Differential Equations with Operator Coefficients: with Applications to Boundary Value Problems for Partial Differential Equations: Springer Monographs in Mathematics

I. Differential Equations with Constant Operator Coefficients.- 1. Power-Exponential Zeros.- 2. Differential Operator Equations in Weighted Sobolev Spaces.- 3. Solutions in a Local Sobolev Space.- 4. Two-Weight L2-Estimates.- II. Differential Equations with Variable Operator Coefficients.- 5. Existence, Uniqueness and “Pointwise” Estimates.- 6. Corollaries of Previous Results Under Special Assumptions on L(t, Dt).- 7. Two-Weight L2-Estimates for Equations with Variable Coefficients.- 8. Connection of Solutions Corresponding to Different Strips.- 9. Applications to the Case of Perturbations Vanishing at Infinity.- 10. Variants and Extensions of the Previous Theory.- III. Asymptotic Theory of Operator Differential Equations.- 11. Complete Asymptotic Expansions Under Exponential and Power Perturbations of A(Dt).- 12. Reduction to a First Order System.- 13. General Asymptotic Representation.- 14. Power-Exponential Asymptotics.- 15. The Case of One Simple Eigenvalue on the Line.- 16. Several Simple Eigenvalues on the Line.- 17. The Case of a Single Multiple Eigenvalue.- A. Holomorphic Operator Functions.- A.1 Introduction.- A.2 Prerequisites on Fredholm Operators.- A.3 Basic Notions of the Spectral Theory of Holomorphic Operator Functions.- A.5 The Local Equivalence of Holomorphic Operator Functions.- A.6 The Smith Form of a Holomorphic Matrix Function.- A.7 The Resolvent of a Holomorphic Matrix Function.- A.8 Fredholm Holomorphic Operator Functions.- A.9 The Adjoint Holomorphic Operator Function.- References.- Index of Notation.- Index of Names.

From the reviews of the first edition:
"The book under review is the first systematic and self-contained presentation of a theory of arbitrary order ordinary differential equations with unbounded operator coefficients in a Hilbert or Banach space … . this is an excellent book, that contains recent results of the topic, deep theoretical results and various applications to PDE-s. It is warmly recommended to specialists in ODE-s, PDE-s, functional analysis." (Jeno Hegedus, Acta Scientiarum Mathematicarum, Vol. 72, 2006)