Theory of Differential Equations

Translator's NotePreface to the Russian EditionChapter I Spaces of Type W 1. Definitions 2. Bounded Operators in Spaces of Type W 3. Fourier Transforms 4. The Case of Several VariablesChapter II Uniqueness Classes for the Cauchy Problem 1. Introduction 2. The Cauchy Problem in a Topological Vector Space 3. The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method 4. The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms 5. Examples 6. The Connection Between the Reduced Order of a System and its Characteristic Roots 7. A Theorem of the Phragmén-Lindelöf Type Appendix 1. Convolution Equations Appendix 2. Equations with Coefficients Which Depend on x Appendix 3. Systems with Elliptic OperatorsChapter III Correctness Classes for the Cauchy Problem 1. Introduction 2. Parabolic Systems 3. Hyperbolic Systems 4. Systems Which are Petrovskii-Correct 5. On the Solutions of Incorrect SystemsChapter IV Generalized Eigenfunction Expansions 1. Introduction 2. Differentiation of Functionals of Strongly Bounded Variation 3. Differentiation of Functionals of Weakly Bounded Variation 4. Existence and Completeness Theorems for the System of Eigenfunctionals 5. Generalized Eigenfunctions of Self-Adjoint Operators 6. The Structure of the Generalized Eigenfunctions 7. Dynamical SystemsNotes and ReferencesBibliographyIndex