A Lyapunov-type inequality for a two-term even-order differential equation (Q2911468)

The authors obtain a Lyapunov-type inequality for a two-term even-order differential equation of the form NEWLINE\[NEWLINE(r_n(t)(r_{n-1}(t)(\dots(r_2(t)(r_1(t)x'')'')''\dots)'')'')''+q(t)x=0, \tag{\(*\)} NEWLINE\]NEWLINE where \(r_k\in C^{2n-2k+2}([a,b],(0,\infty))\), \(k=1,2,\dots,n\), \(q\in C([a,b], \mathbb {R})\).NEWLINENEWLINEThe following result is obtained: if \(x(t)\) is a non-zero solution of \((*)\) satisfying \(x_k(a)=x_k(b)=0,\) \(k=0,1,2,\dots,n\), where the functions \(x_k\) are defined as \(x_0=x\), \(x_1=r_1(t)x_0''\), \(x_2=r_2(t)x_1''\), \dots, \(x_n=r_n(t)x_{n-1}''\), then NEWLINE\[NEWLINE(b-a)^{n+1}\left[\prod_{k=1}^n\int_a^b\frac{dt}{r_k(t)}\right]\cdot\int_a^b|r(t)|dt\;>4^{n+1}.NEWLINE\]