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Symmetry, Stability and Sustainability Issues Concerning Derivations

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Dear Colleagues,

This Special Issue aims to focus on the derivations, various generalized notions of derivation, and the problem of their stability in the Ulam sense. Potential topics include properties and applications of different types of derivations (in the broad sense of the notion), including Lie derivation, Jordan derivations, and various characterizations of derivations by means of functional equations.

We also invite contributions related to the concept of Ulam type stability and concerning subjects such as approximate derivations, asymptotically approximate generalized derivations, and various stability, hyperstability, and superstability issues connected with derivations.

The concept of symmetry often plays an important role in the study of derivations, as can be seen, for example, in the case of biderivations and more generally n-derivations defined on various algebraic structures. Moreover, in the stability results, the distances between the approximate solution and the exact solution of the considered equations are mainly measured by functions that are symmetric in some ways.

Dr. Zbigniew Lesniak
Prof. Dr. Janusz Brzdęk
Prof. Dr. Ajda Fosner
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

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Published Papers (3 papers)

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Research

14 pages, 326 KiB

Open AccessArticle

On Stability of a General n-Linear Functional Equation

by Anna Bahyrycz and Justyna Sikorska

Symmetry 2023, 15(1), 19; https://doi.org/10.3390/sym15010019 - 21 Dec 2022

Cited by 1 | Viewed by 1195

Abstract

Let X be a linear space over

K \in {R, C}

, Y be a real or complex Banach space and

f : X^{n} \to Y

. With some fixed [...] Read more.

Let X be a linear space over

K \in {R, C}

, Y be a real or complex Banach space and

f : X^{n} \to Y

. With some fixed

a_{j i}, C_{i_{1} \dots i_{n}} \in K

(

j \in {1, \dots, n}

i, i_{k} \in {1, 2}

k \in {1, \dots, n}

), we study, using the direct and the fixed point methods, the stability and the general stability of the equation

f (a_{11} x_{11} + a_{12} x_{12}, \dots, a_{n 1} x_{n 1} + a_{n 2} x_{n 2}) = \sum_{1 \leq i_{1}, \dots, i_{n} \leq 2} C_{i_{1} \dots i_{n}} f (x_{1 i_{1}}, \dots, x_{n i_{n}}),

for all

x_{j i_{j}} \in X

(

j \in {1, \dots, n},

i_{j} \in {1, 2}

). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with

m \geq 2

. Full article

(This article belongs to the Special Issue Symmetry, Stability and Sustainability Issues Concerning Derivations)

16 pages, 283 KiB

Open AccessArticle

Near-Fixed Point Results via Ƶ-Contractions in Metric Interval and Normed Interval Spaces

by Muhammad Sarwar, Misbah Ullah, Hassen Aydi and Manuel De La Sen

Symmetry 2021, 13(12), 2320; https://doi.org/10.3390/sym13122320 - 4 Dec 2021

Cited by 1 | Viewed by 1562

Abstract

In this paper, using

α

—admissibility and the concept of simulation functions, some near-fixed point results in the setting of metric interval and normed interval spaces are established. The results have been proved using

Z

-contractions. Full article

(This article belongs to the Special Issue Symmetry, Stability and Sustainability Issues Concerning Derivations)

19 pages, 362 KiB

Open AccessArticle

On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

by Anna Bahyrycz, Janusz Brzdęk, El-sayed El-hady and Zbigniew Leśniak

Symmetry 2021, 13(11), 2200; https://doi.org/10.3390/sym13112200 - 18 Nov 2021

Cited by 11 | Viewed by 2031

Abstract

The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent. Full article

(This article belongs to the Special Issue Symmetry, Stability and Sustainability Issues Concerning Derivations)

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