Let \(\mathcal{I}\) be a skew-symmetric non-degenerate real \(2N\times 2N\) matrix. A real \(2N\times 2N\) matrix \(\mathcal{M}\) is called \(\mathcal{I}\)-symplectic if \(\mathcal{M}^T \cdot \mathcal{I}^{-1}\cdot \mathcal{M}=\mathcal{I}^{-1}\). For an \(\mathcal{I}\)-symplectic matrix \(\mathcal{M}\) with \(\mathcal M^k=I_{2N}\), and a function \(H\in C^2(\mathbb{R}\times\mathbb{R}^{2N})\) with \(H(t+\tau,\mathcal{M}z)=H(t,z)\), the author considers the following \(\mathcal{M}\)-boundary value problem \[ \dot{z}(t)=\mathcal{I}\nabla H\left(t,z(t)\right),\quad z(\tau)=\mathcal{M}z(0). \tag{1} \] In the first part of this paper, it is shown that the \(2m\tau\)-periodic problem for a delay differential system of the form \[ x'(t)=\nabla V(t,x(t-\tau))+\nabla V(t,x(t-2\tau))+ \dotsb +\nabla V(t,x(t-(m-1)\tau)) \] is equivalent to some \(\mathcal{M}\)-boundary value problem of the form (1). The same is proved for the periodic problem for some delay Hamiltonian systems.
Next, the \((\mathcal{I},\mathcal{M})\)-index is defined as a tool for studying the \(\mathcal{M}\)-boundary value problem (1). As applications, existence and multiplicity results for periodic solutions of asymptotically linear delay differential systems and delay Hamiltonian systems are obtained.