Geodesic
Superstability of Pexiderized functional equations arising from distance measures
Kim, Gwang Hui1 ; Lee, Young Whan2
1 Department of Applied Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, Korea
2 Department of Computer Hacking and Information Security, College of Natural Science, Daejeon University, Daejeon, 300-716, Republic of Korea
Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 2, p. 413-423.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we obtain the superstability of the functional equation $f(pr; qs) + g(ps; qr) = \theta(pq; rs)h(p; q)k(r; s)$ for all $p; q; r; s \in G$, where $G$ is an Abelian group, $f; g; h; k$ are functionals on $G^2$, and $\theta$ is a cocycle on $G^2$. This functional equation is a generalized form of the functional equation $f(pr; qs)+f(ps; qr) = f(p; q) f(r; s)$, which arises in the characterization of symmetrically compositive sum-form distance measures and the information measures, and also they can be represented as products of some multiplicative functions and the exponential functional equations. As corollaries, we obtain the superstability of the many functional equations (combination of three variables functions, for example: $f(pr; qs) + g(ps; qr) = \theta(pq; rs)h(p; q)g(r; s))$.
DOI : 10.22436/jnsa.009.02.07
Classification : 39B82, 46S40
Keywords: Distance measure, superstability, multiplicative function, stability of functional equation.

Kim, Gwang Hui 1 ; Lee, Young Whan 2

1 Department of Applied Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, Korea
2 Department of Computer Hacking and Information Security, College of Natural Science, Daejeon University, Daejeon, 300-716, Republic of Korea 
@article{JNSA_2016_9_2_a6,
     author = {Kim, Gwang Hui and Lee, Young Whan},
     title = {Superstability of {Pexiderized} functional equations arising from distance measures},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {413-423},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {2016},
     doi = {10.22436/jnsa.009.02.07},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.02.07/}
}
TY  - JOUR
AU  - Kim, Gwang Hui
AU  - Lee, Young Whan
TI  - Superstability of Pexiderized functional equations arising from distance measures
JO  - Journal of nonlinear sciences and its applications
PY  - 2016
SP  - 413
EP  - 423
VL  - 9
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.02.07/
DO  - 10.22436/jnsa.009.02.07
LA  - en
ID  - JNSA_2016_9_2_a6
ER  - 
%0 Journal Article
%A Kim, Gwang Hui
%A Lee, Young Whan
%T Superstability of Pexiderized functional equations arising from distance measures
%J Journal of nonlinear sciences and its applications
%D 2016
%P 413-423
%V 9
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.02.07/
%R 10.22436/jnsa.009.02.07
%G en
%F JNSA_2016_9_2_a6
Kim, Gwang Hui; Lee, Young Whan. Superstability of Pexiderized functional equations arising from distance measures. Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 2, p. 413-423. doi : 10.22436/jnsa.009.02.07. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.02.07/

[1] Chung, J. K.; Kannappan, Pl.; Ng, C. T.; Sahoo, P. K. Measures of distance between probability distributions, J. Math. Anal. Appl., Volume 138 (1989), pp. 280-292

[2] Hosszú, M. On the functional equation \(f(x + y; z) + f(x; y) = f(x; y + z) + f(y; z)\), Period. Math. Hungarica, Volume 1 (1971), pp. 213-216

[3] Hyers, D. H.; Isac, G.; Th. M. Rassias Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998

[4] Kannappan, P.; Sahoo, P. K. Sum from distance measures between probability distributions and functional equations, Int. J. of Math. & Stat. Sci., Volume 6 (1997), pp. 91-105

[5] Kannappan, Pl.; Sahoo, P. K.; Chung, J. K. On a functional equation associated with the symmetric divergence measures, Utilitas Math., Volume 44 (1993), pp. 75-83

[6] Kim, G. H. A stability of the generalized sine functional equations, J. Math. Anal. Appl., Volume 331 (2007), pp. 886-894

[7] Kim, G. H. On the stability of mixed trigonometric functional equations, Banach J. Math. Anal., Volume 1 (2007), pp. 227-236

[8] Kim, G. H. The stability of the d'Alembert and Jensen type functional equations, J. Math. Anal. Appl., Volume 325 (2007), pp. 237-248

[9] Kim, G. H. On the stability of the pexiderized trigonometric functional equation, Appl. Math. Compu., Volume 203 (2008), pp. 99-105

[10] Kim, G. H.; Lee, Y. H. Boundedness of approximate trigonometric functionas, Appl. Math. Lett., Volume 22 (2009), pp. 439-443

[11] Kim, G. H.; Lee, Y. W. The superstability of the Pexider type trigonometric functional equation, Aust. J. Math. Anal., Volume 7 (2010), pp. 1-10

[12] Kim, G. H.; Sahoo, P. K. Stability of a functional equation related to distance measure - II, Ann. Funct. Anal., Volume 1 (2010), pp. 26-35

[13] Kim, G. H.; P. K. Sahoo Stability of a functional equation related to distance measure - I, Appl. Math. Lett., Volume 24 (2011), pp. 843-849

[14] Lee, Y. W.; G. H. Kim Superstability of the functional equation with a cocycle related to distance measures, J. Ineq. Appl., Volume 2014 (2014), pp. 1-11

[15] Riedel, T.; Sahoo, P. K. On a generalization of a functional equation associated with the distance between the probability distributions, Publ. Math. Debrecen, Volume 46 (1995), pp. 125-135

[16] Tabor, J. Hyers theorem and the cocycle property, Functional equations results and Advances, Kluwer Academic Publ. (2002), pp. 275-290

Cité par Sources :