On functional equations related to additive mappings and isometries

Certain functional equations, related to the problem of characterization of metrics generated by norms, are considered. The solutions of these equations are strongly connected with additive and isometric mappings.


Introduction
Motivation for this research comes, at least partially, from a question when a metric d in a real vector space X is generated by a norm.Šemrl [8] showed that it is so whenever it satisfies the following three properties: d is translation invariant: x, y, z ∈ X; ( H 1 ) algebraic midpoints are metric ones:

Functional equations arising from (H1) and (H2)
The property (H1) leads to the functional equation which, assuming additionally f (0) = 0, is equivalent to Equation (1), also written in the form x,y∈ X, was considered, e.g., in [6,10]. Obviously, if f is additive or (assuming that the domain is a normed space) f is an isometry with f (0) = 0, or if f is a composition of an additive mapping with an isometry vanishing at zero, then f satisfies (1). The class of solutions of (1) contains the class of solutions of the Fischer-Muszély equation (cf. [1,4]) As a matter of fact, the class of solutions of (FM) consists exactly of all odd solutions of (1) (cf. [10,Proposition 2]). If Y is strictly convex, then the Fischer-Muszély equation (FM) is equivalent to the Cauchy functional equation On functional equations related to additive mappings 99 Example 2.1. Let X = R, Y = R 2 with the maximum norm and let f (x) = (x, sin x), x ∈ R. Then f is an odd isometry-hence a solution of (1), but not additive.
Also the oddness appears to be essential, at least for some domains (cf. [6,Remark 6]).
, y 0 , otherwise satisfies (1) but it is neither additive nor odd (no mater whether Y is strictly convex or not). The domain Z can be replaced by any group containing a subgroup of index 2.
Strict convexity and the oddnes of f can be replaced by its surjectivity. Sikorska [9, Theorem 1] has proved, in fact, the following result. Theorem 2.3. Let (X, +) be a group, Y be a real normed space, and let δ and ε be nonnegative constants. Assume that f : (*) If f satisfies the inequality Actually, the above result was proved under the assumption of completeness of Y . However, this assumption is redundant. In fact, takeỸ -the completion of Y andf : X →Ỹ defined byf (x) = f (x) for x ∈ X. The density of Y inỸ and δ-surjectivity of f yields the δ + η-surjectivity off with any η > 0. Thus applying the above theorem (for the complete spaceỸ ) we get Since η is arbitrary, ( * * ) holds. For δ = ε = 0 we get the following characterization (cf. [9, Corollary 1]).

Theorem 2.4.
Let (X, +) be a group and Y be a real normed space. Let f : X → Y be surjective and satisfy ( * ). Then f is a solution of (1) if and only if it is additive.
As a corollary we obtain a Mazur-Ulam type theorem for isometries defined on additive subgroups of normed spaces. Theorem 2.5. Let X, Y be real normed spaces and let (G, +) be a subgroup of the additive group (X, +). Let I : G → Y be a surjective isometry and I(0) = 0. Then I is additive.
Proof. Each isometry I : G → Y vanishing at zero satisfies (1) and ( * ) follows from the commutativity of G. Hence the assertion follows from Theorem 2.4.
Taking G = X we get the classical Mazur-Ulam theorem. Since (FM) implies (1), we derive one more corollary. Theorem 2.6. Let (X, +) be a group, Y be a real normed space and let f : X → Y be a solution of (FM). If f is surjective and satisfies ( * ), then it is additive.
Similar results can be found in [7,Corollary 5] (under more restrictive assumptions) and in [11,Corollary 1] (under the assumption of completeness of Y , however with no commutativity of any kind required). Now, consider the functional equation arising from (H2): Without loss of generality we may assume (2) with the additional condition f (0) = 0. Again, it is easy to see that an additive mapping or an isometry, or a composition of such mappings satisfies (2).
Theorem 2.7. Let X be a uniquely 2-divisible group and let Y be a strictly convex normed space. Then each solution f : X → Y of (2) is an affine mapping (each solution of (2) * is additive).
Proof. Assuming (2), for all x, y ∈ X one has Without the assumption of strict convexity of Y this result is no longer true. For, let X = R and Y = R 2 with the maximum norm and consider Example 2.1.
One can also prove the following characterization of strict convexity (compare similar results related to equation (1)  Proof. One part of the proof follows from Theorem 2.7. For the reverse assume that X is not strictly convex, hence there exist a, b ∈ X, a = b and such that a = b = a+b 2 = 1. Following [2] we define a mapping f : R → X by the formula The assumption upon a, b implies that f is an isometry from R to X. Thus, in particular, f satisfies (2) * but it is not additive.
The form of (2) requires 2-divisibility of the domain. However, one can consider this equation in its equivalent form: with X being a (not necessarily 2-divisible) group. The mapping f : Z → Y defined in Example 2.2 satisfies (1) but not (2) . We end this section with two questions. Do the implications (1) ⇒ (2) and (2) * ⇒ (1) hold true, for X being a 2-divisible group? What is the general solution of (2)?

Functional equation related to (H3)
Assuming that d is embedded in a norm space, condition (H3) takes the form Without loss of generality we assume f (0) = 0 (otherwise we take f −f (0)) and we arrive at the functional equation being a subject of future considerations: Notice that Eq. (3) is equivalent to the system of Eqs. (1) and (2) * , as well as to (1) and f (2x) = 2 f (x) , x ∈ X.
Ger [7] gave the general solution of the Fischer-Muszély equation (FM). Namely, if X is an Abelian group and Y is a real normed space, then f : X → Y satisfies (FM) if and only if, f = I •A where A : X → Z is an additive operator from X into some normed space Z and I : A(X) → Y is an odd isometry.
Following this characterization we give the description of solutions of (3). By R X we mean the set of all real valued mappings defined on X and by B(T, R) the linear space of all bounded mappings from a set T to R, equipped with the supremum norm. Proof. The method of proof is taken from [7]. Let p( From (3) we easily derive i.e., which together with p(2x) = 2p(x), x ∈ X gives (it can be proved by induction) Thus p : X → [0, ∞) is a sublinear and even functional on X. Applying a description of such functionals given in [7, Theorem 1], we get that there exist a nonempty subset T of R X and an additive operator A : X → B(T, R) such that LetX := X/ ker A and defineÂ :X → B(T, R) andf :X → Y bŷ Both mappings are well defined. Moreover,Â is additive and injective. Consider a subgroup G :=Â(X) = A(X) of the group (B(T, R), +). Let I : G → Y be defined by A is a bijection fromX onto G hence for each u ∈ G there exists x u ∈ X such thatÂ([ Vol. 89 (2015) On functional equations related to additive mappings 103 Observe that I is an isometry: Moreover, I(0) =f (Â −1 (0)) =f ([0]) = f (0) = 0 and finally, for an arbitrary x ∈ X we have It is clear that the reverse statement also holds true. For an arbitrary normed space Z, an arbitrary additive mapping A : X → Z and an arbitrary isometry I : A(X) → Y such that I(0) = 0, the composition f := I • A satisfies (3). Indeed, we have for x, y ∈ X: as well as whence (3) follows. Thus we have proved the following characterization. Under some additional assumptions, solutions of (3) must be additive. Proof. It follows from Theorem 2.4 that any surjective solution of (1) (hence also of (3)) must be additive. Since (3) implies (2) * , if Y is strictly convex, the assertion follows from Theorem 2.7.

Concluding remarks
Summing up our considerations we have, in the general case: However, assuming that f is an odd solution of (3), the isometry I from Theorem 3.2 has to be odd as well. It means that f , as a composition of an additive mapping with an odd isometry, is a solution of (FM). Therefore, the class of solutions of (FM) consists exactly of all odd solutions of (3). Thus, with respect to odd solutions, Eqs. Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.