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On non-oscillation of a scalar delay differential equation. (English) Zbl 0890.34059
Summary: For the scalar delay differential equation $$\dot x(t)+ \sum^m_{k=1} A_k(t)x \bigl( h_k(t) \bigr)=0, \quad h_k(t) \le t,$$ a connection between the following properties is established: non-oscillation, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. Explicit non-oscillation and oscillation conditions, a comparison theorem and a criterion for existence of a positive solution are presented. Some of the results are generalized to an impulsive delay differential equation.

34K11Oscillation theory of functional-differential equations
34A37Differential equations with impulses
34K25Asymptotic theory of functional-differential equations