A modified successive projection method for Mann’s iteration process

Abstract

Mann’s iteration process, which is often used to solve the fixed point problems of nonexpansive mappings, has only weak convergence in infinite-dimensional Hilbert spaces. One way to overcome this weakness is the hybrid method proposed by Nakajo and Takahashi in 2003. But the hybrid method is difficult to implement, since the projection operator on an intersection of the domain and two half-spaces in each step has no closed-form formulae in general. A modified successive projection method for Mann’s iteration process proposed in this paper not only has strong convergence, but also is easy to implement. Therefore, our method effectively improves the hybrid method. We also extend our method to solve the nonexpansive semigroup problems. The numerical results show the superiority of the method in this paper.