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 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.

scholarly journals On a general bilinear functional equation

2021 ◽  
Author(s):  
Anna Bahyrycz ◽  
Justyna Sikorska
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
General Linear ◽  
Linear Spaces ◽  
General Linear Equation ◽  
Bilinear Functional

AbstractLet X, Y be linear spaces over a field $${\mathbb {K}}$$ K . Assume that $$f :X^2\rightarrow Y$$ f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all $$x,x_i,y,y_i \in X$$ x , x i , y , y i ∈ X and with $$a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}$$ a i , b i ∈ K \ { 0 } , $$A_i,\,B_i \in {\mathbb {K}}$$ A i , B i ∈ K ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all $$x_i,y_i \in X$$ x i , y i ∈ X ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ), where $$C_1:=A_1B_1$$ C 1 : = A 1 B 1 , $$C_2:=A_1B_2$$ C 2 : = A 1 B 2 , $$C_3:=A_2B_1$$ C 3 : = A 2 B 1 , $$C_4:=A_2B_2$$ C 4 : = A 2 B 2 . We describe the form of solutions and study relations between $$(*)$$ ( ∗ ) and $$(**)$$ ( ∗ ∗ ) .

On generalized hyperstability of a general linear equation

2016 ◽  
Vol 149 (2) ◽  
pp. 413-422 ◽  
Author(s):  
L. Aiemsomboon ◽  
W. Sintunavarat
Keyword(s):  
Linear Equation ◽  
General Linear ◽  
General Linear 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.

A note on stability of the general linear equation

2008 ◽  
Vol 75 (3) ◽  
pp. 267-270 ◽  
Author(s):  
Janusz Brzdȩk ◽  
Andrzej Pietrzyk
Keyword(s):  
Linear Equation ◽  
General Linear ◽  
General Linear Equation
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