Ulam-Hyers stability of undecic functional equation in quasi-$\beta$-normed spaces: Fixed point method

Abstract

In this paper, we acquire the general solution of the undecic functional equation $$\begin{align*} & f(x+6y)-11f(x+5y)+55f(x+4y)-165f(x+3y)+330f(x+2y)\\ & \qquad\qquad -462f(x+y)+462f(x)-330f(x-y)+165f(x-2y)-55f(x-3y)\\ & \quad\qquad\qquad\qquad\qquad\qquad +11f(x-4y)-f(x-5y)=39916800f(y). \end{align*}$$ We also obtain the generalized Ulam-Hyers stability of the above functional equation in quasi-$\beta$-normed spaces using fixed point method. Moreover, we investigate the pertinent stabilities of the above functional equation using sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms as upper bounds. We also present a counter-example for non-stability of the above functional equation in singular case.