On oscillation properties of delay differential equations with positive and negative coefficients.

*(English)*Zbl 1056.34063
J. Math. Anal. Appl. 274, No. 1, 81-101 (2002); corrigendum ibid. 351, No. 2, 819-820 (2009).

Summary: For a scalar delay differential equation

\[ x^{\prime}(t) +a(t)x(h(t))-b(t)x(g(t))=0,\quad t\geq t_{0},\tag{E} \]

where \(a(t)\geq 0\), \(b(t)\geq 0\), \(h(t)\leq t\), \(g(t)\leq t\), a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequalities, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. A comparison theorem and explicit nonoscillation and oscillation results are presented.

\[ x^{\prime}(t) +a(t)x(h(t))-b(t)x(g(t))=0,\quad t\geq t_{0},\tag{E} \]

where \(a(t)\geq 0\), \(b(t)\geq 0\), \(h(t)\leq t\), \(g(t)\leq t\), a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequalities, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. A comparison theorem and explicit nonoscillation and oscillation results are presented.

##### MSC:

34K11 | Oscillation theory of functional-differential equations |

34K12 | Growth, boundedness, comparison of solutions to functional-differential equations |

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\textit{L. Berezansky} et al., J. Math. Anal. Appl. 274, No. 1, 81--101 (2002; Zbl 1056.34063)

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