Functional problem and verifying solutions 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.
It appears that you are asking whether FP = FNP.  That is an open problem.  It is known that FP = FNP if and only if P = NP.  I suspect that most theoreticians expect that it's likely that P != NP, which would imply FP != FNP.  If FP != FNP, there exists a function problem whose solutions can be verified in polynomial time, but where you can't find a solution in polynomial time.  In particularly, any FNP-complete problem -- such as FSAT, say -- will have that property, if FP != FNP.  So, the FNP-complete problems are particularly good candidates for having the property you articulate.  See https://en.wikipedia.org/wiki/Function_problem for an overview.