3-Quadratic Functions in Lipschitz Spaces

2020 ◽  
pp. 127-133
Author(s):  
Ismail Nikoufar
Keyword(s):  
Functional Equations ◽  
Classical Problem ◽  
Quadratic Functions ◽  
Quadratic Functional ◽  
Lipschitz Spaces ◽  
Stability Of Functional Equations ◽  
The Stability

Stability of functional equations is a classical problem proposed by Ulam. In this paper, we prove the stability of the 3-quadratic functional equations in Lipschitz spaces.

scholarly journals On the probabilistic stability of some functional equations

2013 ◽  
Vol 29 (1) ◽  
pp. 125-132
Author(s):  
CLAUDIA ZAHARIA ◽  
◽  
DOREL MIHET ◽  
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Quadratic Functional ◽  
Normed Spaces ◽  
Stability Of Functional Equations ◽  
The Stability ◽  
Stability Results

We establish stability results concerning the additive and quadratic functional equations in complete Menger ϕ-normed spaces by using fixed point theory. As particular cases, some theorems regarding the stability of functional equations in β - normed and quasi-normed spaces are obtained.

scholarly journals A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equation ◽  
Uniqueness Theorem ◽  
Functional Equations ◽  
Additive Functional ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Additive Type ◽  
Stability Of Functional Equations ◽  
The Stability

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

scholarly journals Inner product spaces and quadratic functional equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jae-Hyeong Bae ◽  
Batool Noori ◽  
M. B. Moghimi ◽  
Abbas Najati
Keyword(s):  
Functional Equations ◽  
Quadratic Functions ◽  
Inner Product ◽  
Quadratic Functional ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces ◽  
Restricted Domains ◽  
The Stability

AbstractIn this paper, we introduce the functional equations $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ f ( 2 x − y ) + f ( x + 2 y ) = 5 [ f ( x ) + f ( y ) ] , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + 4 f ( y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + f ( 2 y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 4 [ f ( x ) + f ( y ) ] + [ f ( − x ) + f ( − y ) ] . We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

scholarly journals Stability of Exponential Functional Equations with Involutions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jaeyoung Chung ◽  
Soon-Yeong Chung
Keyword(s):  
Commutative Semigroup ◽  
Functional Equations ◽  
Exponential Functional ◽  
Not Otherwise Specified ◽  
Stability Of Functional Equations ◽  
The Stability

LetSbe a commutative semigroup if not otherwise specified andf:S→ℝ. In this paper we consider the stability of exponential functional equations|f(x+σ(y))-g(x)f(y)|≤ϕ(x)or ϕ(y),|f(x+σ(y))-f(x)g(y)|≤ϕ(x)orϕ(y)for allx,y∈Sand whereσ:S→Sis an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

scholarly journals On the Stability of Quadratic Functional Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Jung Rye Lee ◽  
Jong Su An ◽  
Choonkil Park
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Quadratic Functional ◽  
The Stability ◽  
Fixed Positive Integer ◽  
Rassias Stability

LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

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 General Solution and Stability of Additive-Quadratic Functional Equation in IRN-Space

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
K. Tamilvanan ◽  
Nazek Alessa ◽  
K. Loganathan ◽  
G. Balasubramanian ◽  
Ngawang Namgyel
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Normed Space ◽  
Functional Equations ◽  
Mixed Type ◽  
Research Area ◽  
Quadratic Functional ◽  
Research Papers ◽  
The Stability ◽  
Stability Results

The investigation of the stabilities of various types of equations is an interesting and evolving research area in the field of mathematical analysis. Recently, there are many research papers published on this topic, especially additive, quadratic, cubic, and mixed type functional equations. We propose a new functional equation in this study which is quite different from the functional equations already dealt in the literature. The main feature of the equation dealt in this study is that it has three different solutions, namely, additive, quadratic, and mixed type functions. We also prove that the stability results hold good for this equation in intuitionistic random normed space (briefly, IRN-space).

Sign in / Sign up

Export Citation Format

Share Document