scholarly journals On the Stability Problem in Fuzzy Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
G. Zamani Eskandani ◽  
P. Găvruţa ◽  
Gwang Hui Kim
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Stability Problem ◽  
Cauchy Functional Equation ◽  
Open Problems ◽  
The Stability ◽  
Fuzzy Banach Space

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

Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

2017 ◽  
Vol 60 (1) ◽  
pp. 95-103 ◽  
Author(s):  
Chang-Kwon Choi ◽  
Jaeyoung Chung ◽  
Yumin Ju ◽  
John Rassias
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Functional Equations ◽  
Measure Zero ◽  
Stability Theorem ◽  
Cubic Functional Equation ◽  
Real Normed Space ◽  
Restricted Domains

AbstractLet X be a real normed space, Y a Banach space, and f : X → Y. We prove theUlam–Hyers stability theorem for the cubic functional equationin restricted domains. As an application we consider a measure zero stability problem of the inequalityfor all (x, y) in Γ ⸦ ℝ2 of Lebesgue measure 0.

scholarly journals On the stability problem of quadratic functional equations in 2-Banach spaces

2017 ◽  
pp. 5054-5061
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Stability Problem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Normed Spaces ◽  
The Stability ◽  
Stability Results

In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and G·avruta for the quadratic functional equation:f(2x + y) + f(2x ¡ y) = 2f(x + y) + 2f(x ¡ y) + 4f(x) ¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by thequadratic functional equation in normed spaces to 2-Banach spaces.

scholarly journals Stability of the Exponential Functional Equation in Riesz Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Representation Theorem ◽  
Spectral Representation ◽  
Cauchy Functional Equation ◽  
Exponential Functional ◽  
The Stability ◽  
Class Of Functions

We deal with the stability of the exponential Cauchy functional equationF(x+y)=F(x)F(y)in the class of functionsF:G→Lmapping a group (G, +) into a Riesz algebraL. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

Stability of Quadratic Functional Equation in Fuzzy Banach Space

2013 ◽  
Vol 373-375 ◽  
pp. 1881-1884
Author(s):  
Xiao Jing Zhan ◽  
Pei Sheng Ji
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Method ◽  
Point Method ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Fuzzy Banach Space

In this paper, we investigate the Hyers-Ulam stability of the functional equation ƒ(2x+y)+ƒ(2x-y)=8ƒ(x)+2ƒ(y) in fuzzy Banach space using the fixed point method.

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

2015 ◽  
Vol 93 (2) ◽  
pp. 272-282 ◽  
Author(s):  
JAEYOUNG CHUNG ◽  
JOHN MICHAEL RASSIAS
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Cyclic Subgroup ◽  
Real Banach Space ◽  
Measure Zero ◽  
Stability Theorem ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

scholarly journals Banach Limit in the Stability Problem of a Linear Functional Equation

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Roman Badora ◽  
Janusz Brzdęk
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Banach Limit ◽  
The Stability

AbstractWe present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

scholarly journals Stability results and separations theorems

2021 ◽  
Author(s):  
Roman Badora
Keyword(s):  
Functional Equation ◽  
Separation Theorems ◽  
Cauchy Functional Equation ◽  
Additive Map ◽  
Different Types ◽  
The Stability ◽  
Stability Results

AbstractThe presented work summarizes the relationships between stability results and separation theorems. We prove the equivalence between different types of theorems on separation by an additive map and different types of stability results concerning the stability of the Cauchy functional equation.

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