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].     

Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces

In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation for fixed integers k with k≠0,±1 in the quasi-Banach spaces.     

19.

On the stability of a mixed functional equation deriving from additive,quadratic and cubic mappings     

Li Guang Wang  Kun Peng Xu  Qiu Wen Liu 《数学学报(英文版)》2014,30(6):1033-1049

In this paper, we investigate the general solution and the Hyers–Ulam stability of the following mixed functional equation f(2x + y) + f(2x- y) = 2f(2x) + 2f(x + y) + 2f(x- y)- 4f(x)- f(y)- f(-y)deriving from additive, quadratic and cubic mappings on Banach spaces.     

20.

On a pythagorean-type functional equation with exponent 3 involving finite fields     

J. L. García-Roig  Emma Martín-Gutiérrez 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2000,70(1):85-92

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

On the stability of a mixed functional equation deriving from additive,quadratic and cubic mappings

In this paper, we investigate the general solution and the Hyers–Ulam stability of the following mixed functional equation f(2x + y) + f(2x- y) = 2f(2x) + 2f(x + y) + 2f(x- y)- 4f(x)- f(y)- f(-y)deriving from additive, quadratic and cubic mappings on Banach spaces.     

On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

On the stability of the orthogonal Pexiderized Cauchy equation

We investigate the stability of Pexiderized mappings in Banach modules over a unital Banach algebra. As a consequence, we establish the Hyers-Ulam stability of the orthogonal Cauchy functional equation of Pexider type f₁(x+y)=f₂(x)+f₃(y), x⊥y in which ⊥ is the orthogonality in the sense of Rätz.     

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.     

Homomorphisms of heterogeneous algebras

A construction of all homomorphisms of a heterogeneous algebra into an algebra of the same type is presented. A relational structure is assigned to any heterogeneous algebra, and homomorphisms between these relational structures make it possible to construct homomorphisms between heterogeneous algebras. Homomorphisms of relational structures can be constructed using homomorphisms of algebras that are described in [11].     

Generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation

Using the fixed point method, we prove the generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation.     

Stability of a generalized quadratic functional equation in Schwartz distributions

We consider the Hyers-Ulam stability problem of the generalized quadratic functional equation
uoA＋voB-2woP1 - 2ko P2 =0,
which is a distributional version of the classical generalized quadratic functional equation
f（x＋y）＋g（x - y） - 2h（x） - 2k（y）=0     

Stability of a mixed additive and cubic functional equation in quasi-Banach spaces

In this paper we establish the general solution and investigate the Hyers-Ulam-Rassias stability of the following functional equation

f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+2[f(2x)−2f(x)]     

On the stability of a quartic functional equation

In this paper, we prove the generalized Hyers-Ulam stability for the following quartic functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y).     

On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces

In this paper we investigate the Hyers-Ulam-Rassias stability of the following functional equation:

     

Solution and stability of a cubic functional equation

In this paper, we investigate the general solution and the stability of a cubic functional equation f（x ＋ ny） ＋ f（x - ny） ＋ f（nx） = n^2 f（x ＋ y） ＋ n^2 f（x - y）＋ （n^3 - 2n^2 ＋ 2）f（x）,where n ≥ 2 is an integer. Furthermore, we prove the stability by the fixed point method.     

Asymptotic aspect of the quadratic functional equation in multi-normed spaces

We investigate the Hyers-Ulam stability of the quadratic functional equation for mappings from abelian groups into multi-normed spaces. We also study the stability on a restricted domain and present an asymptotic behavior of the quadratic equation in the framework of multi-normed spaces.     

Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces

In this paper we establish the general solution of the functional equation

f(2x+y)+f(2x−y)=f(x+y)+f(x−y)+2f(2x)−2f(x)     

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

On the stability of Jensen’s functional equation on groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group G such that the Jensen functional equation is stable on G.     

Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation

f(x+ky)+f(x−ky)=k²f(x+y)+k²f(x−y)+2(1−k²)f(x)