scholarly journals Jordan∗-Derivations onC∗-Algebras andJC∗-Algebras

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Jong Su An ◽  
Jianlian Cui ◽  
Choonkil Park
Keyword(s):  
Functional Equation ◽  
Functional Inequality ◽  
Ulam Stability

We investigate Jordan∗-derivations onC∗-algebras and Jordan∗-derivations onJC∗-algebras associated with the following functional inequality‖f(x)+f(y)+kf(z)‖≤‖kf((x+y)/k+z)‖for some integerkgreater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan∗-derivations onC∗-algebras and of Jordan∗-derivations onJC∗-algebras associated with the following functional equationf((x+y)/k+z)=(f(x)+f(y))/k+f(z)for some integerkgreater than 1.

HEAT KERNEL METHOD FOR THE LEVI-CIVITÁ EQUATION IN DISTRIBUTIONS AND HYPERFUNCTIONS

2015 ◽  
Vol 92 (1) ◽  
pp. 77-93
Author(s):  
JAEYOUNG CHUNG ◽  
PRASANNA K. SAHOO
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Kernel Method ◽  
Functional Inequality ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
Positive Real ◽  
Image Position ◽  
Ulam Stability Problem

Let$G$be a commutative group and$\mathbb{C}$the field of complex numbers,$\mathbb{R}^{+}$the set of positive real numbers and$f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$where${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$is a symmetric decreasing function in the sense that${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$for all$0<t_{1}\leq t_{2}$and$0<s_{1}\leq s_{2}$. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$in the space of Gelfand hyperfunctions, where$u,v,w,k$are Gelfand hyperfunctions,$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$, and$\circ$,$\otimes$,${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$and${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.

scholarly journals Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung ◽  
Chang-Kwon Choi ◽  
Jongjin Kim
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Functional Inequality ◽  
Commutative Group ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
General Solutions

LetSandGbe a commutative semigroup and a commutative group, respectively,CandR+the sets of complex numbers and nonnegative real numbers, respectively, andσ:S→Sorσ:G→Gan involution. In this paper, we first investigate general solutions of the functional equationf(x+σy)=f(x)g(y)-g(x)f(y)for allx,y∈S, wheref,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality|f(x+σy)-f(x)g(y)+g(x)f(y)|≤ψ(y)for allx,y∈G, wheref,g:G→Candψ:G→R+.

scholarly journals Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1299
Author(s):  
Soon-Mo Jung ◽  
Ki-Suk Lee ◽  
Michael Th. Rassias ◽  
Sung-Mo Yang
Keyword(s):  
Functional Equation ◽  
Type Equation ◽  
Functional Inequality ◽  
Unit Element ◽  
Interesting Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Ulam Stability ◽  
Approximation Properties ◽  
The Mean

Let X be a commutative normed algebra with a unit element e (or a normed field of characteristic different from 2), where the associated norm is sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, f(x)−g(y)=(x−y)h(sx+ty), where f,g,h:X→X are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove the Hyers-Ulam stability of that functional equation under some additional conditions.

On the stability of a nonic functional equation in quasi-normed spaces

2016 ◽  
Vol 12 (3) ◽  
pp. 4368-4374
Author(s):  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Ulam Stability ◽  
Normed Spaces ◽  
The Stability

In this paper, we extend normed spaces to quasi-normed spaces and prove the generalized Hyers-Ulam stability of a nonic functional equation:$$\aligned&f(x+5y) - 9f(x+4y) + 36f(x+3y) - 84f(x+2y) + 126f(x+y) - 126f(x)\\&\qquad + 84f(x-y)-36f(x-2y)+9f(x-3y)-f(x-4y) = 9 ! f(y),\endaligned$$where $9 ! = 362880$ in quasi-normed spaces.

scholarly journals Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Murali Ramdoss ◽  
Divyakumari Pachaiyappan ◽  
Choonkil Park ◽  
Jung Rye Lee
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Functional Equations ◽  
Mixed Type ◽  
Research Paper ◽  
Fixed Point Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Modular Spaces

AbstractThis research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method.

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.

COMMENT ON "STABILITY OF (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRAS" [M. ESHAGHI GORDJI AND N. GHOBADIPOUR, INT. J. GEOM.METHODS MOD. PHYS. 7(7) (2010) 1–10]

2012 ◽  
Vol 10 (02) ◽  
pp. 1220027
Author(s):  
CHOONKIL PARK ◽  
JUNG RYE LEE ◽  
DONG YUN SHIN ◽  
MADJID ESHAGHI GORDJI
Keyword(s):  
Functional Equation ◽  
Ulam Stability ◽  
C Algebras

Eshaghi Gordji and Ghobadipour proved the Hyers–Ulam stability of (α, β, γ)-derivations on Lie C*-algebras associated with the following functional equation [Formula: see text] Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the conditions and prove the corrected theorems.

scholarly journals Approximate mixed type additive and quartic functional equation

2017 ◽  
Vol 35 (1) ◽  
pp. 43 ◽  
Author(s):  
Abasalt Bodaghi
Keyword(s):  
Functional Equation ◽  
Mixed Type ◽  
Current Work ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Quartic Functional Equation ◽  
All Solutions

In the current work, we introduce a general form of a mixed additive and quartic functional equation. We determine all solutions of this functional equation. We also establish the generalized Hyers-Ulam stability of this new functional equation in quasi-$\beta$-normed spaces.

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