General solution and generalized Ulam - Hyers stability of $r_i-$ type $n$ dimensional quadratic-cubic functional equation in random normed spaces: Direct and fixed point methods

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Authors :

Matina J. Rassias1, M. Arunkumar2, P. Agilan3*

Author Address :

1Department of Statistical Science, University College London, 1-19 Torrington Place, 140, London, WC1E 7HB, UK.
2Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.
3Department of Mathematics, Jeppiaar Institute of Technology, Sriperumbudur, Chennai - 631 604, Tamil Nadu, India.

*Corresponding author.

Abstract :

In this paper, the authors introduce and establish the general solution and generalized Ulam- Hyers stability of a $r_i$ type $n-$ dimensional Quadratic-Cubic functional equation
&sum_{i=0}^{n}left[f(r_{2i}x_{2i}+r_{2i+1}x_{2i+1}) ight] onumber
& = sum_{i=0}^{n}left(sum_{u=0}^{1}
left[ sum_{v=0}^{1}
left(frac{r_{2i}r_{2i+1}(-1)^{u+v}+r_{2i}r_{2i+1}^2(-1)^{u}+r_{2i}^2r_{2i+1}(-1)^{v}}{4} ight) ight.
%%%%%% ight. onumber&left.qquadqquadqquadqquadqquadqquad
left(fleft((-1)^u x_{2i}+(-1)^v x_{2i+1} ight) ight) ight] onumber
&+left(frac{r_{2i}^3+r_{2i}^2-r_{2i}r_{2i+1}^2}{4} ight)f(x_{2i})
+left(frac{r_{2i}^2-r_{2i}^3+r_{2i}r_{2i+1}^2}{4} ight)f(-x_{2i}) onumber
&left.+left(frac{r_{2i+1}^3+r_{2i+1}^2-r_{2i}^2r_{2i+1}}{4} ight)f(x_{2i+1})
+left(frac{r_{2i+1}^2-r_{2i+1}^3+r_{2i}^2r_{2i+1}}{4} ight)f(-x_{2i+1}) ight)
where $r_{2i},r_{2i+1}in R-left{0 ight}$, $left(i=0,1,2 cdots n ight)$ and $n$ is a positive integer in Random normed spaces .

Keywords :

Quadratic functional equation, Cubic functional equation, Mixed functional equation, Generalized Ulam - Hyers stability,fixed point, Random normed spaces.



Article Info :

Received : November 12, 2017; Accepted : December 30, 2017.



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