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 Hyperstability of the k -Cubic Functional Equation in Non-Archimedean Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Youssef Aribou ◽  
Mohamed Rossafi
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
Banach Spaces ◽  
Functional Equations ◽  
Cubic Functional Equation ◽  
Fixed Point Approach ◽  
Fixed Positive Integer

Using the fixed point approach, we investigate a general hyperstability results for the following k -cubic functional equations f k x + y + f k x − y = k f x + y + k f x − y + 2 k k 2 − 1 f x , where k is a fixed positive integer ≥ 2 , in ultrametric Banach spaces.

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 Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Azadi Kenary ◽  
H. Rezaei ◽  
Y. W. Lee ◽  
G. H. Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Functional Equations ◽  
Mixed Type ◽  
Direct Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Fixed Point Methods ◽  
Rassias Stability

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equationf(x+ky)+f(x−ky)=f(x+y)+f(x−y)+(2(k+1)/k)f(ky)−2(k+1)f(y)for fixed integerskwithk≠0,±1in fuzzy Banach spaces.

scholarly journals On the Stability of Quadratic Functional Equations inF-Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Xiuzhong Yang
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
The Stability ◽  
Rassias Stability

The Hyers-Ulam-Rassias stability of quadratic functional equationf(2x+y)+f(2x-y)=f(x+y)+f(x-y)+6f(x)and orthogonal stability of the Pexiderized quadratic functional equationf(x+y)+f(x-y)=2g(x)+2h(y)inF-spaces are proved.

A Fixed Point Approach to the Stability of Quadratic Functional Equations in Modular Spaces

2017 ◽  
Vol 6 (1) ◽  
pp. 171-175
Author(s):  
Seong Sik Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Method ◽  
Quadratic Functional ◽  
Fixed Point Approach ◽  
Modular Spaces ◽  
The Stability ◽  
Positive Integers ◽  
Rassias Stability

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation f(kx + y) + f(kx – y) = 2k2f(x) + 2f(y) for any fixed positive integers k ∈ Ζ+ in modular spaces by using fixed point method.

scholarly journals Random Stability of Quadratic Functional Equations

2019 ◽  
Vol 16 (1) ◽  
pp. 498-507
Author(s):  
Mee Kwang Kang
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Random Normed Spaces ◽  
Fixed Positive Integer

In this paper, we investigate the generalized Hyers-Ulam stability on random -normed spaces associated with the following generalized quadratic functional equation ,where  is a fixed positive integer via two methods

scholarly journals HUR-approximation of an ELTA functional equation

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4311-4328
Author(s):  
A.R. Sharifi ◽  
Azadi Kenary ◽  
B. Yousefi ◽  
R. Soltani
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Linear Mapping ◽  
Stability Theorem ◽  
Normed Spaces ◽  
The Stability ◽  
Rassias Stability

The main goal of this paper is study of the Hyers-Ulam-Rassias stability (briefly HUR-approximation) of the following Euler-Lagrange type additive(briefly ELTA) functional equation ?nj=1f (1/2 ?1?i?n,i?j rixi- 1/2 rjxj) + ?ni=1 rif(xi)=nf (1/2 ?ni=1 rixi) where r1,..., rn ? R, ?ni=k rk?0, and ri,rj?0 for some 1? i < j ? n, in fuzzy normed spaces. The concept of HUR-approximation originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

scholarly journals Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functions ◽  
Normed Spaces

We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.

scholarly journals On the stability of radical septic functional equations

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2229
Author(s):  
Emanuel Guariglia ◽  
Kandhasamy Tamilvanan
Keyword(s):  
Functional Equation ◽  
Approximate Solution ◽  
Vector Space ◽  
Banach Spaces ◽  
Functional Equations ◽  
Normed Vector Space ◽  
Relevant Case ◽  
The Stability ◽  
Normed Vector ◽  
Stability Results

This paper deals with the approximate solution of the following functional equation fx7+y77=f(x)+f(y), where f is a mapping from R into a normed vector space. We show stability results of this equation in quasi-β-Banach spaces and (β,p)-Banach spaces. We also prove the nonstability of the previous functional equation in a relevant case.

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).

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