Representations of Green's function of the bounded solutions problem for a differential-algebraic equation

The equation (Fu')(t)=(Gu)(t)+f(t), t is an element of R, where F and G are bounded linear operators, is considered. It is assumed that infinity is a pole of the resolvent of the pencil lambda (sic) lambda F-G and the spectrum of the pencil is disjoint from the imaginary axis. Under these assumptions, to each free term f bounded on R (in the sense of distributions) there corresponds a unique bounded solution u and u(t)=integral(infinity)(-infinity) G(s)f(t-s) ds. The kernel G is called Green's function. In this paper, the representations of Green's function based on functional calculus in Banach algebras are constructed.