In this paper, we are concerned with the anti-periodic fractional boundary value problem
$$ \begin{aligned} & \bigl({}^{C}D_{a}^{\alpha,\psi}u \bigr) (x)+f \bigl(x,u(x) \bigr)=0,\quad a< x< b, \\ &u(a)+u(b)=0,\qquad u'(a)+u'(b)=0, \end{aligned} $$
(1.1)
where
\((a,b)\in\mathbb{R}^{2}\),
\(a< b\),
\(1<\alpha<2\),
\(\psi\in C^{2}([a,b])\),
\(\psi'(x)>0\),
\(x\in[a,b]\),
\({}^{C}D_{a}^{\alpha,\psi}\) is the
ψ-Caputo fractional derivative of order
α, and
\(f: [a,b]\times\mathbb{R}\to\mathbb{R}\) is a given function. A Lyapunov-type inequality is derived for problem (
1.1). Next, as an application of the obtained inequality, an upper bound of possible eigenvalues of the corresponding problem is obtained.
Let us mention some motivations for studying problem (
1.1). Suppose that
\(u\in C^{2}([a,b])\),
\((a,b)\in\mathbb{R}^{2}\),
\(a< b\), is a nontrivial solution to the boundary value problem
$$ \begin{aligned} &u''(x)+w(x)u(x)=0, \quad a< x< b, \\ &u(a)=0,\qquad u(b)=0, \end{aligned} $$
(1.2)
where
\(w\in C([a,b])\) is a given function. Then (see [
17])
$$ \int_{a}^{b} \bigl\vert w(x) \bigr\vert \,dx > \frac{4}{b-a}. $$
(1.3)
Inequality (
1.3) is known in the literature as Lyapunov’s inequality, which provides a necessary condition for the existence of a nontrivial solution to (
1.2). Many generalizations and extensions of (
1.3) were derived by many authors. In particular, Hartman and Wintner [
9] proved that if
\(u\in C^{2}([a,b])\) is a nontrivial solution to (
1.2), then
$$ \int_{a}^{b} (b-s) (s-a) w^{+}(s)\,ds>b-a, $$
(1.4)
where
$$w^{+}(s)=\max \bigl\{ w(s),0 \bigr\} ,\quad a\leq s\leq b. $$
It can be easily seen that (
1.3) follows from (
1.4). For other results related to Lyapunov-type inequalities, see, for example, [
3,
5,
16,
18,
19,
21] and the references therein. On the other hand, due to the importance of fractional calculus in applications, the study of Lyapunov-type inequalities was extended to fractional boundary value problems by many authors. The first contribution in this direction is due to Ferreira [
6], where the fractional boundary value problem
$$ \begin{aligned} & \bigl(D_{a}^{\alpha}u \bigr) (x)+w(x)u(x)=0,\quad a< x< b, \\ &u(a)=0,\qquad u(b)=0, \end{aligned} $$
(1.5)
with
\(w\in C([a,b])\),
\(1<\alpha<2\) and
\(D_{a}^{\alpha}\) is the Riemann–Liouville fractional derivative of order
α, was studied. The main result in [
6] is the following: If
u is a nontrivial solution to (
1.5), then
$$ \int_{a}^{b} \bigl\vert w(x) \bigr\vert \,dx> \Gamma(\alpha) \biggl(\frac{4}{b-a} \biggr)^{\alpha-1}. $$
(1.6)
Note that in the limit case
\(\alpha=2\), (
1.5) reduces to (
1.2). Moreover, taking
\(\alpha=2\) in (
1.6), we obtain (
1.3). For other works related to Lyapunov-type inequalities for fractional boundary value problems, see, for example, [
4,
7,
8,
10‐
12,
20] and the references therein. In particular, in [
8], the anti-periodic fractional boundary value problem
$$ \begin{aligned} & \bigl({}^{C}D_{a}^{\alpha}u \bigr) (x)+w(x)u(x)=0,\quad a< x< b, \\ &u(a)+u(b)=0,\qquad u'(a)+u'(b)=0, \end{aligned} $$
(1.7)
where
\(w\in C([a,b])\),
\(1<\alpha<2\) and
\({}^{C}D_{a}^{\alpha}\) is the Caputo fractional derivative of order
α, was studied. Note that (
1.7) is a special case of (
1.1) with
\(\psi(x)=x\) and
\(f(x,z)=w(x)z\).
The rest of the paper is organized as follows. In Sect.
2, we recall some basic concepts on fractional calculus and prove some preliminary results. In Sect.
3, a Lyapunov-type inequality is established for problem (
1.1). Moreover, some particular cases are discussed. Next, an application to fractional eigenvalue problems is given. In Sect.
4, we end the paper with some open questions.