Some oscillation problems for a second-order linear delay differential equation. (English) Zbl 0915.34064

The following scalar delay differential equation is considered \[ \ddot x(t)+\sum_{k=1}^m a_k(t) x(g_k(t))=0,\quad g_k(t)\leq t . \] Nonoscillation and properties related to nonoscillation are studied. The main result is that under some natural assumptions for a delay differential equation the following four assertions are equivalent: nonoscillation of solutions to this equation and the corresponding differential inequality, positiveness of the fundamental function, and the existence of a nonnegative solution to a generalized Riccati inequality.
The equivalence of the oscillation properties with a differential inequality is applied to obtain new explicit conditions for nonoscillation and oscillation and to prove some known results in a different way.
A generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the Sturm comparison theorem for a second-order ordinary differential equation. By applying the positiveness of the fundamental function the positive solutions to two nonoscillatory equations are compared. Conditions on the initial function are given implying the positiveness of the corresponding solution.
Reviewer: I.Ginchev (Varna)


34K11 Oscillation theory of functional-differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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[1] Myshkis, A. D., Linear Differential Equations with Retarded Argument (1972), Nauka: Nauka Moscow · Zbl 0261.34040
[2] Norkin, S. B., Differential Equations of the Second Order with Retarded Argument. Differential Equations of the Second Order with Retarded Argument, Transl. Math. Monographs, 31 (1972), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0234.34080
[3] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G., Oscillation Theory of Differential Equations with Deviating Argument (1987), Dekker: Dekker New York/Basel · Zbl 0832.34071
[4] Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations (1991), Clarendon: Clarendon Oxford · Zbl 0780.34048
[5] Erbe, L. N.; Kong, Q.; Zhang, B. G., Oscillation Theory for Functional Differential Equations (1995), Dekker: Dekker New York/Basel
[6] Berezansky, L.; Braverman, E., On non-oscillation of a scalar delay differential equation, Dynamic Systems Appl., 6, 567-580 (1997) · Zbl 0890.34059
[7] Azbelev, N., Zeros of solutions of a second-order linear differential equation with time-lag, Differential Equations, 7, 865-873 (1971) · Zbl 0272.34094
[8] Kantorovich, L. V.; Akilov, G. P., Functional Analysis (1982), Pergamon: Pergamon Oxford · Zbl 0484.46003
[9] Berezansky, L.; Larionov, A. S., Positiveness of the Cauchy matrix of a linear functional-differential equation, Differential Equations, 24, 1221-1230 (1988) · Zbl 0701.34077
[10] Manfoud, W. E., Comparison theorems for delay differential equations, Pacific J. Math., 83, 187-197 (1979) · Zbl 0441.34053
[11] Brands, J. J.A. M., Oscillation theorems for second-order functional differential equations, J. Math. Anal. Appl., 63, 54-64 (1978) · Zbl 0384.34049
[12] Philos, C. G., A comparison result in oscillation theory, J. Pure Appl. Math., 11, 1-7 (1980) · Zbl 0432.34045
[13] Philos, C. G., Oscillatory and asymptotic behavior of the bounded solutions of differential equations with deviating arguments, Hiroshima Math. J., 8, 31-48 (1978) · Zbl 0378.34055
[14] Philos, C. G.; Sficas, Y. G., Oscillatory and asymptotic behavior of second and third order retarded differential equations, Czechoslavak Math. J., 32, 169-182 (1982) · Zbl 0507.34062
[15] Domshlak, Y., Comparison theorems of Sturm type for first and second order differential equations with sign variable deviations of the argument, Ukrain. Mat. Zh., 34, 158-163 (1982) · Zbl 0496.34036
[16] Domshlak, Y., Sturmian Comparison Method in Investigation of Behavior of Solutions for Differential-Operator Equations (1986), Elm: Elm Baku
[17] Györi, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamic Systems Appl., 5, 277-303 (1996) · Zbl 0859.34053
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