A stability analysis of trapezoidal methods for Volterra integral equationa with completely positive kernels

The solution of the Volterra integral equation with completely positive kernel y(t) + int_0^t b(t − s) y(s) ds = u0 + int_0^t b(t − s) g(s) ds, t ⩾ 0, is nonnegative and nonincreasing provided that g is nonincreasing and 0 ⩽ g(t) ⩽ u0 for any t > 0. We prove that under some additional hypotheses this property is inherited by the solution of the recurrence relation resulting from applying the trapezoidal method to this equation.