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].     

Pexider type operators and their norms in <Emphasis Type="Italic">X</Emphasis><Subscript><Emphasis Type="Italic">λ</Emphasis></Subscript> spaces

In this paper, we introduce Pexiderized generalized operators on certain special spaces introduced by Bielecki-Czerwik and investigate their norms.     

On the completeness of normed spaces

Let $(G,+)$ be an abelian group and let $E$ be a normed space. A mapping $f:G\to E$ is called $\epsilon$-quadratic if for a given $\epsilon >0$ it satisfies $6f(x+y)+f(x?y)?2f\left(x\right)?2f\left(y\right)6\le \epsilon$ for all $x,y\in G$. In this work, we show that $E$ is complete if every $\epsilon$-quadratic mapping $f:X\to E$ can be estimated by a quadratic mapping, where $X$ is ${\mathbb{N}}_0$ or a finitely generated free abelian group.     

Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation

In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi- Banach algebras associated with the following Pexiderized Jensen functional equation
f（x＋y/2＋z）-g（x-y/2＋z）=h（y）.
This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias＇ stability theorem that appeared in his paper： On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 （1978）.     

On Approximate Additive–Quartic and Quadratic–Cubic Functional Equations in Two Variables on Abelian Groups

We establish the generalized Hyers–Ulam stability of the additive–quartic and quadratic–cubic mappings with two variables on abelian groups.     

On the stability of a cubic functional equation

In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation
2f（x ＋ 2y） ＋ f（2x - y） = 5f（x ＋ y） ＋ 5f（x - y）＋ 15f（y）
in the spirit of Hyers, Ulam, Rassias and Gavruta.     

Fixed Points and Quartic Functional Equations in β-Banach Modules

Let M?=?{ 1, 2, . . . ,?n?} and let ${\mathcal {V}=\{\,I \subseteq M: 1 \in I\,\}}$ , where n is an integer greater than 1. Denote ${M{\setminus}{I}}$ by I ^c for ${I \in \mathcal {V}.}$ We investigate the solution of the following generalized quartic functional equation $$\begin{array}{ll} \sum\limits_{I \in\mathcal {V}}f\, \left({\sum\limits_{i \in I}}a_ix_i-\sum\limits_{i \in I^c}a_ix_i\right) \, = \,2^{n-2} \sum\limits_{1\leq i < j \leq n}a^2_{i}a^2_{j} \left[f(x_{i}+x_{j})+f(x_{i}-x_{j})\right] \\ \qquad \qquad \qquad \quad\quad\quad \quad\quad\quad\quad +\,2^{n-1} \sum\limits^{n}_{i=1}a^2_{i} \left(a^2_{i}-\sum\limits^{n}_{\substack{{j=1}\\{j\neq i}}}a^2_{j}\right)f(x_{i}) \end{array}$$ in β-Banach modules on a Banach algebra, where ${a_{1},\ldots, a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ with a _? ?≠ ±1 for all ${\ell \in \{1 , 2, \ldots ,\,n-1\}}$ and a _n ?=?1. Moreover, using the fixed point method, we prove the generalized Hyers–Ulam stability of the above generalized quartic functional equation. Finally, we give an example that the generalized Hyers–Ulam stability does not work.