1 Introduction
Let u be a nontrivial solution to the second order differential equation with the Dirichlet boundary condition where \(q:[a,b]\to\mathbb{R}\) is continuous. Then the so-called Lyapunov inequality [1] 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.
$$ u''(t)+q(t)u(t)=0,\quad a< t< b $$
(1.1)
$$ u(a)=u(b)=0, $$
(1.2)
$$ (b-a)\int_{a}^{b}\bigl\vert q(s) \bigr\vert \,ds>4 $$
(1.3)
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 where \(q^{+}(s)\) is the positive part of q, defined as For other generalizations and extensions of the classical Lyapunov’s inequality, we refer to [2‐17] and the references therein.
$$\int_{a}^{b} (b-s) (s-a)q^{+}(s)\,ds>b-a, $$
$$q^{+}(s)=\max\bigl\{ q(s),0\bigr\} . $$
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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 where he proved that if u is a nontrivial continuous solution to the above problem, then In [8], Ferreira obtained a Lyapunov-type inequality for the Caputo fractional boundary value problem where he established that if u is a nontrivial continuous solution to the above problem, then 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 with mixed boundary conditions or For boundary conditions (1.6) and (1.7), two Lyapunov-type inequalities were established respectively as follows: and Rong and Bai [16] established a Lyapunov-type inequality for the above fractional differential equation with the fractional boundary conditions 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 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].
$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 1< \alpha\leq2, \\ & u(a)=u(b)=0, \end{aligned}$$
$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds> \frac{\Gamma(\alpha)\alpha^{\alpha}}{[(\alpha -1)(b-a)]^{\alpha-1}}. $$
(1.4)
$$\begin{aligned} &\bigl({}^{C}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0, \quad a< t< b, 1< \alpha\leq2, \\ & u(a)=u(b)=0, \end{aligned}$$
$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds> \Gamma(\alpha) \biggl(\frac{4}{b-a} \biggr)^{\alpha-1}. $$
(1.5)
$$\bigl({}^{C}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 1< \alpha\leq2 $$
$$ u(a)=u'(b)=0 $$
(1.6)
$$ u'(a)=u(b)=0. $$
(1.7)
$$ \int_{a}^{b} (b-s)^{\alpha-2} \bigl\vert q(s)\bigr\vert \,ds\geq\frac {\Gamma(\alpha)}{\max\{\alpha-1,2-\alpha\}(b-a)} $$
(1.8)
$$\int_{a}^{b} (b-s)^{\alpha-1}\bigl\vert q(s) \bigr\vert \,ds\geq\Gamma(\alpha). $$
$${}^{C}_{a}{D^{\beta}} u(b)=u(a)=0, $$
$$ \int_{a}^{b} (b-s)^{\alpha-\beta-1} \bigl\vert q(s)\bigr\vert \,ds\geq \frac{(b-a)^{-\beta}}{\max \{\frac{1}{\Gamma(\alpha)}-\frac {\Gamma(2-\beta)}{\Gamma(\alpha-\beta)},\frac{\Gamma(2-\beta )}{\Gamma(\alpha-\beta)}, (\frac{2-\alpha}{\alpha-1} )\frac{\Gamma(2-\beta)}{\Gamma(\alpha-\beta)} \}}. $$
(1.9)
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
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.
$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0, \quad a< t< b, 3< \alpha \leq4, \end{aligned}$$
(1.10)
$$\begin{aligned} &u(a)=u'(a)=u''(a)=u''(b)=0, \end{aligned}$$
(1.11)
Let f be a real function defined on \([a,b]\) (\(a< b\)).
Definition 1.1
The integral where \(\alpha>0\), is called the Riemann-Liouville fractional integral of order α, and \(\Gamma(\alpha)\) is the Euler gamma function defined by
$$\bigl({}_{a}I^{\alpha}f\bigr) (t)=\frac{1}{\Gamma(\alpha)} \int _{a}^{t} (t-s)^{\alpha -1}f(s)\,ds, \quad t \in[a,b], $$
$$\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t} \,dt,\quad\alpha>0. $$
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Definition 1.2
The expression where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.
$${}_{a}D^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \biggl(\frac{d}{dt} \biggr)^{n}\int_{a}^{t}\frac{f(s)}{(t-s)^{\alpha-n+1}} \,ds, $$
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
for some constants
\(c_{i}\in\mathbb{R}\), \(i=1,\ldots,n\), \(n=[\alpha]+1\).
$${}_{a}I^{\alpha}{}_{a}D^{\alpha}f(t)=f(t)+c_{1}(t-a)^{\alpha -1}+c_{2}(t-a)^{\alpha-2}+ \cdots+c_{n}(t-a)^{\alpha-n}, $$
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
where
\(G(t,s)\)
is the Green function of problem (1.10)-(1.11) defined as
$$u(t)=\int_{a}^{b} G(t,s)q(s)u(s)\,ds, $$
$$\begin{aligned} G(t,s)=\frac{1}{\Gamma(\alpha)} \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(t-a)^{\alpha-1}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}}-(t-s)^{\alpha-1},& a\leq s\leq t\leq b,\\ \frac{(t-a)^{\alpha-1}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}},& a\leq t\leq s\leq b. \end{array}\displaystyle \right . \end{aligned}$$
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 for some real constants \(c_{i}\), \(i=1,\ldots,4\). Using the boundary conditions \(u(a)=u'(a)=u''(a)=0\), we get immediately The boundary condition \(u''(b)=0\) yields Hence which concludes the proof. □
$$u(t)=c_{1}(t-a)^{\alpha-1}+c_{2}(t-a)^{\alpha-2}+c_{3}(t-a)^{\alpha -3}+c_{4}(t-a)^{\alpha-4}- \frac{1}{\Gamma(\alpha)}\int_{a}^{t} (t-s)^{\alpha-1}q(s)u(s) \,ds $$
$$c_{2}=c_{3}=c_{4}=0. $$
$$c_{1}=\frac{1}{(b-a)^{\alpha-3}\Gamma(\alpha)}\int_{a}^{b} (b-s)^{\alpha -3}q(s)u(s)\,ds. $$
$$u(t)=\frac{(t-a)^{\alpha-1}}{(b-a)^{\alpha-3}\Gamma(\alpha)}\int_{a}^{b} (b-s)^{\alpha-3}q(s)u(s)\,ds-\frac{1}{\Gamma(\alpha)}\int_{a}^{t} (t-s)^{\alpha-1}q(s)u(s)\,ds, $$
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 For \(a\leq t\leq s\leq b\), we have while for \(a\leq s\leq t\leq b\), we have Consequently, the function \(G(t,s)\) is non-decreasing with respect to t, from which it follows that The proof is complete. □
$$\begin{aligned} \partial_{t} G(t,s)=\frac{(\alpha-1)}{\Gamma(\alpha)} \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}}-(t-s)^{\alpha-2},& a\leq s\leq t\leq b,\\ \frac{(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{ (b-a)^{\alpha-3}},& a\leq t\leq s\leq b. \end{array}\displaystyle \right . \end{aligned}$$
$$\frac{\Gamma(\alpha)}{(\alpha-1)} \partial_{t} G(t,s) =\frac {(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{(b-a)^{\alpha-3}}\geq0, $$
$$\begin{aligned} \frac{\Gamma(\alpha)}{(\alpha-1)} \partial_{t} G(t,s) =& \frac {(t-a)^{\alpha-2}(b-s)^{\alpha-3}}{(b-a)^{\alpha-3}}-(t-s)^{\alpha -2} \\ =& \frac{(t-a)^{\alpha-2}((b-a)-(s-a))^{\alpha-3}}{(b-a)^{\alpha -3}}-\bigl((t-a)-(s-a)\bigr)^{\alpha-2} \\ =& (t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}-(t-a)^{\alpha-2} \biggl(1-\frac{s-a}{t-a} \biggr)^{\alpha-2} \\ \geq& (t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}-(t-a)^{\alpha-2} \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha-2} \\ =& (t-a)^{\alpha-2} \biggl[ \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha -3}- \biggl(1-\frac{s-a}{b-a} \biggr)^{\alpha-2} \biggr] \\ \geq& 0. \end{aligned}$$
$$0=G(a,s) \leq G(t,s)\leq G(b,s),\quad(t,s)\in[a,b]\times[a,b]. $$
We have the following Hartman-Wintner-type inequality.
Theorem 2.3
If a nontrivial continuous solution to the fractional boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b} (b-s)^{\alpha-3}(s-a) (2b-a-s)\bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha )\cdot $$
(2.1)
Proof
Let \(\mathcal{B}=C[a,b]\) be the Banach space endowed with the norm It follows from Lemma 2.1 that a solution u to (1.10)-(1.11) satisfies the integral equation Thus, for all \(t\in[a,b]\), we have which yields Since u is nontrivial, then \(\Vert u\Vert _{\infty}\neq0\), so Now, an application of Lemma 2.2 yields from which the inequality in (2.1) follows. □
$$\Vert y\Vert _{\infty}=\max_{a\leq t\leq b}\bigl\vert y(t) \bigr\vert , \quad y\in\mathcal{B}. $$
$$u(t)=\int_{a}^{b} G(t,s)q(s)u(s)\,ds,\quad t \in[a,b]. $$
$$\begin{aligned} \bigl\vert u(t)\bigr\vert \leq& \int_{a}^{b} \bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \bigl\vert u(s) \bigr\vert \,ds \\ \leq& \biggl(\int_{a}^{b} \sup _{a\leq t\leq b}\bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \,ds \biggr)\Vert u\Vert _{\infty}, \end{aligned}$$
$$\Vert u\Vert _{\infty}\leq \biggl(\int_{a}^{b} \sup_{a\leq t\leq b}\bigl\vert G(t,s)\bigr\vert \bigl\vert q(s) \bigr\vert \, ds \biggr)\Vert u\Vert _{\infty}. $$
$$1\leq\int_{a}^{b} \sup_{a\leq t\leq b} \bigl\vert G(t,s)\bigr\vert \bigl\vert q(s)\bigr\vert \,ds. $$
$$1\leq\int_{a}^{b} G(b,s)\bigl\vert q(s)\bigr\vert \,ds, $$
Corollary 2.4
If a nontrivial continuous solution to the fractional boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b} (b-s)^{\alpha-3}(s-a) \bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha )}{2(b-a)}\cdot $$
(2.2)
Proof
From Theorem 2.3, we have Next we note Thus we get which gives the desired inequality (2.2). □
$$\int_{a}^{b} (b-s)^{\alpha-3}(s-a) (2b-a-s) \bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha). $$
$$2b-a-s\leq2(b-a),\quad s\in[a,b]. $$
$$2(b-a)\int_{a}^{b} (b-s)^{\alpha-3}(s-a)\bigl\vert q(s)\bigr\vert \,ds \geq\Gamma(\alpha), $$
We have the following Lyapunov-type inequality.
Corollary 2.5
If a nontrivial continuous solution to the fractional boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &\bigl({}_{a}D^{\alpha}u\bigr) (t)+q(t)u(t)=0,\quad a< t< b, 3< \alpha\leq4, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha)(\alpha-2)^{\alpha -2}}{2(\alpha-3)^{\alpha-3}(b-a)^{\alpha-1}}\cdot $$
(2.3)
Proof
Let Now, we differentiate \(\psi(s)\) on \((a,b)\), and we obtain after simplifications Observe that \(\psi'(s)\) has a unique zero, attained at the point 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 From Corollary 2.4, we have which yields from which inequality (2.3) follows. □
$$\psi(s)=(b-s)^{\alpha-3}(s-a),\quad s\in[a,b]. $$
$$\psi'(s)=(b-s)^{\alpha-4}\bigl[(b-s)-(\alpha-3) (s-a)\bigr]. $$
$$s^{*}=\frac{b+(\alpha-3)a}{\alpha-2}. $$
$$\max_{a\leq s\leq b}\psi(s)=\psi\bigl(s^{*}\bigr)=(\alpha-3)^{\alpha-3} \biggl(\frac{b-a}{\alpha-2} \biggr)^{\alpha-2}. $$
$$\int_{a}^{b} \psi(s)\bigl\vert q(s)\bigr\vert \,ds\geq\frac{\Gamma(\alpha)}{2(b-a)}, $$
$$\int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{\Gamma(\alpha)}{2(b-a)\psi(s^{*})}, $$
Corollary 2.6
If a nontrivial continuous solution to the boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b} (b-s) (s-a) (2b-a-s) \bigl\vert q(s)\bigr\vert \,ds \geq6. $$
(2.4)
Corollary 2.7
If a nontrivial continuous solution to the boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b} (b-s) (s-a)\bigl\vert q(s)\bigr\vert \,ds \geq\frac{3}{b-a}\cdot $$
(2.5)
Corollary 2.8
If a nontrivial continuous solution to the boundary value problem
exists, where
q
is a real and continuous function in
\([a,b]\), then
$$\begin{aligned} &u''''(t)+q(t)u(t)=0,\quad a< t< b, \\ &u(a)=u'(a)=u''(a)=u''(b)=0 \end{aligned}$$
$$ \int_{a}^{b}\bigl\vert q(s)\bigr\vert \,ds \geq\frac{12}{(b-a)^{3}}\cdot $$
(2.6)
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
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 Now, from which we obtain The proof is complete. □
$$\vert \lambda \vert \int_{0}^{1} (1-s)^{\alpha-3}s\,ds\geq\frac{\Gamma(\alpha )}{2}. $$
$$\int_{0}^{1} (1-s)^{\alpha-3}s\,ds=\int _{0}^{1} s^{2-1}(1-s)^{(\alpha -2)-1} \,ds=B(2,\alpha-2), $$
$$\vert \lambda \vert B(2,\alpha-2)\geq\frac{\Gamma(\alpha)}{2}. $$
Acknowledgements
The second author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in drafting this manuscript and giving the main proofs. All authors read and approved the final manuscript.