Fixed point iterations for non-expansive maps and their applications to constrained minimization and feasibility problems in Hilbert spaces

The purpose of this paper is to introduce some fixed point iterative schemes and prove that they converge faster than
other iterations in the literature. This paper introduces three novel modified multistep iterative schemes (A), (B) and (C). Fixed point theorems are proven with these newly introduced multistep iterative schemes for the class of contraction mappings with fixed point and non-expansive mappings respectively. The rate of convergence was demonstrated numerically with the help of Python programs and the results showed that our modified iterative scheme (C) converged in lesser number of iterations than existing iterative schemes in the literature. With the help of well constructed theorems, these modified multistep iterative schemes were applied to constrained minimization and split feasibility problems for the class of non-expansive mappings in real Hilbert spaces.