Essential Fixed Points of Functions and Multifunctions

Abstract

Suppose $(X,d)$ is a compact metric space with the fixed point property and $\,\mathbb{C}\,$ the family of all continuous self maps on $X$ with the topology of uniform convergence. A fixed point $p$ of $ f\in \,\mathbb{C}\,$ is said to be \emph{essential} if functions near $f$ have fixed points near $\,p\,$. A function which has all of its fixed points essential is called an \emph{essential map}. Fort \cite{F} proved that the set of essential maps is residual in $\,\mathbb{C}\,$ and yet the only known examples of essential maps are those with only one fixed point. In this paper working in [0,1], we first characterize essential fixed points and prove some simple results concerning them. Then we characterize essential maps and give algorithms to construct them. We then study essential components introduced by Kinoshita \cite{K}. Next we consider essential fixed points of multifunctions in which case results differ considerably from the case of single valued functions. This also leads us to a study of selections. We conclude with a study of essential fixed points of non expansive functions in Banach spaces. All along we provide examples to illustrate the concepts and their limitations. Our results throw light on what is already known and takes the subject further. Unsolved problems are also mentioned.