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Symmetry
Special Issues
Ulam's Type Stability and Symmetry
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Ulam's Type Stability and Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 10127

Special Issue Editors

Prof. Dr. Liviu Cădariu

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Guest Editor
Department of Mathematics, Politehnica University of Timișoara, Timișoara, Romania
Interests: Ulam’s type stability of functional equations and integral equations; various methods for proving Ulam’s type stability results (direct and fixed point methods); generalized Hyers–Ulam stability in various spaces (Banach, non-Archimedean and quasi-Banach)
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Dorian Popa

E-Mail Website
Guest Editor
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania
Interests: ulam stability; functional equations; functional inclusions; differential equations; set-valued analyses
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ioan Rașa

E-Mail Website
Guest Editor
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania
Interests: ulam stability of operators; functional equations; functional analysis; approximation theory; inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Issue is mainly devoted to investigations connected with the notion of stability, motivated by the well-known problem of S. Ulam, on the approximate homeomorphisms of metric groups, and related issues. Authors are invited to delivery their contributions on: Ulam’s type stability of functional equations and difference equations, differential equations, integral equations and linear operators, stability of set-valued and iterative functional equations, hyperstability and superstability of functional equations, various methods for proving Ulam’s type stability results, generalized Hyers–Ulam stability, stability on restricted domains and in various (metric, Banach, non-Archimedean, fuzzy, quasi-Banach, etc.) spaces, relations between Ulam’s type stability and fixed point results, its applications and connections to other areas of mathematics (e.g. functional analysis, approximation theory, differential equations, nonlinear analysis).

Prof. Dr. Liviu Cadariu
Prof. Dr. Dorian Popa
Prof. Dr. Ioan Rașa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Generalized Hyers–Ulam stability of functional equations
  • Ulam’s type stability of difference equations and differential equations
  • Direct method and fixed point method
  • Hyperstability and superstability
  • Ulam’s type stability of integral equations
  • Ulam’s type stability of linear operators
  • Ulam stability in set-valued analysis
  • Best Ulam constant

Published Papers (6 papers)

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Research

7 pages, 243 KiB  
Article
Ulam Stability of a Second Linear Differential Operator with Nonconstant Coefficients
by Liviu Cădariu, Dorian Popa and Ioan Raşa
Symmetry 2020, 12(9), 1451; https://doi.org/10.3390/sym12091451 - 03 Sep 2020
Cited by 6 | Viewed by 1420
Abstract
In this paper, we obtain a result on Ulam stability for a second order differential operator acting on a Banach space. The result is connected to the existence of a global solution for a Riccati differential equation and some appropriate conditions on the [...] Read more.
In this paper, we obtain a result on Ulam stability for a second order differential operator acting on a Banach space. The result is connected to the existence of a global solution for a Riccati differential equation and some appropriate conditions on the coefficients of the operator. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
10 pages, 1818 KiB  
Article
On Approximate Aesthetic Curves
by Ioana Crăciun, Dorian Popa, Florina Serdean and Lucian Tudose
Symmetry 2020, 12(9), 1394; https://doi.org/10.3390/sym12091394 - 21 Aug 2020
Cited by 3 | Viewed by 2123
Abstract
Symmetry plays an essential role for generating aesthetic forms. The curve is the basic element used by designers to obtain aesthetic forms. A curve with a linear logarithmic curvature graph gradient is called aesthetic curve. The aesthetic value of a curve increases when [...] Read more.
Symmetry plays an essential role for generating aesthetic forms. The curve is the basic element used by designers to obtain aesthetic forms. A curve with a linear logarithmic curvature graph gradient is called aesthetic curve. The aesthetic value of a curve increases when its gradient is close to a straight line. We introduce the notions of approximate aesthetic curves and approximate neutral curves and obtain estimations between the curvature of an approximate aesthetic/neutral curve and an aesthetic curve. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
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7 pages, 232 KiB  
Article
Ulam Stability for the Composition of Operators
by Ana Maria Acu and Ioan Raşa
Symmetry 2020, 12(7), 1159; https://doi.org/10.3390/sym12071159 - 13 Jul 2020
Cited by 2 | Viewed by 1300
Abstract
Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an [...] Read more.
Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C 0 -semigroup ( T t ) t ≥ 0 of operators for which each T t is Ulam stable. We present an open problem concerning the Ulam stability of the members of the Bernstein C 0 -semigroup. Two other possible problems are mentioned at the end of the paper. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
11 pages, 228 KiB  
Article
Ulam’s Type Stability and Generalized Norms
by Laura Manolescu
Symmetry 2020, 12(4), 502; https://doi.org/10.3390/sym12040502 - 01 Apr 2020
Viewed by 1432
Abstract
A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: [...] Read more.
A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm). Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
15 pages, 284 KiB  
Article
Ulam Stability for a Class of Hill’s Equations
by Ryuma Fukutaka and Masakazu Onitsuka
Symmetry 2019, 11(12), 1483; https://doi.org/10.3390/sym11121483 - 05 Dec 2019
Cited by 7 | Viewed by 1830
Abstract
This paper deals with Ulam’s type stability for a class of Hill’s equations. In the two assertions of the main theorem, we obtain Ulam stability constants that are symmetrical to each other. By combining the obtained results, a necessary and sufficient condition for [...] Read more.
This paper deals with Ulam’s type stability for a class of Hill’s equations. In the two assertions of the main theorem, we obtain Ulam stability constants that are symmetrical to each other. By combining the obtained results, a necessary and sufficient condition for Ulam stability of a Hill’s equation is established. The results are generalized to nonhomogeneous Hill’s equations, and then application examples are presented. In particular, it is shown that if the approximate solution is unbounded, then there is an unbounded exact solution. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
10 pages, 270 KiB  
Article
Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem
by Javier Cuesta
Symmetry 2019, 11(9), 1107; https://doi.org/10.3390/sym11091107 - 03 Sep 2019
Viewed by 1509
Abstract
We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation [...] Read more.
We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetry)
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