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18 pages, 751 KiB

Open AccessArticle

Study of Solutions for a Degenerate Reaction Equation with a High Order Operator and Advection

by José Luis Díaz Palencia, Julián Roa González and Almudena Sánchez Sánchez

Mathematics 2022, 10(10), 1729; https://doi.org/10.3390/math10101729 - 18 May 2022

Cited by 2 | Viewed by 1519

The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection [...] Read more.

The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection and non-Lipstchitz spatially heterogeneous reaction. The paper examines the existence of solutions, uniqueness and travelling wave oscillatory properties (also called instabilities). Such oscillatory behaviour may lead to negative solutions in the proximity of zero. A numerical exploration is provided with the following main finding to declare: the solutions keeps oscillating in the proximity of the null stationary solution due to the high order operator, except if the reaction term is quasi-Lipschitz, in which it is possible to define a region where solutions are positive locally in time. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

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17 pages, 311 KiB

Open AccessArticle

Oscillation and Nonoscillatory Criteria of Higher Order Dynamic Equations on Time Scales

by Ya-Ru Zhu, Zhong-Xuan Mao, Jing-Feng Tian, Ya-Gang Zhang and Xin-Ni Lin

Mathematics 2022, 10(5), 717; https://doi.org/10.3390/math10050717 - 24 Feb 2022

Cited by 1 | Viewed by 1424

Abstract

In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillatory solution by employing fixed point [...] Read more.

In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillatory solution by employing fixed point theorem. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

15 pages, 294 KiB

Open AccessArticle

Upper Bounds for the Distance between Adjacent Zeros of First-Order Linear Differential Equations with Several Delays

by Emad R. Attia and George E. Chatzarakis

Mathematics 2022, 10(4), 648; https://doi.org/10.3390/math10040648 - 19 Feb 2022

Cited by 3 | Viewed by 1731

Abstract

The distance between successive zeros of all solutions of first-order differential equations with several delays is studied in this work. Many new estimations for the upper bound of the distance between zeros are obtained. Our results improve many-well known results in the literature. [...] Read more.

The distance between successive zeros of all solutions of first-order differential equations with several delays is studied in this work. Many new estimations for the upper bound of the distance between zeros are obtained. Our results improve many-well known results in the literature. We also obtain some fundamental results for the lower bound of the distance between adjacent zeros. Some illustrative examples are introduced to show the accuracy and efficiency of the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

12 pages, 786 KiB

Open AccessArticle

Oscillation of Solutions to Third-Order Nonlinear Neutral Dynamic Equations on Time Scales

by Yang-Cong Qiu, Kuo-Shou Chiu, Said R. Grace, Qingmin Liu and Irena Jadlovská

Mathematics 2022, 10(1), 86; https://doi.org/10.3390/math10010086 - 27 Dec 2021

Cited by 5 | Viewed by 2399

Abstract

In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging technique. Two examples are shown to illustrate [...] Read more.

In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging technique. Two examples are shown to illustrate the conclusions. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

12 pages, 277 KiB

Open AccessArticle

Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments

by Blanka Baculikova

Mathematics 2021, 9(20), 2552; https://doi.org/10.3390/math9202552 - 12 Oct 2021

Cited by 19 | Viewed by 1687

Abstract

In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form [...] Read more.

In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form

{(r (t) {(y^{'} (t))}^{α})}^{'} = p (t) y^{α} (τ (t)) .

Such differential equation may possesses two types of nonoscillatory solutions either from the class

N_{0}

(positive decreasing solutions) or

N_{2}

(positive increasing solutions). We establish new criteria for

N_{0} = \emptyset

and

N_{2} = \emptyset

provided that delayed and advanced parts of deviating argument are large enough. As a consequence of these results, we provide new oscillatory criteria. The presented results essentially improve existing ones even for a linear case of considered equations. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

23 pages, 349 KiB

Open AccessArticle

New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

by Irena Jadlovská

Mathematics 2021, 9(17), 2089; https://doi.org/10.3390/math9172089 - 29 Aug 2021

Cited by 36 | Viewed by 2335

Abstract

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based [...] Read more.

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

11 pages, 291 KiB

Open AccessArticle

Some New Oscillation Criteria of Even-Order Quasi-Linear Delay Differential Equations with Neutral Term

by Rongrong Guo, Qingdao Huang and Qingmin Liu

Mathematics 2021, 9(17), 2074; https://doi.org/10.3390/math9172074 - 27 Aug 2021

Cited by 4 | Viewed by 1742

Abstract

The neutral delay differential equations have many applications in the natural sciences, technology, and population dynamics. In this paper, we establish several new oscillation criteria for a kind of even-order quasi-linear neutral delay differential equations. Comparing our results with those in the literature, [...] Read more.

The neutral delay differential equations have many applications in the natural sciences, technology, and population dynamics. In this paper, we establish several new oscillation criteria for a kind of even-order quasi-linear neutral delay differential equations. Comparing our results with those in the literature, our criteria solve more general delay differential equations with neutral type, and our results expand the range of neutral term coefficient. Some examples are given to illustrate our conclusions. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

18 pages, 339 KiB

Open AccessArticle

On Sharp Oscillation Criteria for General Third-Order Delay Differential Equations

by Irena Jadlovská, George E. Chatzarakis, Jozef Džurina and Said R. Grace

Mathematics 2021, 9(14), 1675; https://doi.org/10.3390/math9141675 - 16 Jul 2021

Cited by 31 | Viewed by 2650

Abstract

In this paper, effective oscillation criteria for third-order delay differential equations of the form,

{(r_{2} {(r_{1} y^{'})}^{'})}^{'} (t) + q (t) y (τ (t)) = 0

ensuring that any nonoscillatory [...] Read more.

In this paper, effective oscillation criteria for third-order delay differential equations of the form,

{(r_{2} {(r_{1} y^{'})}^{'})}^{'} (t) + q (t) y (τ (t)) = 0

ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results. Full article

(This article belongs to the Special Issue Recent Advances in Oscillation Theory of Differential Equations: Problems, Solutions and Applications)

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