Complete Second Order Linear Differential Equations in Hilbert Spaces: 92 (Operator Theory: Advances and Applications, 92) - Hardcover

Shklyar, Alexander Ya.

Hardcover

ISBN 10: 3764353775 ISBN 13: 9783764353773

Publisher: Birkhäuser, 1997

Synopsis

Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) + Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.

About this title

Synopsis

Hilbert spaces are Banach spaces where the norm is induced by the inner product. Incomplete second order differential equations and first order equations in Banach spaces are a classical part of functional analysis. This text attempts to present a unified systematic theory of second order differential equations (y"(t)+By'(t)+Cy(t)=0), including the well-posedness of the Cauchy, Dirichlet and Neumann problems; boundary conditions ensuring solvability of boundary-value problems; boundary behaviour and the extension of solutions on a finite interval; stabilization and stability of solutions at infinity; and boundary-value problems on a semi-line. The theory is developed in a special but important case, which can be considered as a model. Exhaustive answers to all the posed questions are given, with special emphasis on the effects arising for complete second order equations which do not arise for incomplete second order or first order equations. To achieve this, new results in the spectral theory of pairs of operators and the boundary behaviour of integral transformations have been developed.

Edition details

• Publisher: Birkhäuser
• Publication date: 1997
• Language: English
• ISBN 10: 3764353775
• ISBN 13: 9783764353773
• Binding: Hardcover
• Number of pages: 232