Fixed Point Theorems

Real Analysis · Axiom Academy

Where a graph meets the line y = x , a point maps to itself — and two great theorems say when such a point must exist. 1. A Fixed Point Is Where the Graph Meets y = x Run a point along the graph y = f(x) and watch the diagonal line y = x . At most places the curve sits above or below the diagonal — there f moves the input. But where the curve crosses the diagonal, the output equals the input: that meeting point is a fixed point x^ * with f(x^ * ) = x^ * . a fixed point is fixed by f — the diagonal is the test f(x) = x^ 2 is fixed at x = 0 and x = 1 ; f(x) = -x only at x = 0 ; the identity f(x) = x fixes every point — its graph is the diagonal. 2. Banach: A Contraction Spirals In Call f a contraction on a complete space if some constant shrinks every distance: . The Banach fixed point theorem says such an f has exactly one fixed point, and the iteration x_ n+1 = f(x_n) marches to it from any start. The animation runs the cobweb for — go to the curve, slide to the diagonal, repeat — and you watch it staircase straight onto . 3. The Error Falls Off Geometrically Contraction does more than promise a limit — it sets the speed . Each step multiplies the remaining error by at most k , so after n steps the gap to x^ * is bounded by k^ n times the initial gap. The bars below are the actual errors for the iteration; each is a fraction of the one before, collapsing toward zero. geometric decay — accuracy roughly multiplies each step 4. Brouwer: Existence, Guaranteed

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