On Schauder's 54th Problem in Scottish Book Revisited

Milan Tasković

Mathematica Moravica, Tome 6 (2002) no. 1.

Résumé

The most famous of many problems in nonlinear analysis is Schauder's problem (Scottish book, problem 54) of the following form, that if C is a nonempty convex compact subset of a linear topological space does every continuous mapping $f: C\to C$ has a fixed point? The answer we give in this paper is yes. In this paper we prove that if C is a nonempty convex compact subset of a linear topological space, then every continuous mapping $f: C \to C$ has a fixed point. On the other hand, in this sense, we extend and connected former results of Brouwer, Schauder, Tychonoff, Markoff, Kakutani, Darbo, Sadovskij, Browder, Krasnoselskij, Ky Fan, Reinermann, Hukuhara, Mazur, Hahn, Ryll-Nardzewski, Day, Riedrich, Jahn, Eisenack-Fenske, Idzik, Kirk, Ghöde, Caristi, Granas, Dugundji, Klee and some others.

Détail

Mots-clés : Fixed point theorems, Brouwer's theorem, Schauder's the­orem, Tychonof's theorem, Markoff's theorem, Kakutani's theorem, Sadovskij's theorem, Schauder's Conjecture (Scottish Book, problem 54)