In this paper we consider the following nonlinear plate equations:
$u_{t t} +\Delta \Delta u + m u = \psi(x, u) ,$
$\psi(x, u) = \pm u^3 + O(u^5),\quad \psi(-x,-u)=-\psi(x,u),
\qquad\qquad\qquad $(1)
with Navier boundary conditions in a $n$–dimensional cube, here
$\psi$ is a $C^\infty$
function, and $m$ is a positive parameter. For this equation we construct some
Cantor families of periodic orbits. Our proof is very simple and is based
on contraction mapping principle and on a suitable correspondence between
Lyapunov Schmidt decomposition and averaging theory.
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