Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations

TL;DR

This paper investigates the long-term behavior of both deterministic and stochastic pantograph equations under forcing, establishing conditions for convergence, boundedness, and inheritance of power law decay, with extensions to more general equations.

Contribution

It provides necessary and sufficient conditions for the asymptotic behavior of forced pantograph equations, including extensions to unbounded delays and finite-dimensional cases.

Findings

Solutions converge to equilibrium under specific forcing conditions

Power law decay can be inherited by solutions with appropriate forcing

Extensions include equations with unbounded delay and finite-dimensional systems

Abstract

This paper considers the asymptotic behaviour of deterministically and stochastically forced linear pantograph equations. The asymptotic behaviour is studied in the case when all solutions of the pantograph equation without forcing tend to a trivial equilibrium. In all cases, we give necessary and sufficient conditions on the forcing terms which enable all solutions to converge to the equilibrium of the unforced equation, and which enable solutions to remain bounded. In the deterministic case, we give sharp conditions on forcing terms which enable the solutions of the forced equations to inherit the power law behaviour of the unforced equation, as well as slower rates of decay or growth not present in the unforced equation. Extensions to equations with general unbounded delay, and to finite--dimensional equations are also presented.