A fixed point theorem for contractive closed-valued mappings on metric spaces

Abstract

In this paper, we prove that if \(f\) is a contractive closed-valued mapping on a metric space \((X,d)\) and there exists a weak-contractive pseudo-orbit \(\{x_n\}\) for \(f\) at \(x_0 \in X\) such that both \(\{x_{n_i}\}\) and \(\{x_{n_i+1}\}\) converge for some subsequence \(\{x_{n_i}\}\) of \(\{x_n\}\), then \(f\) has a fixed point, which improves a fixed point theorem for closed-valued mappings by relaxing "contractive orbits" to " weak-contractive pseudo-orbits".