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Functional equations with division and regular operations

Sergey Davidov

Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, Yerevan 0025, Armenia

E-mail Address: davidov@ysu.am

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Aleksandar Krapež

Mathematical Institute of the Serbian Academy of Science and Arts, Knez Mihailova 36, 11001 Belgrade, Serbia

E-mail Address: sasa@mi.sanu.ac.rs

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Yuri Movsisyan

Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, Yerevan 0025, Armenia

E-mail Address: movsisyan@ysu.am

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https://doi.org/10.1142/S179355711850033XCited by:4 (Source: Crossref)

Abstract

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc.54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar.11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych.27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

Communicated by N. Gilbert

Keywords:

AMSC: 03C05, 03C85, 20N05, 20N99, 26B99

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