Hyers-Ulam-Rassias stability of Jensen's equation and its application

The Hyers-Ulam-Rassias stability for the Jensen functional equation is investigated, and the result is applied to the study of an asymptotic behavior of the additive mappings; more precisely, the following asymptotic property shall be proved: Let and be a real normed space and a real Banach space, respectively. A mapping satisfying is additive if and only if as .

     

Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

Making use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation

     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

Stability of a functional equation deriving from cubic and quadratic functions

In this paper we establish the general solution of the functional equation 6f(x+y)−6f(x−y)+4f(3y)=3f(x+2y)−3f(x−2y)+9f(2y) and investigate the Hyers-Ulam-Rassias stability of this 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 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)     

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 the generalized quadratic and quartic type functional equation in non-Archimedean fuzzy normed spaces

In this paper, we prove some stability results concerning the generalized quadratic and quartic type functional equation in the context of non-Archimedean fuzzy normed spaces in the spirit of Hyers-Ulam-Rassias. As applications, we establish some results of approximately generalized quadratic and quartic type mapping in non-Archimedean normed spaces. Also, we show that the assumption of the non-Archimedean absolute value of $2$ is less than $1$ cannot be omitted in our corollaries. The results improve and extend some recent results.     

Stability of a Jensen type equation in the space of generalized functions

We reformulate and solve the stability problem of a Jensen type functional equation

     

The generalized Hyers-Ulam-Rassias stability of a cubic functional equation

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability for a cubic functional equation f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+12f(x).     

Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation

Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C＊-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces：f（2∑j=1^n-1 xj＋xn）＋f（2∑j=1^n-1 xj-xn）＋4∑j=1^n-1f（xj）=16f（∑j=1^n-1 xj）＋2∑j=1^n-1（f（xj＋xn）＋f（xj-xn）     

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.     

Stability of Jensen equations in the space of generalized functions

Making use of heat kernel, we prove stabilities of the Jensen and Jensen-Pexider equations in a space of generalized functions like the spaces of tempered distributions and Fourier hyperfunctions.     

Stability problem of Ulam for generalized forms of Cauchy functional equation

Let G₁ be a vector space and G₂ a Banach space. In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for a generalized form

     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.     

Stability of a cubic functional equation in fuzzy normed space

In this paper, the authors investigate the general solution of a new cubic functional equation \(\begin{equation*} 3f(x+3y)-f(3x+y)=12[f(x+y)+f(x-y)]+80f(y)-48f(x) \end{equation*}\) and discuss its generalized Hyers - Ulam - Rassias stability in Banach spaces and stability in fuzzy normed spaces.     

Stability of functional equations in the spaces of distributions and hyperfunctions

Making use of the fundamental solution of the heat equation we prove the stability theorems of quadratic functional equation and d'Alembert equation in the spaces of Schwartz distributions and Sato hyperfunctions.     

Solution to Kim-Rassias's question on stability of generalized Euler-Lagrange quadratic functional equations in quasi-Banach spaces

By using Aoki-Rolewicz Theorem on p-normalizing a quasi-normed space, we prove stability results for Euler-Lagrange quadratic functional equations in quasi-Banach spaces. These results improve stability results and give the answer to Kim-Rassias's question.     

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)