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Oscillation Theory for Differential Equations

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 7827

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Special Issue Editors

Prof. Dr. Jozef Džurina

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Guest Editor

Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Kosice, 904200 Košice, Slovakia
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling in physical/social/life sciences

Dr. Irena Jadlovská

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Guest Editor

Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling in physical/social/life sciences
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Dear Colleagues,

Initiated by the seminal work by Sturm in 1836, the oscillation theory of differential equations has been recognized as an important area of mathematical analysis. The essence of the theory lies in establishing conditions for the existence or non-existence of oscillatory and/or non-oscillatory solutions with a prescribed asymptotic behavior, in studying the laws of zero distribution, as well as in revealing the relationship between the oscillatory properties of solutions and corresponding oscillatory processes in systems of a diversified physical nature. Despite the intense research interest, however, development on the subject is still far from complete.

This Special Issue of the journal Mathematics focuses on the latest achievements of research known as the oscillation theory of differential equations, understood in the broad sense. It covers oscillation and non-oscillation results for ordinary, functional, impulsive, fractional, and partial differential equations, as well as their subsequent applications in science and technology.

The twofold purpose of this Special Issue is to bring together the state-of-the-art of theoretical research and recent progress in the study of applied problems.

The Special Issue is expected to welcome high-quality contributions from leading experts and researchers actively working in the field.

We look forward to receiving your valuable research papers.

Prof. Dr. Jozef Džurina
Dr. Irena Jadlovská
Guest Editors

Manuscript Submission Information

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Keywords

Existence or non-existence of oscillatory (non-oscillatory) solutions
Asymptotic behavior of solutions
Zeros distribution
Ordinary, functional, impulsive, fractional, and partial differential equations

Published Papers (5 papers)

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Research

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18 pages, 340 KiB

Open AccessArticle

Oscillation Criteria for Third-Order Nonlinear Neutral Dynamic Equations with Mixed Deviating Arguments on Time Scales

by Zhiyu Zhang, Ruihua Feng, Irena Jadlovská and Qingmin Liu

Mathematics 2021, 9(5), 552; https://doi.org/10.3390/math9050552 - 05 Mar 2021

Cited by 4 | Viewed by 1237

Abstract

Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the inequality technique, new oscillation criteria are established, which improve and generalize related results in the literature. Several examples are given to illustrate the main results. Full article

(This article belongs to the Special Issue Oscillation Theory for Differential Equations)

13 pages, 296 KiB

Open AccessArticle

Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations

by Zuzana Pátíková and Simona Fišnarová

Mathematics 2021, 9(3), 235; https://doi.org/10.3390/math9030235 - 25 Jan 2021

Cited by 4 | Viewed by 1172

Abstract

We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered. Full article

(This article belongs to the Special Issue Oscillation Theory for Differential Equations)

8 pages, 261 KiB

Open AccessArticle

Existence of Nonnegative Solutions of Linear Autonomous Functional Differential Equations

by Mihály Pituk

Mathematics 2020, 8(7), 1098; https://doi.org/10.3390/math8071098 - 05 Jul 2020

Cited by 1 | Viewed by 1411

Abstract

It is shown that if we exclude the existence of nontrivial small solutions, then a linear autonomous functional differential equation has a nontrivial nonnegative solution if and only if it has a nonnegative eigenfunction. Full article

(This article belongs to the Special Issue Oscillation Theory for Differential Equations)

9 pages, 260 KiB

Open AccessFeature PaperArticle

New Oscillation Results for Third-Order Half-Linear Neutral Differential Equations

by K. S. Vidhyaa, John R. Graef and E. Thandapani

Mathematics 2020, 8(3), 325; https://doi.org/10.3390/math8030325 - 02 Mar 2020

Cited by 8 | Viewed by 1663

Abstract

The main purpose of this paper is to obtain criteria for the oscillation of all solutions of a third-order half-linear neutral differential equation. The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations. An additional result is obtained by using a Riccati transformation technique. Examples are provided to show the importance of the main results. Full article

(This article belongs to the Special Issue Oscillation Theory for Differential Equations)

Review

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12 pages, 291 KiB

Open AccessReview

A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

by Mahmoud Abdel-Aty, Musa E. Kavgaci, Ioannis P. Stavroulakis and Nour Zidan

Mathematics 2020, 8(9), 1492; https://doi.org/10.3390/math8091492 - 03 Sep 2020

Viewed by 1321

Abstract

This paper deals with the oscillation of the first-order differential equation with several delay arguments

x^{'} (t) + \sum_{i = 1}^{m} p_{i} (t) x (τ_{i} (t)) = 0, t \geq t_{0},

where the functions [...] Read more.

This paper deals with the oscillation of the first-order differential equation with several delay arguments

x^{'} (t) + \sum_{i = 1}^{m} p_{i} (t) x (τ_{i} (t)) = 0, t \geq t_{0},

where the functions

p_{i}, τ_{i} \in C ([t_{0}, \infty), R^{+}),

for every

i = 1, 2, \dots, m, τ_{i} (t) \leq t

for

t \geq t_{0}

and

{lim}_{t \to \infty} τ_{i} (t) = \infty

. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples. Full article

(This article belongs to the Special Issue Oscillation Theory for Differential Equations)

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