Nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems

In this paper, we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q ¨ i ( t ) = Φ i − 1 ′ ( t , q i − 1 ( t ) − q i ( t ) ) − Φ i ′ ( t , q i ( t ) − q i + 1 ( t ) ) , i ∈ Z , where Φ i ( t , x ) = − ( α i / 2 ) | x | 2 + V i ( t , x ) is T-periodic in t for T > 0 and satisfies Φ i + N = Φ i for some N ∈ N , q i ( t ) is the state of the i-th particle. Assume that α i = 0 for some i ∈ Z and V i ′ ( t , x ) denoting the derivative of V i respect to x is asymptotically linear with x both at origin and at infinity. We would like to point out that this system is resonant both at origin and at infinity and not studied up to now. Based on some new results concerning the precise computations of the critical groups, for a given m ∈ Z , we obtain the existence of nontrivial periodic solutions satisfying q i + m N ( t + T ) = q i ( t ) for all t ∈ R and i ∈ Z under some additional conditions.