On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

Asymptotic solutions of forced nonlinear second order differential equations and their extensions

Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented

Improved estimates for nonoscillatory phase functions

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations play an important role in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.Comment: arXiv admin note: text overlap with arXiv:1409.438

Asymptotically polynomial solutions of difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form $$\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n$$ are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation

A fixed-point approach for decaying solutions of difference equations

A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}\Delta\bigl (a_{n}\Phi(\Delta x_{n})\bigr)+b_{n}\Phi(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive integer and the sequences $a,b,$ are positive. We deal with a particular type of decaying solutions of (\ref{*}), that is the so-called intermediate solutions (see below for the definition) . In particular, we prove the existence of these type of solutions for (\ref{*}) by reducing it to a suitable boundary value problem associated to a difference equation without deviating argument. Our approach is based on a fixed point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future researches complete the paper.Comment: accepted for publication on Philosophical Transactions of the Royal Society A. Issue: Topological degree and fixed point theories in differential and difference equations Editors: Maria Patrizia Pera and Marco Spadin

On the oscillatory behaviour of stochastic delay equations

This is an investigation of the causes of oscillatory behaviour in solutions of stochastic delay differential equations. Delay equations are used to study phenomena in which some part of the history of the system determines its evolution. Real-world interactions are often characterised by inefficiency and such equations are therefore widely used in applications. Real-world processes are also subject to interference in the form of random external perturbations or feedback noise. This interference can have a dramatic effect on the qualitative behaviour of these processes and so should be included in the mathematical analysis. Specifically, we consider the roles played by delayed feedback and noise perturbation in the onset of oscillation around an equilibrium solution. To this end, we consider a nonlinear equation with fixed delay, and a linear equation with asymptotically vanishing delay. Where necessary, results guaranteeing the global existence and uniqueness of solutions are presented. To facilitate the analysis of the linear equation, we present two difference schemes that are designed to mimic the oscillatory behaviour of its solutions. The first, a discretisation on a uniform mesh, is unsuccessful. We determine the reasons for this failure, and design a successful scheme based on this analysis. These choices allow the empirical manipulation of the relative involvement of the delay in the behaviour of solutions. In this way, and by comparison with the known qualitative behaviour of the corresponding deterministic delay differential equation, a picture of the mechanisms underlying oscillatory behaviour can be developed