Stability of the discretized pantograph differential equation

In this paper we study discretizations of the general pantograph equation y’(t) = ay(t) + by(θ(t)) + cy’(∅(t)) , t≥0, y(0)=y0where a , b , c , and y0are complex numbers and where θ and ∅ a re strictly increasing functions on the nonnegative reals with θ(0) = ∅(0) = 0 and θ(t) < t, ∅(t)< t for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a , b , c and the stepsize which imply that the solution sequence ynn=0∞’s DOunded or that it tends to zero algebraically, as a negative power of n. © 1993 American Mathematical Society.