Sections in functional equations

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Aequationes Mathematicae

Sections in functional equations ˘ianu Dan M. Da

˘ and Cristina Mˆındrut ¸a

Abstract. We develop a technique for solving some equations having composite functions as solutions and, on the other hand, a very general procedure for generating new functional equations with predetermined composite solutions. Mathematics Subject Classification. Primary 39B52; Secondary 39B22, 39A70. Keywords. Section method, Radical type equation, Arithmetically homogeneous function.

1. Argument Radical type equations (see [1] and the papers referred to therein) allow for a simple solution technique—partial or complete. For instance, to solve the equation √ (1.1) f ( n xn + y n ) = f (x) + f (y) for all x, y ∈ R in the unknown f : R → R, we ﬁrst note that the equation can be written as f ◦ g (g (x) + g (y)) = f (x) + f (y) for all x, y ∈ R, (1.2) √ n where g : R → R, g (x) := xn and g : g (R) → R, g (xn ) := xn , and that g is a section (a right inverse) of the function g : R → g (R) , g (x) := g (x) (i.e. g ◦ g = idg(R) ). We associate to (1.2) the equation ρ (u + v) = ρ (u) + ρ (v) for all u, v ∈ g (R) ,

(1.3)

where ρ : g (R) → R is the unknown function. It is easy to see that f is a solution of (1.1) if and only if f := ρ ◦ g, where ρ is a solution of the Cauchy equation (1.3).

˘ianu, C. Mˆındrut ˘ D. M. Da ¸a

AEM

This technique is applicable to a large class of functional equations. Take, for instance, the equation f (x + y − Tn (x) − Tn (y)) = f (x) + f (y) for all x, y ∈ A, where A is the set of all functions R → R of class C

(1.4)

n+1

, Tn (x) is the Maclaurin (i) n polynomial of degree n associated to x ∈ A (i.e. Tn (x) (t) := i=0 x i!(0) ti ), and f : A → R is the unknown function. We remark that, deﬁning g : A → B, g (x) := x(n+1) and g : g (A) → A, g (u) := un , where B is the set of continuous functions and un is the result of the recurrence t t u0 (t) := u(τ )dτ and ui (t) := ui−1 (τ )dτ for i = 1, . . . , n, 0

0

we have

g x(n+1) := x − Tn (x) and g (g (u)) = u for all u ∈ g (A) .

Hence g is a section of the function g : A → g (A) , g (x) := g (x) , and Eq. (1.4) can be rewritten in the form f ◦ g (g (x) + g (y)) = f (x) + f (y) for all x, y ∈ A.

(1.5)

Moreover, associating to (1.5) the equation ρ (u + v) = ρ (u) + ρ (v) for all u, v ∈ g (A) ,

(1.6)

where ρ : g (A) → R is the unknown function, it is easy to see that f is a solution to (1.4) if and only if f := ρ ◦ g, where ρ is a solution of equation (1.6). We further expand the context by developing solution techniques for equations of types (1.1) and (1.4) using what we call the “section method”. We show that a large class of equations in which some function-section pairs can be highlighted [as in (1.2) and (1.5)] can be—partially or completely—solved by simpler equations [such as equations (1.3) and (1.6)], and conversely, equations with known solutions can generate more complex equations with predeﬁned solutions by inserting some function-section pairs. Finally, as applications to the section me

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