Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation

Abstract

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-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in p-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in p-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.

Stability results for 2D Navier–Stokes equations with unbounded delay

“…We would also like to mention that for particular unbounded variable delays, the exponential stability of stationary solutions for delayed ODE cannot be obtained (cf. [1]). However, inspired in those results, we will be able to obtain some polynomial stability for problem (P) in the case of proportional variable delays.…”

“…In fact, even for simple ordinary differential equations with unbounded variable delay, for instance, the pantograph equation, in which the delay term is given by ρ(t) = (1 − λ)t with 0 < λ < 1, the exponential stability of stationary solution cannot be reached. But, fortunately, in this simple case the polynomial stability of stationary solution can be proved, see [27,26,1] for details. Enlightened by [1], we show that it is still possible to prove the polynomial stability of stationary solution to Navier-Stokes equations with proportional delay, which is a particular case of unbounded variable delay.…”

The Global Solutions and Moment Boundedness of Stochastic Multipantograph Equations

“…Fan et al [9] have given the sufficient conditions of existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for (2). Appleby and Buckwar [10] have studied the asymptotic growth and decay properties of solutions of the linear stochastic pantograph equation with multiplicative noise. For more literatures we refer the interested reader to [11][12][13].…”

“…Properties of the analytic solution of (1) and (2) as well as numerical methods have been studied by several authors, for example, Lü and Cui [14], Iserles [15,16], Liu et al [17], and 2 Journal of Control Science and Engineering Appleby and Buckwar [10]. A more general form than (1) is the multipantograph equation; it readṡ ( ) = ( , ( ) , ( 1 ) , ( 2 ) , .…”

The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

“…SPDEs can be regarded as a pantograph differential equation perturped by Brownian motion. For example, existence and stability of solution to SPDEs are given in [21][22][23]. Various efficient computational methods are obtained and their convergence and stability have been studied by many authors [24][25][26][27][28][29].…”