Research

18 pages, 341 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Applications of Banach Limit in Ulam Stability

by Roman Badora, Janusz Brzdęk and Krzysztof Ciepliński

Symmetry 2021, 13(5), 841; https://doi.org/10.3390/sym13050841 - 10 May 2021

Cited by 11 | Viewed by 1787

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. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

21 pages, 325 KiB

Open AccessFeature PaperArticle

On the Stability of a Generalized Fréchet Functional Equation with Respect to Hyperplanes in the Parameter Space

by Janusz Brzdȩk, Zbigniew Leśniak and Renata Malejki

Symmetry 2021, 13(3), 384; https://doi.org/10.3390/sym13030384 - 27 Feb 2021

Cited by 1 | Viewed by 1150

Abstract

We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends on the values of [...] Read more.

We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends on the values of the coefficients of the equation and the form of the control function. In the proofs of the main results, we use a fixed point theorem to get an exact solution of the equation close to a given approximate solution. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

16 pages, 271 KiB

Open AccessFeature PaperArticle

Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral

by Sebastian Wójcik

Symmetry 2020, 12(12), 2104; https://doi.org/10.3390/sym12122104 - 17 Dec 2020

Viewed by 1703

Abstract

It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance [...] Read more.

It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

17 pages, 342 KiB

Open AccessArticle

On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

by Mariusz Bajger, Janusz Brzdęk, El-sayed El-hady and Eliza Jabłońska

Symmetry 2020, 12(12), 1974; https://doi.org/10.3390/sym12121974 - 29 Nov 2020

Viewed by 1350

Abstract

Let S denote the unit circle on the complex plane and

★ : S^{2} \to S

be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup

(S, ★)

is [...] Read more.

Let S denote the unit circle on the complex plane and

★ : S^{2} \to S

be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup

(S, ★)

is isomorphic to the group

(S, \cdot)

; thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from

(S, \cdot)

into

(S, ★)

and obtain, in this way, the following description of operation ★:

x ★ y = F (F^{- 1} (x) \cdot F^{- 1} (y))

for

x, y \in S

. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

9 pages, 235 KiB

Open AccessFeature PaperArticle

Normal Toeplitz Operators on the Fock Spaces

by Jongrak Lee

Symmetry 2020, 12(10), 1615; https://doi.org/10.3390/sym12101615 - 29 Sep 2020

Viewed by 1390

Abstract

We characterize normal Toeplitz operator on the Fock spaces

F^{2} (C)

. First, we state basic properties for Toeplitz operator

T_{φ}

on

F^{2} (C)

. Next, we study the normal Toeplitz operator

T_{φ}

on [...] Read more.

We characterize normal Toeplitz operator on the Fock spaces

F^{2} (C)

. First, we state basic properties for Toeplitz operator

T_{φ}

on

F^{2} (C)

. Next, we study the normal Toeplitz operator

T_{φ}

on

F^{2} (C)

in terms of harmonic symbols

φ

. Finally, we characterize the normal Toeplitz operators

T_{φ}

with non-harmonic symbols acting on

F^{2} (C)

. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

17 pages, 327 KiB

Open AccessArticle

Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

by Pshtiwan Othman Mohammed, Thabet Abdeljawad and Artion Kashuri

Symmetry 2020, 12(9), 1503; https://doi.org/10.3390/sym12091503 - 12 Sep 2020

Cited by 25 | Viewed by 2171

Abstract

There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of [...] Read more.

There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

12 pages, 793 KiB

Open AccessArticle

Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

by Pshtiwan Othman Mohammed, Thabet Abdeljawad, Shengda Zeng and Artion Kashuri

Symmetry 2020, 12(9), 1485; https://doi.org/10.3390/sym12091485 - 09 Sep 2020

Cited by 32 | Viewed by 2171

Abstract

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, [...] Read more.

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the

\overset{˘}{σ}

-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a

\overset{˘}{σ}

-convex function with respect to an increasing function involving the

φ

-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

8 pages, 255 KiB

Open AccessFeature PaperArticle

Ulam Stability of a Functional Equation in Various Normed Spaces

by Krzysztof Ciepliński

Symmetry 2020, 12(7), 1119; https://doi.org/10.3390/sym12071119 - 06 Jul 2020

Cited by 4 | Viewed by 1846

Abstract

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in [...] Read more.

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

14 pages, 307 KiB

Open AccessArticle

New Generalized Hermite-Hadamard Inequality and Related Integral Inequalities Involving Katugampola Type Fractional Integrals

by Ohud Almutairi and Adem Kılıçman

Symmetry 2020, 12(4), 568; https://doi.org/10.3390/sym12040568 - 05 Apr 2020

Cited by 8 | Viewed by 2877

Abstract

In this paper, a new identity for the generalized fractional integral is defined. Using this identity we studied a new integral inequality for functions whose first derivatives in absolute value are convex. The new generalized Hermite-Hadamard inequality for generalized convex function on fractal [...] Read more.

In this paper, a new identity for the generalized fractional integral is defined. Using this identity we studied a new integral inequality for functions whose first derivatives in absolute value are convex. The new generalized Hermite-Hadamard inequality for generalized convex function on fractal sets involving Katugampola type fractional integral is established. This fractional integral generalizes Riemann-Liouville and Hadamard’s integral, which possess a symmetric property. We derive trapezoid and mid-point type inequalities connected to this generalized Hermite-Hadamard inequality. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

12 pages, 255 KiB

Open AccessArticle

On Extendability of the Principle of Equivalent Utility

by Małgorzata Chudziak and Marek Żołdak

Symmetry 2020, 12(1), 42; https://doi.org/10.3390/sym12010042 - 24 Dec 2019

Cited by 1 | Viewed by 1504

Abstract

An insurance premium principle is a way of assigning to every risk a real number, interpreted as a premium for insuring risk. There are several methods of defining the principle. In this paper, we deal with the principle of equivalent utility under the [...] Read more.

An insurance premium principle is a way of assigning to every risk a real number, interpreted as a premium for insuring risk. There are several methods of defining the principle. In this paper, we deal with the principle of equivalent utility under the rank-dependent utility model. The principle, generated by utility function and probability distortion function, is based on the assumption of the symmetry between the decisions of accepting and rejecting risk. It is known that the principle of equivalent utility can be uniquely extended from the family of ternary risks. However, the extension from the family of binary risks need not be unique. Therefore, the following problem arises: characterizing those principles that coincide on the family of all binary risks. We reduce the problem thus to the multiplicative Pexider functional equation on a region. Applying the form of continuous solutions of the equation, we solve the problem completely. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

10 pages, 750 KiB

Open AccessArticle

Hermite–Hadamard and Fejér Inequalities for Co-Ordinated (F,G)-Convex Functions on a Rectangle

by Małgorzata Chudziak and Marek Żołdak

Symmetry 2020, 12(1), 13; https://doi.org/10.3390/sym12010013 - 19 Dec 2019

Cited by 2 | Viewed by 1611

Abstract

We introduce the notion of a co-ordinated

(F, G)

-convex function defined on an interval in

R^{2}

and we prove the Hermite–Hadamard and Fejér type inequalities for such functions. [...] Read more.

We introduce the notion of a co-ordinated

(F, G)

-convex function defined on an interval in

R^{2}

and we prove the Hermite–Hadamard and Fejér type inequalities for such functions. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)

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