$

For other generalizations and extensions of the classical Lyapunov’s inequality, we refer to [2–17] and the references therein.

Recently, some Lyapunov-type inequalities for fractional boundary value problems have been obtained. In [9], Ferreira established a Lyapunov-type inequality for a differential equation that depends on the Riemann-Liouville fractional derivative, i.e., for the boundary value problem

$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 1< \alpha\leq2, \\ & u(a)=u(b)=0, \end{aligned}$$

where he proved that if u is a nontrivial continuous solution to the above problem, then

$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds> \frac{\Gamma(\alpha)\alpha^{\alpha}}{[(\alpha -1)(b-a)]^{\alpha-1}}. $$
(1.4)

In [8], Ferreira obtained a Lyapunov-type inequality for the Caputo fractional boundary value problem

$$\begin{aligned} &\bigl({}^{C}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0, \quad a< t< b, 1< \alpha\leq2, \\ & u(a)=u(b)=0, \end{aligned}$$

where he established that if u is a nontrivial continuous solution to the above problem, then

$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds> \Gamma(\alpha) \biggl(\frac{4}{b-a} \biggr)^{\alpha-1}. $$
(1.5)

Observe that if we set \(\alpha=2\) in (1.4) or (1.5), one can obtain the classical Lyapunov inequality (1.3). In [11], Jleli and Samet studied the fractional differential equation

$$\bigl({}^{C}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 1< \alpha\leq2 $$

with mixed boundary conditions

$$ u(a)=u'(b)=0 $$
(1.6)

or

$$ u'(a)=u(b)=0. $$
(1.7)

For boundary conditions (1.6) and (1.7), two Lyapunov-type inequalities were established respectively as follows:

$$ \int_{a}^{b} (b-s)^{\alpha-2} \bigl\vert q(s)\bigr\vert \,ds\geq\frac {\Gamma(\alpha)}{\max\{\alpha-1,2-\alpha\}(b-a)} $$
(1.8)

and

$$\int_{a}^{b} (b-s)^{\alpha-1}\bigl\vert q(s) \bigr\vert \,ds\geq\Gamma(\alpha). $$

Rong and Bai [16] established a Lyapunov-type inequality for the above fractional differential equation with the fractional boundary conditions

$${}^{C}_{a}{D^{\beta}} u(b)=u(a)=0, $$

where \(0<\beta\leq1\) and \(1<\alpha\leq\beta+1\). They established the following result: if a nontrivial continuous solution to the above fractional boundary value problem exists, then

$$ \int_{a}^{b} (b-s)^{\alpha-\beta-1} \bigl\vert q(s)\bigr\vert \,ds\geq \frac{(b-a)^{-\beta}}{\max \{\frac{1}{\Gamma(\alpha)}-\frac {\Gamma(2-\beta)}{\Gamma(\alpha-\beta)},\frac{\Gamma(2-\beta )}{\Gamma(\alpha-\beta)}, (\frac{2-\alpha}{\alpha-1} )\frac{\Gamma(2-\beta)}{\Gamma(\alpha-\beta)} \}}. $$
(1.9)

Observe that if \(\beta=1\), then (1.9) reduces to the Lyapunov-type inequality (1.8). For other related works, we refer to [18–21].

In all the above cited works, the fractional order α belongs to \((1.2]\). In this paper, we are concerned with the problem of finding new Lyapunov-type inequalities for the fractional boundary value problem

$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0, \quad a< t< b, 3< \alpha \leq4, \end{aligned}$$
(1.10)
$$\begin{aligned} &u(a)=u'(a)=u''(a)=u''(b)=0, \end{aligned}$$
(1.11)

where \({}_{a}D^{\alpha}\) is the standard Riemann-Liouville fractional derivative of fractional order α and \(q:[a,b]\to\mathbb{R}\) is a continuous function. As an application, we obtain a lower bound for the eigenvalues of the corresponding problem.

Let f be a real function defined on \([a,b]\) (\(a< b\)).

Definition 1.1

The integral

$$\bigl({}_{a}I^{\alpha}f\bigr) (t)=\frac{1}{\Gamma(\alpha)} \int _{a}^{t} (t-s)^{\alpha -1}f(s)\,ds, \quad t \in[a,b], $$

where \(\alpha>0\), is called the Riemann-Liouville fractional integral of order α, and \(\Gamma(\alpha)\) is the Euler gamma function defined by

$$\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t} \,dt,\quad\alpha>0. $$

Definition 1.2

The expression

$${}_{a}D^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \biggl(\frac{d}{dt} \biggr)^{n}\int_{a}^{t}\frac{f(s)}{(t-s)^{\alpha-n+1}} \,ds, $$

where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.

The following lemma is crucial in finding an integral representation of the fractional boundary value problem (1.10)-(1.11).

Lemma 1.3

Assume that \(f\in C(a,b)\cap L(a,b)\) with a fractional derivative of order \(\alpha>0\) that belongs to \(C(a,b)\cap L(a,b)\). Then

$${}_{a}I^{\alpha}{}_{a}D^{\alpha}f(t)=f(t)+c_{1}(t-a)^{\alpha -1}+c_{2}(t-a)^{\alpha-2}+ \cdots+c_{n}(t-a)^{\alpha-n}, $$

for some constants \(c_{i}\in\mathbb{R}\), \(i=1,\ldots,n\), \(n=[\alpha]+1\).

For more details on fractional calculus, we refer the reader to [22–24].

2 Main results

The following lemmas will be needed.

Lemma 2.1

We have that \(u\in C[a,b]\) is a solution to the boundary value problem (1.10)-(1.11) if and only if u satisfies the integral equation

$$u(t)=\int_{a}^{b} G(t,s)q(s)u(s)\,ds, $$

where \(G(t,s)\) is the Green function of problem (1.10)-(1.11) defined as

$$\begin{aligned} G(t,s)=\frac{1}{\Gamma(\alpha)} \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(t-a)^{\alpha-1}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}}-(t-s)^{\alpha-1},& a\leq s\leq t\leq b,\\ \frac{(t-a)^{\alpha-1}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}},& a\leq t\leq s\leq b. \end{array}\displaystyle \right . \end{aligned}$$

Proof

From Lemma 1.3, \(u\in C[a,b]\) is a solution to the boundary value problem (1.10)-(1.11) if and only if

$$u(t)=c_{1}(t-a)^{\alpha-1}+c_{2}(t-a)^{\alpha-2}+c_{3}(t-a)^{\alpha -3}+c_{4}(t-a)^{\alpha-4}- \frac{1}{\Gamma(\alpha)}\int_{a}^{t} (t-s)^{\alpha-1}q(s)u(s) \,ds $$

for some real constants \(c_{i}\), \(i=1,\ldots,4\). Using the boundary conditions \(u(a)=u'(a)=u''(a)=0\), we get immediately

$$c_{2}=c_{3}=c_{4}=0. $$

The boundary condition \(u''(b)=0\) yields

$$c_{1}=\frac{1}{(b-a)^{\alpha-3}\Gamma(\alpha)}\int_{a}^{b} (b-s)^{\alpha -3}q(s)u(s)\,ds. $$

Hence

$$u(t)=\frac{(t-a)^{\alpha-1}}{(b-a)^{\alpha-3}\Gamma(\alpha)}\int_{a}^{b} (b-s)^{\alpha-3}q(s)u(s)\,ds-\frac{1}{\Gamma(\alpha)}\int_{a}^{t} (t-s)^{\alpha-1}q(s)u(s)\,ds, $$

which concludes the proof. □

Lemma 2.2

The function G defined in Lemma  2.1 satisfies the following property:

$$0\leq G(t,s)\leq G(b,s)=\frac{(b-s)^{\alpha-3}(s-a)(2b-a-s)}{\Gamma (\alpha)},\quad(t,s)\in[a,b]\times[a,b]. $$

Proof

We start by fixing an arbitrary \(s\in(a,b]\). Differentiating \(G(t,s)\) with respect to t, we get

$$\begin{aligned} \partial_{t} G(t,s)=\frac{(\alpha-1)}{\Gamma(\alpha)} \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}}-(t-s)^{\alpha-2},& a\leq s\leq t\leq b,\\ \frac{(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}},& a\leq t\leq s\leq b. \end{array}\displaystyle \right . \end{aligned}$$

For \(a\leq t\leq s\leq b\), we have

$$\frac{\Gamma(\alpha)}{(\alpha-1)} \partial_{t} G(t,s) =\frac {(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{(b-a)^{\alpha-3}}\geq0, $$

while for \(a\leq s\leq t\leq b\), we have

$$\begin{aligned} \frac{\Gamma(\alpha)}{(\alpha-1)} \partial_{t} G(t,s) =& \frac {(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{(b-a)^{\alpha-3}}-(t-s)^{\alpha -2} \\ =& \frac{(t-a)^{\alpha-2}((b-a)-(s-a))^{\alpha-3}}{(b-a)^{\alpha -3}}-\bigl((t-a)-(s-a)\bigr)^{\alpha-2} \\ =& (t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}-(t-a)^{\alpha-2} \biggl(1-\frac{s-a}{t-a} \biggr)^{\alpha-2} \\ \geq& (t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}-(t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha-2} \\ =& (t-a)^{\alpha-2} \biggl[ \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}- \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha-2} \biggr] \\ \geq& 0. \end{aligned}$$

Consequently, the function \(G(t,s)\) is non-decreasing with respect to t, from which it follows that

$$0=G(a,s) \leq G(t,s)\leq G(b,s),\quad(t,s)\in[a,b]\times[a,b]. $$

The proof is complete. □

We have the following Hartman-Wintner-type inequality.

Theorem 2.3

If a nontrivial continuous solution to the fractional boundary value problem

$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b} (b-s)^{\alpha-3}(s-a) (2b-a-s)\bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha )\cdot $$
(2.1)

Proof

Let \(\mathcal{B}=C[a,b]\) be the Banach space endowed with the norm

$$\Vert y\Vert _{\infty}=\max_{a\leq t\leq b}\bigl\vert y(t) \bigr\vert , \quad y\in\mathcal{B}. $$

It follows from Lemma 2.1 that a solution u to (1.10)-(1.11) satisfies the integral equation

$$u(t)=\int_{a}^{b} G(t,s)q(s)u(s)\,ds,\quad t \in[a,b]. $$

Thus, for all \(t\in[a,b]\), we have

$$\begin{aligned} \bigl\vert u(t)\bigr\vert \leq& \int_{a}^{b} \bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \bigl\vert u(s) \bigr\vert \,ds \\ \leq& \biggl(\int_{a}^{b} \sup _{a\leq t\leq b}\bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \,ds \biggr)\Vert u\Vert _{\infty}, \end{aligned}$$

which yields

$$\Vert u\Vert _{\infty}\leq \biggl(\int_{a}^{b} \sup_{a\leq t\leq b}\bigl\vert G(t,s)\bigr\vert \bigl\vert q(s) \bigr\vert \, ds \biggr)\Vert u\Vert _{\infty}. $$

Since u is nontrivial, then \(\Vert u\Vert _{\infty}\neq0\), so

$$1\leq\int_{a}^{b} \sup_{a\leq t\leq b} \bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \,ds. $$

Now, an application of Lemma 2.2 yields

$$1\leq\int_{a}^{b} G(b,s)\bigl\vert q(s)\bigr\vert \,ds, $$

from which the inequality in (2.1) follows. □

Corollary 2.4

If a nontrivial continuous solution to the fractional boundary value problem

$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b} (b-s)^{\alpha-3}(s-a) \bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha )}{2(b-a)}\cdot $$
(2.2)

Proof

From Theorem 2.3, we have

$$\int_{a}^{b} (b-s)^{\alpha-3}(s-a) (2b-a-s) \bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha). $$

Next we note

$$2b-a-s\leq2(b-a),\quad s\in[a,b]. $$

Thus we get

$$2(b-a)\int_{a}^{b} (b-s)^{\alpha-3}(s-a)\bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha), $$

which gives the desired inequality (2.2). □

We have the following Lyapunov-type inequality.

Corollary 2.5

If a nontrivial continuous solution to the fractional boundary value problem

$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha)(\alpha-2)^{\alpha -2}}{2(\alpha-3)^{\alpha-3}(b-a)^{\alpha-1}}\cdot $$
(2.3)

Proof

Let

$$\psi(s)=(b-s)^{\alpha-3}(s-a),\quad s\in[a,b]. $$

Now, we differentiate \(\psi(s)\) on \((a,b)\), and we obtain after simplifications

$$\psi'(s)=(b-s)^{\alpha-4}\bigl[(b-s)-(\alpha-3) (s-a)\bigr]. $$

Observe that \(\psi'(s)\) has a unique zero, attained at the point

$$s^{*}=\frac{b+(\alpha-3)a}{\alpha-2}. $$

It is easily seen that \(s^{*}\in(a,b)\), \(\psi'(s)>0\) on \((a,s^{*})\), and \(\psi'(s)<0\) on \((s^{*},b)\). We conclude that

$$\max_{a\leq s\leq b}\psi(s)=\psi\bigl(s^{*}\bigr)=(\alpha-3)^{\alpha-3} \biggl(\frac{b-a}{\alpha-2} \biggr)^{\alpha-2}. $$

From Corollary 2.4, we have

$$\int_{a}^{b} \psi(s)\bigl\vert q(s)\bigr\vert \,ds\geq\frac{\Gamma(\alpha)}{2(b-a)}, $$

which yields

$$\int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha)}{2(b-a)\psi(s^{*})}, $$

from which inequality (2.3) follows. □

Corollary 2.6

If a nontrivial continuous solution to the boundary value problem

$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b} (b-s) (s-a) (2b-a-s) \bigl\vert q(s)\bigr\vert \,ds \geq6. $$
(2.4)

Proof

Inequality (2.4) follows from Theorem 2.3 with \(\alpha=4\). □

Corollary 2.7

If a nontrivial continuous solution to the boundary value problem

$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b} (b-s) (s-a)\bigl\vert q(s)\bigr\vert \,ds \geq\frac{3}{b-a}\cdot $$
(2.5)

Proof

Inequality (2.5) follows from Corollary 2.4 with \(\alpha=4\). □

Corollary 2.8

If a nontrivial continuous solution to the boundary value problem

$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$

exists, where q is a real and continuous function in \([a,b]\), then

$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{12}{(b-a)^{3}}\cdot $$
(2.6)

Proof

Inequality (2.6) follows from Corollary 2.5 with \(\alpha=4\). □

3 Application

In this section, we give an application of the Hartman-Wintner-type inequality (2.2) for the eigenvalue problem

$$\begin{aligned} &\bigl(_{0}D^{\alpha}u\bigr) (t)+\lambda u(t)=0, \quad0< t< 1, 3< \alpha \leq4, \end{aligned}$$
(3.1)
$$\begin{aligned} &u(0)=u'(0)=u''(0)=u''(1)=0. \end{aligned}$$
(3.2)

Theorem 3.1

If λ is an eigenvalue to the fractional boundary value problem (3.1)-(3.2), then

$$\vert \lambda \vert \geq\frac{\Gamma(\alpha)}{2B(2,\alpha-2)}, $$

where B is the beta function defined by

$$B(x,y)=\int_{0}^{1} s^{x-1}(1-s)^{y-1} \,ds, \quad x,y>0. $$

Proof

Let λ be an eigenvalue to (3.1)-(3.2). Then there exists \(u=u_{\lambda}\), a nontrivial solution to (3.1)-(3.2). An application of Corollary 2.4 yields

$$\vert \lambda \vert \int_{0}^{1} (1-s)^{\alpha-3}s\,ds\geq\frac{\Gamma(\alpha )}{2}. $$

Now,

$$\int_{0}^{1} (1-s)^{\alpha-3}s\,ds=\int _{0}^{1} s^{2-1}(1-s)^{(\alpha -2)-1} \,ds=B(2,\alpha-2), $$

from which we obtain

$$\vert \lambda \vert B(2,\alpha-2)\geq\frac{\Gamma(\alpha)}{2}. $$

The proof is complete. □