Common Fixed Points for a Countable Family of Quasi-Contractive Mappings on a Cone Metric Space with the Convex Structure

Abstract

In this paper, we consider a countable family of surjective mappings $\{T_n\}_{n \in\mathbb{N}}$ satisfying certain quasi-contractive conditions. We also construct a convergent sequence $\{x_n\}_{n \in \mathbb{N}}$ by the quasi-contractive conditions of $\{T_n\}_{n \in\mathbb{N}}$ and the boundary condition of a given complete and closed subset of a cone metric space $X$ with convex structure, and then prove that the unique limit $x^{*}$ of $\{x_n\}_{n \in \mathbb{N}}$  is the unique common fixed point of $\{T_n\}_{n \in \mathbb{N}}$. Finally, we will give more generalized common fixed point theorem for mappings $\{T_{i,j}\}_{i,j \in \mathbb{N}}$. The main theorems in this paper generalize and improve many known common fixed point theorems for a finite or countable family of mappings with quasi-contractive conditions.