Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. This is applied to investigate homomorphisms between quasi-Banach algebras. The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias' stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300].     

Fixed points and stability of the Cauchy–Jensen functional inequality in fuzzy Banach algebras

Using the fixed point method, we prove the Hyers–Ulam stability of the Cauchy–Jensen functional inequality in fuzzy Banach algebras.     

Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.     

Fixed points and additive $${\rho}$$-functional equations

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

     

Isomorphisms Between Quasi-Banach Algebras

In this paper, the author proves the Hyers-Ulam-Rassias stability of homo-morphisms in quasi-Banach algebras. This is used to investigate isomorphisms between quasi-Banach algebras.     

Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. This is applied to investigate isomorphisms between quasi-Banach algebras.     

Generalized quadratic mappings of r-type in several variables

Let X,Y be vector spaces. It is shown that if an even mapping satisfies f(0)=0, and

(∗)     

C *-Ternary 3-Homomorphisms on C *-Ternary Algebras

In this paper, we investigate the Ulam-Hyers stability of C ^*-ternary algebra 3-homomorphisms for the functional equation $$f(x_1 + x_2 + x_3, y_1 + y_2 + y_3, z_1 + z_2 + z_3) = \sum_{1\leq i,j,k\leq 3} f(x_i, y_j, z_k)$$ in C ^*-ternary algebras.     

3-Variable Additive {\rho} -Functional Inequalities and Equations

In this paper, we introduce and investigate additive \({\rho}\) -functional inequalities associated with the following additive functional equations $$\begin{array}{lll} \,\,\,\,\,\,\, f(x+y+z) - f(x)-f(y)-f(z) \,\,\,\, = 0 \\ 2f \left(\frac{x+y}{2}+z \right) - f(x)-f(y)-2f(z) = 0 \\ \,\,2f \left(\frac{x+y+z}{2} \right) - f(x)-f(y)-f(z) = 0\end{array}$$ Furthermore, we prove the Hyers–Ulam stability of the additive \({\rho}\) -functional inequalities in complex Banach spaces and prove the Hyers–Ulam stability of additive \({\rho}\) -functional equations associated with the additive \({\rho}\) -functional inequalities in complex Banach spaces.