Complete Second Order Linear Differential Equations in Hilbert Spaces

✍ Scribed by Alexander Ya. Shklyar (auth.)

• Publisher: Birkhäuser Basel
• Year: 1996
• Tongue: English
• Leaves: 224
• Series: Operator Theory Advances and Applications 92
• Edition: 1
• Category: Library

✦ 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.

✦ Table of Contents

Front Matter....Pages i-xii
Introduction....Pages 1-1
Front Matter....Pages 3-3
Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems....Pages 5-16
Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B ....Pages 17-33
The Cauchy problem....Pages 35-54
Boundary-value problems on a finite segment....Pages 55-74
Front Matter....Pages 75-77
Boundary behaviour of an integral transform R ( t ) as t → 0 depending on the sub-integral measure....Pages 79-99
Initial data of solutions....Pages 101-120
Front Matter....Pages 121-122
The general form of weak solutions....Pages 123-132
Fatou-Riesz property....Pages 133-142
Extension of weak solutions....Pages 143-152
Stability and stabilization of weak solutions....Pages 153-170
Front Matter....Pages 171-172
The Dirichlet problem on a half-line....Pages 173-180
The Neumann problem on a half-line....Pages 181-190
Back Matter....Pages 191-220

✦ Subjects

Mathematics, general