Integral Equations, Volterra Equations, and the Remarkable Resolvent: Contractions

with special accent on the case in which a(t) is unbounded. We use contraction mappings to establish close relations between a(t) and ∫ t 0 R(t, s)a(s)ds. This work gives us a fundamental understanding of the nature of R(t, s). It establishes numerous elementary boundedness results including some from a new point of view. And it tells us that one of our long-held basic assumptions is very incomplete. For more than one hundred years investigators have taken the view that, for a well behaved kernel C(t, s), the solution follows a(t): if a(t) is bounded, the solution x(t) is bounded; if a(t) is L, then x is L; if a(t) is periodic, x approaches a periodic function. Indeed, the author, himself, has formally stated this in a number of papers. A more accurate view may be that ∫ t 0 R(t, s)a(s)ds follows a(t) and, hence, there is the occasional appearance that x is following a(t), particularly when a(t) is bounded. But when a(t) is unbounded, we have a much clearer perception. Investigators spent much time in the 19th century devising methods of solving differential and integral equations in closed form. Although there are still vigorous areas of