A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

Abstract

In this paper, a quasi-periodically forced nonlinear beam equation ${u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}$ with hinged boundary conditions is considered, where μ > 0, ${\varepsilon}$ is a small positive parameter, ${\phi}$ is a real analytic quasi-periodic function in t with a frequency vector ω = (ω ₁,ω ₂ . . . , ω _m ), and the nonlinearity h is a real analytic odd function of the form ${h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} \neq0, \bar{r} \in {\mathbb {N}}.}$ The above equation admits a quasi-periodic solution.