Loop-Transformations (input feedforward or output feedback) can be used for passivity analysis, also in the case when considering a static (memoryless) nonlinearity. This is described for example in Nonlinear Systems (Khalil, page 267). Let
$$ y_s = f(u) \tag{1} $$
be such a nonlinearity (for example, $f$ can be a saturation function). I understand the input feedforward case, which is
$$ y_s = f(u) - k_1 u \tag{2} $$
But for output feedback, the equation gets
$$ y_s = f(u + k_2 y_s) \tag{3} $$
So the problem is now that $(3)$ isn't an explicit equation anymore for $y_f$. So, how could I compute $y_s$ now, for example in a simulation? Do I have to provide an initial condition for $y_s$? If yes, which value should I choose?
Question: How to deal in general with loop transformations that lead to implicit equations that can't be isolated?