7.4 Misc.

If $$\Phi$$ is just nonexpansive (the constant $$c$$ above is one), this is no longer true, but replacing $$\Phi$$ by $$\alpha \Phi + (1 - \alpha) I$$ for $$\alpha \in (0,1)$$ we get Krasnoselskii-Mann iterates of the form $\theta_n = \alpha \Phi(\theta_{n-1}) + (1 - \alpha) \theta_{n-1}$ that will converge to a fixed point of $$\Phi$$ provided it has one.

Iteration, fixed points, convergence criteria. Ref to Nonlinear Parameter Optimization Using R Tools.