Applications to Fixed Point Theorems

Abstract

Let E be a closed, bounded, convex subset of a Banach space X and f: E ⟶ E. Consider the iteration scheme defined by $\bar{x_{0}} = x_{0} \in E$, $\bar{x}_{n+1} = f(x_{n}),\ x_{n} =\sum \limits _{ k=0}^{n}a_{nk}\bar{x_{k}},\ n \geq 1$, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f. During the past few years several mathematicians have obtained fixed point results using Mann and other iteration schemes for certain classes of infinite matrices. In this chapter, we present some results using such schemes which are represented as regular weighted mean methods. Results of this chapter appeared in [20, 40, 82] and [84].