Exercise · ZW-FIXED-POINT-ITERATION-004 · ★☆☆ Basic

What does the fixed-point theorem state about the existence of a fixed point?

Choose one option.

Show answer
  • A. That if \( g \) maps \( [a,b] \) into itself and is continuous, at least one fixed point exists in \( [a,b] \)

Explanation

The theorem guarantees at least one fixed point under continuity and the interval-mapping condition. The option claiming exactly two fixed points is a common mistake, since the number of fixed points depends on \( g \).

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Review the concept: Fixed-point iteration

Iterative method that approximates a solution of the equation \( x = g(x) \) by generating the sequence \( x_{n+1} = g(x_n) \) from an initial value \( x_0 \).

\[x_{n+1} = g(x_n), \quad |g'(p)| < 1 \Rightarrow \text{convergencia local}\]

More practice problems: fixed-point iteration

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