The stability of functional equation min{f(x + y), f(x  y)} = f(x)  f(y)
 Barbara Przebieracz^{1}Email author
https://doi.org/10.1186/1029242X201122
© Przebieracz; licensee Springer. 2011
Received: 23 February 2011
Accepted: 21 July 2011
Published: 21 July 2011
Abstract
In this paper, we prove the stability of the functional equation min {f(x + y), f(x  y)} = f(x)  f(y) in the class of real, continuous functions of real variable.
MSC2010: 39B82; 39B22
Keywords
1. Introduction
where G is an abelian group and f : G → ℝ. The first two of them are satisfied by f(x) = a(x), where a: G → ℝ is an additive function; moreover, the first one characterizes the absolute value of additive functions. The solutions of Equation (1.2) are appointed by Volkmann during the Conference on Inequalities and Applications in Noszwaj (Hungary, 2007), under the assumption that f : ℝ → ℝ is a continuous function. Namely, we have
Theorem 1.1 (Jarczyk and Volkmann [2]). Let f : ℝ → ℝ be a continuous function satisfying Equation (1.2). Then either there exists a constant c ≥ 0 such that f(x) = cx, x ∈ ℝ, or f is periodic with period 2p given by f(x) = cx with some constant c > 0, x ∈ [p, p].
Actually, it is enough to assume continuity at a point, since this implies continuity on ℝ, see [2]. Moreover, some measurability assumptions force continuity. Baron in [3] showed that if G is a metrizable topological group and f : G → ℝ is Baire measurable and satisfies (1.2) then f is continuous. Kochanek and Lewicki (see [4]) proved that if G is metrizable locally compact group and f : G → ℝ is Haar measurable and satisfies (1.2), then f is continuous.
As already mentioned in [2], Kochanek noticed that every function f defined on an abelian group G which is of the form f = g ∘ a, where g : ℝ → ℝ is a solution of (1.2) described by Theorem 1.1 and a: G → ℝ is an additive function, is a solution of Equation (1.2).
Solutions of the Equation (1.3), according to [1], with the additional assumption that G is divisible by 6, are either f ≡ 0 or f = exp(a), where a: G → ℝ is an additive function. Without this additional assumption, however, we have the following remark (see [5]).
if and only if

f ≡ 0
or

f = exp ∘a, for some additive function a
or
The result concerning stability of (1.1) was presented by Volkmann during the 45th ISFE in BielskoBiala (Poland, 2007) (for the proof see [2]) and superstability of (1.3) was proved in [6].
In this paper, we deal with the stability of Equation (1.2) in the class of continuous functions from ℝ to ℝ.
2. Main Result
We are going to prove
that is, f is "close" to the solution F(x) = cx of (1.2).
We will write instead of α  β  ≤ δ to shorten the notation.
Notice that
In the following lemma, we list some properties of functions satisfying (2.1) in more general settings.
The last inequality together with (2.5) finishes the proof of (v). □
which is impossible. Therefore, (2.7) implies . Now, it is enough to put p = z'' + x''.
whence either or . Let us consider the first possibility, the second is analogous. From min{f (y  2x), 2M) = min{f (y  2x), we deduce that f(y  2x) ≤ 2δ. But 2δ ≥ f(y  2x) ≥ min {f(y  2x), which contradicts (2.9) and, thereby, ends the proof of (2.8).
Thereby,
Since f restricted to (0, ∞) is 19δ approximately additive, there is an additive function a: (0, ∞) → ℝ such that (see [7]). Moreover, since f is continuous, a(x) = cx for some positive c. Assertions (ii) and (iii) of Lemma 2.1 finish the proof of (2.2). □
Remark. Kochanek noticed (oral communication) that we can decrease easily 21δ appearing in (2.2) to 19δ, by repeating the consideration from the proof, which we did for positive real halfline, for the negative real halfline. We would obtain , for x > 0, , and , for x < 0, where c, c' are some positive constants. But, since , x ∈ ℝ, we can deduce that c' = c.
Declarations
Acknowledgements
This paper was supported by University of Silesia (Stability of some functional equations). The author wishes to thank prof. Peter Volkmann for valuable discussions. The author gratefully acknowledges the many helpful suggestions of an anonymous referee.
Authors’ Affiliations
References
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