John A. D. Appleby
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View article: Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs
Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs Open
This is the first of a two-part paper which determines necessary and sufficient conditions on the asymptotic behaviour of forcing functions so that the solutions of additively pertubed linear differential equations obey certain growth or d…
View article: Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations
Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations Open
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…
View article: Solution space characterisation of perturbed linear functional and integrodifferential Volterra convolution equations: Cesàro limits
Solution space characterisation of perturbed linear functional and integrodifferential Volterra convolution equations: Cesàro limits Open
In this article we discuss the requirements needed in order to characterise the solution space of perturbed linear integro-differential Volterra convolution equations. We highlight in general how the pointwise behaviour of perturbation fun…
View article: Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases
Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases Open
In this article we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are $p$-summable along with when their continuous time counterparts, perturbed linear stochastic Volterra integ…
View article: Solution space characterisation of perturbed linear Volterra integrodifferential convolution equations: The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si4.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> case
Solution space characterisation of perturbed linear Volterra integrodifferential convolution equations: The case Open
In this paper we characterise the Lp stability of the following perturbed linear Volterra integrodifferential convolution equation, x′(t)=∫[0,t]ν(ds)x(t−s)+f(t),t≥0,where ν is a finite signed Borel measure. Additionally we provide a framew…
View article: Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations
Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations Open
In this paper we investigate the mean square asymptotic stability of a perturbed scalar linear stochastic functional differential equation. Specifically, we are able to give necessary and sufficient conditions on the forcing terms for conv…
View article: Solution Space Characterisation of Perturbed Linear Volterra Integrodifferential Convolution Equations: the Lp case
Solution Space Characterisation of Perturbed Linear Volterra Integrodifferential Convolution Equations: the Lp case Open
In this paper we characterise the Lp stability of perturbed linear Volterra integrodifferential convolution equations. Additionally we provide a framework which points to necessary and sufficient conditions on the forcing function that ens…
View article: Growth and fluctuation in perturbed nonlinear Volterra equations
Growth and fluctuation in perturbed nonlinear Volterra equations Open
View article: Growth rates of solutions of superlinear ordinary differential equations
Growth rates of solutions of superlinear ordinary differential equations Open
View article: Exact and memory-dependent decay rates to the non-hyperbolic equilibrium of differential equations with unbounded delay and maximum functional
Exact and memory-dependent decay rates to the non-hyperbolic equilibrium of differential equations with unbounded delay and maximum functional Open
In this paper, we obtain the exact rates of decay to the non--hyperbolic equilibrium of the solution of a functional differential equation with maxima and unbounded delay. We study the convergence rates for both locally and globally stable…
View article: Hartman-Wintner growth results for sublinear functional differential equations
Hartman-Wintner growth results for sublinear functional differential equations Open
In this paper, we determine rates of growth to infinity of scalar autonomous nonlinear functional and Volterra differential equations. In these equations, the right-hand side is a positive continuous linear functional of a nonlinear functi…
View article: Sufficient Conditions for Polynomial Asymptotic Behaviour of the\n Stochastic Pantograph Equation
Sufficient Conditions for Polynomial Asymptotic Behaviour of the\n Stochastic Pantograph Equation Open
This paper studies the asymptotic growth and decay properties of solutions of\nthe stochastic pantograph equation with multiplicative noise. We give\nsufficient conditions on the parameters for solutions to grow at a polynomial\nrate in $p…
View article: Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation
Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation Open
This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-t…
View article: On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations
On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations Open
In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear ord…
View article: Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity
Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity Open
In this paper we consider the rate of convergence of solutions of a scalar ordinary differential equation which is a perturbed version of an autonomous equation with a globally stable equilibrium. Under weak assumptions on the nonlinear me…