Artículo de publicación ISIIn this paper we establish the existence of mild solutions for a non-autonomous abstract semi-linear second order differential equation submitted to nonlocal initial conditions. Our approach relies on the existence of an evolution operator for the corresponding linear equation and the properties of the Hausdorff measure of non-compactness.Partially supported by: CONICYT under grant FONDECYT 1130144 and DICYT-USACH, FONDECYT 1110090 and MECESUP PUC 0711

Artículo de publicación ISIIn this paper, we give a necessary and sufficient conditions for the existence and uniqueness of periodic solutions of inhomogeneous abstract fractional differential equations with delay. The conditions are obtained in terms of R-boundedness of operator-valued Fourier multipliers determined by the abstract model.FONDECYT 1100485 FONDECYT de Iniciacion 1107504

MSC: 45N05 47D06 45J05 47N20 34G20 Keywords: k-regular sequence M-boundedness R-boundedness UMD spaces Lebesgue-Bochner spaces Besov spaces Triebel-Lizorkin spaces Strong and mild well-posedness Operator-valued Fourier multiplierOperator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces L p , the Besov spaces B s p,q (and related spaces such as the Hölder-Zygmund spaces C s ) and the Triebel-Lizorkin spaces F s p,q . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integrodifferential equations.

Maximal regularity for an integro-differential equation with infinite delay on periodic vectorvalued Besov spaces is studied. We use Fourier multipliers techniques to characterize periodic solutions solely in terms of spectral properties on the data. We study a resonance case obtaining a compatibility condition which is necessary and sufficient for the existence of periodic solutions.

We study abstract equations of the form λu t u t c 2 Au t c 2 μAu t f t , 0 < λ < μ which is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of α; β; γ -regularized families, which is a particular case of a; kregularized families, and characterize maximal regularity in L p -spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given. Hindawi Publishing Corporation