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Croatian Operational Research Review, Vol. 8 No. 1, 2017

Original scientific paper

Characterizing fi xed points

Sanjo Zlobec ; Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6

Fulltext: english, pdf (127 KB) pages 351-356 downloads: 213* cite
APA 6th Edition
Zlobec, S. (2017). Characterizing fi xed points. Croatian Operational Research Review, 8 (1), 351-356.
MLA 8th Edition
Zlobec, Sanjo. "Characterizing fi xed points." Croatian Operational Research Review, vol. 8, no. 1, 2017, pp. 351-356. Accessed 21 May 2019.
Chicago 17th Edition
Zlobec, Sanjo. "Characterizing fi xed points." Croatian Operational Research Review 8, no. 1 (2017): 351-356.
Zlobec, S. (2017). 'Characterizing fi xed points', Croatian Operational Research Review, 8(1), pp. 351-356. doi:
Zlobec S. Characterizing fi xed points. Croatian Operational Research Review [Internet]. 2017 [cited 2019 May 21];8(1):351-356. doi:
S. Zlobec, "Characterizing fi xed points", Croatian Operational Research Review, vol.8, no. 1, pp. 351-356, 2017. [Online]. doi:

A set of sufficient conditions which guarantee the existence of a point x⋆ such that f(x⋆) = x⋆ is called a "fixed point theorem". Many such theorems are named after well-known mathematicians and economists.
Fixed point theorems are among most useful ones in applied mathematics, especially in economics and game theory. Particularly important theorem in these areas is Kakutani's fixed point theorem which ensures existence of fixed point for point-to-set mappings, e.g., [2, 3, 4]. John Nash developed and applied Kakutani's ideas to prove the existence of (what became known as) "Nash equilibrium" for finite games with mixed strategies for any number of players. This work earned him a Nobel Prize in Economics that he shared with two mathematicians. Nash's life was dramatized in the movie "Beautiful Mind" in 2001. In this paper, we approach the system f(x) = x differently. Instead of studying existence of its solutions our objective is to determine conditions which are both necessary and sufficient that an arbitrary point x⋆ is a fixed point, i.e., that it satisfies f(x⋆) = x⋆. The existence of solutions for continuous function f of the single variable is easy to establish using the Intermediate Value Theorem of Calculus. However, characterizing fixed points x⋆, i.e., providing answers to the question of finding both necessary and sufficient conditions for an arbitrary given x⋆ to satisfy f(x⋆) = x⋆, is not simple even for functions of the single variable. It is possible that constructive answers do not exist. Our objective is to find them.
Our work may require some less familiar tools. One of these might be the "quadratic envelope characterization of zero-derivative point" recalled in the next section. The results are taken from the author's current research project "Studying the Essence of Fixed Points". They are believed to be original. The author has received several feedbacks on the preliminary report and on parts of the project which can be seen on Internet [9].

single variable calculus; fixed point; fundamental theorem of calculus; quadratic envelope characterization of zero-derivative point; primal and dual characterizations of fixed points

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