628 research outputs found
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
where 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 of the equation, there
exists a function whose Fourier transform decays as and
which represents solutions of the differential equation with accuracy on the
order of . 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 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
. 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 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 sgn is a positive
integer and the sequences 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
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