# Scalar Delay Differential Equations on Semiaxis with Positive and Negative Coefficients

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## Abstract

In Chap. 3, nonoscillation of equations with several delays and coefficients of different signs is considered. Unlike equations with positive coefficients, the existence of a positive solution in this case does not imply positivity of the fundamental function, as the first example of the chapter demonstrates. Also, nonoscillatory solutions do not necessarily tend to zero. There have been many mistakes made when studying such equations, one of them made by the authors in their paper in the Journal of Mathematical Analysis and Applications published in 2002. A corrected result on the relation of nonoscillation and positivity of the fundamental function is included in this chapter. In addition, the chapter presents comparison results, explicit nonoscillation conditions, discussion on the asymptotic properties of nonoscillatory solutions and the analysis of the equation with one delay term and an oscillating coefficient. For such an equation, examples demonstrate that even if the positive part of the coefficient “prevails”, this still does not guarantee that nonoscillatory solutions tend to zero.

## Keywords

Scalar Delay Differential Equation Nonoscillation Conditions Nonoscillatory Solution Oscillatory Coefficients Nonoscillation Criteria## References

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