Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.


Introduction
In [2, section 1] the author mentions that the original result by Hyers in [5] holds true for commutative semigroups in general. Thus we have the following result.

Theorem 1. (Forti and Hyers) Let S be a commutative semigroup and X be a
Banach spaces and let f : S → X be a function such that for some ε > 0 Then for every x ∈ S the limit α(x) = lim n→∞ f (2 n x) 2 n exists, the function α is additive and (2) Moreover α is the unique additive function satisfying the last inequality.
In the proof it is shown that the sequence ( f (2 n x) 2 n ) n∈N0 is a Cauchy sequence, and thus convergent, since X is assumed to be a Banach space. But one might ask whether this assumption is necessary to find some additive α such that (2) holds.
It was shown in [3] that the completeness of X is in some sense necessary.

Theorem 2. (Forti and Schwaiger)
Let G be an abelian group with an element of infinite order and X a normed space. Assume that for all f : G → X and all ε > 0 Eq. (1) implies the existence of an additive function α : G → X such that for some δ > 0. Then X is a Banach space.
For G = Z this has been proved earlier in [6] and for G as in the above theorem by a different method in [7,Theorem 12.7].
Condition (1) has been weakened by replacing ε by suitable functions ϕ depending on x and y with the result that the bound in (2) also depends on x, y and ϕ. Many of these results (see [2, pp. 9-11]) are special cases of a very general result (see [1]) for equations of the form g[F (x, y)] = H[g(x), g(y)] with the bounding function ϕ ≥ 0 such that Recently Ludwig Reich posed the question whether a result similar to Theorem 2 would hold true in this more general situation. In what follows this will be answered in the affirmative.

A stability result
The conditions from [1] are in the special case of the Cauchy equation contained in the conditions below when p = 2.

Theorem 3. Let X be a Banach space and S an Abelian semigroup. Assume
Let moreover p ∈ N be greater than 1, put and assume Then there is an additive function a : S → X such that This a is unique among all additive for all x ∈ S (and some c ≥ 0 depending possibly on b) and may be defined by a(x) = lim n→∞ In particular this holds true for q = p. Applying (6) to and Writing f j (x) := f (p j x)/p j , x ∈ S, j ∈ N 0 , the above equation reads as This may be used to show that the sequence (f j (x)) j∈N0 is a Cauchy sequence since and since the latter series is a tail of the series Φ p (x): Since X is complete the function a : Moreover a is additive: Since S is Abelian we may estimate which tends to 0 for n → ∞ by (11). Thus d(x) = 0 for all x, i. e., a = b. Remark 1. 1. As mentioned before the case p = 2 results in a well known result (see [1] and additionally the references in [2] and also [4]). For p > 2 the result seems to be new in the archimedean case. For stability investigations where the target space of the function involved is a Banach space over the non archimedean valued field Q p some results may be found in [8]. But of course the question arises whether there are functions ϕ such that ϕ p satisfies the conditions of the theorem for some p > 2, but ϕ 2 does not. An example of such ϕ is, for example, the following. x, y if p = 3. But (ii) and thus also (i) is not satisfied for p = 2 since ϕ 2 (2 n · 1, 2 n · 1)/2 n = 2 n for all n. 2. Obviously the Theorem may be applied in the particular case when ϕ = ε > 0 is constant.

Some characterizations of completeness by stability, the archimedean case
Consider an integer p > 1. Let S be a commutative semigroup and put then, as mentioned in (11), lim m→∞ Φ ϕ,m (x) = 0. Let K ⊆ F be a non-empty cone. I.e., K + K ⊆ K, [0, ∞) · K ⊆ K (and K = ∅). If X is a normed space over Q with completion X c , then the set is a subspace of X S . In the sequel we will indicate the Cauchy difference by γ f (x, y). Motivated by the proof of Theorem 3 and certain results of [7] let p n . This mapping is well defined and linear. Theorem 4. Using the notation from above, ker(α) is equal to Proof. If f belongs to the kernel of α, by an application of Theorem 3.
Conversely, if f belongs to C K , then thus α(f ) = 0.

Theorem 5.
If there exists some ϕ ∈ K such that ψ = ψ ϕ belongs to K, and if ϕ(x, x) > 0 for all x ∈ S, then α is surjective.
Proof. Consider some a ∈ Hom(S, X c ) and some ϕ ∈ K such that ψ ∈ K and ϕ(x, x) > 0 for all x ∈ S. Since X = X c we may find for each x ∈ S some value f (x) ∈ X such that f (x) − a(x) < ϕ(x, x). This defines a function f : S → X. Then γ f (x, y) = (f (x + y, x + y) − a(x + y)) − (f (x, x) − a(x)) − (f (y, y) − a(y)) and, therefore, γ f (x, y) ≤ ϕ(x + y, x + y) + ϕ(x, x) + ϕ(y, y) = ψ(x, y). From ψ ∈ K we derive that f ∈ C. By Theorem 3 there exists a unique b ∈ Hom(S, and a belongs to the image of α.
Remark 2. Using the notation from above we have provided that there exists some ϕ ∈ K such that ψ = ψ ϕ belongs to K and that ϕ(x, x) > 0 for all x ∈ S. Theorem 6. For each ϕ ∈ F the function ψ = ψ ϕ belongs to F.
Proof. Let ϕ ∈ F and N ∈ N. Then ψ satisfies (i) since and (ii) since ψ(p n x, p n y) p n = ϕ(p n (x + y), p n (x + y)) + ϕ(p n x, p n x) + ϕ(p n y, p n y) p n → n→∞ 0 which finishes the proof. Now we can characterize completeness of X in the following way: Theorem 7. Consider S = N 0 , an integer p > 1, X a normed space over Q, and ϕ ∈ F = F p satisfying ϕ(n, n) > 0 for all n ∈ N 0 . If for each function f : N 0 → X satisfying Proof. Consider some x 0 ∈ X c , an element of the closure of X, ϕ ∈ F satisfying ϕ(n, n) > 0 for all n ∈ N 0 , and a ∈ Hom(N 0 , X c ) given by a(n) = nx 0 . For each n ∈ N 0 there exists a value f (n) ∈ X such that f (n) − a(n) < ϕ(n, n).

Corollary 1.
Consider S = N 0 , an integer p > 1, X a normed space over Q, and ε > 0. If for each function f : N 0 → X satisfying Proof. This is the special case of Theorem 7 for ϕ = ε/3 since then ψ = ψ ϕ = ε,

Some characterizations of completeness by stability, the non-archimedean case
In a similar way the situation of non-archimedean normed spaces can be studied.
In [8] the following theorem can be found. It is a reformulation of Theorem 3.
Theorem 8. Let X be a Banach space over (Q, | | p ), p a prime, and S an Abelian semigroup which is uniquely divisible by p, i.e., the mapping S x → px =: π(x) ∈ S is bijective. Assume that ϕ : S × S → [0, ∞) and f : S → X satisfy together with and x/p n = π −n (x).
Then there is an additive function a : S → X such that This a is unique among all additive where ϕ p is given by (14). If for m ≥ 1 we denote by Φ ϕ,m (x) the tail and again lim m→∞ Φ ϕ,m (x) = 0.
Let K ⊆ F be a non-empty cone. If X is a normed space over (Q, | | p ), p a prime, with completion X c , and S is an Abelian semigroup which is uniquely divisible by p, then the set For np z = p −m , m ∈ N 0 we obtain |np z | p = p m , and which by Theorem 9 and (ii) from Theorem 8 tends to 0 for m → ∞. Hence, Corollary 2. Consider a prime p, S as in Theorem 12, X a normed space over (Q, | | p ), and ε > 0.
If for each function f : S → X satisfying Proof. This is the special case of Theorem 12 for ϕ = ε/3 since then ψ = ψ ϕ = ε, ψ p = (p − 1)ε, and Φ ψ = (p − 1)ε ∞ n=1 p −(n−1) = pε. Remark 4. Theorem 12 holds true also in the case when S = {np z | n ∈ N 0 , z ∈ Z} is replaced by S = Q since in this case as well the additive functions defined on S are determined uniquely by their values at 1.

Remark 5. The
Hyers sequence (p n f (x/p n )) n∈N0 plays an important role when constructing an additive function close to f . Now it may be asked whether the sequence (q n f (x/q n )) n∈N0 also converges when q = p.
First we analyze the consequences of the fact that (q n f (x/q n )) n∈N0 converges when p and q are relatively prime. Let p be a prime, S an Abelian semigroup which is uniquely divisible by p, and q be an integer coprime with p. Moreover assume that ϕ : S × S → [0, ∞) and f : S → X, X a Banach space over (Q, | | p ), satisfy γ f (x, y) ≤ ϕ(x, y), x, y ∈ S. From the proof of Theorem 8 we get in a similar way as (9) and (10) in the proof of Theorem 3 that and