In: Tatra Mountains Mathematical Publications, vol. 34, no. 3
Rok, strany: 2006, 289 - 306
In recent years, the investigation of functional equations containing the iterates of unknown functions have resulted in the development of a number of new methods and a new chapter in the regularity theory of functional equations. It is well-known that the solutions of the classical Cauchy equation are either very regular (infinitely many times differentiable) or very irregular (i.e., their graphs are everywhere dense on the plane). Thus, very weak regularity assumptions of a solution of the Cauchy equation force its very strong regularity properties. A situation for composite functional equations (that is for equations containing the iterates of unknown functions) is not so extreme. However, in a great deal of situations, starting with monotonicity and continuity properties of the solutions, their local Lipschitz continuity, differentiability everywhere but at countably many points, or further regularity properties can be proved. The proofs require the Lebesgue's theorem about the almost everywhere differentiability of monotonic functions, the Bernstein–Doetsch theorem on convex functions, the theorems of Steinhaus and Piccard concerning the algebraic sum of two sets of positive Lebesgue measure or of the second Baire category. Having proved the sufficient differentiability properties of the unknown functions, a functional equation can be differentiated with respect to the variables, and the composite terms can be eliminated. Thus, for derivatives of unknown functions, we obtain a new functional equation that does not involve the iterates of unknown functions, and, as a result, it can be solved using more standard methods of the theory of functional equations.
Páles, Z. 2006. Regularity problems and results concerning composite functional equations. In Tatra Mountains Mathematical Publications, vol. 34, no.3, pp. 289-306. 1210-3195.
Páles, Z. (2006). Regularity problems and results concerning composite functional equations. Tatra Mountains Mathematical Publications, 34(3), 289-306. 1210-3195.