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Lyapunov-type inequality for higher-order differential equations. (English) Zbl 1019.34039
Here, the differential equation of \(n\)th order \(D^ny+yf(t,y)=Q(t)\) is examined, with \(Dy=\frac {1}{r_1(t)}y'\), \(D^i y=\frac {1}{r_i(t)} (D^{i-1}y)'\), \(i=2,\ldots ,n\), \(r_n\equiv 1\), \(r_i\in C (I,(0,\infty))\), \(Q\in C(I,\mathbb{R})\), \(f\in C(I\times \mathbb{R},\mathbb{R})\). A Lyapunov-type inequality of the form \[ \begin{aligned} 4& \leq \left (\int _a^b r_1(s_1) ds_1\right)\left (\int _a^b\left [ r_2(s_1)\biggl |\int _{\alpha _1}^{s_1}r_3(s_2)\biggl |\int _{\alpha _2}^{s_2} r_4(s_3)\ldots \right .\right . \\ & \qquad \biggl |\int _{\alpha _{n-3}}^{s_{n-3}}r_{n-1}(s_{n-2})\biggl |\int _{\alpha _{n-2}} ^{s_{n-2}} W(s_{n-1},M) ds_{n-1}\biggl |ds_{n-2}\biggl |\ldots \biggl |ds_2\biggl |+ \\ & \qquad \left .\left .+\frac {1}{M}r_2(s_1)\biggl |\int _{\alpha _1}^{s_1} r_3(s_2)\biggl |\ldots \biggl |\int _{\alpha _{n-2}}^{s_{n-2}} Q(s_{n-1}) ds_{n-1}\biggl |ds_{n-2}\biggl |\ldots \biggl |ds_2\biggl |\right ]ds_1\right) \end{aligned} \] is proved, where \(|f(t,y)|\leq W(t,|y|)\) and \(W\) is increasing in the second variable. Using this inequality, a criterion for the disconjugacy of the equation \(D^n y+p(t)=0\) on the interval \([a,b]\) is derived, where \(p\in C([a,b],\mathbb{R})\), and sufficient conditions are stated, under which oscillatory solutions to (1) converge to zero. Further, for the sequence \(\{t_m\}\) of zeros of an oscillatory solution to \(D^n y+p(t)y=0\), \(t\geq 0\), where \(p\in L^\sigma ([0,\infty),\mathbb{R})\), sufficient conditions for the property \(\lim _{m\to \infty }(t_{m+k}-t_m)=\infty \) are formulated.

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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