scholarly journals Stability and Superstability of Generalized (, )-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
M. Eshaghi Gordji ◽  
M. B. Ghaemi ◽  
G. H. Kim ◽  
Badrkhan Alizadeh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Cauchy Functional Equation ◽  
Cauchy’S Functional Equation ◽  
Additive Cauchy Functional Equation

Let be an algebra, and let , be ring automorphisms of . An additive mapping is called a -derivation if for all . Moreover, an additive mapping is said to be a generalized -derivation if there exists a -derivation such that for all . In this paper, we investigate the superstability of generalized -derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.

scholarly journals A Fixed Point Approach to Superstability of Generalized Derivations on Non-Archimedean Banach Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
M. B. Ghaemi ◽  
Badrkhan Alizadeh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Banach Algebras ◽  
Generalized Derivations ◽  
Cauchy Functional Equation ◽  
Fixed Point Approach

We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.

scholarly journals On Hyperstability of the Cauchy Functional Equation in n-Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1886
Author(s):  
Janusz Brzdęk ◽  
El-sayed El-hady
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Main Tool ◽  
Normed Spaces ◽  
Cauchy Functional Equation ◽  
Additive Cauchy Functional Equation

We present some hyperstability results for the well-known additive Cauchy functional equation f(x+y)=f(x)+f(y) in n-normed spaces, which correspond to several analogous outcomes proved for some other spaces. The main tool is a recent fixed-point theorem.

Approximate generalized derivations close to derivations in Lie C*-algebras

2015 ◽  
Vol 21 (1) ◽  
Author(s):  
Madjid Eshaghi ◽  
Sadegh Abbaszadeh
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras

AbstractWe apply a fixed point theorem to prove that there exists a unique derivation close to an approximately generalized derivation in Lie

scholarly journals A New Fixed Point Theorem and a New Generalized Hyers-Ulam-Rassias Stability in Incomplete Normed Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1117
Author(s):  
Maryam Ramezani ◽  
Ozgur Ege ◽  
Manuel De la Sen
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Fixed Point Method ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Normed Spaces ◽  
Rassias Stability

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results.

A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

2013 ◽  
Vol 89 (1) ◽  
pp. 33-40 ◽  
Author(s):  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Functional Equations ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Real Numbers ◽  
Formula One ◽  
Image Position

AbstractWe prove a hyperstability result for the Cauchy functional equation$f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function$f$, mapping a normed space${E}_{1} $into a normed space${E}_{2} $, and for all real numbers$r, s$with$r+ s\gt 0$one of the following two conditions must be valid:$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

scholarly journals Orthogonal stability of the generalized quadratic functional equations in the sense of Rätz

2019 ◽  
Vol 52 (1) ◽  
pp. 523-530
Author(s):  
Laddawan Aiemsomboon ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Algebra ◽  
Point Theorem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Banach Module ◽  
Unital Banach Algebra

AbstractLet (X, ⊥) be an orthogonality module in the sense of Rätz over a unital Banach algebra A and Y be a real Banach module over A. In this paper, we apply the alternative fixed point theorem for proving the Hyers-Ulam stability of the orthogonally generalized k-quadratic functional equation of the formaf(kx + y) + af(kx - y) = f(ax + ay) + f(ax - ay) + \left( {2{k^2} - 2} \right)f(ax)for some |k| > 1, for all a ɛ A1 := {u ɛ A||u|| = 1} and for all x, y ɛ X with x⊥y, where f maps from X to Y.

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 Brzdęk’s fixed point method for the generalised hyperstability of bi-Jensen functional equation in (2,β)-Banach spaces

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4897-4910
Author(s):  
Iz-Iddine El-Fassi
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Fixed Point Method ◽  
Point Method ◽  
Jensen Functional Equation ◽  
Jensen Functional ◽  
The Fixed Point Theorem

Using the fixed point theorem [12, Theorem 1] in (2,?)-Banach spaces, we prove the generalized hyperstability results of the bi-Jensen functional equation 4f(x + z/2; y + w/2) = f (x,y) + f (x,w) + f (z,y) + f (y,w). Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. The method we use here can be applied to various similar equations in many variables.

Ulam stability of some functional inclusions for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5489-5495 ◽  
Author(s):  
Janusz Brzdęk ◽  
Magdalena Piszczek
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Ulam Stability ◽  
Nonlinear Operators ◽  
The Family ◽  
Fixed Positive Integer

We show that some multifunctions F : K ? n(Y), satisfying functional inclusions of the form ? (x,F(?1(x)),..., F(?n(x)))? F(x)G(x), admit near-selections f : K ? Y, fulfilling the functional equation ? (x,f (?1(x)),..,, f(?n(x)))= f(x), where functions G : K ? n(Y), ?: K x Yn ? Y and ?1,..., ?n ? KK are given, n is a fixed positive integer, K is a nonempty set, (Y,?) is a group and n(Y) denotes the family of all nonempty subsets of Y. Our results have been motivated by the notion of Ulam stability and some earlier outcomes. The main tool in the proofs is a very recent fixed point theorem for nonlinear operators, acting on some spaces of multifunctions.

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