Optimal error bounds for non-expansive fixed-point iterations in normed spaces

This paper investigates optimal error bounds and convergence rates for general Mann iterations computing fixed-points of non-expansive maps. We look that achieve the smallest fixed-point residual after n steps, by minimizing a worst-case bound $$\Vert x^n-Tx^n\Vert \le R_n$$ derived from nested family transport problems. prove this is tight so $$R_n$$ yields iterations. Inspired numerical results we identify attain rate $$R_n=O(1/n)$$ , which also show to be best possible. In particular, classical Halpern iteration achieves several alternative stepsizes, determine analytically stepsizes residuals at every step n, with $$R_n\approx \frac{4}{n+4}$$ . affine maps, get exactly $$R_n=\frac{1}{n+1}$$ Finally, Krasnosel’skiĭ–Mann $$\varOmega (1/\sqrt{n})$$ present evidence suggesting even extended variants cannot reach faster rate.