Is there a functional problem for which there is an algorithm that can decide if a solution is a solution or not in polynomial time but we can't find a solution in polynomial time?
Let FP be the class of functional problems.
Update: Every problem in FP, say $a$, has a correspondent decisione problem $a'$ (the associated decision problem $a'$ is the problem where we ask if a possible solution is a solution), I ask if there is a functional problem $a$ for which $a'$ is in P but $a$ is NOT in FP.
In this link: Hamiltonian cycle, verifying and finding
we have instead a functional problem HC (hamiltonian cycle) for which HC' is P and HC is in FP.
Intuitively I think that the kind of problems I search, should be 'more' the problem of kind of HC. But maybe the hard part, given $a$ (with $a'$ in P), is to prove that $a$ is NOT in FP.