We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T * -extension of a nilpotent Lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive description in terms of symplectic quadratic double extensions.

We describe the asymptotic behaviour and the stability properties of the solutions to the nonlinear second-order difference equation; n $ 0;for all values of the real parameters a, b, and any initial condition ðx 21 ; x 0 Þ [ R 2 .

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In this paper we show the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times. ᮊ