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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.

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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##### References:
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