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Quadratic functionals: positivity, oscillation, Rayleigh's principle

Werner Kratz (1998)

Archivum Mathematicum

In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix...

Quadratic functionals with a variable singular end point

Zuzana Došlá, PierLuigi Zezza (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper we introduce the definition of coupled point with respect to a (scalar) quadratic functional on a noncompact interval. In terms of coupled points we prove necessary (and sufficient) conditions for the nonnegativity of these functionals.

Qualitative theory of half-linear second order differential equations

Ondřej Došlý (2002)

Mathematica Bohemica

Some recent results concerning properties of solutions of the half-linear second order differential equation ( r ( t ) Φ ( x ' ) ) ' + c ( t ) Φ ( x ) = 0 , Φ ( x ) : = | x | p - 2 x , p > 1 , ( * ) are presented. A particular attention is paid to the oscillation theory of ( * ) . Related problems are also discussed.

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