Lyapunov-type inequalities for odd order linear differential equations (Q2826978)

The authors obtain Lyapunov-type inequalities for odd order linear differential equations NEWLINE\[NEWLINE\begin{aligned} & x^{(2n+1)}(t) + (-1)^{n-1} q(t)x(t) = 0, \;t \in [a,b], \\ & x^{(i+1)}(a) = x^{(i+1)}(b)=0, \;i = 0,1,\ldots,n-1, \;x(c) = 0\end{aligned} NEWLINE\]NEWLINE where \(c \in [a,b],\) \(n \in \mathbb{N}\) and \(q \in C([a,b],\mathbb{R}).\) More precisely, they prove that if the previous problem has a nontrivial solution, then NEWLINE\[NEWLINE \displaystyle \int_a^b | q(t) | \;dt > \displaystyle \frac{2^{2n}(2n-1)!}{(b-a)^{2n}S_n}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE S_n = \displaystyle \sum_{j=0}^{n-1} \displaystyle \sum_{k=0}^{j} 2^{2k-2j} \displaystyle \frac{(n-1+j)!}{j!(n-1)!}\displaystyle \frac{j!}{k!(j-k)!} B(n+1,n+k-j), NEWLINE\]NEWLINE where \(B(\alpha,\beta) = \displaystyle \int_0^1 z^{\alpha-1}(1-z)^{\beta -1} \;dz\) is the Beta function for \(\alpha,\beta > 0.\)NEWLINENEWLINEThe proof uses the Green's function for even order linear boundary value problems. By using the Fredholm alternative theorem, a criterion for the existence and uniqueness of solutions to the corresponding nonhomogeneous linear boundary value problems of odd order is provided. Finally, two corollaries for the case \(n=1\) are given.