Fully-geometric mesh one-leg methods for the generalized pantograph equation: Approximating Lyapunov functional and asymptotic contractivity

Motivated by recent stability results on one-step methods, especially Runge-Kutta methods, for the generalized pantograph equation (GPE), in this paper we study the stability of one-leg multistep methods for these equations since the one-leg methods have less computational cost than Runge-Kutta methods. To do this, a new stability concept, Gq(q¯)-stability defined for variable stepsizes one-leg methods with the stepsize ratio q which is an extension of G-stability defined for constant stepsizes one-leg methods, is introduced. The Lyapunov functional of linear system is obtained and numerically approximated. It is proved that a Gq(q¯)-stable fully-geometric mesh one-leg method can preserve the decay property of the Lyapunov functional for any q∈[1,q¯]. The asymptotic contractivity, a new stability concept at vanishing initial interval, is introduced for investigating the effect of the initial interval approximation on the stability of numerical solutions. This property and the bounded stability of Gq(q¯)-stable one-leg methods for linear and nonlinear problems are analyzed. A numerical example which further illustrates our theoretical results is provided.