Martin Bohner

Dynamic equations on time scales: a survey

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. In this paper we give an introduction to the time scales calculus. We also present various properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of first order. Several examples and applications, among them an insect population model, are considered. We then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients. Finally, we consider self-adjoint equations and, more generally, so-called symplectic systems, and present several results on the positivity of quadratic functionals.

Basic Calculus on Time Scales and some of its Applications

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.

Discrete Oscillation Theory

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Sturm-Liouville eigenvalue problems on time scales

For Sturm-Liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish Rayleighs principle. Our results not only unifly the corresponding theories for differential and difference equations, but are also new in the discrete case.

Nonoscillation and Oscillation: Theory for Functional Differential Equations

This book summarizes the qualitative theory of differential equations with or without delays, collecting recent oscillation studies important to applications and further developments in mathematics, physics, engineering, and biology. The authors address oscillatory and nonoscillatory properties of first-order delay and neutral delay differential equations, second-order delay and ordinary differential equations, higher-order delay differential equations, and systems of nonlinear differential equations. The final chapter explores key aspects of the oscillation of dynamic equations on time scales-a new and innovative theory that accomodates differential and difference equations simultaneously.

Impulsive differential equations

This paper deals with the periodic solutions problem for impulsive differential equations. By using Lyapunovs second method and the contraction mapping principle, some conditions ensuring the existence and global attractiveness of unique periodic solutions are derived, which are given from impulsive control and impulsive perturbation points of view. As an application, the existence and global attractiveness of unique periodic solutions for Hopfield neural networks are discussed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed results.

PARTIAL DIFFERENTIATION ON TIME SCALES

Let \(n\in {\mathbb {N}}\) be fixed. For each \(i\in \{1,2,\ldots ,n\}\), we denote by \({\mathbb {T}}_i\) a time scale.

Oscillation criteria for perturbed nonlinear dynamic equations

In this paper, we discuss the oscillatory behavior of a certain nonlinear perturbed dynamic equation on time scales. We establish some new oscillation criteria for such dynamic equations and supply examples.

On the oscillation of second-order half-linear dynamic equations1

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation when or . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases.

Multiple Integration on Time Scales

Let \({\mathbb {T}}_i\), \(i\in \{1,2,\ldots ,n\}\), be time scales. For \(i\in \{1,2,\ldots ,n\}\), let \(\sigma _i\), \(\rho _i\), and \(\varDelta _i\) denote the forward jump operator, the backward jump operator, and the delta differentiation, respectively, on \({\mathbb {T}}_i\).