Under some $L^p$ -norms ($1\leq p \leq \infty$) assumptions for the derivative of the restoring force, the exact multiplicity, the stability and instability, decay rate and the bifurcation results of $T$-periodic solutions for damped Duffing equation are discussed. The class of the restoring force is extended, comparing with the class of $L^\infty$-norm condition. The tools include the global bifurcation theorem, the connections between degree theory and local index of periodic solutions of R. Ortega, the estimates for the periodic eigenvalues and the results of the rotation number of Hill’s equation by $L^p$ -norms ($1\leq p \leq \infty$) of M. Zhang.
