Functional equations and group substitutions (original) (raw)

Aequationes Mathematicae, to appear. SOLVING FUNCTIONAL EQUATIONS VIA FINITE SUBSTITUTIONS

2013

In this paper, we study single variable functional equations that involve one unknown function and a finite set of known functions that form a group under the operation of composition. The main theorems give sufficient conditions for the existence and uniqueness of a (local) solution and also stability-type result for the solution. In the proofs, beside the standard methods of classical analysis, some group theoretical tools play a key role.

Functional equations on finite groups of substitutions

Expositiones Mathematicae, 2012

Motivated by some investigations of Babbage, we study a class of single variable functional equations. These are functional equations involving one unknown function and a finite set of known functions that form a group under the operation of composition. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. In the proof of the main result, the Implicit Function Theorem and, when the stabilizer is nontrivial, the Global Existence and Uniqueness Theorem play a key role.

Solving functional equations via finite substitutions

Aequationes mathematicae, 2013

In this paper, we study single variable functional equations that involve one unknown function and a finite set of known functions that form a group under the operation of composition. The main theorems give sufficient conditions for the existence and uniqueness of a (local) solution and also stability-type result for the solution. In the proofs, beside the standard methods of classical analysis, some group theoretical tools play a key role.

On a Composite Functional Equation

Demonstratio Mathematica, 2003

We determine all continuous functions / : (0, oo)-• (0, oo) satisfying the functional equation /(xG(/(s))) = /(z)G (/(*)) where G is continuous and strictly increasing function such that 1 € G((0,oo)).

An alternative Cauchy functional equation on a semigroup

Aequationes mathematicae, 2013

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation f (xy) − f (x) − f (y) ∈ {0, 1} where f : S → R, S is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215-226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions f : H → R of the alternative functional equation on

On a generalized functional equation of Abel

Publicationes Mathematicae, 2002

We present some results concerning the following generalization of a functional equation of Abel ψ (xf (y) + yg(x)) = ϕ(x) + ϕ(y). With f = g we get the original Abel's equation that was mentioned explixitly by D. Hilbert in the second part of his fifth problem. The present generalization implies many applications in the theory of functional equations, particularly those dealing with determination of parametrized subsemigroups. We solve the equation in the class of continuous real functions defined in an interval containing 0.