A combinatorial lemma and its applications

• Published in: Journal of inequalities and applications Vol. 2016; no. 1; pp. 1 - 22
• Main Author: Mackowiak, Piotr
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 31-03-2016; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Approximation; Browder fixed point theorem; Combinatorial analysis; combinatorial methods; continuum of zeros; Continuums; Demand; equilibrium; Existence theorems; fixed point; Inequalities; Kakutani fixed point theorem; Mapping; Mathematics; Mathematics and Statistics; Theorem proving; combinatorial methods; Kakutani fixed point theorem; continuum of zeros; fixed point; 91B02; zero of a map; Sperner lemma; Browder fixed point theorem; equilibrium; 91B50; 54H25
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-016-1043-y

In this paper, we present a generalization of a combinatorial lemma we stated and proved in a recent work. Then we apply the generalized lemma to prove: (1) a theorem on the existence of a zero for an excess demand mapping, (2) the existence of a continuum of zeros for a parameterized excess demand mapping, (3) Sperner’s lemma on labelings of triangulations. Proofs of these results are constructive: they contain algorithms (based on the combinatorial lemma) for the computation of objects of interest or, at least, of their approximations.