Article
Full entry |
PDF
(1.3 MB)
Feedback
PDF
(1.3 MB)
Feedback
Summary:
Sufficient conditions are given under which the sequence of the absolute values of all local extremes of $y^{[i]}$, $i\in \lbrace 0,1,\dots , n-2\rbrace $ of solutions of a differential equation with quasiderivatives $y^{[n]}=f(t,y^{[0]},\dots , y^{[n-1]})$ is increasing and tends to $\infty $. The existence of proper, oscillatory and unbounded solutions is proved.
References:
[1] M. Bartušek: Monotonicity theorem for second-order non-linear differential equations. Arch. Math. XVI (1980), no. 3, 127–136.
[2] M. Bartušek: The asymptotic behaviour of solutions of the differential equation of the third order. Arch. Math. XX (1984), no. 3, 101–112. MR 0784861
[3] M. Bartušek: The asymptotic behaviour of oscillatory solutions of the equation of the fourth Order. Arch. Math. 21 (1985), no. 2, 93–104. MR 0817551
[4] M. Bartušek: On oscillatory solution of the differential equation of the $n$-th order. Arch. Math. 22 (1986), no. 3, 145–156. MR 0868130
[5] M. Bartušek: Asymptotic Properties of Oscillatory Solutions of Differential Equations of the $n$-th Order. FOLIA FSN Univ. Masaryk. Brunensis, Math. 3, Masaryk Univ. Brno, 1992. MR 1271586
[6] M. Bartušek: Oscillatory criteria for nonlinear $n$-th order differential equations with quasi-derivatives. Georgian Math. J. 3 (1996), no. 4, 301–314. DOI 10.1007/BF02256721 | MR 1397813
[7] Š. Belohorec: Monotone and oscillatory solutions of a class of nonlinear differential equations. Math. Čas. 19 (1969), no. 3, 169–187. MR 0304773 | Zbl 0271.34045
[8] I. Bihari: Oscillation and monotonicity theorems concerning non-linear differential equations of the second order. Acta Math. Acad. Sci. Hung. IX (1958), no. 1–2, 83–104. DOI 10.1007/BF02023866 | MR 0095321
[9] D. Bobrowski: Some properties of oscillatory solutions of certain differential equations of second order. Ann. Soc. Math. Polonae XI (1967), 39–48. MR 0218668 | Zbl 0166.35002
[10] K.M. Das: Comparison and monotonicity theorems for second order non-linear differential equations. Acta Math. Acad. Sci. Hung. 15 (1964), no. 3–4, 449–456. DOI 10.1007/BF01897153 | MR 0176155
[11] I. Foltyńska: On certain properties of oscillatory solutions of the second order nonlinear differential equation (Polish). Fasc. Math. 4 (1969), 57–64.
[12] I.T. Kiguradze: Some Singular Boundary-Value Problems for Ordinary Differential Equations. Izd. Tbiliss. Univ., Tbilisi, 1975. (Russian) MR 0499402
[13] I.T. Kiguradze: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Nauka, Moscow, 1990. (Russian)
Search
Partner of
