scholarly journals Stability of a Quartic Functional Equation in Restricted Domains

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung ◽  
Yu-Min Ju
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Stability Problem ◽  
Measure Zero ◽  
Stability Theorem ◽  
Real Normed Space ◽  
Quartic Functional Equation ◽  
Restricted Domains

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

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.

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 The new investigation of the stability of mixed type additive-quartic functional equations in non-Archimedean spaces

2020 ◽  
Vol 53 (1) ◽  
pp. 174-192
Author(s):  
Anurak Thanyacharoen ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Normed Space ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Group ◽  
Ulam Stability ◽  
Quartic Functional Equation ◽  
The Stability

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

TWO NEW GENERALISED HYPERSTABILITY RESULTS FOR THE DRYGAS FUNCTIONAL EQUATION

2017 ◽  
Vol 95 (2) ◽  
pp. 269-280 ◽  
Author(s):  
LADDAWAN AIEMSOMBOON ◽  
WUTIPHOL SINTUNAVARAT
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Nonempty Subset ◽  
Image Position

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation $$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$ where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.

scholarly journals Differential Equations on Functions from R into Real Banach Space

2013 ◽  
Vol 21 (4) ◽  
pp. 261-272
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Normed Space ◽  
Real Banach Space ◽  
Riemann Integral ◽  
Differentiable Functions ◽  
Real Normed Space

Abstract In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in [30].

scholarly journals Generalized Stability of Euler-Lagrange Quadratic Functional Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Hark-Mahn Kim ◽  
Min-Young Kim
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
General Solution ◽  
Rational Numbers ◽  
Stability Theorem ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Generalized Stability

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equationf(ax+by)+af(x-by)=(a+1)b2f(y)+a(a+1)f(x), in(β,p)-Banach space, wherea,bare fixed rational numbers such thata≠-1,0andb≠0.

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