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Approximation of a Linear Autonomous Differential Equation with Small Delay
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A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

Institute of Mathematics, University of Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt am Wörthersee, Austria
Symmetry 2019, 11(11), 1332; https://doi.org/10.3390/sym11111332
Received: 30 September 2019 / Revised: 16 October 2019 / Accepted: 22 October 2019 / Published: 24 October 2019
We consider linear differential equations with variable delay of the form x ( t ) + p ( t ) x ( t τ ( t ) ) = 0 , t t 0 , where p : [ t 0 , ) [ 0 , ) and τ : [ t 0 , ) ( 0 , ) are continuous functions, such that t τ ( t ) (as t ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t A ( t ) 1 e holds , where A : = ( t ) t τ ( t ) t p ( s ) d s . Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ , condition B > 1 / e implies that all solutions of the above delay differential equation are oscillatory. View Full-Text
Keywords: oscillation; delay differential equation; variable delay; deviating argument; non-monotone argument; slowly varying function oscillation; delay differential equation; variable delay; deviating argument; non-monotone argument; slowly varying function
MDPI and ACS Style

Garab, Á. A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. Symmetry 2019, 11, 1332.

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