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Complete Second Order Linear Differential Equations in Hilbert Spaces

  • Book
  • © 1997

Overview

Authors:
  1. Alexander Ya. Shklyar

    1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

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  • The prerequisite for its understanding is the knowledgeof a first course in functional analysis and operator theory...
  • The monograph is a valuable reference book for postgraduate students and research mathematicians in analysis and differential equations." - M.-C. Anisiu

Part of the book series: Operator Theory: Advances and Applications (OT, volume 92)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      • Alexander Ya. Shklyar
      Pages 1-1

  4. Initial Data of Solutions

    1. Front Matter

      Pages 75-77
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    2. Boundary behaviour of an integral transform R(t) as t → 0 depending on the sub-integral measure

      • Alexander Ya. Shklyar
      Pages 79-99

    3. Initial data of solutions

      • Alexander Ya. Shklyar
      Pages 101-120

  5. Extension, Stability, and Stabilization of Weak Solutions

    1. Front Matter

      Pages 121-122
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    2. The general form of weak solutions

      • Alexander Ya. Shklyar
      Pages 123-132

    3. Fatou-Riesz property

      • Alexander Ya. Shklyar
      Pages 133-142

    4. Extension of weak solutions

      • Alexander Ya. Shklyar
      Pages 143-152

    5. Stability and stabilization of weak solutions

      • Alexander Ya. Shklyar
      Pages 153-170

  6. Boundary-Value Problems on a Half-Line

    1. Front Matter

      Pages 171-172
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    2. The Dirichlet problem on a half-line

      • Alexander Ya. Shklyar
      Pages 173-180

    3. The Neumann problem on a half-line

      • Alexander Ya. Shklyar
      Pages 181-190

  7. Back Matter

    Pages 191-220
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Keywords

About this book

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.

Authors and Affiliations

  • Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    Alexander Ya. Shklyar

Bibliographic Information

  • Book Title: Complete Second Order Linear Differential Equations in Hilbert Spaces

  • Authors: Alexander Ya. Shklyar

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-0348-9187-5

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Verlag 1997

  • Hardcover ISBN: 978-3-7643-5377-3Published: 18 February 1997

  • Softcover ISBN: 978-3-0348-9940-6Published: 05 October 2011

  • eBook ISBN: 978-3-0348-9187-5Published: 06 December 2012

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: XII, 220

  • Topics: Functional Analysis, Analysis

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