Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Książka - szczegóły

Tytuł książki

## Algebraic and analytic properties of solutions of abstract differential equations

### Seria

Rozprawy Matematyczne tom/nr w serii: 41 wydano: 1964

### Abstrakty

EN

CONTENTS
INTRODUCTION............................................................................................................................... 3
Chapter I. ALGEBRAIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIAL
EQUATIONS
§ 1. Ordinary abstract differential equations
1. Taylor’s formula for an abstract derivative.......................................................................... 4
2 π-solutions................................................................................................................................. 5
§ 2. Fundamental system of solving operations in linear spaces and algebras
1. Operational independence and solving operations.......................................................... 8
2. One linear differential equation of the first order................................................................. 9
3. A system of linear differential equations of the first order............................................... 11
4. Linear differential equations of order n.............................................................................. 15
5. Partial derivatives..................................................................................................................... 18
6. Linear partial differential equations...................................................................................... 20
7. Wroński's fundamentality criteria in algebras................................................................. 24
8. Examples................................................................................................................................ 25
§ 3. Universal spaces of analytic elements
1. Introduction............................................................................................................................. 26
2. The space $C_N(ℬ)$........................................................................................................... 27
3. Multiplications, superposition and convolution of elements
of $C_N(ℬ)$.................................................................................................................................. 29
4. The space $C_N^m(ℬ)$ of analytic functions of many multipliers................................. 32
6. Examples.................................................................................................................................. 33
Chapter II. ANALYTIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIAL
EQUATIONS
§ 4. Existence, uniqueness and continuity of solutions
1. Regular operations in $K_Z$-linear spaces....................................................................... 35
2. The well-defined problem of solution of an abstract differential equation.................... 37
3. Examples................................................................................................................................... 41
§ 5. Analytic elements
1. Introduction.............................................................................................................................. 43
§ 6. The separation of variables
1. The separation of variables.................................................................................................. 46
2. Examples................................................................................................................................. 49
§ 7. Summation theorem
1. The Kojima-Schur and the Toeplitz theorems................................................................. 52
2. Euler’s theorems..................................................................................................................... 64
3. Newton’s interpolation formulas........................................................................................ 55
4. Examples................................................................................................................................. 59
REFERENCES............................................................................................................................ 61

Warszawa

### Seria

Rozprawy Matematyczne tom/nr w serii: 41

63

### Opis fizyczny

Rozprawy Matematyczne, Tom XLI

wydano
1964

autor

### Bibliografia

• [1] S. Bellert, On foundations of operational calculus, Bull. Acad. Polon. Sci., Cl. III, 5 (1957), pp. 855-858.
• [2] I. S. Berezin, N. P. Żidkow, Numerical methods (in Russian), Moskva 1959.
• [3] G. Birkhoff, Lattice theory, New York 1948 (in Russian), Moskva 1952.
• [4] R. Bittner, Operational calculus in linear spaces, Studia Math. 20 (1961), pp. 1-18.
• [5] R. Bittner, On certain axiomatics for the operational calculus, Bull. Acad. Polon. Sci., Cl. III, 7 (1959), pp. 1-9.
• [6] R. Bittner, On a new definition of polynomials, Bull, Acad. Polon. Sci., Cl. III, 9 (1961), pp. 79-84.
• [7] R. Bittner, Algebraic properties of linear derivative equations in linear spaces, Bull. Acad. Polon. Sci., Cl. III, 9 (1961), pp. 133-139.
• [8] R. Bittner, Algebraic properties of linear derivative equations in algebras, Bull. Acad. Polon. Sci., Cl. III, 9 (1961), pp. 141-142.
• [9] R. Bittner, Universal spaces for analytic elements, Bull. Acad. Polon. Sci., Cl. III, 10 (1962), pp. 441-443.
• [10] R. Bittner, Non linear derivative equations in $X_Z$-spaces, Bull. Acad. Polon. Sci., Cl. III, 10 (1962), pp. 475-478.
• [11] R. Bittner, Summation theorems for analytic elements, Bull. Acad. Polon. Sci., Cl. III, 10 (1962), pp. 437-440.
• [12] R. Bittner, Abstrakcyjne równania różniczkowe cząstkowe o współczynnikach przemiennych, Zeszyty Naukowe Wydziału Mat. Fiz. Chem. WSP w Gdańsku 1 (1961), pp. 5-10.
• [13] R. Bittner, Wzór sumacyjny Eulera a postać wykładnicza elementów analitycznych, Zeszyty Naukowe Wydziału Mat. Fiz. Chem. WSP w Gdańsku 2 (1962), pp. 5-11.
• [14] R. Bittner, A substitution of a multiplier, Zeszyty Naukowe Wydziału Mat. Fiz. Chem. WSP w Gdańsku (in print).
• [15] A. D. Booth, Numerical methods, London 1957.
• [16] A. S. Householder, Principles of numerical analysis, New York 1953.
• [17] Ch. Jordan, Calculus of finite differences, New York 1947.
• [18] L. V. Kantorovitch, Lineare halbgeordnete Räume, Mat. Sbornik 2 (44) (1937), pp. 121-136.
• [19] L. V. Kantorovitch, B. Z. Vulich, A. G. Pinskier, Functional analysis in semi-ordered spaces (in Russian), Moskva 1950.
• [20] M. Krzyżański, Równania różniczkowe cząstkowe rzędu drugiego. Warszawa 1957.
• [21] c. Kuratowski, Topologie, vol. I, Warszawa 1958.
• [22] M. Kwapisz, B. Palczewski, W. Pawelski, Sur l'existence el l'unicité des solutions des certaines équations différentielles du type $u_{xyz} = f(x, y, z, u, u_x, u_v, u_z, u_{xy}, u_{xz}, u_{yz}$, Ann. Polon. Math. 11 (1961), pp. 75-106.
• [23] II, Levy, F. Lessman, Finite différence équations, London 1959.
• [24] K. Maliński, Równania różniczkowe liniowe jednorodne w przestrzeniach liniowych, Zeszyty Naukowe Wydziału Mat. Fiz. Chem. WSP w Gdańsku 2 (1962), pp. 13-21.
• [25] J. G. Mikusiński, Sur certains espaces abstraits, Fund, Math. 36 (1949), pp. 125-130.
• [26] J. G. Mikusiński, Sur lesfondements du calcul opératoire, Studia Math. 11 (1950), pp. 41-70.
• [27] J. G. Mikusiński, Le calcul opérationnel d'intervalle fini, Studia Math. 15 (1956), pp. 225-251.
• [28] J. G. Mikusiński, Extensions de l'espace linéaire aver, dérivation. Studia Math. 16 (1957), pp. 156-172.
• [29] J. G. Mikusiński, Sur l'espace linéaire avec dérivation, Studia Math. 16 (1957), pp. 113-123.
• [30] J. G. Mikusiński, Sur les solutions linéairement indépendantes des équations différentielles à coefficients constants. Studia Math. 16 (1957), pp. 41-47.
• [31] J. G. Mikusiński, Sur les théorèmes d'unicité et le nombre de solutions linéairement indépendantes, Studia Math. 16 (1957), pp. 96-98.
• [32] J. G. Mikusiński, Operational calculus, Warsaw 1958.
• [33] N. Nicolescu, Probléme de l'analycité par rapport à un opérateur linéaire, Studia Math. 16 (1958), pp. 353-363.
• [34] A. Patocka, O pewnych dystrybucjach skończonego rzędu w przestrzeniach liniowych (in preparation).
• [35] W. Pogorzelski, Równania całkowe i ich zastosowania II, Warszawa 1958.
• [36] R. Sikorski, On Mikusiński's algebraical theory of differential equations, Studia Math. 16 (1957), pp. 230-236.
• [37] W. Słowikowski, A generalization of Mikusiński's operational calculus, Bull. Acad. Polon. Sci., Cl. III, 4 (1956), pp. 643-647.
• [38] W. Słowikowski, A generalization of the theory of distribution, Bull. Acad. Polon. Sci., Cl. III, 3 (1955), pp. 3-6.
• [39] J. P. Steffensen, Interpolation, New York 1950.
• [40] Z. Szmydt, Sur un nouveau type de problèmes pour un système d'équations différentielles hyperboliques du second ordre à deux variables indépendantes, Bull. Acad. Polon. Sci., Cl. III, 4 (1956), pp. 67-72.
• [41] V. Volterra, J. Péres, Leçons sur la composition et les fonctions permutables, Paris 1924.
• [42] B. Z. Vulich, Introduction to the theory of semi-ordered spaces (in Russian), Moskva 1961.
• [43] T. Ważewski, Sur un procédé de prouver la convergence des approximations successives sans utilisation des séries des comparison, Bull. Acad, Polon. Sci. 8 (1960), pp. 47-52.
• [44] E. Whittaker, G. Robinson, The calculus of observations, London 1948.

 EN