On Halpern-type iterations: are inertial terms necessary for strong convergence?

Inertial-type iterations were originally introduced to accelerate the weak convergence of iterative methods for various classes of problems. Inspired by these successes, recent studies have extended inertial techniques to Halpern-type schemes to enhance strong convergence performance. In this paper, we establish a structural equivalence between non-inertial and inertial Halpern-type iterations: strong convergence of the error-free scheme guarantees strong convergence of the scheme with error terms. This connection shows that many existing inertial Halpern-type schemes can be unified and derived within a non-inertial framework under weaker assumptions. Furthermore, numerical experiments on split and convex feasibility problems show that inertial terms need not accelerate convergence: in the configurations tested, the non-inertial iteration often attains a smaller error with fewer iterations and less computation time, although the comparison depends on the anchor, the initial points, and the inertial parameter. Finally, we provide a refined perspective on stopping criteria for split feasibility problems, clarifying the applicability of conditions previously discussed in the literature (e.g., Cholamjiak et al. [ J. Fixed Point Theory Appl. 20 (1):42, 2018]) and proposing a robust alternative for Halpern-type approximations.