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Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems

Volker Reitmann (2011)

Mathematica Bohemica

Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.

Relations between weighted Orlicz and B M O φ spaces through fractional integrals

Eleonor Ofelia Harboure, Oscar Salinas, Beatriz E. Viviani (1999)

Commentationes Mathematicae Universitatis Carolinae

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator I α maps weak weighted Orlicz - φ spaces into appropriate weighted versions of the spaces B M O ψ , where ψ ( t ) = t α / n φ - 1 ( 1 / t ) . This generalizes known results about boundedness of I α from weak L p into Lipschitz spaces for p > n / α and from weak L n / α into B M O . It turns out that the class of weights corresponding to I α acting on weak - L φ for φ of lower type equal or greater than n / α , is the same as the one solving the problem for weak...

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