ABSTRACT
In this book, the main method of the study of exponential stability for linear functional-differential equations is based on the Bohl-Perron theorem. To apply this method one has to prove that for any bounded on a semiaxes right-hand side of the given linear functional differential equation all solutions and their derivatives of the first and the second order are also bounded on the semi-axes. To show it there are several approaches. The first one is called in the book a priori estimation method and is used in the chapter.
By this method, a solution of the given equation and it's the first and the second derivatives are estimated on any finite interval on the semi-axes. If these estimates are don't depend on the interval then by Bohl-Perron theorem the equation is exponentially stable. To estimate solutions and their derivatives one use here also integral inequalities for solutions of non-oscillation equations of the first and the second order delay differential equations, matrix inequalities and properties of M-matrices.
