ABSTRACT
This chapter deals with a connection between nonoscillation and stability problems for a scalar linear delay differential equation of the second order with damping terms. Explicit conditions on the positivity of the fundamental function of this class of equations are obtained by construction and analysis of a generalized Riccati inequality. It was proved in the chapter that equations with positive fundamental function (under some natural additional conditions) are exponentially stable and for the fundamental functions, an integral inequality holds. This inequality used to obtain explicit exponential stability for a general linear differential equation of the second order with damping terms by two different approaches. In the first one, a differential equation is transformed into a linear operator equation, which considered in a functional space on a semi-axes. Solvability conditions of this equation and an application of Bohl-Perron theorem give explicit stability tests for delay differential equations. This inequality is also used in the second approach to obtain an a priori estimate for solutions of a general linear differential equation of the second order with damping terms. Then one applies in the chapter a priori estimation method which based on a matrix inequality for a solution of and its first and second derivatives and Bohl-Perron theorem. In the chapter were also considered integro-differential equations and equations with distributed delays. The last section of the chapter includes some open problems and topics for future research.
