Superstability of Generalized Multiplicative Functionals
© Takeshi Miura et al. 2009
Received: 2 March 2009
Accepted: 20 May 2009
Published: 16 June 2009
It seems that the stability problem of functional equations had been first raised by S. M. Ulam (cf. [1, Chapter VI]). "For what metric groups is it true that an -automorphism of is necessarily near to a strict automorphism? (An -automorphism of means a transformation of into itself such that for all .)" D. H. Hyers  gave an affirmative answer to the problem: if and is a mapping between two real Banach spaces and satisfying for all , then there exists a unique additive mapping such that for all . If, in addition, the mapping is continuous for each fixed , then is linear. This result is called Hyers-Ulam stability of the additive Cauchy equation . J. A. Baker [3, Theorem ] considered stability of the multiplicative Cauchy equation : if and is a complex valued function on a semigroup such that for all , then is multiplicative, or for all . This result is called superstability of the functional equation . Recently, A. Najdecki [4, Theorem ] proved the superstability of the functional equation : if , is a (real or complex valued) functional from a commutative semigroup and is a mapping from into itself such that for all , then holds for all , or is bounded.
2. Main Result
Thus, . Now it is easy to see that . Consequently, if is bounded, then for all . The constant is the best possible since for satisfies for each . It should be mentioned that the above proof is essentially due to P. Šemrl [5, Proof of Theorem and Proposition ] (cf. [6, Proposition ]).
If satisfies (2.2) for some , then by quite similar arguments to the proof of Theorem 2.1, we can prove that for all , or for all . Thus, Theorem 2.1 is still true under the weaker condition (2.16) instead of (2.2). This was pointed out by the referee of this paper. The condition (2.16) is related to that introduced by Kannappan .
Let be a ring homomorphism from into itself, that is, and for each . It is well known that there exist infinitely many such homomorphisms on (cf. [8, 9]). If is not identically , then we see that for every , the field of all rational real numbers. Thus, if we consider the case when , a nonzero ring homomorphism, and , then satisfies the conditions (a), (b), and (c).
The authors would like to thank the referees for valuable suggestions and comments to improve the manuscript. The first and fourth authors were partly supported by the Grant-in-Aid for Scientific Research.
- 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
- Baker JA: The stability of the cosine equation. Proceedings of the American Mathematical Society 1980,80(3):411–416. 10.1090/S0002-9939-1980-0580995-3MathSciNetView ArticleMATHGoogle Scholar
- Najdecki A: On stability of a functional equation connected with the Reynolds operator. Journal of Inequalities and Applications 2007, 2007:-3.Google Scholar
- Šemrl P: Non linear perturbations of homomorphisms on The Quarterly Journal of Mathematics. Series 2 1999, 50: 87–109. 10.1093/qjmath/50.197.87View ArticleMATHGoogle Scholar
- Jarosz K: Perturbations of Banach Algebras, Lecture Notes in Mathematics. Volume 1120. Springer, Berlin, Germany; 1985:ii+118.Google Scholar
- Kannappan Pl: On quadratic functional equation. International Journal of Mathematical and Statistical Sciences 2000,9(1):35–60.MathSciNetMATHGoogle Scholar
- Charnow A: The automorphisms of an algebraically closed field. Canadian Mathematical Bulletin 1970, 13: 95–97. 10.4153/CMB-1970-019-3MathSciNetView ArticleMATHGoogle Scholar
- Kestelman H: Automorphisms of the field of complex numbers. Proceedings of the London Mathematical Society. Second Series 1951, 53: 1–12. 10.1112/plms/s2-53.1.1MathSciNetView ArticleMATHGoogle Scholar
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