Anashkin O. Development of Lyapunov's second method in the theory of stability for differential and functional differential equations

In the framework of Lyapunov second method the following main problems are studied: the problem on -stability for ordinary differential equations and functional differential equations with integrally continuous right hand part with respect to parameter , the problem on Lyapunov stability for ordinary differential equations, functional differential equations and delay difference equations. New sufficient conditions on stability, asymptotic stability and instability are obtained. These conditions admit the usage of non-decresent Lyapunov functions and fuctionals. This new approach essencially enlarges a class of appropriate Lyapunov functions. It gives an opportunity to simplify a procedure of the Lyapunov function construction for some concrete equations. New results are applied for investigation of conditions of the parametric resonance appearance in some linear systems with delay.