DOI: https://doi.org/10.11568/kjm.2016.24.2.297

### A fixed point approach to the stability of the functional equation related to distance measures

#### Abstract

In this paper, by using fixed point theorem, we obtain the stability of the following functional equations

\begin{align*}

f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)f(p,q)h(r,s) \\

f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)g(p,q)h(r,s),

\end{align*}

where $G$ is a commutative semigroup, $\theta : G^{4} \rightarrow \mathbb{R}_{k}$ a function and $f,g,h$ are functionals on $G^{2}$.

#### Keywords

#### Subject classification

39B82, 39B52.#### Sponsor(s)

#### Full Text:

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J. Brzdek, A. Najdecki and B. Xu, Two general theorems on superstability of functional equations, Aequationes Math., Doi: 10.1007/s00010-014-0266-6.[1]. (Google Scholar)

J. K. Chung, P. Kannappan, C. T. Ng and P. K. Sahoo, Measures of distance between probability distributions, J. Math. Anal. Appl. 138 (1989), 280–292. (Google Scholar)

J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society 74 (1968), 305–309. (Google Scholar)

M. Hosszu ́, On the functional equation f(x+y,z)+f(x,y) = f(x,y+z)+f(y,z), Periodica Math. Hungarica 1 (3) (1971), 213–216. (Google Scholar)

Pl. Kannappan and P. K. Sahoo, Sum form distance measures between probability distributions and functional equations, Int. J. of Math. & Stat. Sci. 6 (1997), 91–105. (Google Scholar)

Pl. Kannappan, P. K. Sahoo and J. K. Chung, On a functional equation associated with the symmetric divergence measures, Utilitas Math. 44 (1993), 75–83. (Google Scholar)

G. H. Kim, The Stability of the d’Alembert and Jensen type functional equations, Jour. Math. Anal & Appl. 325 (2007), 237–248. (Google Scholar)

Pl. Kannappan and G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog. Crac. Stud. Math. 1 (2001), 49–58. (Google Scholar)

G. H. Kim, On the Stability of the pexiderized trigonometric functional equation, Appl. Math. Compu. 203 (2008), 99–105. (Google Scholar)

G. H. Kim and Y.H. Lee, The superstability of the Pexider type trigonometric functional equation, Math. Ineq. & Appl., submitted. (Google Scholar)

G. H. Kim and Y.H. Lee, Boundedness of approximate trigonometric functional equations, Appl. Math. Lett. 331 (2009), 439–443. (Google Scholar)

G. H. Kim, On the Stability of trigonometric functional equations, Ad. Diff. Eq. Vol 2007, Article ID 90405, (2007). (Google Scholar)

G. H. Kim and Sever S. Dragomir, On the Stability of generalized d’Alembert and Jensen functional equation, Intern. Jour. Math. & Math. Sci., Article ID 43185, DOI 10.1155 (2006), 1–12. (Google Scholar)

G. H. Kim and Y. W. Lee Superstability of Pexiderized functional equations arising from distance measures, J. Nonlinear Sci. Appl. 9 (2016), 413–423. (Google Scholar)

G. H. Kim and P. K. Sahoo, Stability of a Pexider type functional equation related to distance measures, Jour. Math. Ineq. 9 (4) (2015), 11691179. (Google Scholar)

G. H. Kim and P. K. Sahoo, Stability of a functional equation related to distance measure - I, Appl. Math. Lett. 24 (2011), 843–849. (Google Scholar)

G. H. Kim and P. K. Sahoo, Stability of a functional equation related to distance measure - II, Ann. Funct. Anal. 1 (2010) 26–35. (Google Scholar)

Y. W. Lee and G. H. Kim Superstability of the functional equation with a cocycle related to distance measures, Math. Ineq. & Appl., (2014), 2014:393 doi:10.1186/1029-242X-2014-393 (Google Scholar)

Y. W. Lee and G. H. Kim Superstability of the functional equation related to distance measures, Jour. Ineq. & Appl. 20152015:352, DOI: 10.1186/s13660-015-0880-4 (Google Scholar)

T. Riedel and 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. (Google Scholar)

T. Riedel and P. K. Sahoo, On two functional equations connected with the characterizations of the distance measures, Aequationes Math. 54 (1998), 242–263. (Google Scholar)

P. K. Sahoo, On a functional equation associated with stochastic distance measures, Bull. Korean Math. Soc. 36 (1999), 287–303. (Google Scholar)

T. Riedel and 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. (Google Scholar)

J. Tabor, Hyers theorem and the cocycle property, Fumctional equations-Results and Advaces, Kluwer Academic Publ.(Z. Dar ́oczy and Z. P ́ales), (2002), 275-290. (Google Scholar)

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