Scalar Delay Differential Equations on Semiaxes
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Chapter 2 deals with nonoscillation properties of scalar differential equations with positive coefficients and a finite number of delays. There are several monographs and a lot of papers on oscillation, however, there are not so many results on nonoscillation, especially in monographs on the oscillation theory. One of the aims of this chapter is to consider nonoscillation together with other relevant problems: differential inequalities, comparison results, solution estimations, sufficient conditions for positivity of solutions of the initial value problem, stability, slowly oscillating solutions. The second purpose is to derive some nonoscillation methods which will be used for other classes of functional differential equations. In particular, a solution representation formula is applied here, so the most important nonoscillation property is the positivity of the fundamental function of the considered equation.
Other results of this chapter include explicit oscillation conditions and a discussion of the well-known constants 1 and 1/e which are usually used in oscillation and nonoscillation conditions, and of the case when the values of the computed parameter are between the two constants.
KeywordsScalar Delay Differential Equation Nonoscillation Conditions Non-oscillatory Method Nonoscillatory Solution Oscillation Results
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