Generalized Stabilities of Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations in Quasi-β-Normed Spaces

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