Multiple homoclinic orbits for second order discrete Hamiltonian systems without symmetric condition

摘要

In this paper, we investigate the second order self-adjoint discrete Hamiltonian system Δ [ p ( n ) Δ u ( n − 1 ) ] − L ( n ) u ( n ) + λ a ( n ) ∇ G ( u ( n ) ) + μ b ( n ) ∇ F ( u ( n ) ) = 0 $Delta[p(n)Delta u(n-1)]-L(n)u(n)+lambda a(n)abla G(u(n))+mu b(n)abla F(u(n))=0$ , where p , L : Z → R N × N $p,L:mathbb{Z}ightarrowmathbb{R}^{Nimes N}$ are both positive definite for all n ∈ Z $ninmathbb{Z}$ , and no symmetric condition on G and F is needed. We establish two new criteria to guarantee that the above system has at least two nontrivial homoclinic solutions or infinitely many homoclinic solutions via critical point theory.