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Title: On non-oscillation on semi-axis of solutions of second order deviating differential equations (English)
Author: Labovskiy, Sergey
Author: Alves, Manuel
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 143
Issue: 4
Year: 2018
Pages: 355-376
Summary lang: English
Category: math
Summary: We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations \begin {equation*} u''(x)+\sum _i p_i(x) u'(h_i(x))+\sum _i q_i(x) u(g_i(x)) = 0 \end {equation*} without the delay conditions $h_i(x),g_i(x)\le x$, $i=1,2,\ldots $, and $$ u''(x)+\int _0^{\infty }u'(s){\rm d}_sr_1(x,s)+\int _0^{\infty } u(s){\rm d}_sr_0(x,s) = 0. $$ (English)
Keyword: non-oscillation
Keyword: deviating non-delay equation
Keyword: singular boundary value problem
MSC: 34C10
MSC: 34K10
MSC: 34K11
idZBL: Zbl 06997371
idMR: MR3895261
DOI: 10.21136/MB.2017.0025-17
Date available: 2018-11-29T09:23:09Z
Last updated: 2020-07-01
Stable URL:
Reference: [1] Agarwal, R. P., Berezansky, L., Braverman, E., Domoshnitsky, A. I.: Nonoscillation Theory of Functional Differential Equations with Applications.Springer, Berlin (2012). Zbl 1253.34002, MR 2908263, 10.1007/978-1-4614-3455-9
Reference: [2] Azbelev, N. V.: Zeros of solutions of a second-order linear differential equation with time-lag.Differ. Equations 7 (1973), 865-873. Zbl 0272.34094, MR 0289893
Reference: [3] Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F.: Introduction to the Theory of Functional Differential Equations. Methods and Applications.Contemporary Mathematics and Its Applications 3. Hindawi Publishing Corporation, New York (2007). Zbl 1202.34002, MR 2319815
Reference: [4] Berezansky, L., Braverman, E.: Some oscillation problems for a second order linear delay differential equation.J. Math. Anal. Appl. 220 (1998), 719-740. Zbl 0915.34064, MR 1614948, 10.1006/jmaa.1997.5879
Reference: [5] Domoshnitskij, A. I.: Extension of Sturm's theorem to apply to an equation with time-lag.Differ. Equations 19 (1983), 1099-1105 translation from Differ. Uravn. 19 1983 1475-1482. Zbl 0538.34038, MR 0718547
Reference: [6] Hartman, P.: Ordinary Differential Equations.Birkhäuser, Basel (1982). Zbl 0476.34002, MR 0658490
Reference: [7] Kamenev, I. V.: Necessary and sufficient conditions for the disconjugacy of the solutions of a second order linear equation.Differ. Uravn. 12 (1976), 751-753 Russian. Zbl 0335.34016, MR 0412516
Reference: [8] Kondrat'ev, V. A.: Sufficient conditions for non-oscillatory or oscillatory nature of solutions of equation $y''+ p (x) y = 0$.Dokl. Akad. Nauk SSSR 113 (1957), 742-745 Russian. Zbl 0088.06104, MR 0091393
Reference: [9] Krasnosel'skij, M. A., Zabreiko, P. P., Pustylnik, E. I., Sobolevskii, P. E.: Integral Operators in Spaces of Summable Functions.Monographs and Textbooks on Mechanics of Solids and Fluids. Noordhoff International Publishing, Leyden (1976). Zbl 0312.47041, MR 0385645
Reference: [10] Kreĭn, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space.Usp. Mat. Nauk 3 (1948), 3-95 Russian. Zbl 0030.12902, MR 0027128
Reference: [11] Labovskii, S.: Little vibrations of an abstract mechanical system and corresponding eigenvalue problem.Funct. Differ. Equ. 6 (1999), 155-167. Zbl 1041.34050, MR 1733234
Reference: [12] Labovskii, S., Shindiapin, A.: On existence of nontrivial solution of a singular functional differential equation.Funct. Differ. Equ. 5 (1998), 183-194. Zbl 1050.34518, MR 1681191
Reference: [13] Labovskij, S. M.: A condition for the nonvanishing of the Wronskian of a fundamental system of solutions of a linear differential equation with a delayed argument.Differ. Equations 10 (1975), 316-319. Zbl 0315.34082, MR 0380049
Reference: [14] Labovskij, S. M.: Constancy of the sign of the Wronskian of a fundamental system, of Cauchy's function, and of Green's function of a two-point boundary-value problem for an equation with delay.Differ. Equations 11 (1976), 1328-1335. Zbl 0347.34052, MR 0397115
Reference: [15] Labovskij, S. M.: Positive solutions of linear functional-differential equations.Differ. Equations 20 (1984), 428-434 translation from Differ. Uravn. 20 1984 578-584. Zbl 0593.34064, MR 0742813
Reference: [16] Labovskij, S. M.: Positive solutions of a two-point boundary-value problem for a singular linear functional equation.Differ. Equations 24 (1988), 1116-1123 translation from Differ. Uravn. 24 1988 1695-1704. Zbl 0675.34034, MR 0972847
Reference: [17] Labovskiy, S.: On monotone solutions of a linear functional differential equation.Reports of The Extended Sessions of a Seminar of The I. N. Vekua Institute of Applied Mathematics, vol. 3, 1990, pp. 102-105.
Reference: [18] Labovskiy, S.: On existence of positive on semi-axis solutions for a second order deviating differential equations.Int. Miniconf. Qualitative Theory of Differential Equations and Applications, Moscow, 2013, pp. 190-207.


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