Fischer–Muszély functional equation almost everywhere

AbstractWe show that, under suitable assumptions, a function f from a group (G, +) into a real or complex inner product space $${(H, (\cdot \vert \cdot))}$$(H,(·|·)), satisfying the Fischer–Muszély functional equation
$$\parallel f(x + y)\parallel\, \, =\, \parallel f(x) + f(y)\parallel$$‖f(x+y)‖=‖f(x)+f(y)‖for all pairs (x, y) off a sufficiently small (negligible) subset of G2 has to be almost everywhere equal to an additive map from G into H, i.e. the set $${\{x \in G: f(x) \neq a(x)\}}$${x∈G:f(x)≠a(x)} is small (negligible) in G. Small sets in G and G2 are defined in an axiomatic way. Several corollaries illustrating some consequences of this result are presented.

In the last half-century the functional equation (FM) f(x + y) = f(x) + f(y) was attracting much attention from numerous mathematicians. It has extensively been studied, among others, by Fischer and Muszély [6] (see also an earlier version [5] in Hungarian, and [4]), Dhombres [3], Aczél and Dhombres [1], Berruti and Skof [2], Skof [15], Ger [9], [10], Ger and Koclȩga [12] and Schöpf [14], who presented various partial results for mappings with values in some special Banach spaces like inner product spaces, strictly convex spaces or in general normed linear spaces but with scalar domains. In [11] the general solution of the equation mentioned was given. J. Tabor Jr. has shown in [16] that the Fischer-Muszély equation is Hyers-Ulam stable in the class of surjective mappings.
The main goal of the present paper is to exhibit another stability property: we shall show that under suitable assumptions a function satisfying the which like in [7], forces the equality Since M is supposed to be a member of Ω(J ), there exists a set U ∈ J such that for every x ∈ G \ U the section M [x] falls into J .
Let N stand for the set-theoretical union of the following seven sets: and Each one of these seven sets yields a member of the ideal Ω(J ). Indeed, this is obvious for the first three sets as well as, by the invariance assumptions, for the set M 4 . To check that M 1 ∈ Ω(J ) note that for every x / ∈ 1 2 U ∈ J the section Similarly, since for every x / ∈ U ∈ J the section we infer that M 2 ∈ Ω(J ). Finally, for every x / ∈ U ∈ J the section which shows that M 3 ∈ Ω(J ). Consequently, the union N of all the sets spoken of yields a member of the ideal Ω(J ) as well. Now, fix arbitrarily a pair (x, y) ∈ G 2 \ N. Then: because (x, y) / ∈ M ; 2. f (2x) = 2f (x) and f (2y) = 2f (y) because of (1) and the fact that because (x, y) / ∈ M 1 which forces the pair (2x, y) to stay off the set M ; because (x, y) / ∈ M 2 which forces the pair (x, 2y) to stay off the set M ; 210 R. Ger AEM because (x, y) / ∈ M 4 which forces the pair (x + y, y) to stay off the set M ; Relations 3. and 4. jointly with 2. imply that whereas a similar conclusion can be drawn from relations 5. and 6. jointly with 2. By means of 1., after squaring both sides of (2) and (3), by a simple calculation, we derive the equalities respectively, which immediately imply that Along the same lines as in the paper [6] of P. Fischer and Gy.Muszély, from the trivial equality with the aid of 1. and (4) we derive the relationship This clearly forces the additivity relation that remains valid for all pairs (x, y) ∈ G 2 \ N , i.e. Ω(J )-almost everywhere in G 2 . Now, it remains to apply a de Bruijn's type result from [8]: there exists a unique additive function a : G → H such that the equality Thus the proof has been completed. Remark 1. The leading idea of the proof above was to run along the lines of the proof presented in [6] treating it as the obstacle race. However, the set of obstacles, although basically caused by the fact that the validity of the (FM) equation is postulated merely almost everywhere, was enlarged by another one; namely, close to the bottom of page 199 in [6] the authors write: If we interchange the variables x and y in the equation (16) we get which is wrong; actually, we get then and not (17) because of the lack of the commutativity of the domain semigroup. In what follows we shall present a few corollaries illustrating some consequences of the theorem just proved. Proof. Let J stand for the p.l.i. ideal of all bounded subsets of the space X. Clearly, any bounded set and, in particular, any ball M : = B((0, 0), r) in the product space X 2 yields a member of Ω(J ). Assume that Put T 1 (x, y) := (y, x) and T 2 (x, y) := (y, x − y), (x, y) ∈ X 2 . The images T 1 (M ) and T 2 (M ) are contained in M and √ 5M , respectively, so that they stay in Ω(J ). Moreover, 1 2 U is bounded for any bounded set U . Finally, since the set E := {x ∈ X : x ≤ r} belongs to J and for every is satisfied. Thus all the assumptions of the Theorem are fulfilled which ends the proof. , (x, y) ∈ G 2 yield homeomorphic selfmappings of G 2 we infer that both the images T 1 (M ) and T 2 (M ) stay in Ω(J ). Moreover since, by assumption, the map G x −→ 1 2 x ∈ G is a homeomorphism of G onto itself, the set 1 2 U is of the first Baire category provided that so is U . To finish the proof it remains to apply the Theorem.

Corollary 4. Let (Z, +) be the additive group of all integers and let (H, (·|·))
be an inner product space. If a sequence (a n ) n∈Z of elements of the space H satisfies the Fischer-Muszély equation a n+m = a n + a m (5) for all but finite set of pairs (n, m) ∈ Z 2 , then there exists a unique vector c ∈ H such that a n = nc for all but finite number of integers n.  {(n, kn) ∈ Z 2 : n / ∈ E} that is contained in Z 2 \ M. Thus all the assumptions of the Theorem are fulfilled which implies the existence of a unique additive map a : Z → H such that the set {n ∈ Z : a(n) = a n } is finite. Since, obviously, a(n) = na(1), n ∈ Z, we get the equality a n = nc for all but a finite number of integeres n, with a unique c := a(1) ∈ H, as claimed.
Remark 2. As it is, the formulation of the Theorem leaves room for improvements. For instance, it would be desirable to have: • the group considered replaced by a semigroup; • the inner product space replaced by a strictly convex one; • the assumption removed. Unfortunately, at present none of these three wishes can be accomplished because of the proof technique applied.