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- Open Access
Superstability of the functional equation related to distance measures
- Young Whan Lee^{1}Email authorView ORCID ID profile and
- Gwang Hui Kim^{2}
https://doi.org/10.1186/s13660-015-0880-4
© Lee and Kim 2015
- Received: 26 April 2015
- Accepted: 30 October 2015
- Published: 6 November 2015
Abstract
Keywords
- information measure
- distance measure
- superstability
- multiplicative function
- stability of functional equation
MSC
- 39B82
- 39B52
1 Introduction
Baker et al. in [1] introduced that if f satisfies the stability inequality \(|E_{1}(f)-E_{2}(f)|\leq \varepsilon\), then either f is bounded or \(E_{1}(f)=E_{2}(f)\). This is now frequently referred to as superstability. Baker [2] also proved the superstability of the cosine functional equation (also called the d’Alembert functional equation).
Suppose \(f : I^{2} \to\mathbb{R}\) satisfies (DM) for all \(p,q, r, s \in I\). Thenwhere \(M_{1} , M_{2} : \mathbb{R} \to\mathbb{C}\) are multiplicative functions. Further, either \(M_{1}\) and \(M_{2}\) are both real or \(M_{2}\) is the complex conjugate of \(M_{1}\). The converse is also true.$$f(p, q) = M_{1} (p) M_{2} (q) + M_{1} (q) M_{2} (p), $$
The above equation (DM) characterized by distance measures can be considered by characterization of a symmetrically compositive sum-form information measurable functional equation.
For other functional equations with the information measure, the interested reader should refer to [6–9] and [10–12].
2 Results
In this section, we investigate the superstability of the pexiderized equation related to (IM).
Theorem 1
Proof
Let g be an unbounded solution of inequality (2.1). Then there exists a sequence \(\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}\) in \(G^{n}\) such that \(0 \neq|g(Z_{m})|\rightarrow\infty\) as \(m\rightarrow\infty\).
Theorem 2
Proof
Assume that there exists a sequence \(\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}\) in \(G^{n}\) such that \(\lim_{m to \infty} |h(Z_{m})|=\infty\) with \(|h(Z_{m})|\neq0\) for each m.
By using (2.5), (2.6) and (2.7), let us go through the same procedure as in Theorem 1, then we arrive at the required result. □
Corollary 1
Corollary 2
Proof
Thus if f is bounded, then g is bounded. Hence, by Theorem 1, in the case g is unbounded, g also is a solution of (IM).
From a similar calculation as that in Theorem 1 and Theorem 2, we obtain the required result. □
Corollary 3
Proof
From a similar calculation as that in Corollary 2 we obtain the required result. □
Corollary 4
Corollary 5
3 Discussion
For example, let \(X=(x _{1} ,x _{2} , \ldots,x _{n} )\) and \(Y=(y _{1}, y _{2} , \ldots,y _{n} )\). And define \(f(X)=f(x _{1}, x _{2}, \ldots,x _{n} ):= \sum_{i=1} ^{n} \frac{1}{ x _{i}}\). Then f is a solution of the above equation. Thus our results are not limited. We expect to know the general solution of it.
4 Conclusions
Also the pexiderized functional equation of the above equation satisfies the property of superstability.
Declarations
Acknowledgements
This research was supported by the Daejeon University Fund (2014).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.
Authors’ Affiliations
References
- Baker, J, Lawrence, J, Zorzitto, F: The stability of the equation \(f(x+y) = f(x)f(y)\). Proc. Am. Math. Soc. 74, 242-246 (1979) MATHMathSciNetGoogle Scholar
- Baker, JA: The stability of the cosine equation. Proc. Am. Math. Soc. 80, 411-416 (1980) MATHView ArticleGoogle Scholar
- Chung, JK, Kannappan, PL, Ng, CT, Sahoo, PK: Measures of distance between probability distributions. J. Math. Anal. Appl. 138, 280-292 (1989) MATHMathSciNetView ArticleGoogle Scholar
- Kim, GH, Sahoo, PK: Stability of a functional equation related to distance measure - I. Appl. Math. Lett. 24, 843-849 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Kim, GH, Sahoo, PK: Stability of a functional equation related to distance measure - II. Ann. Funct. Anal. 1, 26-35 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Daróczy, Z: Generalized information functions. Inf. Control 16, 36-51 (1970) MATHView ArticleGoogle Scholar
- Ebanks, BR, Sahoo, P, Sander, W: Characterization of Information Measures, vol. X. World Scientific, Singapore (1997) MATHGoogle Scholar
- Gselmann, E: Recent results on the stability of the parametric fundamental equation of information. Acta Math. Acad. Paedagog. Nyházi. 25, 65-84 (2009) MATHMathSciNetGoogle Scholar
- Gselmann, E, Maksa, G: Some functional equations related to the characterizations of information measures and their stability. In: Rassias, TM (ed.) Handbook of Functional Equations: Stability Theory, pp. 199-241. Springer, New York (2015) Google Scholar
- Kullback, S: Information Theory and Statistics. Wiley Publication in Mathematical Statistics, vol. XVII. Wiley, New York (1959) MATHGoogle Scholar
- Riedel, T, Sahoo, PK: On two functional equations connected with the characterizations of the distance measures. Aequ. Math. 54, 242-263 (1998) MathSciNetView ArticleGoogle Scholar
- Shannon, CE: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379-423, 623-656 (1948) MATHMathSciNetView ArticleGoogle Scholar