The spectrum of functions defined on R+; Application to functional equations
Suppose that M is a closed subspace of L-infinity (R+, X), where X is a complex Banach space. We define the M-spectrum sigma(M) (u) of a function u is an element of L-infinity(R+,X). Certain conditions will be supposed on both M and sigma(M)(u) to insure the existence of u is an element of M. We prove that if u is uniformly continuous and bounded, such that sigma(M) (u) is at most countable and, for every lambda is an element of sigma(M)(u), the function e(-i lambda t)u(t) is ergodic, then u is an element of M. We apply this result to the integro-differential operator equation a(u'(t)-Au(t))+beta integral(t)(0) u(t-s) d mu(s)=f(t), t >= 0, where the free term f is an element of M boolean AND C-ub (R+, X) and alpha, beta is an element of C. Here, A is the generator of a C-0-semigroup of linear bounded operators defined on X, D(A) is the domain of A, and mu is a bounded Borel measure on R+. Certain conditions will be imposed to guarantee the existence of solutions in the class M.
theory of almost periodic punctions harmonic analysis