The Axiom of Infinite Choice
Mathematica Moravica, Vol. 16-1 (2012), 1–32.
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Abstract. In this paper we present the Axiom of Infinite Choice: Given any set $P$, there exist at least countable choice functions or there exist at least finite choice functions.
This paper continues the study of the Axiom of Choice 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. Tasković [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.
Keywords. The Axiom of Infinite Choice, The Axiom of Choice, Zermelo's Axiom of Choice, Lemma of Infinite Maximality, Zorn's lemma, Restatements of the Axiom of Infinite Choice, Choice functions, Foundation of the Fixed Point Theory, Geometry of the Axiom of Infinite Choice, Axioms of Infinite Choice for Points and Apices.
On a result of Jachymski, Matkowski, and Świątkowski
Mathematica Moravica, Vol. 16-1 (2012), 33–35.
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Abstract. We prove on a new manner, via Monotone Principle (of fixed point), a result of Jachymski, Matkowski, and Świątkowski [Journal of Applied Analysis, 1 (1995), 125-134, Theorem 1, p. 130].
Keywords. Fixed point, Nonlinear conditions for fixed points, topological spaces, Monotone Principle of Fixed Point, TCS-convergence, $d$-Cauchy completeness.
Inequalities of General Convex Functions and Applications
Mathematica Moravica, Vol. 16-1 (2012), 37–116.
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Abstract. This paper presents very characteristic illustrations of transversal sets (as upper, lower and middle) via general convex functions. Since the general convex functions are defined by a functional inequality, it is not surprising that this notation will lead to a number od important inequalities. This fact is connected de facto with the notation of transversal sets.