Stability of additivity and fixed point methods
© Brzdęk; licensee Springer. 2013
Received: 30 July 2013
Accepted: 4 October 2013
Published: 8 November 2013
We show that the fixed point methods allow to investigate Ulam’s type stability of additivity quite efficiently and precisely. Using them we generalize, extend and complement some earlier classical results concerning the stability of the additive Cauchy equation.
(As usual, denotes the family of all functions mapping a set into a set .) Roughly speaking, -stability of equation (1) means that every approximate (in the sense of (2)) solution of (1) is always close (in the sense of (3)) to an exact solution to (1).
Let us mention that this type of stability has been a very popular subject of investigations for the last nearly fifty years (see, e.g., [1–10]). The main motivation for it was given by S.M. Ulam (cf. [4, 11]) in 1940 in his talk at the University of Wisconsin, where he presented, in particular, the following problem.
then a homomorphism exists with for ?
Hyers  published a partial answer to it, which can be stated as follows.
Quite often we describe that result of Hyers simply saying that Cauchy functional equation (1) is Hyers-Ulam stable (or has the Hyers-Ulam stability).
In the next few years, Hyers and Ulam published some further stability results for polynomial functions, isometries and convex functions in [12–15]. Let us mention yet that now we are aware of an earlier (than that of Hyers) result concerning such stability that is due to Pólya and Szegö [, Teil I, Aufgabe 99] (see also [, Part I, Ch. 3, Problem 99]) and reads as follows (ℕ stands for the set of positive integers).
For every real sequence with , there is a real number ω such that . Moreover, .
The next theorem is considered to be one of the most classical results.
It was motivated by Th.M. Rassias (see [18–20]) and is composed of the outcomes in [18, 21, 22]. Note that Theorem 1 with yields the result of Hyers and it is known (see ; cf. also [23, 24]) that for an analogous result is not valid. Moreover, it was shown in  that estimation (5) is optimum for in the general case.
Theorem 1 has a very nice simple form. However, recently, it was shown in  that it can be significantly improved; namely, in the case , each satisfying (4) must actually be additive and the assumption of completeness of is not necessary in such a situation. So, taking into account that result in , we can reformulate Theorem 1 in the following way.
Theorem 2 Let and be two normed spaces and and be fixed real numbers. Let be a mapping satisfying (4). If and is complete, then there exists a unique additive function such that (5) holds. If , then f is additive.
The second statement of Theorem 2, for , can be described as the φ-hyperstability of the additive Cauchy equation for . Unfortunately, such result does not remain valid if we restrict the domain of f, as the following remark shows it.
1 The main result
In this paper we prove the following complement to Theorem 1, which covers also the situation described in Remark 1 (see Remark 2).
It is easily seen that Theorem 3 yields the subsequent corollary.
which is sharper than (5) for .
for any real , any bounded function with , and any such that for , (for instance, we can take for , with some additive ).
In some cases, estimation (9) provided in Corollary 1 is optimum as the subsequent example shows. Unfortunately, this is not always the case, because the possibly sharpest such estimation we have in Theorem 2 for .
Actually, the calculations are very elementary, but for the convenience of readers, we provide them.
2 Auxiliary result
The proof of Theorem 3 is based on a fixed point result that can be easily derived from [, Theorem 2] (cf. [, Theorem 1] and ). Let us mention that [, Theorem 2] was already used, in a similar way as here, for the first time in  for proving some stability results for the functional equation of p-Wright affine functions, next in [26, 31] in proving hyperstability of the Cauchy equation, and (very recently) also for investigations of stability and hyperstability of some other equations in [32–34] (the Jensen equation, the general linear equation, and the Drygas functional equation, respectively).
The fixed point approach to Ulam’s type stability was proposed for the first time in  (cf.  for a generalization; see also ) and later applied in numerous papers; for a survey on this subject, we refer to .
X is a nonempty set and is a complete metric space.
are given maps.
- (H3)is an operator satisfying the inequality
- (H4)is an operator defined by
Now we are in a position to present the above mentioned fixed point result following from [, Theorem 2].
Moreover, for .
3 Proof of Theorem 3
for every , , . Consequently, for each , also (H3) is valid with and .
for some (the case is trivial, so we exclude it here). Observe yet that T and are solutions to equation (16) for all .
This implies (7) with ; clearly, equality (22) means the uniqueness of T as well.
Remark 3 Note that from the above proof we can derive a much stronger statement on the uniqueness of T than the one formulated in Theorem 3. Namely, it is easy to see that is the unique additive mapping such that (19) holds with some .
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