scholarly journals Stability in Generalized Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Young-Su Lee
Keyword(s):  
Functional Equation ◽  
Additive Functional ◽  
Generalized Functions ◽  
Heat Kernels ◽  
Tempered Distributions ◽  
Additive Functional Equation ◽  
Independent Variables ◽  
Space Of Distributions ◽  
The Stability ◽  
Fourier Hyperfunctions

We consider the following additive functional equation with -independent variables: in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.

scholarly journals Generalized Hyers–Ulam Stability of the Additive Functional Equation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 76 ◽  
Author(s):  
Yang-Hi Lee ◽  
Gwang Kim
Keyword(s):  
Functional Equation ◽  
Additive Function ◽  
Additive Functional ◽  
Stability Theorem ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Stability Theorems ◽  
The Stability

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.

scholarly journals Stability of a Bi-Additive Functional Equation in Banach Modules Over aC⋆-Algebra

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Won-Gil Park ◽  
Jae-Hyeong Bae
Keyword(s):  
Functional Equation ◽  
Borel Function ◽  
Additive Functional ◽  
Additive Functional Equation ◽  
Banach Modules ◽  
The Stability

We solve the bi-additive functional equationf(x+y,z−w)+f(x−y,z+w)=2f(x,z)−2f(y,w)and prove that every bi-additive Borel function is bilinear. And we investigate the stability of a bi-additive functional equation in Banach modules over a unitalC⋆-algebra.

scholarly journals Local stability of the additive functional equation and its applications

2003 ◽  
Vol 2003 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Soon-Mo Jung ◽  
Byungbae Kim
Keyword(s):  
Functional Equation ◽  
Large Class ◽  
Local Stability ◽  
Unbounded Domains ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Jensen’S Functional Equation ◽  
Restricted Domains ◽  
The Stability

The main purpose of this paper is to prove the Hyers-Ulam stability of the additive functional equation for a large class of unbounded domains. Furthermore, by using the theorem, we prove the stability of Jensen's functional equation for a large class of restricted domains.

The stability of an additive functional equation in menger probabilistic φ-normed spaces

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
D. Miheţ ◽  
R. Saadati ◽  
S. Vaezpour
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theory ◽  
Stability Result ◽  
Point Theory ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
Probabilistic Normed Spaces ◽  
The Stability

AbstractWe establish a stability result concerning the functional equation: $\sum\limits_{i = 1}^m {f\left( {mx_i + \sum\limits_{j = 1,j \ne i}^m {x_j } } \right) + f\left( {\sum\limits_{i = 1}^m {x_i } } \right) = 2f\left( {\sum\limits_{i = 1}^m {mx_i } } \right)} $ in a large class of complete probabilistic normed spaces, via fixed point theory.

On the stability of some functional equations in Menger φ-normed spaces

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Dorel Miheţ ◽  
Reza Saadati
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
The Stability ◽  
Stability Results

AbstractRecently, the authors [MIHEŢ, D.—SAADATI, R.—VAEZPOUR, S. M.: The stability of an additive functional equation in Menger probabilistic φ-normed spaces, Math. Slovaca 61 (2011), 817–826] considered the stability of an additive functional in Menger φ-normed spaces. In this paper, we establish some stability results concerning the cubic, quadratic and quartic functional equations in complete Menger φ-normed spaces via fixed point theory.

scholarly journals On the Stability of -Derivations and Lie -Algebra Homomorphisms on Lie -Algebras: A Fixed Points Method

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Fixed Points ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
The Stability ◽  
Algebra Homomorphisms ◽  
Stability Results

We investigate new generalized Hyers-Ulam stability results for -derivations and Lie -algebra homomorphisms on Lie -algebras associated with the additive functional equation:

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