FOM: recursive definitions

[Peter Manolios]

I am interested in any references that relate to the following
question:

Under what conditions is a recursive equation satisfiable by some
total function?

Notice, that I am interested in total functions.  If I were interested
in partial functions, then the first recursion theorem from recursive
function theory would be a good reference.  However, that result is
not applicable here.  To see why, consider the following recursive
equation.

  f(x) = f(x)+1

The above is "monotone" as per the requirements of the first recusion
theorem, thus, there is a least partial function that satisfies the
equation, namely the nowhere defined function.  However, there are no
total functions which satisfy the above equation.

Any references would be appreciated.  

Pete Manolios