Real Analysis Exchange

Essential Fixed Points of Functions and Multifunctions

I. Del Prete, S. Naimpally, and M. di Iorio

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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.

Article information

Real Anal. Exchange, Volume 25, Number 1 (1999), 369-382.

First available in Project Euclid: 5 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20] 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] Secondary54B20 54C35: Function spaces [See also 46Exx, 58D15]

fixed point essential fixed point map essential map non expansive map multifunction selection


Del Prete, I.; di Iorio, M.; Naimpally, S. Essential Fixed Points of Functions and Multifunctions. Real Anal. Exchange 25 (1999), no. 1, 369--382.

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