The defining property of an integral equation with resolvent R ( t , s ) is the relation between a ( t ) and ∫ 0 t R ( t , s ) a ( s ) d s for functions a ( t ) in a given vector space. We study the behaviour of a solution of an integral equation: x ( t ) = a 1 ( t ) + a 2 ( t ) − ∫ 0 t C ( t , s ) x ( s ) d s when a 1 ( t ) is periodic, C ( t + T , s + T ) = C ( t , s ) , while a 2 ( t ) is typified by ( t + 1 ) β with 0 < β < 1 . There is a resolvent, R ( t , s ) , so that x ( t ) = a 1 ( t ) + a 2 ( t ) − ∫ 0 t R ( t , s ) [ a 1 ( s ) + a 2 ( s ) ] d s . We show that the integral ∫ 0 t R ( t , s ) a 2 ( s ) d s 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 < ∞ . 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 ∫ 0 t R ( t , s ) a ( s ) d s 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 ) d s ] where k is constant. The integral ∫ 0 t R ( t , s ) sin ( s + 1 ) β d s is an L p approximation to sin ( t + 1 ) β for 0 < β < 1 , but contraction mappings show us that precisely at β = 1 that approximation fails and sin ( t + 1 ) − ∫ 0 t R ( t , s ) sin ( s + 1 ) d s approaches a nontrivial periodic function.