Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance

We are concerned with the existence of unbounded orbits of the mapping $$ \left\{\begin{array}{rl} &\theta_1=\theta+2\pi+\d{1}{\rho}\mu(\theta)+o(\rho^{-1}),\\[3mm] &\rho_1=\rho+c-\mu'(\theta)+o(1), \quad\rho\to\infty, \end{array} \right. $$ where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, $\mu(\theta)$ is non-negative (or non-positive) and $\mu(\theta)$ has finitely many degenerate zeros in $[0, 2\pi]$. We prove that every orbit of the given mapping goes to infinity in the future or in the past for $\rho$ large enough. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$($=\int_0^xf(s)ds$), $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.