## Real Analysis Exchange

### 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.

#### Article information

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

Dates
First available in Project Euclid: 5 January 2009

https://projecteuclid.org/euclid.rae/1231187611

Mathematical Reviews number (MathSciNet)
MR1758893

Zentralblatt MATH identifier
1015.54021

#### Citation

Del Prete, I.; di Iorio, M.; Naimpally, S. Essential Fixed Points of Functions and Multifunctions. Real Anal. Exchange 25 (1999), no. 1, 369--382. https://projecteuclid.org/euclid.rae/1231187611

#### References

• L. E. J. Brouwer, Uber Abbildungen von Mannigfaltigkeiten, Math. Ann. 71(1912), 97–115.
• F. E. Browder, Convergence of approximants to fixed points of non expansive non linear mappings in Banach spaces, Arch. Rat. Mech. Anal. 24(1967a), 82–90.
• A. M. Bruckner, Stability in the family of $\omega$-limit sets of continuous self-maps of the interval, Real Analysis Exchange, 22(1997), 52–75.
• M. K. Fort, Jr, Essential and non essential fixed points, Amer. J. Math. 72(1950), 315–322.
• G. Gabor, On the classification of fixed points, Math. Japonica, v.40, n.2, (1994), 361–369.
• Jiang, Jia-He, Essential fixed points of the multivalued mappings, Sci. Sinica, v.11, n.3, (1962), 293–298.
• Jiang, Jia-he, Essential component of the set of fixed points of the multivalued mappings and its application to the theory of games, Sci. Sinica, v.12, n.7, (1963), 951–964.
• S. Kinoshita, On essential components of the set of fixed points, Osaka Math. J. 4(1952), 19–22.
• T. Kok-Keong, Y. Jian, Y. Xian-Zhi, The stability of coincident points for multivalued mappings, Nonlinear Analysis, Theory, Methods and Applications, vol. 25, No. 2, (1995) 163–168.
• B. O'Neill. Essential sets and fixed points, Amer. J. Math. 75(1953), 497–509.
• W. Strother, On an open question concerning fixed points, Proc. Amer. Math. Soc. 4(1953), 988–993.
• W. Strother, Fixed points, fixed sets, and M-retracts, Duke. Math. J. 22(1955), 551–556.
• H. F. Senter, W. J. Dotson, Approximating fixed points of non expansive mappings, Proc. Amer. Math. Soc. 44(1974), 375–390.
• Y. Yonezawa, On f.p.p and f*.p.p of some not locally connected continua, Fund. Math. 139(1991), 91–98.