Cauchy type functional equations related to some associative rational functions

Abstract L. Losonczi [4] determined local solutions of the generalized Cauchy equation f(F (x, y)) = f(x) + f(y) on components of the definition of a given associative rational function F. The class of the associative rational function was described by A. Chéritat [1] and his work was followed by paper [3] of the author. The aim of the present paper is to describe local solutions of the equation considered for some singular associative rational functions.


Introduction
By an associative function on a nonempty set A usually we understand a binary operation F : A × A → A satisfying for all x, y, z ∈ A an equation F (x, F (y, z)) = F (F (x, y), z). (1) Rational functions, which are defined as elements of a field of fractions of polynomials in two variables have a form of a quotient of two polynomials in two variables.
Since the natural domain of such a function is usually not a rectangular A × A, the associativity is defined by a conditional form of (1). We propose the following definition. for all (x, y, z) ∈ R 3 such that (x, y), (y, z), (x, F (y, z)), (F (x, y), z) ∈ D. An associative rational function is often called an associative operation. [146]

Katarzyna Domańska
In [3] (Theorem 2) are included, among others, as associative the following functions (with natural domains in question) where α, β ∈ R. Let us consider the functional equation where f : I → R is an unknown function, I ⊂ R is an interval, F : D → I is a given rational associative function and (x, y) ∈ D ⊂ R 2 .
In [4] the local solutions of the generalized Cauchy equation (E) were determined by L. Losonczi for the following operations of the class of operations of the form (2), They are so-called local solutions only, i.e. such that equation (E) is satisfied not necessarily on the whole domain of the definition of the operation F but only on some its subsets. For example, the components of the domain D might happen to be such subsets although, in some cases, even they may prove to be "to large". The aim of the present paper is to describe local solutions of equations where α ∈ R, β ∈ R.

Main results
We will start with (E1) and its trivial solutions. First we will consider the case β = 0.

Theorem 1
The only solution f : R → R of the functional equation is the constant function f (x) = 0 for x ∈ R.
Cauchy type functional equations Proof. Assume that f : R → R is a solution of (E1.1). Substituting y = 0 in (E1.1) we obtain This means that f is of the form where c ∈ R. By setting and, therefore, c = 0. This means that f ≡ 0. Obviously, the constant function f ≡ 0 is a solution of (E1.1).
A description of the solutions of (E1) in the case β = 0 is given by the following

Theorem 2
If a function f : R → R is a solution of the functional equation then there exists a constant c ∈ R such that Conversely, for any constant c ∈ R the above function satisfies (3).
Proof. Assume that f : R → R is a solution of (3). Let x = 0 be arbitrary fixed and let y = 0, then by (3) we obtain This means that f is of the form (4), where c ∈ R.
In order to check that f given by (4) satisfies equation (3) fix arbitrarily a pair (x, y) such that x = −y and consider the following three cases In the case (i) we have f (x) = 0, f (y) = c and xy x+y = 0, consequently f xy x+y = c. The case (ii) is similar. In the case (iii) we have f (x) = f (y) = 0 and xy x+y = 0, which give f xy x+y = 0. [148]

Katarzyna Domańska
It seems natural to consider equation (E1) on the components of the domain of definition of rational operation (2). It turns out that equation (E1) considered on the components of the domain of definition of function (2) admits nonzero solutions which, however, have no more than two values. We proceed with a description of local solutions of (E1). We will describe non-trivial solutions on the components of the set We will focus on the case β > 0. For β < 0 reasoning is similar. Namely, Theorem 3 for β < 0 may be proved similarly to Theorem 4 with β > 0 and analogously Theorem 4 for β < 0 may be proved similarly to Theorem 3 with β > 0. More precisely, if β < 0, β = −α, α > 0 and

Theorem 3
The general local solution f : R → R of the functional equation where c ∈ R is an arbitrary constant.
for every x ≤ −β. Observe that the implication holds and therefore, we get f (x) = f (y) for any x, y ≤ −β, whence f (x) = c for every x ≤ −β, where c ∈ R is an arbitrary constant. Consequently f is of the form (5) where c ∈ R is arbitrary.
Obviously, each function of the form as above is a solution of (E1.2).

Cauchy type functional equations
[149] Theorem 4 The general local solution f : R → R of the functional equation where c, d ∈ R are arbitrary constants.
It is not difficult to check that f given by (6) is a solution of (E1.3).
In order to obtain non trivial solutions of (E1) we exclude the lines x = 0, y = 0, x = −β, y = −β from the set D where operation (2) is considered. We will describe solutions on these subsets of D which seem to be characteristic. Still, for the same reason as before, we will focus on the condition β > 0.
Because in the above proof the function g : (1, +∞) → R is exponential, writing H instead of g we obtain

Remark
The general local solution f : (0, +∞) → R of the functional equation (E1.4) can be written in the form where H : (1, +∞) → R is an arbitrary continuous exponential function.
Analogously we can proof the following result.

Theorem 7
If the function f : (−∞, 0) \ {−β} → R is a solution of the functional equation then there exist an additive function A : R → R and a constant c ∈ R such that Conversely, for any additive function A : R → R and any constant c ∈ R the function f given by (8) is a solution of equation (E1.6).
It is easy to check that (8) is a solution of (E1.6). Thus, the proof is completed. [152]

Katarzyna Domańska
Finally we prove Theorem 8 If the function f : (−∞, −β)∪(0, +∞) → R is a solution of the functional equation then there exist an additive function A : R → R and a constant c ∈ R such that Conversely, for any additive function A : R → R and a constant c ∈ R the function f given by (11) satisfies equation (E1.7).