Integral equations, large and small forcing functions: Periodicity

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 behavior of a solution of an integral equation x(t) = a1(t) + a2(t) − ∫ t 0 C(t, s)x(s)ds when a1(t) is periodic, C(t+ T, s+ T ) = C(t, s), while a2(t) is typified by (t+ 1) β with 0 < β < 1. There is a resolvent, R(t, s), so that x(t) = a1(t) + a2(t) − ∫ t 0 R(t, s)[a1(s) + a2(s)]ds. We show that the integral ∫ t 0 R(t, s)a2(s)ds so closely approximates a2(t) that the only trace of that large function, a2(t), in the solution is an L -function, p < ∞. In short, that large function a2(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-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 −∞ C(t, s)ds ] where k is constant. The integral ∫ t 0 R(t, s) sin(s+1)ds is an L approximation to sin(t+1) for 0 < β < 1, but contraction mappings show us that precisely at β = 1 that approximation fails and sin(t + 1)− ∫ t 0 R(t, s) sin(s+ 1)ds approaches a nontrivial periodic function.