Projection Algorithms for Fixed Point Problems

We study projection algorithms for finding fixed points of nonexpansive mappings in a Hilbert space. In previous work, the well-definedness of the iterates depended on the nonemptiness of the fixed point set. We first show that the existence of the iterates generated by a general projection algorithm is independent of the existence of fixed points. Next, we introduce a multi-step inertial projection algorithm and establish results on existence, strong convergence and iteration complexity. Our results generalise and improve projection algorithms discussed in the literature.