© Soon-Mo Jung. 2009
Received: 2 July 2009
Accepted: 5 November 2009
Published: 8 November 2009
In 1940, Ulam gave a wide-ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems . Among those was the question concerning the stability of homomorphisms.
In the following year, Hyers affirmatively answered in his paper  the question of Ulam for the case where and are Banach spaces.
for any (see [3, Definition ]).
This terminology is also applied to the case of other functional equations. It should be remarked that we can find in the books [4–7] a lot of references concerning the stability of functional equations (see also [8–18]).
In this paper, we will solve the functional equation
2. General Solution to (1.7)
In this section, let be either a real vector space if or a complex vector space if . In the following theorem, we investigate the general solution of the functional equation (1.7).
for all and . If we substitute for in (2.3) and divide the resulting equations by , respectively, , and if we substitute for in the resulting equations, then we obtain the equations in (2.3) with in place of , where . Therefore, the equations in (2.3) are true for all and .
which completes the proof.
It should be remarked that the functional equation (1.7) is a particular case of the linear equation with and . Moreover, a substantial part of proof of Theorem 2.1 can be derived from theorems presented in the books [19, 20]. However, the theorems in [19, 20] deal with solutions of the linear equation under some regularity conditions, for example, the continuity, convexity, differentiability, analyticity and so on, while Theorem 2.1 deals with the general solution of (1.7) without regularity conditions.
3. Hyers-Ulam Stability of (1.7)
We can prove the Hyers-Ulam stability of the functional equation (1.7) as we see in the following theorem.
On the other hand, it also follows from (3.1) that
By (3.9) and (3.15), we have
By (3.24) and (3.25), we have
for any , that is, for all . Therefore, we conclude that for any . (The presented proof of uniqueness of is somewhat long and involved. Indeed, the referee has remarked that the uniqueness can be obtained directly from [21, Proposition ].)
The functional equation (1.7) is a particular case of the linear equations of higher orders and the Hyers-Ulam stability of the linear equations has been proved in [21, Theorem ]. Indeed, Brzdęk et al. have proved an interesting theorem, from which the following corollary follows (see also [22, 23]):
Hence, the estimation (3.2) of Theorem 3.1 is better in these cases than the estimation (3.28).
As we know, is the Fibonacci sequence. So if we set and in Theorems 2.1 and 3.1, then we obtain the same results as in [24, Theorems , , and ].
The author would like to express his cordial thanks to the referee for useful remarks which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Moszner Z: On the stability of functional equations. Aequationes Mathematicae 2009,77(1–2):33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
- Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, Fla, USA; 2003.MATHGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser Boston, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Baker J, Lawrence J, Zorzitto F: The stability of the equation . Proceedings of the American Mathematical Society 1979,74(2):242–246.MathSciNetMATHGoogle Scholar
- Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar
- Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
- Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- Ger R, Šemrl P: The stability of the exponential equation. Proceedings of the American Mathematical Society 1996,124(3):779–787. 10.1090/S0002-9939-96-03031-6MathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153. 10.1007/BF01830975MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias stability of functional equations. Dynamic Systems and Applications 1997,6(4):541–565.MathSciNetMATHGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar
- Székelyhidi L: On a theorem of Baker, Lawrence and Zorzitto. Proceedings of the American Mathematical Society 1982,84(1):95–96.MathSciNetView ArticleMATHGoogle Scholar
- Kuczma M: Functional Eequations in a Single Variable, Monografie Matematyczne. Volume 46. PWN—Polish Scientific Publishers, Warsaw, Poland; 1968:383 pp..Google Scholar
- Kuczma M, Choczewski B, Ger R: Iterative Functional Equations, Encyclopedia of Mathematics and Its Applications. Volume 32. Cambridge University Press, Cambridge, UK; 1990:xx+552.View ArticleMATHGoogle Scholar
- Brzdęk J, Popa D, Xu B: Hyers-Ulam stability for linear equations of higher orders. Acta Mathematica Hungarica 2008,120(1–2):1–8. 10.1007/s10474-007-7069-3MathSciNetView ArticleMATHGoogle Scholar
- Brzdęk J, Popa D, Xu B: Note on nonstability of the linear recurrence. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2006, 76: 183–189. 10.1007/BF02960864View ArticleMathSciNetMATHGoogle Scholar
- Trif T: Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficients. Nonlinear Functional Analysis and Applications 2006,11(5):881–889.MathSciNetMATHGoogle Scholar
- Jung S-M: Hyers-Ulam stability of Fibonacci functional equation. Bulletin of the Iranian Mathematical Society, In press Bulletin of the Iranian Mathematical Society, In press
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.