On the stability of a sum form functional equation related to entropies of type (\(\alpha,\beta\))
Volume 14, Issue 3, pp 168--180
http://dx.doi.org/10.22436/jnsa.014.03.06
Publication Date: November 19, 2020
Submission Date: July 15, 2020
Revision Date: October 18, 2020
Accteptance Date: October 22, 2020
-
1241
Downloads
-
2755
Views
Authors
Dhiraj Kumar Singh
- Department of Mathematics, Zakir Husain Delhi College (University of Delhi), Jawaharlal Nehru Marg, Delhi 110002, India.
Shveta Grover
- Department of Mathematics, University of Delhi, Delhi 110007, India.
Abstract
In this paper, we discuss the stability of the sum form functional equation
\[
\sum\limits _{i=1}^{n}\sum\limits _{j=1}^{m}f(p_{i} q_{j} )
=\sum\limits _{i=1}^{n}g(p_{i}) \sum\limits _{j=1}^{m}f(q_{j} )+\sum\limits _{i=1}^{n}f(p_{i}) \sum\limits _{j=1}^{m}q_{j}^{\beta }
\]
for all complete probability distributions \((p_1,\ldots,p_n)\in \Gamma_n\), \((q_1,\ldots,q_m)\in \Gamma_m\),
\(n\ge 3\), \(m\ge 3\) are fixed integers, \(f\), \(g\) are real valued mappings each having the domain \(I=[0,1]\) and \(\beta\) is a fixed positive real power such that \(\beta \neq 1\), \(0^\beta:=0\), \(1^\beta:=1\).
Share and Cite
ISRP Style
Dhiraj Kumar Singh, Shveta Grover, On the stability of a sum form functional equation related to entropies of type (\(\alpha,\beta\)), Journal of Nonlinear Sciences and Applications, 14 (2021), no. 3, 168--180
AMA Style
Singh Dhiraj Kumar, Grover Shveta, On the stability of a sum form functional equation related to entropies of type (\(\alpha,\beta\)). J. Nonlinear Sci. Appl. (2021); 14(3):168--180
Chicago/Turabian Style
Singh, Dhiraj Kumar, Grover, Shveta. "On the stability of a sum form functional equation related to entropies of type (\(\alpha,\beta\))." Journal of Nonlinear Sciences and Applications, 14, no. 3 (2021): 168--180
Keywords
- Stability
- additive mapping
- logarithmic mapping
- multiplicative mapping
- bounded mapping
- entropies of type \((\alpha,\beta)\)
MSC
References
-
[1]
J. Aczél, Lectures on functional equations and their applications, Academic Press, New York-London (1966)
-
[2]
M. Behara, P. Nath, Information and entropy of countable measurable partitions. I, Kybernetika (Prague), 10 (1974), 491--503
-
[3]
Z. Daróczy, L. Losonczi, Über die Erweiterung der auf einer Punktmenge additiven Funktionen, Publ. Math. Debrecen, 14 (1967), 239--245
-
[4]
J. Havrda, F. Charvát, Quantification method of classification processes. Concept of structural $\alpha$-entropy, Kybernetika (Prague), 3 (1967), 30--35
-
[5]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
-
[6]
P. Kannappan, An application of a differential equation in information theory, Glasnik Mat. Ser. III, 14 (1979), 269--274
-
[7]
P. Kannappan, On a generalization of sum form functional equation. III, Demonstratio Math., 13 (1980), 749--754
-
[8]
I. Kocsis, On the stability of a sum form functional equation of multiplicative type, Acta Acad. Paedagog. Agriensis Sect. Math. (N.S.), 28 (2001), 43--53
-
[9]
I. Kocsis, G. Maksa, The stability of a sum form functional equation arising in information theory, Acta Math. Hungar., 79 (1998), 39--48
-
[10]
L. Losonczi, G. Maksa, The general solution of a functional equation of information theory, Glasnik Mat. Ser. III, 16 (1981), 261--268
-
[11]
L. Losonzi, G. Maksa, On some functional equations of the information theory, Acta Math. Acad. Sci. Hungar., 39 (1982), 73--82
-
[12]
G. Maksa, On the stability of a sum form equation, Results Math., 26 (1994), 342--347
-
[13]
P. Nath, On some functional equations and their applications, Publ. Inst. Math. (Beograd) (N.S.), 20 (1976), 191--201
-
[14]
P. Nath, D. K. Singh, On a sum form functional equation related to entropies of type $(\alpha,\beta)$, Asian-Eur. J. Math., 6 (2013), 13 pages
-
[15]
P. Nath, D. K. Singh, On a sum form functional equation related to various nonadditive entropies in information theory, Tamsui Oxf. J. Inf. Math. Sci., 30 (2014), 23--43
-
[16]
P. Nath, D. K. Singh, On an equation related to nonadditive entropies in information theory, Math. Appl. (Brno), 6 (2017), 31--41
-
[17]
P. Nath, D. K. Singh, On the stability of a functional equation, Palestine J. Math., 6 (2017), 573--579
-
[18]
D. K. Singh, P. Dass, On a functional equation related to some entropies in information theory, J. Discrete Math. Sci. Cryptogr., 21 (2018), 713--726
-
[19]
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York-London (1960)
-
[20]
G. S. Young, The linear functional equation, Amer. Math. Monthly, 65 (1958), 37--38