Integral equations, large and small forcing functions: Periodicity

Abstract: The defining property of an integral equation with resolvent is the relation between and for functions in a given vector space. We study the behaviour of a solution of an integral equation: when is periodic, , while is typified by with . There is a resolvent, , so that We show that the integral so closely approximates that the only trace of that large function, , in the solution is an -function, . In short, that large function 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 -function. The noteworthy property here is that with great precision the integral can duplicate vector spaces of functions both large and small, both monotone and oscillatory; however, it cannot duplicate a given nontrivial periodic function other than where is constant. The integral is an approximation to for , but contraction mappings show us that precisely at that approximation fails and approaches a nontrivial periodic function. [Copyright &y& Elsevier]