Asymptotic behavior of solutions to a perturbed ODE (Q2469426)

The author proves existence results for the following class of boundary value problems on infinite intervals \[ x''(t)+2f(t)x'+\beta(t)x+g(t,x)=0 \text{ a.e. } t\in \mathbb R_+, \] \[ x(\infty)=x'(\infty)=0, \] where \(f,\beta\in C(\mathbb R_+), \;g\in C(\mathbb R_+\times\mathbb R),\) and \(x(\infty)=\lim_{t\to\infty}x(t),\;x'(\infty)=\lim_{t\to\infty}x'(t).\) The proofs are based on the Schauder-Tychonoff fixed point theorem. One example is presented.