Semilinear evolution equations of second order via maximal regularity.
Maximal regularity for flexible structural systems in Lebesgue spaces.
Maximal regularity of the discrete harmonic oscillator equation.
Almost automorphic solutions of difference equations.
Spectral properties of cosine operator functions.
Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces
We use Fourier multiplier theorems to establish maximal regularity results for a class of integro-differential equations with infinite delay in Banach spaces. Concrete equations of this type arise in viscoelasticity theory. Results are obtained for periodic solutions in the vector-valued Lebesgue and Besov spaces. An application to semilinear equations is considered.
Maximal regularity of delay equations in Banach spaces
We characterize existence and uniqueness of solutions for an inhomogeneous abstract delay equation in Hölder spaces. The main tool is the theory of operator-valued Fourier multipliers.
Periodic solutions of degenerate differential equations in vector-valued function spaces
Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multiplier theorems, we obtain necessary and sufficient conditions for the existence and uniqueness of periodic solutions to the equation d/dt(Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our results are obtained in the scales of periodic Besov and periodic Lebesgue vector-valued spaces.
Linear dynamics of semigroups generated by differential operators
During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss...
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