Homoclinic solutions for a discrete periodic Hamiltonian system with perturbed terms (Q6624049)

The authors establish sufficient conditions for the existence of at least one non-trivial homoclinic solution and their multiplicity to a periodic discrete Hamiltonian system with perturbed terms of the form \N\[\N-\Delta [p(n)\Delta u (n-1)]+L(n)u(n)= \nabla F(n,u(n))+h(n),\N\]\Nwhere \(n\in \mathbb Z, u\in \mathbb R^N, p,L:\mathbb Z \to \mathbb R^{N\times N},\) are \(T\)-periodic, and \(F:\mathbb Z\times \mathbb R^N\to \mathbb R\), \(h\in l^1\setminus \{ 0\}\). Homoclinic solution means that it satisfies the following boundary condition \N\[\N\lim_{|n|\to +\infty}u(n)=0.\N\]\NNew conditions are proposed to relax positive definiteness of the potential function and superquadraticity of the nonlinearity \(F\) both at origin and at infinity which are typically required.