6 - Some aspects of functional equations

Abstract

This chapter discusses some important concepts from functional equations like regularity properties, uniqueness results, restricted domains, extension theorems, inequalities, and differentiability. These concepts are also applied to associative functions. The main aim for mathematicians working on functional equations would always be to characterize all triangular norms (t-norms) without any further condition. The problem, to describe explicitly all these systems of functional equations, is still unsolved. But much more can be said if some additional regularity property like continuity is assumed. One typical (and natural) way to solve functional equations is to reduce it to a differential equation, and then to solve this differential equation, to get a solution of the original functional equation. The trick of improving the regularity of functions in functional equations can be used rather often, even for complicated functional equations containing more than one unknown function. It is easy to see that the family of Hamacher t-norms satisfies the functional equation. Regularity theorems of the type “Lower regularity implies higher regularity” are much more complicated in the case of the associativity equation than in the case of the Cauchy equation.