scholarly journals Nonlinear fuzzy stability of a functional equation related to a characterization of inner product spaces via fixed point technique

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1067-1080
Author(s):  
Zhihua Wang ◽  
Prasanna Sahoo
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Method ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Fixed Point Technique ◽  
The Stability ◽  
Fuzzy Banach Space

Using the fixed point method, we prove some results concerning the stability of the functional equation 2n?i=1 f(xi-1/2n 2n?j=1 xj)=2n?i=1 f (xi)-2nf(1/2n 2n?i=1 xi) where f is defined on a vector space and taking values in a fuzzy Banach space, which is said to be a functional equation related to a characterization of inner product spaces.

scholarly journals The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Khodaei
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Vector Space ◽  
Fixed Point Method ◽  
Inner Product ◽  
Product Spaces ◽  
Real Vector ◽  
Fixed Integer ◽  
Inner Product Spaces ◽  
Fuzzy Approximation

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer holds for all The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation which is said to be a functional equation associated with inner product spaces.

scholarly journals Hyperstability of the Fréchet Equation and a Characterization of Inner Product Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Anna Bahyrycz ◽  
Janusz Brzdęk ◽  
Magdalena Piszczek ◽  
Justyna Sikorska
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Fréchet Equation ◽  
New Inequalities

We prove some stability and hyperstability results for the well-known Fréchet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.

scholarly journals Stability of a generalization of the Fréchet functional equation

2015 ◽  
Vol 14 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Renata Malejki
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Function Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
New Inequalities

AbstractWe prove some stability and hyperstability results for a generalization of the well known Fréchet functional equation, stemming from one of the characterizations of the inner product spaces. As the main tool we use a fixed point theorem for some function spaces. We end the paper with some new inequalities characterizing the inner product spaces.

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 the Fréchet Equation in Quasi-Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 490
Author(s):  
Sang Og Kim
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Inner Product ◽  
Main Role ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Fréchet Equation

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-Banach spaces. In the proof, a fixed point theorem due to Dung and Hang, which is an extension of a fixed point theorem in Banach spaces, plays a main role.

scholarly journals Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 295-303
Author(s):  
Gwang Hui Kim ◽  
Iz-iddine El-Fassi ◽  
Choonkil Park
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Quadratic Functional Equation ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Class Of Functions

AbstractWe have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

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