Superstability of Pexiderized functional equations arising from distance measures
Gwang Hui Kim
- Department of Applied Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, Korea.
Young Whan Lee
- Department of Computer Hacking and Information Security, College of Natural Science, Daejeon University, Daejeon, 300-716, Republic of Korea.
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: \(f(pr; qs) + g(ps; qr) = \theta(pq; rs)h(p; q)g(r; s))\).
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Gwang Hui Kim, Young Whan Lee, Superstability of Pexiderized functional equations arising from distance measures, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 2, 413--423
Kim Gwang Hui, Lee Young Whan, Superstability of Pexiderized functional equations arising from distance measures. J. Nonlinear Sci. Appl. (2016); 9(2):413--423
Kim, Gwang Hui, Lee, Young Whan. "Superstability of Pexiderized functional equations arising from distance measures." Journal of Nonlinear Sciences and Applications, 9, no. 2 (2016): 413--423
- Distance measure
- multiplicative function
- stability of functional equation.
J. K. Chung, Pl. Kannappan, C. T. Ng, P. K. Sahoo, Measures of distance between probability distributions, J. Math. Anal. Appl., 138 (1989), 280-292.
M. Hosszú, On the functional equation \(f(x + y; z) + f(x; y) = f(x; y + z) + f(y; z)\), Period. Math. Hungarica, 1 (1971), 213-216.
D. H. Hyers, G. Isac, Th. M. Rassias , Stability of Functional Equations in Several Variables, Birkhäuser, Boston (1998)
P. Kannappan, P. K. Sahoo, Sum from distance measures between probability distributions and functional equations, Int. J. of Math. & Stat. Sci., 6 (1997), 91-105.
Pl. Kannappan, P. K. Sahoo, J. K. Chung, On a functional equation associated with the symmetric divergence measures, Utilitas Math., 44 (1993), 75-83.
G. H. Kim, A stability of the generalized sine functional equations, J. Math. Anal. Appl., 331 (2007), 886-894.
G. H. Kim, On the stability of mixed trigonometric functional equations, Banach J. Math. Anal., 1 (2007), 227-236.
G. H. Kim, The stability of the d'Alembert and Jensen type functional equations, J. Math. Anal. Appl., 325 (2007), 237-248.
G. H. Kim, On the stability of the pexiderized trigonometric functional equation, Appl. Math. Compu., 203 (2008), 99-105.
G. H. Kim, Y. H. Lee, Boundedness of approximate trigonometric functionas, Appl. Math. Lett., 22 (2009), 439-443.
G. H. Kim, Y. W. Lee, The superstability of the Pexider type trigonometric functional equation, Aust. J. Math. Anal., 7 (2010), 10 pages.
G. H. Kim, P. K. Sahoo, Stability of a functional equation related to distance measure - II, Ann. Funct. Anal., 1 (2010), 26-35.
G. H. Kim, P. K. Sahoo , Stability of a functional equation related to distance measure - I, Appl. Math. Lett., 24 (2011), 843-849.
Y. W. Lee, G. H. Kim , Superstability of the functional equation with a cocycle related to distance measures, J. Ineq. Appl., 2014 (2014), 11 pages.
T. Riedel, P. K. Sahoo, On a generalization of a functional equation associated with the distance between the probability distributions, Publ. Math. Debrecen, 46 (1995), 125-135.
J. Tabor, Hyers theorem and the cocycle property, Functional equations results and Advances, Kluwer Academic Publ., (2002), 275-290.