Resumen de On the Asymptotic Stability for Nonlinear Oscillators with Time-Dependent Damping

László Hatvani

• The equation x′′+h(t,x,x′)x′+f(x)=0(x∈R,xf(x)≥0,t∈[0,∞))is considered, where the damping coefficient h allows an estimate a(t)|x′|αw(x,x′)≤h(t,x,x′)≤b(t)W(x,x′).Sufficient conditions on the lower and upper control functions a, b are given guaranteeing that along every motion the total mechanical energy tends to zero as t→∞. The key condition in the main theorem is of the form ∫0∞a(t)ψ(t;a,b)dt=∞,which is required for every member ψ of a properly defined family of test functions. In the second part of the paper corollaries are deduced from this general result formulated by explicit analytic conditions on a, b containing certain integral means. Some of the corollaries improve earlier theorems even for the linear case.