Second-Order Delay Differential Equations with Damping Terms

Chapter 8 deals with nonoscillation problems for a scalar linear delay differential equation of the second order including explicitly a term with the first derivative which is usually called “a damping term”. Most of publications deal with equations not containing the term with the first derivative; for these equations, positivity of the coefficients and a solution on the semiaxis implies that its derivative is nonnegative. This fact is very important, and it is employed in most investigations on second-order delay differential equations. If the first derivative is included in the equation explicitly, i.e. the equation contains the damping term, then a sign of a solution does not uniquely define the sign of its derivative, which makes the study of oscillation properties of the equations with the damping term more complicated. This is the reason why such equations are much less studied than equations without the damping term. The main result of the chapter is the following: if a generalized Riccati inequality has a nonnegative solution, then the differential equation has a positive solution with a nonnegative derivative, and the fundamental function of this equation is positive. If the damping term is not delayed, this immediately yields that the following four properties are equivalent: nonoscillation of solutions of this equation and the corresponding differential inequality, positivity of the fundamental function and existence of a nonnegative solution of the generalized Riccati inequality. The generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be considered as a natural generalization of the well-known Sturm comparison theorem for a second-order ordinary differential equation. The chapter also contains explicit nonoscillation conditions which are obtained by substituting specific solutions of the generalized Riccati inequality.