Global stability condition for collocation methods for Volterra integral equations of the second kind (Q916329)

Consider the linear Volterra integral equation with degenerate kernel, \[ (*)\quad y(t)=g(t)+\int^{t}_{0}\sum^{n}_{i=1}a_ i(t)b_ i(s)y(s)ds,\quad t\geq 0, \] with continuous data g, \(a_ i\), and \(b_ i\). The authors show that if g and \(\sum^{n}_{i=1}| a_ i(t)|\) are bounded on \([0,\infty)\), and if \(b_ i\in L^ 1[0,\infty)\), then the solution of (*) remains bounded. Moreover, this boundedness property is inherited by the collocation solution in the space of piecewise continuous polynomial spline functions of degree m-1, for any stepsize \(h>0\) for which the collocation solution is uniquely defined.