On the approximate solutions of non-linear functional equations under mild differentiability conditions

1991 ◽  
Vol 58 (1-2) ◽  
pp. 3-7 ◽  
Author(s):  
I. K. Argyros
Keyword(s):  
Functional Equations ◽  
Linear Functional ◽  
Approximate Solutions ◽  
Linear Functional Equations ◽  
Non Linear

ON HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

2015 ◽  
Vol 92 (2) ◽  
pp. 259-267 ◽  
Author(s):  
DONG ZHANG
Keyword(s):  
Functional Equation ◽  
Exact Solutions ◽  
Normed Space ◽  
Functional Equations ◽  
Linear Functional ◽  
Approximate Solutions ◽  
Linear Functional Equation ◽  
Several Variables ◽  
Linear Functional Equations

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.

scholarly journals Multi-variable quaternionic spectral analysis

2021 ◽  
Vol 41 (3) ◽  
pp. 335-379
Author(s):  
Ilwoo Cho ◽  
Palle E.T. Jorgensen
Keyword(s):  
Spectral Analysis ◽  
Functional Equations ◽  
Linear Functional ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Linear Functional Equations ◽  
Non Linear

In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii).

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