Single-valued signals, multi-valued signals, and fixed point of contractive signals

Abstract

Recent research conducted by Nadler presented the concept of multi-valued contractive signals. There can be only one fixed point for the contraction signal of a complete metric space F into itself, according to the Banach contractions signal concept. Nadler presented a theorem that is very similar to this one for multi-valued contraction signals. Fixed points do occur, therefore multiple-valued contraction signals on two-dimensional metric spaces are studied in this paper. Although we acquired some new conclusions from our investigation, the results obtained are inferior to those obtained by Iseki’s theorems. We investigate the equivalence of the existence of fixed points of single-valued and multi-valued signals for particular classes of signals by proving some equivalence theorems for the completeness of 2-metric spaces. These theorems show that the existence of fixed points for single-valued signals is equivalent to the existence of fixed points for multi-valued signals.