On the Stability Problem in Fuzzy Banach Space

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space

The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer ≥2,∑=1‖∑−(1/)=1‖2=∑=1‖‖2∑−‖(1/)=1‖2 holds for all 1,…,∈. The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation ∑=1(∑−(1/)=1∑)==1(∑)−((1/)=1) which is said to be a functional equation associated with inner product spaces

Stability of a functional equation deriving from cubic and quartic functions

In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation &4(f(3x+y)+f(3x-y))=-12(f(x+y)+f(x-y)) &+12(f(2x+y)+f(2x-y))-8f(y)-192f(x)+f(2y)+30f(2x)

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

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