Algebraic and analytic properties of solutions of abstract differential equations

R. Bittner

GDML_Books, (1964), p.

Résumé

CONTENTSINTRODUCTION............................................................................................................................... 3Chapter I. ALGEBRAIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIALEQUATIONS§ 1. Ordinary abstract differential equations1. Taylor’s formula for an abstract derivative.......................................................................... 42 π-solutions................................................................................................................................. 5§ 2. Fundamental system of solving operations in linear spaces and algebras1. Operational independence and solving operations.......................................................... 82. One linear differential equation of the first order................................................................. 93. A system of linear differential equations of the first order............................................... 114. Linear differential equations of order n.............................................................................. 155. Partial derivatives..................................................................................................................... 186. Linear partial differential equations...................................................................................... 207. Wroński’s fundamentality criteria in algebras................................................................. 248. Examples................................................................................................................................ 25§ 3. Universal spaces of analytic elements1. Introduction............................................................................................................................. 262. The space $C_N(ℬ)$........................................................................................................... 273. Multiplications, superposition and convolution of elementsof $C_N(ℬ)$.................................................................................................................................. 294. The space $C_N^m(ℬ)$ of analytic functions of many multipliers................................. 326. Examples.................................................................................................................................. 33Chapter II. ANALYTIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIALEQUATIONS§ 4. Existence, uniqueness and continuity of solutions1. Regular operations in $K_Z$-linear spaces....................................................................... 352. The well-defined problem of solution of an abstract differential equation.................... 373. Examples................................................................................................................................... 41§ 5. Analytic elements1. Introduction.............................................................................................................................. 43§ 6. The separation of variables1. The separation of variables.................................................................................................. 462. Examples................................................................................................................................. 49§ 7. Summation theorem1. The Kojima-Schur and the Toeplitz theorems................................................................. 522. Euler’s theorems..................................................................................................................... 643. Newton’s interpolation formulas........................................................................................ 554. Examples................................................................................................................................. 59REFERENCES............................................................................................................................ 61

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