scholarly journals Applications of Banach Limit in Ulam Stability

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 841
Author(s):  
Roman Badora ◽  
Janusz Brzdęk ◽  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Single Variable ◽  
Cauchy Equation ◽  
Banach Limit

We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation.

scholarly journals Banach Limit in the Stability Problem of a Linear Functional Equation

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Roman Badora ◽  
Janusz Brzdęk
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Banach Limit ◽  
The Stability

AbstractWe present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

scholarly journals Stability of Functional Equations in Dislocated Quasi-Metric Spaces

2018 ◽  
Vol 32 (1) ◽  
pp. 215-225 ◽  
Author(s):  
Beata Hejmej
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Linear Functional ◽  
Metric Spaces ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Cauchy Functional Equation ◽  
Complete Metric Spaces ◽  
Stability Of Functional Equations ◽  
Quasimetric Space

Abstract We present a result on the generalized Hyers-Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.

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 Stability of linear functional equations in Banach modules

2004 ◽  
Vol 35 (1) ◽  
pp. 29-36
Author(s):  
Chun-Gil Park
Keyword(s):  
Functional Equation ◽  
Banach Algebra ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Linear Functional Equations ◽  
Banach Modules ◽  
Unital Banach Algebra ◽  
Rassias Stability

We prove the Hyers-Ulam-Rassias stability of the linear functional equation in Banach modules over a unital Banach algebra.

scholarly journals On Ulam’s Type Stability of the Linear Equation and Related Issues

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Janusz Brzdęk ◽  
Krzysztof Ciepliński ◽  
Zbigniew Leśniak
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Survey Paper ◽  
Ulam’S Type Stability ◽  
Stability Results

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

scholarly journals A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Soon-Mo Jung ◽  
Michael Th. Rassias
Keyword(s):  
Functional Equation ◽  
Vector Space ◽  
Linear Functional ◽  
Fibonacci Numbers ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Third Order

Given a vector spaceX, we investigate the solutionsf:R→Xof the linear functional equation of third orderfx=pfx-1+qfx-2+rf(x-3), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.

scholarly journals Hyperstability of Some Functional Equations on Restricted Domain

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Anna Bahyrycz ◽  
Jolanta Olko
Keyword(s):  
Functional Equations ◽  
Linear Functional ◽  
Stability Result ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Restricted Domain ◽  
Stability Studies ◽  
Control Functions ◽  
Restricted Domains ◽  
Fréchet Equation

The paper concerns functions which approximately satisfy, not necessarily on the whole linear space, a generalization of linear functional equation. A Hyers-Ulam stability result is proved and next applied to give conditions implying the hyperstability of the equation. The results may be used as tools in stability studies on restricted domains for various functional equations. We use the main theorem to obtain a few hyperstability results of Fréchet equation on restricted domain for different control functions.

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