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Special Issue "Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory"

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

Deadline for manuscript submissions: 31 December 2023 | Viewed by 11561

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Special Issue Editors

Dr. Ahmed A. El-Deeb

E-Mail Website
Guest Editor

Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
Interests: dynamic inequalities; differential equations; theory of time scale; fractional calculus; delay differential equations; difference equations; oscillation and stability theory nonlinear dynamics

Dr. Clemente Cesarano

E-Mail Website
Guest Editor

Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; lie algebra
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Special Issue Information

Dear Colleagues,

Natural nonlinear phenomena may be observed in many areas. Precisely, in physics, nonlinearity is presented in fluid dynamics, nonlinear optics, and so on. As is well known, in some situations, it is not necessary to obtain exact solutions of the initial value problems which model the natural phenomena, but it is sufficient to have enough knowledge and information about the qualitative properties of the solutions; on the other hand, in other situations, analytically, we cannot obtain exact solutions for our problem, so dynamic inequalities such as Hardy inequality, Ostrowski inequality, Opial inequality, and Gronwall inequality involving functions of one and more than one independent variables, which provide explicit bounds on unknown functions, play a fundamental role in the development of qualitative theory and can be used as handy tools in the study of existence, uniqueness, oscillation, stability, and other qualitative properties of the solutions of certain dynamic equations on time scales.

Currently, the application of dynamic inequalities on time scales is a subject of strong interest. Therefore, the main aim of this Special Issue is to focus on all new aspects of recent developments in the theory of dynamic inequalities and fractional calculus on time scales, including numerical examples, as well as on contributions related to the symmetry approach to applications of inequalities in the field of dynamic equations on time scales.

It is the purpose of this Special Issue to collate original and significant contributions dealing with inequalities on time scales and applications to dynamic equations. We also welcome researchers in related fields to contribute their recent results to this Special issue.

Dr. Ahmed A. El-Deeb
Dr. Clemente Cesarano
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

dynamic inequalities on time scales
dynamical systems based upon fractional calculus
Hardy inequalities on time scales
Opial inequalities on time scales
Gronwall inequalities on time scales
fractional dynamic inequalities
existence and uniqueness
stability and numerical methods
oscillation behavior of half-linear dynamic equations
time scales calculus

Published Papers (14 papers)

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Research

Open AccessArticle

Novel Integral Inequalities on Nabla Time Scales with C-Monotonic Functions

and

Symmetry 2023, 15(6), 1248; https://doi.org/10.3390/sym15061248 - 12 Jun 2023

Cited by 2 | Viewed by 417

Abstract

Through the paper, we present several inequalities involving C-monotonic functions with

C \geq 1,

on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying [...] Read more.

Through the paper, we present several inequalities involving C-monotonic functions with

C \geq 1,

on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for our results, when

T = R,

we obtain the continuous analouges of inequalities that had previously been proved in the literature. When

T = N

, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of

C

-monotonic functions are critical in determining the best way to solve inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

A Variety of Weighted Opial-Type Inequalities with Applications for Dynamic Equations on Time Scales

Barakah Almarri

Samer D. Makarish

and

Ahmed A. El-Deeb

Symmetry 2023, 15(5), 1039; https://doi.org/10.3390/sym15051039 - 08 May 2023

Viewed by 472

Abstract

Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Existence Results of Periodic Solutions to First-Order Neutral Differential Equations on Time Scales

Qiaoling Zhang

Bo Du

and

Xueping Hu

Symmetry 2022, 14(11), 2405; https://doi.org/10.3390/sym14112405 - 14 Nov 2022

Viewed by 621

Abstract

The purpose of this paper is to study the existence of periodic solutions for the first-order nonlinear neutral differential equation on time scales. Burton–Krasnoselskii’s fixed point theorem will be sufficiently general for application to the considered equation. An example has been carried out to show our results. It should be pointed out that the problem of periodic solutions is one of the current hot topics in the study of dynamic equations, which contains rich symmetry ideas and methods. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

Waleed Abuelela

Ahmed A. El-Deeb

and

Dumitru Baleanu

Symmetry 2022, 14(11), 2257; https://doi.org/10.3390/sym14112257 - 27 Oct 2022

Viewed by 565

Abstract

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Some New Inverse Hilbert Inequalities on Time Scales

Ahmed A. El-Deeb

Samer D. Makharesh

and

Barakah Almarri

Symmetry 2022, 14(11), 2234; https://doi.org/10.3390/sym14112234 - 25 Oct 2022

Cited by 1 | Viewed by 609

Abstract

Several inverse integral inequalities were proved in 2004 by Yong. It is our aim in this paper to extend these inequalities to time scales. Furthermore, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Our results are proved using some algebraic inequalities, inverse Hölder’s inequality and inverse Jensen’s inequality on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Third-Order Neutral Differential Equations with Damping and Distributed Delay: New Asymptotic Properties of Solutions

and

Symmetry 2022, 14(10), 2192; https://doi.org/10.3390/sym14102192 - 18 Oct 2022

Cited by 3 | Viewed by 771

Abstract

In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Diamond-α Hardy-Type Inequalities on Time Scales

Ahmed A. El-Deeb

and

Jan Awrejcewicz

Symmetry 2022, 14(10), 2047; https://doi.org/10.3390/sym14102047 - 30 Sep 2022

Cited by 3 | Viewed by 872

Abstract

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond-

α

dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy [...] Read more.

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond-

α

dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy inequalities for several functions by using the diamond-

α

dynamic integrals on time scales. The results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. Our main results extend existing results of the integral and discrete Hardy-type inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Fractional Leindler’s Inequalities via Conformable Calculus

and

Symmetry 2022, 14(10), 1958; https://doi.org/10.3390/sym14101958 - 20 Sep 2022

Cited by 2 | Viewed by 607

Abstract

In this paper, some fractional Leindler and Hardy-type inequalities and their reversed will be proved by using integration by parts and Hölder inequality on conformable fractional calculus. As a special case, some classical integral inequalities will be obtained. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new fractional inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

On Some Dynamic (ΔΔ)^∇- Gronwall–Bellman–Pachpatte-Type Inequalities on Time Scales and Its Applications

Ahmed A. El-Deeb

Alaa A. El-Bary

and

Jan Awrejcewicz

Symmetry 2022, 14(9), 1902; https://doi.org/10.3390/sym14091902 - 11 Sep 2022

Cited by 3 | Viewed by 943

Abstract

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Dynamic Hardy–Copson-Type Inequalities via (γ,a)-Nabla-Conformable Derivatives on Time Scales

and

Symmetry 2022, 14(9), 1847; https://doi.org/10.3390/sym14091847 - 05 Sep 2022

Cited by 2 | Viewed by 909

Abstract

We prove new Hardy–Copson-type

(γ, a)

-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller’s chain rule, the integration by parts formula, and the dynamic Hölder inequality on time scales. When

γ = 1

, [...] Read more.

We prove new Hardy–Copson-type

(γ, a)

-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller’s chain rule, the integration by parts formula, and the dynamic Hölder inequality on time scales. When

γ = 1

, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we obtain new continuous and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Some Generalizations of (∇∇)^Δ–Gronwall–Bellman–Pachpatte Dynamic Inequalities on Time Scales with Application

Ahmed A. El-Deeb

Alaa A. El-Bary

and

Jan Awrejcewicz

Symmetry 2022, 14(9), 1823; https://doi.org/10.3390/sym14091823 - 02 Sep 2022

Viewed by 894

Abstract

As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of the main results. The inequalities proved here can be used in the analysis as handy tools to study the stability, boundedness, existence, uniqueness and oscillation behavior for some kinds of partial dynamic equations on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Some Generalizations of Novel (Δ∇)^Δ–Gronwall–Pachpatte Dynamic Inequalities on Time Scales with Applications

Ahmed A. El-Deeb

and

Dumitru Baleanu

Symmetry 2022, 14(9), 1806; https://doi.org/10.3390/sym14091806 - 31 Aug 2022

Viewed by 668

Abstract

We established several novel inequalities of Gronwall–Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partial delay dynamic equation. The Leibniz integral rule on time scales has been used in the technique of our proof. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

Ostrowski–Trapezoid–Grüss-Type on (q, ω)-Hahn Difference Operator

Ahmed A. El-Deeb

and

Jan Awrejcewicz

Symmetry 2022, 14(9), 1776; https://doi.org/10.3390/sym14091776 - 26 Aug 2022

Cited by 4 | Viewed by 816

Abstract

We use two parameters for functions whose second (q, ω)-derivatives are bounded in order to prove several recent extensions of the Ostrowski inequality and its companion inequalities on (q, ω)-Hahn difference operator. Furthermore, we procure some q-integral and continuous inequalities as special cases of the main results as well these generalizations. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

Open AccessArticle

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

and

Symmetry 2022, 14(8), 1653; https://doi.org/10.3390/sym14081653 - 10 Aug 2022

Cited by 9 | Viewed by 1106

Abstract

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy. Full article

(This article belongs to the Special Issue Advances in Dynamic Inequalities on Time Scales and Their Applications to Qualitative Theory)

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