Splitting lemma in the infinity and a strong resonant problem with periodic nonlinearity

In this paper we study the periodic boundary problem: −u″−u+g(u)=h(t),u(0)=u(2π),u′(0)=u′(2π)−u″−u+g(u)=h(t),u(0)=u(2π),u′(0)=u′(2π) with periodic nonlinearity gg. We proved that the problem has infinite solutions. The method we used is a variational reduction method of T. Bartsch and S.J. Li.