Overview

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.

Full Product Details

Author:   Alexander Ya. Shklyar
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1997 ed.
Volume:   92
Dimensions:   Width: 15.50cm , Height: 1.40cm , Length: 23.50cm
Weight:   1.120kg
ISBN:  

9783764353773

ISBN 10:   3764353775
Pages:   220
Publication Date:   18 February 1997
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I. Well-posedness of boundary-value problems.- to Part I.- 1. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems.- 2. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B.- 3. The Cauchy problem.- 4. Boundary-value problems on a finite segment.- II. Initial data of solutions.- to Part II.- 5. Boundary behaviour of an integral transform R(t) as t ? 0 depending on the sub-integral measure.- 6. Initial data of solutions.- III. Extension, stability, and stabilization of weak solutions.- to Part III.- 7. The general form of weak solutions.- 8. Fatou-Riesz property.- 9. Extension of weak solutions.- 10. Stability and stabilization of weak solutions.- IV. Boundary-value problems on a half-line.- to Part IV.- 11. The Dirichlet problem on a half-line.- 12. The Neumann problem on a half-line.- Commentaries on the literature.- List of symbols.

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