Harmonic analysis and asymptotic behavior solutions to the abstract Cauchy problem

Let C-ub(J,X) denote the Banach space of all uniformly continuous bounded functions defined on J is an element of {R(+),R} with values in a Banach space X. Let F be a class from C-ub(J,X). We introduce a spectrum sp(F)(phi) of a function phi is an element of C-ub(R,X) with respect to F. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on R(+) and all bounded uniformly continuous mild solutions on R to the abstract Cauchy problem (*) omega'(t) = A omega(t) + phi(t), omega(0) = x, phi is an element of F, where A is the generator of a C-0-semigroup T(t) of bounded operators. If phi = 0 and sigma(A) boolean AND iR is countable, all bounded uniformly continuous mild solutions on R(+) to (*) are studied. We prove the boundedness and uniform continuity of all mild solutions on R(+) in the cases (i) T(t) is a uniformly exponentially stable C-0-semigroup and phi is an element of C-ub(R,X); (ii) T(t) is a uniformly bounded analytic C-0-semigroup, phi is an element of C-ub(R,X) and sigma(A) boolean AND i sp(phi) = empty set. Under the condition (i) if the restriction of phi to R(+) belongs to F = F(R(+),X), then the solutions belong to F. In case (ii) if the restriction of phi to R(+) belongs to F = F(R(+),X) and T(t) is almost periodic, then the solutions belong to F. The existence of mild solutions on R to (*) is also discussed.