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

The equation x '' + h(t, x, x')x' + f (x) = 0 (x is an element of R, xf (x) >= 0, t is an element of[0, infinity)) is considered, where the damping coefficient h allows an estimate a(t)vertical bar x'vertical bar(alpha) w(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 -> infinity. The key condition in the main theorem is of the form integral(infinity)(0) a(t) psi (t; a, b) dt = infinity, which is required for every member psi 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.