Morse index and Maslov-type index of the discrete Hamiltonian system

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of case $\omega=1$ in \cite{RoS1}, \cite{RoS2} and \cite{Maz1} to case $\omega \in {\bf U}$ via direct method which is different from that of \cite{RoS1}, \cite{RoS2} and \cite{Maz1}. Moreover well-posedness of the splitting numbers $S_{\omega}$ of the discrete Hamiltonian system is proven, thus index iteration theories in \cite{Bot1} and \cite{Lon4} are also valid for the discrete Hamiltonian system case.