FNP complexity class
The Hamiltonian Cycle Problem asks to decide, given a graph $G$, whether $G$ is Hamiltonian. This is the decision version. The function version(FNP) asks to actually find a Hamiltonian cycle in $G$ (if any).

I suggest looking at FNP, and related classes (such as NPMV and NPSV) in the Complexity Zoo.

In addition, a quick search in Google Books reveals many books in which you may find more info.

Please be more specific if you have more questions in mind.