Integral equations, large forcing, strong resolvents

We consider an integral equation x(t) = a(t) − R t 0 C(t, s)x(s)ds, a resolvent R(t, s), and a variation of parameters formula x(t) = a(t)− R t 0 R(t, s)a(s)ds. We present three general results concerning the behavior of solutions based on the signs of C(t, s) and its derivatives, in conjunction with the magnitude of a(t) or of a ′ (t). The first result shows that (x(t) − a(t))2 is bounded even when a(t) is unbounded. The second result, under a different set of conditions on the derivatives of C, shows that x(t) is bounded when a ′ is bounded. Classical theory favors the idea that x(t) follows a(t) when the kernel is nice. In three recent papers we disputed this, providing many results in which R t 0 R(t, s)a(s)ds faithfully duplicates a(t), even when a(t) is unbounded, resulting in x(t) being bounded and, most interestingly, bearing absolutely no relation to a(t). Most of this set of results was based on certain smallness conditions on C(t, s). By taking into account sign conditions on C and its derivatives we find that both views can be defended even when a(t) is unbounded. There is now ample evidence that there are two strikingly different theories about the relation of the solution to a(t). These should be very important and rewarding areas for investigation.