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In this paper the realization problems for the KreΘn-Langer class N ΞΊ of matrixvalued functions are being considered. We found the criterion when a given matrix-valued function from the class N ΞΊ can be realized as linear-fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii-Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Ξ ΞΊ with indefinite metric. We specify three subclasses of the class N ΞΊ (R) of all realizable matrix-valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Ξ ΞΊ. Alternatively we show that the class N ΞΊ (R) can be realized as transfer matrix-functions of some canonical impedance systems with self-adjoint main operators in rigged spaces Ξ ΞΊ . The case of scalar functions of the class N ΞΊ (R) is considered in details and some examples are presented.
π SIMILAR VOLUMES
In this paper, we employ a well-known fixed point theorem for cones to study the existence of positive periodic solutions to the n-dimensional system x + A(t)x = H(t)G(x). Moreover, the eigenvalue intervals for x + A(t)x = Ξ»H(t)G(x) are easily characterized.