Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911172570120192 |
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| author | Appleby, John A. D. Lawless, Emmet |
| author_facet | Appleby, John A. D. Lawless, Emmet |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16435 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations Appleby, John A. D. Lawless, Emmet Classical Analysis and ODEs Probability 34K06, 34K12, 34K20, 34K25, 34K50, 60H10 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. |
| title | Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations |
| topic | Classical Analysis and ODEs Probability 34K06, 34K12, 34K20, 34K25, 34K50, 60H10 |
| url | https://arxiv.org/abs/2410.16435 |