Harmonic analysis and asymptotic behavior of solutions to the abstract cauchy problem

Abstract

LetC ub ( $$\mathbb{J}$$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $$\mathbb{J}$$ 2 e {ℝ+, ℝ} with values in a Banach spaceX. Let ℱ be a class fromC ub( $$\mathbb{J}$$ ,X). We introduce a spectrumspℱ(φ) of a functionφ eC ub (ℝ,X) with respect to ℱ. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ℝ to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ e ℱ, whereA is the generator of aC 0-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩iℝ is countable, all bounded uniformly continuous mild solutions on ℝ+ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ℝ+ in the cases (i)T(t) is a uniformly exponentially stableC 0-semigroup andφ eC ub(ℝ,X); (ii)T(t) is a uniformly bounded analyticC 0-semigroup,φ eC ub (ℝ,X) andσ(A) ∩i sp(φ) = O. Under the condition (i) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), then the solutions belong to ℱ. In case (ii) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), andT(t) is almost periodic, then the solutions belong to ℱ. The existence of mild solutions on ℝ to (*) is also discussed.