In this paper, we establish the product formula for the fixed point index on product cone, andthen, as applications, consider the existence, nonexistence and multiplicity of positive solutions for a second-order ordinary differential system with parameters. The discussion is based on the product formula and thefundamental properties of the fixed point index.
In this paper we consider the first order discrete Hamiltonian systems x1(n + 1)-x1(n) = -Hx2 (n, x(n)), x2(n)-x2(n-1) = Hx1 (n, x(n)), where x(n) =( x1(n) x2(n)) ∈ R2N , H(n1, Z)∈1/2S(n)z·z+R(n1 z) is periodic in n and superlinear as |z| →∞. We prove the existence and infinitely many (geometrically distinct) homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.
