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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:
[1] DAHIYA R. S., SINGH B.: A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equation. J. Math. Phys. Sci. 7 (1973), 163-170. · Zbl 0239.34033
[2] ELIASON S. B.: A Liapunov inequality for a certain second order nonlinear differential equation. J. London Math. Soc. 2 (1970), 461-466. · Zbl 0201.11802
[3] ELIASON S. B.: Liapunov type inequalities for certain second order functional differential equations. SIAM J. Appl. Mat. 27 (1974), 180-199. · Zbl 0292.34077
[4] ELIASON S. B. : Distance between zeros of certain differential equations having delayed arguments. Ann. Mat. Pura Appl. 106 (1975), 273-291. · Zbl 0316.34081
[5] HARTMAN P.: Ordinary Differential Equations. Wiley, New York, 1964. · Zbl 0125.32102
[6] LIAPUNOV A. M.: Probleme general de la stabilitie du mouvement. Ann. of Math. Stud. 17, Princeton Univ. Press, Princeton, NJ, 1949.
[7] PACHPATTE B. G.: On Liapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195 (1995), 527-536. · Zbl 0844.34014
[8] PARHI N., PANIGRAHI S.: On Liapunov-type inequality for third-order differential equations. J. Math. Anal. Appl. 233 (1999), 445-460. · Zbl 0932.34030
[9] PARHI N., PANIGRAHI S.: On Liapunov-type inequality for delay-differential equations of third order. Czechoslovak Math. J. · Zbl 1023.34069
[10] PATULA W. T.: On the distance between zeros. Proc. Amer. Math. Soc. 52 (1975 , 247-251. · Zbl 0305.34050
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