Characterization of t-affine differences and related forms

In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form tf(x)+(1-t)f(y)-f(tx+(1-t)y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} tf(x)+(1-t)f(y)-f(tx+(1-t)y), \end{aligned}$$\end{document}where t∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in (0,1)$$\end{document} is a given number. We give a general solution of the functional equation associated with this problem.


Introduction
The well-known cocycle functional equation F (x + y, z) + F (x, y) = F (x, y + z) + F (y, z) and its applications have a long history in connection with many areas of mathematics, as discussed for example in [1,[5][6][7]. It has occurred in different fields, including homological algebra, the Dehan theory of polyhedra, statistics and information theory. The general solution of the cocycle functional equation on abelian groups has been known for about half a century (see [1,6,7]). It turns out that a function F : G → H (where G and H stand for abelian and a divisible abelian group, respectively) is a solution to the system of functional equations F (x + y, z) + F (x, y) = F (x, y + z) + F (y, z), x, y, z ∈ G F (x, y) = F (y, x), x,y ∈ G 986 A. Olbryś AEM if and only if the function F is representable as a Cauchy difference i.e. it has the form F (x, y) = f (x) + f (y) − f (x + y), x,y ∈ G for some one-place function f : G → H. The above characterization have been proved first by Erdös in [6] for real functions, then by Jessen, Karpf and Thourup in [7] on abelian groups. An interesting method for finding the symmetric solutions of the cocycle functional equation on commutative semigroups was given by B. Ebanks in [4]. A similar characterization for differences of the form where f is arbitrary function and λ and μ are given parameters has been given by Ebanks in [3]. In particular, he has characterized a Jensen differences as a special case λ = 2, μ = 1 2 . The above form is called Jensen difference because it vanishes exactly when f is a solution of the Jensen functional equation. In our main result we generalize the following statement which is a particular case of result proved by Bruce Ebanks [3,Corollary 7] Theorem 1. Let G be a uniquely 2-divisible abelian group, and let X be a rational vector space. Then a map Δ : G × G → X satisfies conditions: A key tool we are going to use in the proof of our main result is the following theorem from [5,Theorem 3.3]. This theorem gives a general solution of the most general form of the cocycle functional equation.

Theorem 2.
Let G be an abelian group and X a rational vector space. The general solution is given by Vol. 95 (2021) Characterization of t-affine differences and related forms 987 where A 1 , B 1 , C 1 : G × G → X are additive in the first variable, B 2 : G × G → X is additive in its second variable, and f i : G → X (i = 1, . . . , 11) is arbitrary.

Results
Let t ∈ (0, 1) be a fixed number. Throughout this paper X and Y stand for linear spaces over the field K such that Q(t) ⊆ K ⊆ R, where Q(t) is the smallest field containing a singleton {t}. Clearly, Q ⊆ Q(t), where Q denotes the field of rational numbers. The assumptions about X and Y will not be repeated in the sequel. The purpose of the present paper is to characterize a difference of the form where t ∈ (0, 1) is a given number. Let us recall that a function f : respectively. The above difference we will call a t-affine difference because it vanishes exactly when f is a t-affine function (see [8] for more information about t-affine and t-convex functions). It can be easily seen that the t-affine difference has the following two properties: Furthermore, let us observe that a f satisfies the following functional equation: Indeed, for arbitrary u, x, y, v ∈ X we get In our main result we show that conditions (i)-(iii) characterize exactly those maps ω : X × X × {t, 1 − t} → [0, ∞) which can be expressed as a t-affine difference a f for some t-convex function f : X → R. In [9] we have proved the following result in this spirit.
Then there exists a function h : D → R such that if and only if for all x, y, u, v ∈ D the map ω satisfies the functional equation: The following theorem gives a general solution of the functional equation corresponding to equation (iii).

Theorem 4. The general solution
is given by where d, r : X → Y are additive functions satisfying the condition c ∈ Y is an arbitrary constant and n : X → Y is an arbitrary function.
Proof. It is easy to verify that ω given by (3) with (4) is a solution of (2). Conversely, suppose that (2) holds. Put y = 0 in (2) to get Vol. 95 (2021) Characterization of t-affine differences and related forms 989 This equation can be rewritten in the form We can rewrite the above equation in the form where, Choosing the fourth line of the solution in Theorem 2, we see that and finally by putting α := t 1−t we obtain is an additive map with respect to the i-th variable, for i = 1, 2 and f, g, h : X → Y are arbitrary. Now, we substitute (6) into (2) and obtain after rearrangement Put u = 0 in (7) to obtain Subtracting (8) from (7) we get We see that the right hand side of (9) is independent of y. Therefore the lefthand side left unchanged upon setting y = 0. That is, after some calculation, we get Setting u, v, y, x in the palace of αtu, tv, (1 − t)y, and tx, respectively, we obtain Vol. 95 (2021) Characterization of t-affine differences and related forms 991 From this we see that for arbitrary fixed u, v, x ∈ X the map is additive, consequently for all u, v, x, y, z ∈ X. Putting z := y and v := x + y we get for any x, y, u ∈ X, hence the function x − → A 1 (u, x) is a polynomial function of second order for arbitrary u ∈ X. Therefore, by a result of Djoković [2], this map can be written as the sum of a constant, an additive map and the diagonalization of a symmetric bi-additive map i.e.
It is easy to observe that a 0 , a 1 , a 2 are additive with respect to the first variable. By a similar argument we deduce that is a diagonalization of a symmetric bi-additive map. It is not hard to check that b 0 , b 1 , b 2 are additive with respect to y. Inserting (11) and (12) back into (10) and simplifying we get hence, a 2 = b 2 , in particular; a 2 is a 3-additive and symmetric function.

(u, x)+γ(u, x, x)+γ(u, u, x) = p(x+u)+p(0)−p(u)−p(x), u,x ∈ X. (16)
Since the right hand side of (16) is symmetric in x and u, the function b is bi-additive and symmetric. Observe that the function δ : X → Y given by the formula is additive. Indeed, for arbitrary u, x ∈ X by virtue of (16), bi-additivity and symmetry of b, the 3-additivity and symmetry of γ we have Therefore on account of (13) and (17) f has the form where c 1 = −p(0) = f (0) − th(0). Now, we return to equation (7). Put y = 0 in (7) to obtain Subtracting the resulting equation from (7), we get Vol. 95 (2021) Characterization of t-affine differences and related forms 993 Eq. (19). Having the forms (11) and (12) in mind we obtain Replace in the above equation αx by x and define the functionsb, l, q : X → Y by the formulas Similarly as before one can check that the function k : X → Y given by the formula is additive, moreover, we get the representation where = a 0 (αx) + a 1 (αx, y) + a 2 (αx, y, y) + b 0 (y) + b 1 (αx, y) +c(αx, y) +c +a 2 (αx, y, y) + a 2 (αx, αx, y) Vol. 95 (2021) Characterization of t-affine differences and related forms 995 −a 2 (αx, αx, y) − a 2 (αx, y, y) Finally, set u = y = 0 in (2) and substitute the new form of ω into Eq. (2) to obtain after rearrangement Put βx and βv in the place of x and v, respectively, for β ∈ Q \ {0}. Since any additive function is Q-homogeneous we get consequently, This completes the proof of the theorem.

Remark 1.
It follows from the Proof of Theorem 4 that it is also true in the case where t ∈ R \ {0, 1}.
As an immediate consequence of the previous theorem we obtain the following corollary.
if and only if it has the form where d : X → Y is an additive function and f : X → Y is an arbitrary map.
Proof. Assume that ω satisfies Eq.
(2) and vanishes on the diagonal. On account of Theorem 4 ω has the form wherec ∈ Y is a constant, d, r : X → Y are additive and f : X → Y is arbitrary. Since the t-affine difference vanishes on the diagonal, Because d + r is an additive map, by putting x = 0 we getc = 0 and consequently In order to present our next result we need some kind of symmetry of ω and this leads us to the consideration of ω as a map of three variables. The following theorem generalizes Corollary 8 from [3] in the case where G is a linear space.
vanishes on the diagonal and is symmetric i.e.
ω(x, y, t) = ω(y, x, 1 − t), x,y ∈ X, Since By putting we can rewrite the above equation in the form It follows from the above identity and the additivity of m that Putting z = 0 and replacing tx by x and (1 − t)y by y we have p(x) + p(y) = p(x + y) + p(0), x,y ∈ X, so subtracting 2p(0) from the both sides of this equation we get (p(x) − p(0)) + (p(y) − p(0)) = (p(x + y) − p(0)), x,y ∈ X.
Therefore, a function r : X → Y given by the formula is additive and consequently, This implies that ω(x, y, s) = a(x − y) + a g (x, y, t), s= t a(y − x) + a g (x, y, 1 − t), s = 1 − t. The proof of the theorem is completed. Now, we present a particular case of Theorem 9 from [10] (for D = X) which we are going to use in our last result.  s)z, s), for all x, z ∈ X and s ∈ {t, 1 − t}. Then for arbitrary c ∈ R there exists a t-concave function g y : X → R such that g y (y) = c, g y (x) ≤ c, x ∈ X, and ω(x, z, t) = g y (tx + (1 − t)z) − tg y (x) − (1 − t)g y (z), x,z ∈ X.
Our main result reads as follows Proof. It is easy to see that an affine difference a f satisfies conditions (a)-(c) from Theorem 6. Conversely, assume that ω is symmetric, vanishes on the diagonal and satisfies the functional Eq. (22). On account of Theorem 5 there exist a function h : X → R and an additive function a : X → R such that ω(x, y, s) = a((2s − 1))(x − y)) + a h (x, y, s), x,y ∈ X, s ∈ {t, 1 − t}.
As it can be easily checked the function ω of the above form satisfies conditions (a)-(c) from Theorem 6 then there exists a concave function g : X → R such that ω(x, y, t) = g(tx + (1 − t)y) − tg(x) − (1 − t)g(y), x,y ∈ X. Vol. 95 (2021) Characterization of t-affine differences and related forms 999 Finally, by putting f := −g we see that f is t-convex, moreover, ω(x, y, t) = a f (x, y, t), x,y ∈ X.