The defining property of an integral equation with resolvent R(t, s) is the relation between a(t) and t 0 R(t, s)a(s)ds for functions a(t) in a given vector space. We study the behaviour of a solution of an integral equation:
We show that the integral t 0 R(t, s)a 2 (s)ds so closely approximates a 2 (t) that the only trace of that large function, a 2 (t), in the solution is an L p -function, p < β. In short, that large function a 2 (t) has essentially no long-term effect on the solution which turns out to be the sum of a periodic function, a function tending to zero, and an L p -function. The noteworthy property here is that with great precision the integral t 0 R(t, s)a(s)ds can duplicate vector spaces of functions both large and small, both monotone and oscillatory; however, it cannot duplicate a given nontrivial periodic function a(t) other than k 1 + t -β C(t, s)ds where k is constant. The integral t 0 R(t, s) sin(s + 1) Ξ² ds is an L p approximation to sin(t + 1) Ξ² for 0 < Ξ² < 1, but contraction mappings show us that precisely at Ξ² = 1 that approximation fails and sin(t + 1) -t 0 R(t, s) sin(s + 1)ds approaches a nontrivial periodic function.