@ABond Sure, you can make the F[i] be something else.
Below, F[i] is assigned the solution y(t) from sol[i].
sys:= diff(y(t),t) = cos( 1/2*y(t)-x(t)-Pi^2 ):
for i from 1 to N do
sol[i]:=dsolve([newsys, y(0)=10*sqrt(2)/2], numeric,
Is that what you mean?
Option remember is being put on the F[i], since otherwise this whole approach gets very slow as N the number of iterations gets even moderately high (say, >6). There may be an altogether different way to do it (viz. your other posts about saving interpolants), but this seems at least simple.