On Some Fixed Point Theorems for Generalized Contractive Mappings in an Euclidean Space $\mathbb{R}^n$

Abstract

In this paper a couple of fixed point theorems for contraction and Kannan mappings are proved in an Euclidean space $\mathbb{R}^n$ via calculus method and using the max/mini principle. It is shown that though our approach to Banach and Kannan mappings is different from the constructive one, we are not far away from the usual method. Actually one can take an arbitrary point $x_0\in \mathbb{R}^n$ and can define a sequence $\{x_n\}$ of iterates of the mapping under consideration. Then it is shown that the sequence converges to a fixed point geometrically.