Solution, extensions and applications of the Schauder's 54th problem in Scottish book

This paper presents the Axiom of Infinite Choice: Given any set P, there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Taskovic [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice! This paper presents applications of the Axiom of Infinite Choice to the Fredholm and Leray-Schauder theory. In this sense, I give a solution and some extensions of Schauder's problem (in Scottish book, problem 54). This paper presents some new mathematical n-person games. In the theory of n-person games, there have been some further developments in the direction of transversal games and mathematical alternative theory