Complete Second Order Linear Differential Equations in Hilbert Spaces

1. Front Matter

   Pages i-xii

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2. Introduction

   1. Introduction

      • Alexander Ya. Shklyar

      Pages 1-1

3. Well-Posedness of Boundary-Value Problems

   1. Front Matter

      Pages 3-3

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   2. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems

      • Alexander Ya. Shklyar

      Pages 5-16

   3. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B

      • Alexander Ya. Shklyar

      Pages 17-33

   4. The Cauchy problem

      • Alexander Ya. Shklyar

      Pages 35-54

   5. Boundary-value problems on a finite segment

      • Alexander Ya. Shklyar

      Pages 55-74

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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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.

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

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

  Alexander Ya. Shklyar

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