Here, the following boundary value problem for Hamiltonian systems is studied
where the function is called Hamiltonian and is a symplectic -matrix. Special attention is given to the case in which the Hamiltonian , besides being measurable on , is convex and continuously differentiable with respect to . The basic assumption is that the Hamiltonian satisfies the following growth condition:
Let and . There exist positive constants , and functions such that
for all and a.e. . The main result assures that under suitable bounds on and the functions , the problem above has at least a solution that belongs to . Such a solution corresponds, in the duality, to a function that minimizes the dual action restricted to a subset of .