Convergence rate of the Halpern iterations with possibly distinct anchor and initial guess

All existing convergence rate estimates of Halpern iterations are established only for the case where the anchor coincides with the initial guess. This paper aims to investigate the convergence rate of general Halpern iterations, where the anchor and the initial guess may not necessarily be the same. We present tight convergence rate estimates for both predetermined and adaptive anchoring parameters. These results generalize existing related work.