Notes on Multiple Periodic Solutions for Second-order Discrete Hamiltonian System

By a new orthogonal direct sum decomposition $E_M=Y\oplus Z$, which $Z$ is related to $\mathrm{\Delta }{u}_i(i=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},M)$, and a new functional $I(u)$, the method in [2] is improved to obtain new multiple periodic solutions with negativity hypothesis on $F$ for a second-order discrete Hamiltonian system. Moreover, we exhibit an instructive example to make our result more clear, which hasn’t been solved by the known results.