Stability results and separations theorems

The presented work summarizes the relationships between stability results and separation theorems. We prove the equivalence between different types of theorems on separation by an additive map and different types of stability results concerning the stability of the Cauchy functional equation.


Introduction
One of the first stability results in the theory of functional equations is the result of Pólya and Szegö from the year 1924 (see [15]). However, Ulam's question (see [18]) is taken as the beginning of the theory of the stability of functional equations. In 1940 Ulam formulated a question concerning the stability of the Cauchy functional equation a(x + y) = a(x) + a (x) in other words, on the stability of additive functions or the stability of homomorphisms of groups. Less than a year later, Hyers (see [10]) gives the answer to Ulam's question proving (in essence) the following Theorem 1.1. (Hyers) If (S, +) is a commuttative semigroup and X is a Banach space, then for each function f : S → X satisfying with some ε ≥ 0 there is an additive function a : S → X fulfilling Just over 25 years later, we find the first attempts to transfer the Hahn-Banach separation theorem (or the Mazur-Orlicz theorem) to the case of functionals defined on groups (semigroups). One of the most popular results belonging to this trend is Kranz's theorem (see [12], here the finite version, with functions with values in R) Theorem 1.2. (Kranz) If (S, +) is a commutative semigroup, p : S → R is subadditive, i.e.
q(x + y) ≥ q(x) + q(y), x,y ∈ S and q(x) ≤ p(x), x ∈ S, then there exists an additive function a : S → R which separates p and q, i.e.

q(x) ≤ a(x) ≤ p(x), x ∈ S.
In the mid-eighties of the twentieth century, it was widely known that stability theorems could be derived from separation theorems. A. Smajdor was the first to notice that separability theorems can also be obtained from stability theorems (see [7], Remark 2). The same, a separation theorem as a consequence of a stability result, can be found in Badora's paper [1]. In the paper of Cabello Sánchez [4] we find an analysis of the relationship between stability in the sense of Yost and separation theorems. Then in many works we find both stability theorems derived from separation theorems (e.g. Páles [14]) and separation theorems as consequences of stability results (e.g. [1]). Many of these results have not been published to this day and this work is to systematize our knowledge of the relationship between theorems about separation by an additive map and the results regarding the stability of the Cauchy functional equation.
From Hyers' theorem every commutative semigroup has property (H). In [17] Székelyhidi transferred this property to amenable semigroups. More information about which group (semigroup) has property (H) can be found in Forti's paper [5] .
We say that a semigroup S has property (G-K) if for every functions ϕ, ψ : there exists an additive function a : S → R satisfying In the paper by Gajda and Kominek [7] we find that every amenable semigroup and every weakly commutative semigroup has property (G-K). The equivalence of properties (H) and (G-K) shows the following Theorem 2.1. Let S be a semigroup. The following conditions are equivalent: (i) S has property (H); (ii) S has property (G-K).
Proof. The proof of the implication "(i)⇒(ii)" was credited to A. Smajdor ( [7], Remark 2) however, the author of this work does not know the original proof of this fact by A. Smajdor. Here we suggest the following: assume that ϕ and ψ satisfy (2.3), (2.4), (2.5) and let Then, for x, y ∈ S we have This allows us to define the function h : S → R by the formula Evidently, Moreover, and so, We can use these inequalities to estimate the Cauchy difference of the function h Hence, the function h satisfies inequality (2.1) with ε = 2k. Our assumption guarantees the existence of an additive map a : For x ∈ S and n ∈ N we have But for the additive map a: a(x n ) = n · a(x), for superadditive ϕ: ϕ(x n ) ≥ n · ϕ(x) and for subadditive ψ: ψ(x n ) ≤ n · ψ(x). Therefore, n which gives (2.7) and ends the proof of this implication.
For the proof of the implication "(ii)⇒(i)" assume that a map f : Our assumption guarantees the existence of an additive function a : S → R fulfilling In the vector case, the analogue of assumption (H) will be the following condition.
We say that a semigroup S has property (G-G1) if for every normed space Y , every bounded, convex, closed subset A of Y containing zero and every there exists an additive function a : S → Y such that This type of stability result (for an Abelian semigroup S and a Banach space Y ) was shown by Gajda and Ger in [6] (see Corollary 1).
We replace the separation property (G-K) by the following selection assumption.
We say that a semigroup S has property (G-G2) if for every normed space Y and every multifunction F : S → cc(Y ), where by cc(Y ) we denote the collection of all nonempty closed convex subsets of Y , fulfilling (F is a subadditive multifunction) there exists an additive function a : S → Y which is a selection of F , which means that Gajda and Ger in [6] proved such a selection theorem for an Abelian semigroup S and a Banach space Y (see Theorem 1).
The next theorem gives the equivalence of conditions (G-G1) and (G-G2).
Theorem 2.2. Let S be a semigroup. The following conditions are equivalent: Proof. For the proof of the implication "(I)⇒(II)" first, for x ∈ S, we select one element f (x) from a non-empty set F (x). Next, we observe that assumption (2.12) guarantees the existence of a positive r > 0 such that (by B(0, r) we denote the closed ball with center zero and radius r). Hence, for every x, y ∈ S we have which means that the map f satisfies (2.9) with A = B(0, 2r). By assumption (I) there exists an additive function a : S → Y fulfilling (2.10). Namely, But a is an additive map. So, and, as a result, which shows that Let us fix x ∈ S. Then f (x) ∈ F (x) and, by the subadditivity of F and the convexity of F (x), Hence, which ends the proof of "(I)⇒(II)". The implication "(II)⇒(I)", in fact, was noticed in [6]. Having a function f fulfilling (2.9) with some bounded, convex, containing zero, closed subset A of a normed space Y we define a multifunction F : S → cc(Y ) putting Then F is a subadditive multifunction satisfying (2.12) with sup{ diamF (x) : x ∈ S } = diamA < ∞ and the existence (by (II)) of an additive function a : S → Y , which is a selection of F , implies the existence of an additive function a : S → Y such that condition (2.10) holds true, which ends the proof.

The Ger stability
In [8] Ger suggested replacing the constant function from the Ulam problem by some function of one of the variables and proved that any Abelian semigroup has the following property: Let (S, ·) be a semigroup. We say that S has property (G) iff for every function f : S → R and every ρ : there exists an additive function a : S → R such that Also, in [14] Páles proved that every Abelian semigroup has the following separation property.
We say that a semigroup (S, ·) has property (P) iff for any functions ϕ, ψ : S → R there exists an additive map a : S → R fulfilling if and only if there exists a function f : S → R such that Here we have only the following implication (the truth of the reverse implication is an open problem): The situation is slightly different in the case of groups. In [2] (see also [5]) Badora distinguished a class G of groups. Namely, we say that a group (G, ·) belongs to the class G iff for every subadditive map γ : G → R. i.e. γ satisfies (2.4) there exists an additive function a : G → R such that (3.5) In [2] Badora proved that every amenable and every weakly commutative group belongs to the class G. Moreover, he proved that the Hyers stability theorem holds true for groups from the class G.
For groups we have the following: Proof. The implication "(a)⇒(b)" we have proved previously. Now, we will prove the implication "(b)⇒(c)". In the beginning we note that and if e is a neutral element of G, then Therefore,

From the first inequality and (2.4) we have
and as a result we get Now we can apply our assumption to the function γ and ρ(x) = γ(x) + γ(x −1 ) to obtain the additive function a : G → R fulfilling (3.2). Therefore, and hence we have By putting which is the expected inequality for a. It remains to show "(c)⇒(a)". In the proof of this implication assuming condition (3.3) with some additive function a : G → R, just taking f = a we get condition (3.4). So, we just need to show that assuming (c) from condition (3.4) (with some f ) we get condition (3.3) (with some additive a). Assume that f, ϕ, ψ : G → R and (3.4) holds true. We can define γ 1 : G → R by the formula Then, by (3.4), Moreover the function γ 1 is subadditive. Indeed, for x, y ∈ G we have The group G belongs to the family G. Hence, there is an additive map a 1 : So, we can also define a function γ 2 : G → R by The map γ 2 is superadditive Then and −γ 2 is subadditive. Our assumption implies the existence of an additive function a 2 : G → R such that Let a = −a 2 . Then by (3.6) we have But γ 1 (x) ≤ ψ(x), x ∈ G. We obtain, which ends the proof.
In the vector-valued case (see Badora, Ger, Páles [3]) we have: We say that a semigroup (S, ·) has property (Gv) iff for every linear space Y , every function f : S → Y and every R : there exists an additive function a : S → Y such that Proof. (B) is a consequence of (A) because if Φ : S → 2 Y \ {∅} is a multifunction and there is an additive function a : S → Y , such that condition (3.9) is satisfied, then taking f = a we notice that f and Φ satisfy condition (3.10).
If instead, f : S → Y is a function satisfying condition (3.10), then condition (3.7) with the function R = Φ − f is satisfied. By assumption, there exists an additive function a : S → Y satisfying (3.8), but that means that a satisfies condition (3.9).
In the proof of the implication "(A) is a consequence of (B)" we just take Φ = R + f .

The Yost stability
Following Yost's problem (see Lima and Yost [13], see also Ger [9]) We say that a semigroup (S, ·) has property (Y) iff for every functions f : S → R, γ : S → [0, +∞) satisfying there is an additive mapping a : S → R fulfilling Back to Kranz's theorem We say that a semigroup (S, ·) has property (K) iff for any functions p, q : S → R such that p is subadditive (2.4), q is superadditive (2.3), and condition (2.5) is satisfied there exists an additive function a : S → R satisfying (2.7). Proof. In the proof of the implication "(α) ⇒ (β)" for subadditive p and superadditive q such that (2.5) holds true we define f, γ : S → R putting Then γ is subadditive and non-negative. Moreover, for every x, y ∈ S, by the subadditivity of p we have and by the superadditivity of q which gives (4.1). By (Y) we obtain the existence of an additive map a : S → R fulfilling (4.2) which gives the desired inequality For the proof of the opposite implication "(β) ⇒ (α)" with f : S → R and γ : S → [0, +∞) fulfilling (4.1) we define p, q : S → R as follows Then, by inequality (4.1), p is subadditive, q is superadditive and by the nonnegativity of γ (2.5) is also fulfilled. Condition (β) implies the existence of an additive function satisfying (2.7) which leads to and ends the proof.
Going to the vector-valued case we replace condition (Y) by the original Yost "near additivity".
We say that a semigroup (S, ·) has property (Yv) iff for every normed space Y and functions f : there is an additive map a : S → Y fulfilling and the Kranz selection theorem by the following additive selection property first studied by W. Smajdor in [16].
We say that a semigroup (S, ·) has property (Kv) iff for every normed space Y every subadditive multifunction F : has an additive selection a : S → Y a(x) ∈ F (x), x ∈ S. (4.6) In this case we have only the following implication (the truth of the reverse implication is an open problem, also with stronger assumptions on the function F , such as, bounded, closed and convex values of F ):

Theorem 4.2. If a semigroup S has property (Kv) then S has property (Yv).
Proof. Let Y be a normed space and let f : S → Y , γ : S → [0, +∞) satisfy (4.3). Inequality (4.3) may be written as γ(xy)) , x,y ∈ S, which leads to the following inclusion for all x, y ∈ S. But for closed balls we have B(v 1 , which can be equivalently written as follows Hence, the multifunction F : S → 2 Y \ {∅} defined by  Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.
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