Title: Generalized Stabilities of Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations in Quasi-β-Normed Spaces
Author(s): John Michael Rassias, Krishnan Ravi, Beri Venkatachalapathy Senthil Kumar
Pages: 167-174
Cite as:
John Michael Rassias, Krishnan Ravi, Beri Venkatachalapathy Senthil Kumar, Generalized Stabilities of Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations in Quasi-β-Normed Spaces, Int. J. Anal. Appl., 14 (2) (2017), 167-174.

Abstract


The aim of this paper is to investigate generalized Ulam-Hyers stabilities of the following Euler-Lagrange-Jensen-$(a,b)$-sextic functional equation

$$

f(ax+by)+f(bx+ay)+(a-b)^6\left[f\left(\frac{ax-by}{a-b}\right)+f\left(\frac{bx-ay}{b-a}\right)\right]\\

= 64(ab)^2\left(a^2+b^2\right)\left[f\left(\frac{x+y}{2}\right)+f\left(\frac{x-y}{2}\right)\right]\\

+2\left(a^2-b^2\right)\left(a^4-b^4\right)[f(x)+f(y)]

$$

where $a\neq b$, such that $k\in \mathbb{R}$; $k=a+b\neq 0,\pm1$ and $\lambda=1+(a-b)^6-2\left(a^6+b^6\right)-62(ab)^2\left(a^2+b^2\right)\neq 0$, in quasi-$\beta$-normed spaces by using fixed point method. In particular, we prove generalized stabilities involving the sum of powers of norms, product of powers of norms and the mixed product-sum of powers of norms of the above functional equation in quasi-$\beta$-normed spaces by using fixed point method. A counter-example for a singular case is also indicated.

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