- Research Article
- Open Access
Stability of a Bi-Jensen Functional Equation II
© Kil-Woung Jun et al. 2009
Received: 7 October 2008
Accepted: 24 January 2009
Published: 5 February 2009
Găvruta  provided a further generalization of Rassias' theorem in which he replaced the bound by a general function.
2. Stability of a Bi-Jensen Functional Equation
Jun et al.  established the basic properties of a bi-Jensen mapping in the following lemma.
Now we have the stability of a bi-Jensen mapping.
for all . Then, , and satisfy (2.2), (2.3), (2.4) for all . In addition, satisfy (2.5) for all and also satisfy (2.5) for all . But we get . Hence, the condition is necessary to show that the mapping F is unique.
We have another stability result applying for several cases.
Applying Theorems 2.2–2.7, we easily get the following corollaries.
Applying Theorems 2.6 and 2.7, we obtain the desired result.
- Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1968.MATHGoogle 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(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle 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
- Găvruta 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
- Bae J-H, Park W-G: On the solution of a bi-Jensen functional equation and its stability. Bulletin of the Korean Mathematical Society 2006,43(3):499–507.MathSciNetView ArticleMATHGoogle Scholar
- Jun K-W, Jung I-S, Lee Y-H: Stability of a bi-Jensen functional equation. preprint preprintGoogle Scholar
- Jun K-W, Lee Y-H, Oh J-H: On the Rassias stability of a bi-Jensen functional equation. Journal of Mathematical Inequalities 2008,2(3):363–375.MathSciNetView ArticleMATHGoogle Scholar
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