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Complete Second Order Linear Differential Equations in Hilbert Spaces

  • Alexander Ya. Shklyar

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

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction

    1. Alexander Ya. Shklyar
      Pages 1-1
  3. Well-Posedness of Boundary-Value Problems

  4. Initial Data of Solutions

    1. Front Matter
      Pages 75-77
    2. Alexander Ya. Shklyar
      Pages 101-120
  5. Extension, Stability, and Stabilization of Weak Solutions

    1. Front Matter
      Pages 121-122
    2. Alexander Ya. Shklyar
      Pages 123-132
    3. Alexander Ya. Shklyar
      Pages 133-142
    4. Alexander Ya. Shklyar
      Pages 143-152
    5. Alexander Ya. Shklyar
      Pages 153-170
  6. Boundary-Value Problems on a Half-Line

    1. Front Matter
      Pages 171-172
    2. Alexander Ya. Shklyar
      Pages 173-180
    3. Alexander Ya. Shklyar
      Pages 181-190
  7. Back Matter
    Pages 191-220

About this book

Introduction

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.

Keywords

Finite Hilbert space calculus equation function functional analysis integral transform partial differential equations proof theorem

Authors and affiliations

  • Alexander Ya. Shklyar
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

Bibliographic information