The successive contraction method for fixed points of pseudo-contractive mappings

In this paper, a new general iterative method is proposed for finding a fixed point of a Lipschitz continuous pseudo-contractive mapping defined on a closed convex subset of a real Hilbert space. Since each iteration of our method can be reduced to finding the fixed point of a strict contraction, our iterative method is called the successive contraction method. We give exact and inexact versions of the successive contraction method and prove their weak convergence, respectively. The main advantage of our method is that its convergence does not require the compactness assumption on the operators under consideration, which is quite different from the existing algorithms such as the Ishikawa iterative method. This superiority is also demonstrated by comparing the numerical performances of our method and the Ishikawa iteration process. © 2019 Applied Set-Valued Analysis and Optimization.