Popoviciu Type Equations on Cylinders

Using a correspondence between the Popoviciu type functional equations and the Fréchet equation we investigate the solutions of the Popoviciu type functional equations on cylinders.


Introduction
In 1965 Popoviciu [18] proved that if I ⊂ R is an interval then a continuous function f : I → R is convex if and only if f satisfies the following inequality 3F x + y + z 3 + F (x) + F (y) + F (z) ≥ 2 F x + y 2 + F y + z 2 + F z + x 2 for x, y, z ∈ I. (1) Inequality (1) is known as the Popoviciu inequality (cf. e.g. [17]). In [18] it has been also proved that the only continuous solutions f : R → R of the equation satisfied for all x, y, z ∈ R, are the affine functions. This result has been generalized by Trif [21], who has proved that if X and Y are real linear spaces, then a function f : X → Y satisfies Eq. (2) for all x, y, z ∈ X if and only if there exist an additive function A : X → Y and a B ∈ Y such that f (x) = A(x) + B 334 M. Chudziak Results. Math. for x ∈ X. Stability of Eq. (2) has been investigated in [3]. Solutions and stability problem for the following generalizations of Eq. (2) have been studied, among others, by: • Lee [15], Smajdor [20], Trif [22]: (3) • Lee [16]: • Lee and Lee [14], Smajdor [19]: Popoviciu type functional equations on groups have been investigated in [4] and [5]. In this paper we consider Eq. (2) on cylinders. More precisely, given two groups (G, +) and (H, +) and nonempty subsets A and B of G, we deal with the equation The paper is organized as follows. In the next section we show that there is a natural correspondence between the solutions of (6) and the solution of the Fréchet equation on cylinders, that is equation of the form In Sects. 3 and 4 we deal with the solution of (7) and (6), respectively. In the last section we present some comments and remarks concerning the solutions of (7). Our results are motivated by the recent papers [1] and [2], where the solutions of the d'Alembert functional equation on cylinders have been considered.
A similar problem for the Cauchy equation has been earlier studied in [6] and [7].

Correspondence Between Popoviciu Type Equations and Fréchet Equation
A crucial role in our considerations will play the following result describing a correspondence between the Popoviciu type functional equations and the Fréchet equation. In order to formulate the result let us recall that, given a positive integer k, a semigroup (G, +) is said to be (uniquely) divisible by k provided for every y ∈ G there exists a (unique) x ∈ G such that kx = y. Mf and Proof. Assume that F : G → H satisfies (6). Let b := F (0). Then, applying (6) with x = y = z = 0, we get (8). Furthermore, setting in (6) Hence, for every x, y, z ∈ G, we have so making use of (6), we get Results. Math.

Fréchet Equation on Cylinders
According to Theorem 2.1, the Popoviciu type functional equations are closely related to Eq. (7), being the Fréchet equation on a cylinder. It is remarkable that the Fréchet equation known also as the Deeba equation (cf. [8,11]), is strictly related to the problem of characterization of inner product spaces. It is well known that a normed space (X, ) is an inner product space if and only if the function f : X → [0, ∞) given by f (x) = x 2 for x ∈ X, satisfies Eq. (19) for all x, y, z ∈ X. The solutions of (19) in a more general setting have been considered in [11] (see also [12]). Various aspects of stability problem for (19) have been studied in [8][9][10].
The following result will play an important role in the considerations of this section.

Proposition 3.1. Let (G, +) be a commutative semigroup, (H, +) be a commutative group. Assume that A and B are nonempty subsets of G and a function f : G → H satisfies Eq. (7). Then the following two sets
are subsemigroups of (G, +) containing A and B, respectively. Moreover, if (G, +) is a group then the sets P 1 (B) and P 2 (A) are subgroups of (G, +).
Proof. From (7) it follows that A ⊂ P 1 (B) and B ⊂ P 2 (A), so P 1 (B) and P 2 (A) are nonempty. Let y 1 , y 2 ∈ P 1 (B). Then, applying (7), we get whence y 1 + y 2 ∈ P 1 (B). Thus P 1 (B) is a subsemigroup of (G, +). The same arguments show that P 2 (B) is a subsemigroup of (G, +). Now, assume that (G, +) is a group. As previously, we present the proof only for P 1 (B). Since we have already proved that P 1 (B) is a subsemigroup of (G, +), it is enough to show that −y ∈ P 1 (B) for every y ∈ P 1 (B). To this end fix a y ∈ P 1 (B). Note that taking in (7) x = 0, we get f (0) = 0. Therefore, setting in (7) x = −y, we obtain On the other hand, in view of (7), for every (x, z) ∈ G × B, we have Vol. 67 (2015) Popoviciu Type Equations on Cylinders 339 and so Thus, making use of (20), for every (x, z) ∈ G × B, we get This means that −y ∈ P 1 (B).
where G(A) and G(B) are sub(semi)groups of (G, +) generated by A and B, respectively.
From Corollary 3.2 we derive the following result.

Popoviciu Type Equations on Cylinders
Applying the results of the previous sections, we obtain the following two theorems concerning the solutions of the Popoviciu type equations on cylinders.
where, as previously, G(A) and G(B) are subsemigroups (subgroups) of (G, +) generated by A and B, respectively.
From Theorem 4.1 we derive the following result. = N F x + y n + F y + z n + F z + x n for x, y, z ∈ G.