Integral equations, large and small forcing functions: Periodicity

The defining property of an integral equation with resolvent R(t,s)R(t,s) is the relation between a(t)a(t) and ∫0tR(t,s)a(s)ds for functions a(t)a(t) in a given vector space. We study the behaviour of a solution of an integral equation: x(t)=a1(t)+a2(t)−∫0tC(t,s)x(s)ds when a1(t)a1(t) is periodic, C(t+T,s+T)=C(t,s)C(t+T,s+T)=C(t,s), while a2(t)a2(t) is typified by (t+1)β(t+1)β with 0<β<10<β<1. There is a resolvent, R(t,s)R(t,s), so that x(t)=a1(t)+a2(t)−∫0tR(t,s)[a1(s)+a2(s)]ds. We show that the integral ∫0tR(t,s)a2(s)ds so closely approximates a2(t)a2(t) that the only trace of that large function, a2(t)a2(t), in the solution is an LpLp-function, p<∞p<∞. In short, that large function a2(t)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 LpLp-function. The noteworthy property here is that with great precision the integral ∫0tR(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)a(t) other than k[1+∫−∞tC(t,s)ds] where kk is constant. The integral ∫0tR(t,s)sin(s+1)βds is an LpLp approximation to sin(t+1)βsin(t+1)β for 0<β<10<β<1, but contraction mappings show us that precisely at β=1β=1 that approximation fails and sin(t+1)−∫0tR(t,s)sin(s+1)ds approaches a nontrivial periodic function.