Abstract
Let G be a commutative semigroup, or ℂ and . Generalizing the stability of the functional equation with bounded difference (Najdecki in J. Inequal. Appl. 2007:79816, 2007), we prove the stability of the above functional equation with unbounded differences. We also give a more precise description for bounded components of .
MSC:39B82.
Similar content being viewed by others
1 Main results
Throughout this paper, is a commutative semigroup with an identity e, ℝ the set of real numbers, ℂ the set of complex numbers, or ℂ, , and and are given functions. For , we define . A function is said to be an involution if and for all . A function is called an exponential function provided that for all .
Generalizing the result of Ger and Šemrl [1], Najdecki [2] proved the stability of the functional equation
in the class of functions . The particular cases of (1.1) are the exponential equation (see Aczél and Dhombres [3] and Baker [4]) and the equation
for all , where (see Brzdȩk [5], Brzdȩk, Najdecki and Xu [6] and Chudziak and Tabor [7] for related equations). As mentioned in [2, 5], (1.2) arises in averaging theory applied to the turbulent fluid motion and is connected with the Reynolds operator (see Marias [8]), the averaging operator and the multiplicatively symmetric operator (see [3]). Moreover, the equation (1.2) is connected with a description of some associative operations, i.e., the binary operation defined by is associative if and only if f satisfies (1.2) (see [5] for more details). We also refer the reader to Belluot, Brzdȩk and Ciepliński [9] and Brzdȩk and Ciepliński [10] for some recent developments on the issues of stability and superstability for functional equations.
The main result of Najdecki [2] is the following.
Theorem 1.1 Let , satisfy
for all with any norm in . Then there exist ideals such that , PF is bounded and QF satisfies (1.1), where , are the natural projections.
In this paper, generalizing the above result we consider the functional inequalities
for all with any norm in (see [6] for related results).
Throughout this paper we denote
where .
Theorem 1.2 Let , satisfy (1.4) for all with any norm in . Assume that one of the following two conditions is fulfilled.
-
(i)
g is an involution,
-
(ii)
for each , there exists a sequence , (possibly depending on j) such that
(1.6)
Then there exist ideals such that , PF is bounded and QF satisfies (1.1), where , are the natural projections. Moreover, is exponential provided g is bijective.
Remark The case (ii) of Theorem 1.2 includes Theorem 1.1.
Theorem 1.3 Let , satisfy (1.5) for all with any norm in . Assume that g is an involution. Then there exist ideals such that , PF is bounded, QF satisfies (1.1), where , are the natural projections.
If we replace by the usual norm on defined by
we can estimate PF (in Theorem 1.2 and Theorem 1.3) as follows.
Theorem 1.4 The following two statements are valid.
-
(a)
If , satisfies (1.4), then PF satisfies
(1.7)
for all , where denotes the number of the elements of L. In particular, if and G is a group, then PF satisfies either
for all , or
for all .
-
(b)
If , satisfies (1.5), then PF satisfies (1.7). In particular if G is a group, g is surjective and , then PF satisfies (1.8) or (1.9).
2 Proofs
Let and be given. We first consider the stability of the functional equation
in the class of functions , i.e., we investigate both bounded and unbounded functions satisfying the functional inequalities
for all .
Lemma 2.1 Assume that is an involution and is an unbounded function satisfying the inequality (2.2). Then f is exponential and satisfies (2.1). In particular, if G is 2-divisible, then f has the form
for all , where is an exponential function.
Proof Choose a sequence , , such that as . Putting , , in (2.2), dividing the result by and letting we have
for all . Multiplying both sides of (2.5) by and using (2.2) and (2.5) we have
for all . Thus, f is an exponential function, say . From (2.2) and (2.6) we have
for all . Since f is unbounded, from (2.7) we have
for all . Replacing y by in (2.6) and using (2.8) we get the equation (2.1). In particular, if G is 2-divisible, then we can write
for all . This completes the proof. □
Lemma 2.2 Let be an unbounded function satisfying (2.2). Assume that there exists a sequence , , satisfying
Then f satisfies (2.1).
Proof Note that (2.10) implies
Putting , , in (2.2) and dividing the result by we have
for all , . Letting in (2.11) we have
for all . Multiplying both sides of (2.12) by and using (2.2) and (2.12) we have
for all . This completes the proof. □
Lemma 2.3 Assume that g is bijective and is an unbounded function satisfying the inequality (2.2). Then is an exponential function.
Proof Choose a sequence , , such that as . Putting , , in (2.2), dividing the result by , replacing y by and letting we have
for all . Multiplying both sides of (2.14) by and using (2.2) and (2.14) we have
for all . Thus, is an exponential function. This completes the proof. □
Proof of Theorem 1.2 Since every two norms in are equivalent, from (1.4) there exists such that
for all and all . For the case (i), by Lemma 2.1, satisfies (2.1) for all . For the case (ii), by Lemma 2.2, satisfies (2.1) for all . Let , . Then it follows that , PF is bounded and QF satisfies (1.1). If g is bijective, then by Lemma 2.3, are exponential function for all , which implies is an exponential function. This completes the proof. □
Lemma 2.4 Assume that is an involution and is an unbounded function satisfying the inequality (2.3). Then f satisfies (2.1). In particular, if G is 2-divisible, then f has the form
for all , where is an exponential function.
Proof Choose a sequence , , such that as . Putting , , in (2.3), dividing the result by and letting we have
Putting in (2.3) and replacing y by in the result we have
for all . Multiplying both sides of (2.18) by and using (2.3), (2.18), and (2.19) we have
for all . Putting in (2.20) we have
for all . From (2.19) and (2.21) we have
for all . Since f is unbounded, from (2.22) we have . Thus, f satisfies (2.1). This completes the proof. □
Proof of Theorem 1.3 From (1.5), as in (2.16) there exists such that
for all , . Applying Lemma 2.4 to (2.23) for each we find that satisfies (2.1) for all , which implies that QF satisfies (1.1). This completes the proof. □
Now, we investigate bounded functions satisfying each of (2.2) and (2.3) (see [4, 11–13] for bounded solutions of an exponential functional equation).
Lemma 2.5 Let be a bounded function satisfying (2.2). Then f satisfies
for all . In particular, G is a group and let , then f satisfies either
for all , or
for all .
Proof Let . Using the triangle inequality with (2.2) we have
for all . Taking the supremum of the left hand side of (2.27) with respect to we get for all . Thus, we have
for all . From (2.28) we have
for all . Solving the inequality (2.29) we get (2.24). Now, we assume that G is a group. Replacing x by in (2.2) and using the triangle inequality we have
for all . Taking the supremum of the left hand side of (2.30) with respect to we get for all . Thus, we have
for all . From (2.28) and (2.31) we have
for all . For each fixed , solving the inequality (2.32) we get
or
Now, assume that there exist a bounded function f and such that
Then from (2.31) we have
On the other hand, from (2.35) we have
which contradicts (2.36). Thus, f satisfies (2.25) for all , or it satisfies (2.26) for all . This completes the proof. □
Lemma 2.6 Let be a bounded function satisfying (2.3). Then f satisfies (2.24) for all . In particular, if G is a group and g is surjective, then f satisfies (2.25) for all , or satisfies (2.26) for all .
Proof Using the triangle inequality with (2.3) we have
for all . Taking the supremum of the left hand side of (2.37) with respect to we get for all . Thus, we have
for all . From (2.38) we get (2.24) as in the proof of Lemma 2.5. We assume that G is a group. For given , choosing such that , putting in (2.3) and using the triangle inequality we have
for all . Taking the supremum of the left hand side of (2.39) we get for all . Thus, we have
for all . Now, the remaining parts of the proof are the same as those of Lemma 2.5. □
Proof of Theorem 1.4 From Lemma 2.5 and Lemma 2.6, for each we have
for all . Thus, from (2.41) we have
for all , which gives (1.7). Now, if , say we have
for all . Thus, the inequalities (1.8) and (1.9) follow immediately from (2.25) and (2.26). This completes the proof. □
Author’s contributions
The author is the only person who is responsible to this work.
References
Ger R, Šemrl P: The stability of exponential equation. Proc. Am. Math. Soc. 1996, 124: 779–787. 10.1090/S0002-9939-96-03031-6
Najdecki A: On stability of functional equation connected with the Reynolds operator. J. Inequal. Appl. 2007., 2007: Article ID 79816
Aczél J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, New York; 1989.
Baker JA: The stability of cosine functional equation. Proc. Am. Math. Soc. 1980, 80: 411–416. 10.1090/S0002-9939-1980-0580995-3
Brzdȩk J: On solutions of a generalization of the Reynolds functional equation. Demonstr. Math. 2008, 41: 859–868.
10.1007/s0010-014-0266-6
Chudziak J, Tabor J: On the stability of the Gołąb-Schinzel functional equation. J. Math. Anal. Appl. 2005, 302: 196–200. 10.1016/j.jmaa.2004.07.053
Marras Y:Sur l’équation fonctionnelle . Bull. Cl. Sci., Acad. R. Belg. 1969, 55: 779–787. 5e Série
Brillouët-Belluot N, Brzdȩk J, Ciepliński K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012., 2012: Article ID 716936
Brzdȩk J, Ciepliński K: Hyperstability and superstability. Abstr. Appl. Anal. 2013., 2013: Article ID 401756
Albert M, Baker JA: Bounded solutions of a functional inequality. Can. Math. Bull. 1982, 25: 491–495. 10.4153/CMB-1982-071-9
Chung J: On an exponential functional inequality and its distributional version. Can. Math. Bull. 2012. 10.4153/CMB-2014-012-x
Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.
Acknowledgements
The author is very thankful to the referees for valuable suggestions that improved the presentation of the paper. This work was supported by Basic Science Research Program through the National Foundation of Korea (NRF) funded by the Korea Government (no. 2012R1A1A008507).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chung, J. Stability of a functional equation connected with Reynolds operator. Adv Differ Equ 2014, 158 (2014). https://doi.org/10.1186/1687-1847-2014-158
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-158