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# Existence of bounded uniformly continuous mild solutions on $\Bbb{R}$ of evolution equations and some applications

08/2011;

Source: arXiv

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**ABSTRACT:**LetC ub ( \mathbbJ\mathbb{J} , X) denote the Banach space of all uniformly continuous bounded functions defined on \mathbbJ\mathbb{J} 2 ε {ℝ+, ℝ} with values in a Banach spaceX. Let ℱ be a class fromC ub( \mathbbJ\mathbb{J} ,X). We introduce a spectrumspℱ(φ) of a functionφ εC 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 ε 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 εC ub(ℝ,X); (ii)T(t) is a uniformly bounded analyticC 0-semigroup,φ εC ub (ℝ,X) andσ(A) ∩i sp(φ) = Ø. 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.Semigroup Forum 12/1997; 54(1):58-74. · 0.38 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Suppose =[α, ∞) for someα∈ or = and letXbe a Banach space. We study asymptotic behavior of solutions on of neutral system of equations with values inX. This reduces to questions concerning the behavior of solutions of convolution equations (*)H∗Ω=b, whereH=(Hj, k) is anr×rmatrix,Hj, k∈′L1,b=(bj) andbj∈′(, X), for 1⩽j, k⩽r. We prove that ifΩis a bounded uniformly continuous solution of (*) withbfrom some translation invariant suitably closed class , thenΩbelongs to , provided, for example, that det Hhas countably many zeros on andc0⊄X. In particular, ifbis (asymptotically) almost periodic, almost automorphic or recurrent,Ωis too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equationΩ(t+1)−Ω(t)=b(t) with =X= andbStepanoff almost periodic.Journal of Differential Equations 10/1998; · 1.57 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For solutions y of a linear difference-differential system on a halfline or the reals it is shown that if y and f belong to a class of Banach-space valued functions, then all derivatives y (j) are in too, The main assumption on is here: If and all differences for then for with Examples are the almost periodic asymptotic ap, almost automorphic, Eberlein weakly ap, Stepanoff ap, pseudo ap, various classes of ergodic functions, weighted L p , C k , uniformly continuous functions, O(w). The assumption can be weakened to y an element of some “mean” class where for Also proofs of various results on mean classes needed here and of for the mentioned above are supplied.Journal of Difference Equations and Applications 09/2004; 10(11):1005-1023. · 0.86 Impact Factor

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