Remarks on hyperstability of the Cauchy functional equation

We present some simple observations on hyperstability for the Cauchy equation on a restricted domain. Namely, we show that (under some weak natural assumptions) functions that satisfy the equation approximately (in some sense), must be actually solutions to it. In this way we demonstrate in particular that hyperstability is not a very exceptional phenomenon as it has been considered so far. We also provide some simple examples of applications of those results.


Introduction
In this paper N, Z, Q, R and C denote the sets of positive integers, integers, rationals, reals and complex numbers, respectively; N 0 := N ∪ {0} and R + := [0, ∞). Moreover, X and Y always stand for normed spaces (unless clearly stated otherwise) and U ⊂ X is nonempty.
In what follows we say that a function f mapping U into a set Z, endowed with a binary operation + : Z 2 → Z, is additive on U provided it satisfies the conditional Cauchy functional equation x, y ∈ U, x + y ∈ U ; (1.1)
Then it is easily seen that which yields x, y ∈ X, x|y = 0, i = 1, 2.
It is easily seen that if g has the form described either by (A) or by (B), then it fulfils (2.1).

Proposition 2.2.
Let dim X > 2 and g : X → Y . Suppose that there are positive real numbers p and L 0 with Then the following two statements are valid.
(α) If p = 1 or X is not a real inner product space, then g is additive.
(β) If X is a real inner product space and p = 1, then there exist an additive mapping a : X → Y and a vector z 0 ∈ Y such that z 0 ≤ 2L 0 and Proof. Note that (2.4) yields If X is not a real inner product space, then it follows from [39] that the even part of g is identically equal zero. This means that g is odd and consequently it is additive in view of [38,Theorem,p. 270]. So it remains to consider the case where the norm in X comes from some real inner product ·|· . Then (2.4) takes form (2.1) with L = 4L 0 and it is enough to use Proposition 2.1.

Proposition 2.3.
Let dim X > 2 and let g : X → Y . Suppose that there are functions η, μ : R → R with μ(0) = 0 and Then g is additive.
Proof. Taking x = y in (2.6) we obtain that x, y ∈ X, x = y .

Some further hyperstability results
Given A, B : X → X, for the simplicity of notations we write AB := A • B and define the mapping A + B : It is easily seen that, in the particular case where A is additive (i.e., A(x+y) = A(x) + A(y) for every x, y ∈ X), we have (with U = X) Now, we are in a position to show the following result.
Let p ∈ R + be such that one of the following two conditions is valid: Then every function g : U → Y for which there exists L ∈ R + such that is additive on U .
Proof. In view of (3.2), from (3.3) (with x replaced by D(x) and y = C(x)) we obtain First consider the case of (a). Then U = E(U ) and (3.4) yields Vol. 86 (2013)

Hyperstability of the Cauchy equation 259
The proof is by induction. Clearly the case n = 0 is just (3.3). So fix l ∈ N 0 and assume that (3.6) holds true with n = l. Then, by (3.5), Since, according to (3 Thus we have proved that (3.6) is valid for each n ∈ N 0 . Since κ < 1, letting n → ∞ in (3.6) we obtain that g is additive on U .
Next assume that (b) holds. From (3.4) we deduce that We show by induction that, for each The case n = 0 is trivial. Take l ∈ N 0 and assume that (3.8) is valid for n = l. Then, by (3.7), Thus we have proved by induction that (3.8) is valid for each n ∈ N 0 . Since η < 1, letting n → ∞ in (3.8) we obtain that g is additive on U .
Remark 3.2. Observe that condition (3.1) in Theorem 3.1 is valid for instance when D = C n with some n ∈ N 0 or Dx = γx for x ∈ X with some γ ∈ Q (because C is assumed to be additive). is fulfilled. In some of these cases we can derive some complementary stability and hyperstability results from the subsequent proposition, which follows easily from [7, Theorem 1] (cf. also [21]).
Suppose that Y is complete and there is ε ∈ {−1, 1} such that 2 ε V ⊂ V and (3.9) Then there exists a unique F : V → Y that is additive on V and Proposition 3.4 yields in particular the subsequent two corollaries generalizing the results of Hyers [22], Aoki [2], Rassias [32,33] and Gajda [20] (see also [34,Theorem 1] Corollary 3.5. Let Y be complete, g : U → Y , δ, L 1 , L 2 ∈ R + , q, r ∈ (−∞, 1), and 2U ⊂ U . Suppose that there exist L ∈ R + , p ∈ (0, 1), and C : U → X such that for every x, y ∈ U\{0} with x + y ∈ U\{0}. Then there exists a unique function G : U → Y that is additive on U and satisfies  ∞), and U ⊂ 2U . Suppose that there exist L ∈ R + , p ∈ (1, ∞), and C : U → X such that for every x, y ∈ U with x + y ∈ U . Then there exists a unique function G : U → Y that is additive on U and satisfies Proof. It is enough to use Proposition 3.4 with V := U , ε = −1 and Note that Corollaries 3.5 and 3.6 with δ = L 1 = L 2 = 0 supply additional two hyperstability results, which cannot be deduced from Theorem 3.1.

J. Brzdȩk AEM
Remark 3.7. In connection with the statements of Theorem 3.1 and Corollaries 3.5 and 3.6 there arises the natural question when a function that is additive on U can be extended to an additive function f : X → Y . Some information on investigations of this issue can be found in [1], [28, Theorem 1.1, Ch. XVIII]) and [36,Chapter 4] (see also [37, pp. 143-4] for some extension procedure). Below we provide one more result concerning this problem, which corresponds somewhat to the outcomes in [1]. (Let us recall that I ⊂ 2 X is an ideal provided A ∪ B ∈ I and 2 A ⊂ I for every A, B ∈ I).
Assume that there exists an ideal I ⊂ 2 X such that X ∈ I, X \U ∈ I and Then there is a unique additive f : Proof. It is easy to deduce from [6, Lemma 1] that Clearly, by (3.12), Thus we have proved that (3.14) According to (3.13), we may define f : X → Y by: First we show that f is an extension of h. To this end fix z ∈ U and u ∈ U ∩ (U − z). Then (3.11) yields Next we prove that f is additive. Take z, w ∈ X. According to (3.13), Then To complete the proof it remains to show the uniqueness of f . So, suppose that f 1 : X → Y is additive and f 1 (x) = h(x) for x ∈ U . Take z ∈ X and a, b ∈ U with z = a − b. Then (d) X = R n with some n ∈ N and I is the family of all subsets of X that are of finite Lebesgue measure. (e) X is a Polish space and I is the σ-ideal of Christensen zero subsets of X (see, e.g., [18]).

Some consequences
In what follows, given I ⊂ 2 X and f, g : X → Y , we say that f = g I-almost everywhere (abbreviated to I-a.e.) in X if there is a set T ∈ I such that f (x) = g(x) for every x ∈ X \T . Moreover we write αT := {αx : x ∈ T } for T ⊂ X and α ∈ R. The next theorem is a consequence of some previous results in this paper. (An ideal I ⊂ 2 X is a σ-ideal provided n∈N T n ∈ I for every family of sets {T n } n∈N ⊂ I).
Theorem 4.1. Let g : X → Y and I ⊂ 2 X be a σ-ideal such that (3.12) holds and Assume that one of the following two conditions is fulfilled.
(i) There exist T ∈ I, c, d ∈ R, cd(c + d) = 0, and reals L > 0 and p > 1 such that (ii) There exist T ∈ I, C : X → X with C(2x) = 2C(x) for x ∈ X, and positive reals L and p = 1 such that Then there is a unique additive operator f : X → Y with f = g I-a.e. in X.
Proof. First assume that (i) holds. Define C, D : X → X by: C(x) = cx and D(x) = dx for x ∈ X. Write X T := X \T , It is easily seen that X \U ∈ I, cU = U , dU = U and (c + d)U = U . Further, if cd > 0, then |c + d| = |c| + |d| and consequently if cd < 0, then (without loss of generality, because (4.2) is symmetric with regard to x and y) we can assume that |d| = |c| + |d + c| and consequently This means that one of conditions (a) and (b) of Theorem 3.1 is valid and consequently g is additive on U . Hence, by Proposition 3.8, there is an additive operator f : X → Y with g(x) = f (x) for x ∈ U . The uniqueness of f also follows from Proposition 3.8. If (ii) holds, then we write U := n∈Z 2 n (X \T ).
Clearly 2U = U . Let W be the Banach space that is the completion of Y . Then we can consider g to be a mapping from X into W . Hence, by Corollaries 3.5 (when p < 1) and 3.6 (when p > 1) with δ = L 1 = L 2 = 0, g is additive on U . Now it is enough to apply Proposition 3.8 analogously as before. The next two corollaries provide two further examples of simple applications of Theorem 3.1, which correspond to some results in [3,5,[11][12][13][14][15][16][17]24] concerning the inhomogeneous Cauchy equation and the cocycle equation. Then the conditional functional equation g(x + y) = g(x) + g(y) + G(x, y) x, y ∈ U, x + y ∈ U, (4.4)