scholarly journals Additive and Fréchet functional equations on restricted domains with some characterizations of inner product spaces

2021 ◽  
Vol 7 (3) ◽  
pp. 3379-3394
Author(s):  
Choonkil Park ◽  
◽  
Abbas Najati ◽  
Batool Noori ◽  
Mohammad B. Moghimi ◽  
...  
Keyword(s):  
Functional Equations ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Asymptotic Behaviors ◽  
Inner Product Spaces ◽  
Restricted Domains ◽  
Different Types

<abstract><p>In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.</p></abstract>

scholarly journals General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Margherita Fochi
Keyword(s):  
Functional Equations ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Restricted Domain ◽  
Product Spaces ◽  
General Solutions ◽  
Inner Product Spaces ◽  
Restricted Domains

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

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 Functional Equations Related to Inner Product Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Choonkil Park ◽  
Won-Gil Park ◽  
Abbas Najati
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Real Vector ◽  
Inner Product Spaces

LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

ON THE HYPERSTABILITY OF A PEXIDERISED -QUADRATIC FUNCTIONAL EQUATION ON SEMIGROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 459-470 ◽  
Author(s):  
IZ-IDDINE EL-FASSI ◽  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Quadratic Functional Equation ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Image Position

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$, where the map $\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of $G$ with $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all $x\in G$. We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the $\unicode[STIX]{x1D70E}$-quadratic and $\unicode[STIX]{x1D70E}$-Drygas equations.

scholarly journals Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jae-Young Chung ◽  
Dohan Kim ◽  
John Michael Rassias
Keyword(s):  
Asymptotic Behavior ◽  
Functional Equations ◽  
Functional Inequality ◽  
Ulam Stability ◽  
Asymptotic Behaviors ◽  
Stability Problems ◽  
Restricted Domains

We consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general restricted domains. The main purpose of this paper is to find the restricted domains for which the functional inequality satisfied in those domains extends to the inequality for whole domain. As consequences of the results we obtain asymptotic behavior of the equations.

scholarly journals Intuitionistic Fuzzy Stability of Functional Equations Associated with Inner Product Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Zhihua Wang ◽  
Themistocles M. Rassias
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces ◽  
Intuitionistic Fuzzy ◽  
Stability Of Functional Equations ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Stability Results

In intuitionistic fuzzy normed spaces, we investigate some stability results for the functional equation which is said to be a functional equation associated with inner products space.

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