A fixed point approach to the hyperstability of the general linear equation in β-Banach spaces

Analysis ◽  
2018 ◽  
Vol 38 (3) ◽  
pp. 115-126
Author(s):  
Iz-iddine EL-Fassi ◽  
Samir Kabbaj ◽  
Abdellatif Chahbi
Keyword(s):  
Fixed Point ◽  
Linear Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
General Linear ◽  
General Linear Equation ◽  
Fixed Point Approach ◽  
Nonlinear Anal ◽  
Stability Of Functional Equations ◽  
The Fixed Point Theorem

AbstractThe purpose of this paper is first to reformulate the fixed point theorem (see Theorem 1 of [J. Brzdȩk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 2011, 17, 6728–6732]) in β-Banach spaces. We also show that this theorem is a very efficient and convenient tool for proving the hyperstability results of the general linear equation in β-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it.

Non-Archimedean hyperstability of Cauchy–Jensen functional equations on a restricted domain

2018 ◽  
Vol 24 (2) ◽  
pp. 155-165
Author(s):  
Iz-iddine EL-Fassi
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Metric Spaces ◽  
Fixed Point Approach ◽  
Nonlinear Anal ◽  
Stability Of Functional Equations ◽  
Jensen Functional ◽  
Empty Subset ◽  
The Stability ◽  
Fixed Positive Integer

Abstract Let X be a normed space, {U\subset X\setminus\{0\}} a non-empty subset, and {(G,+)} a commutative group equipped with a complete ultrametric d that is invariant (i.e., {d(x+z,y+z)=d(x,y} ) for {x,y,z\in G} ). Under some weak natural assumptions on U and on the function {\gamma\colon U^{3}\to[0,\infty)} , we study the new generalized hyperstability results when {f\colon U\to G} satisfies the inequality d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)% \biggr{)}\leq\gamma(x,y,z) for all {x,y,z\in U} , where {\frac{x+y}{\alpha}+z\in U} and {\alpha\geq 2} is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.

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 a new class of functional equations satisfied by polynomial functions

2021 ◽  
Author(s):  
Timothy Nadhomi ◽  
Chisom Prince Okeke ◽  
Maciej Sablik ◽  
Tomasz Szostok
Keyword(s):  
Linear Equation ◽  
Linear Combination ◽  
Functional Equations ◽  
Classical Result ◽  
Systematic Approach ◽  
Polynomial Function ◽  
General Linear ◽  
Polynomial Functions ◽  
New Class ◽  
General Linear Equation

AbstractThe classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation $$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$ F ( x + y ) - F ( x ) - F ( y ) = y f ( x ) + x f ( y ) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

A NOTE ON THE GENERALISED HYPERSTABILITY OF THE GENERAL LINEAR EQUATION

2017 ◽  
Vol 96 (2) ◽  
pp. 263-273 ◽  
Author(s):  
LADDAWAN AIEMSOMBOON ◽  
WUTIPHOL SINTUNAVARAT
Keyword(s):  
Fixed Point ◽  
Linear Equation ◽  
Linear Functional ◽  
Fixed Point Theory ◽  
Point Theory ◽  
General Linear ◽  
Linear Functional Equation ◽  
Normed Spaces ◽  
General Linear Equation ◽  
Fixed Point Methods

Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.

scholarly journals Ulam stability of functional equations in 2-Banach spaces via the fixed point method

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Krzysztof Ciepliński
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Functional Equations ◽  
Fixed Point Method ◽  
Point Method ◽  
Ulam Stability ◽  
Several Variables ◽  
Stability Of Functional Equations ◽  
Special Cases ◽  
The Stability

AbstractUsing the fixed point method, we prove the Ulam stability of two general functional equations in several variables in 2-Banach spaces. As corollaries from our main results, some outcomes on the stability of a few known equations being special cases of the considered ones will be presented. In particular, we extend several recent results on the Ulam stability of functional equations in 2-Banach spaces.

A fixed point approach to stability of functional equations

2011 ◽  
Vol 74 (17) ◽  
pp. 6728-6732 ◽  
Author(s):  
Janusz Brzdȩk ◽  
Jacek Chudziak ◽  
Zsolt Páles
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Approach ◽  
Stability Of Functional Equations
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