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.