Oscillatory Behavior of Second-Order Nonlinear Neutral Differential Equations

Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/143614 Open Access We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.

Qualitative Theory of Differential Equations, Difference Equations, and Dynamic Equations on Time Scales

We are pleased to present this special issue. This volume reflects an increasing interest in the analysis of qualitative behavior of solutions to differential equations, difference equations, and dynamic equations on time scales. Numerous applications arising in the engineering and natural sciences call for the development of new efficient methods and for the modification and refinement of known techniques that should be adjusted for the analysis of new classes of problems. The twofold goal of this special issue is to reflect both the state-of-the-art theoretical research and important recent advances in the solution of applied problems.

Oscillation of fourth-order quasilinear differential equations

We study oscillatory behavior of a class of fourth-order quasilinear differential equations without imposing restrictive conditions on the deviated argument. This allows applications to functional differential equations with delayed and advanced arguments, and not only these. New theorems are based on a thorough analysis of possible behavior of nonoscillatory solutions; they complement and improve a number of results reported in the literature. Three illustrative examples are presented.

Project-based learning in a computer modelling course

AbstractThe paper reports authors’ experience of implementing educational projects in a computer modelling course offered to the students majoring in “Secondary Education (Computer Science)” at Ternopil Volodymyr Hnatiuk National Pedagogical University. We analyze approaches to teaching mathematical and computer modelling such as: integration of modelling tasks, naturalistic case study, using of role-playing games, possibilities of STEM-education, motivation and positive attitude to modelling training, etc. Then we illustrate the implementation of the project to study the population dynamics of the grape snail Helix pomatia. The implementation of the project splits into several stages: form…

Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameterσassociated with the intensity of harvesting i…

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/395368 Open Access We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

Global Non-monotonicity of Solutions to Nonlinear Second-Order Differential Equations

We study behavior of solutions to two classes of nonlinear second-order differential equations with a damping term. Sufficient conditions for the first derivative of a solution x(t) to change sign at least once in a given interval (in a given infinite sequence of intervals) are provided. These conditions imply global non-monotone behavior of solutions.

Asymptotic behavior of an odd-order delay differential equation

Published version of an article in the journal: Boundary Value Problems. Also available from the publisher at: http://dx.doi.org/10.1186/1687-2770-2014-107 Open Access We study asymptotic behavior of solutions to a class of odd-order delay differential equations. Our theorems extend and complement a number of related results reported in the literature. An illustrative example is provided.

On asymptotic behavior of solutions to higher-order sublinear Emden–Fowler delay differential equations

Abstract We study asymptotic behavior of solutions to a class of higher-order sublinear Emden–Fowler delay differential equations. Our theorems improve several results reported recently in the literature. Two examples are provided to illustrate the importance and advantages of new criteria.

Oscillation of second-order nonlinear differential equations with damping

Abstract We study oscillatory properties of solutions to a class of nonlinear second-order differential equations with a nonlinear damping. New oscillation criteria extend those reported in [ROGOVCHENKO, Yu. V.—TUNCAY, F.: Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008), 208–221] and improve a number of related results.

Oscillation criteria for even-order neutral differential equations

Abstract We study oscillatory behavior of solutions to a class of even-order neutral differential equations relating oscillation of higher-order equations to that of a pair of associated first-order delay differential equations. As illustrated with two examples in the final part of the paper, our criteria improve a number of related results reported in the literature.

Mathematical Modelling with Biology Undergraduates: Balancing Task Difficulty and Level of Support

We report on extra-curricular activities with biology undergraduates focusing our attention on the selection of mathematical modelling tasks with different levels of cognitive demand and the level of teacher’s guidance during students’ collaborative work on the tasks.

Oscillation results for second-order nonlinear neutral differential equations

Published version of an article in the journal: Advances in Difference Equations. Also available from the publisher at: http://dx.doi.org/10.1186/1687-1847-2013-336 Open Access We obtain several oscillation criteria for a class of second-order nonlinear neutral differential equations. New theorems extend a number of related results reported in the literature and can be used in cases where known theorems fail to apply. Two illustrative examples are provided.

Nonlinear Functional Difference Equations with Applications

Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales

and Applied Analysis 3 thank Guest Editors Josef Dibĺik, Alexander Domoshnitsky, Yuriy V. Rogovchenko, Felix Sadyrbaev, and Qi-Ru Wang for their unfailing support with editorial work that ensured timely preparation of this special edition. Tongxing Li Josef Dibĺik Alexander Domoshnitsky Yuriy V. Rogovchenko Felix Sadyrbaev Qi-Ru Wang

‘I’m still making dots for them’: mathematics lecturers’ views on their mathematical modelling practices

Using Cultural-Historical Activity Theory, we analyze lecturers’ views on the aims and teaching practices of mathematical modelling (MM) education in Norway and England. We aim to expose the tensio...

Oscillation of second-order neutral differential equations

Author's version of an article in the journal: Funkcialaj Ekvacioj. Also available from the publisher at: http://www.math.kobe-u.ac.jp/~fe/

On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

Abstract By using comparison principles, we analyze the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Due to less restrictive assumptions on the coefficients of the equation and on the deviating argument τ , our criteria improve a number of related results reported in the literature.

Oscillation theorems for second-order nonlinear neutral delay differential equations

Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/594190 Open Access We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.

Joy of Mathematical Modelling: A Forgotten Perspective?

We argue the relevance of including an affective perspective in the mathematical modelling education research and emphasise its importance for the teaching and learning of mathematical modelling at all levels, especially at the university. Our argument is supported by a recent survey of mathematics lecturers’ views on mathematical modelling, several follow-up interviews, and a review of literature on mathematical modelling that relates to enjoyment, pleasure, and appreciation. Findings from the survey and the follow-up interviews indicate that there is a group of practitioners who hold strong views on the importance of enjoyment in doing and teaching mathematical modelling.

Functional Differential and Difference Equations with Applications

and Applied Analysis 3 solutions to a class of nonlocal boundary value problems for linear homogeneous secondorder functional differential equations with piecewise constant arguments are obtained. The last but not the least, this issue features a number of publications that report recent progress in the analysis of problems arising in various applications. In particular, dynamics of delayed neural network models consisting of two neurons with inertial coupling were studied, properties of a stochastic delay logistic model under regime switching were explored, and analysis of the permanence and extinction of a single species with contraception and feedback controls was conducted. Other applie…