Question

We showed that if g(x) ∈ C¹[a,b], g(x) ∈ [a,b] for all x ∈ [a,b], and |g′(x)| ≤ c < 1 for all x ∈ [a,b], a unique fixed point exist in [a,b].

1) Using the same argument to demonstrate that if we replace the condition |g′(x)| ≤ c < 1 by g′(x) ≤ c < 1, a unique fixed point still exists.

2) However, the condition g′(x) ≤ c < 1 cannot guarantee the convergence of the fixed-point iteration. Consider the function g(x) = 1 − x² on [0, 1]. Find an initial guess to generate a sequence using fixed-point iteration, which does not converge.

Answer 1