STABILITY OF THE DISCRETIZED PANTOGRAPH DIFFERENTIAL-EQUATION

In this paper we study discretizations of the general pantograph equat ion y'(t) = ay(t) + by(theta(t)) + cy'(phi(t)), t greater-than-or-equa l-to 0, y(0) = y0, where a , b , c , and yo are complex numbers and wh ere theta and phi are strictly increasing functions on the nonnegative reals with theta(0) = phi(0) = 0 and theta(t) < t, phi(t) < t for pos itive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify c onditions on a , b , c and the stepsize which imply that the solution sequence {y(n)}n=0 infinity is bounded or that it tends to zero algebr aically, as a negative power of n.