LAUSR

The equation $$\begin{aligned} x''+h(t,x,x')x'+f(x)=0 \qquad (x\in \mathbb {R},\ xf(x)\ge 0,\ t\in [0,\infty )) \end{aligned}$$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 $$\begin{aligned} a(t)|x'|^\alpha w(x,x')\le h(t,x,x')\le b(t) W(x,x'). \end{aligned}$$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 $$t\rightarrow \infty $$t→∞. The key condition in the main theorem is of the form $$\begin{aligned} \int _0^\infty a(t)\psi (t;a,b)\,{\mathrm{d}}t=\infty , \end{aligned}$$∫0∞a(t)ψ(t;a,b)dt=∞,which is required for every member $$\psi $$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.