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Some functional differential equations

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Pelczar A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction............................................................................................................................................................................... 5 Chapter 0. PRELIMINARIES 0.1. (Preliminary remarks and notation)............................................................................................................................. 9 0.2. (Notation — continuation).............................................................................................................................................. 10 0.3. (Notation and some definitions).................................................................................................................................. 10 0.4. (Statement of problems; definition of solutions of differential functional equations)......................................... 12 0.5. (Equivalence of problems: differential and integral; definition of solutions of integral equations).................. 14 Chapter I. EXISTENCE AND UNIQUENESS OF SOLUTIONS AND THE CONVERGENCE OF SUCCESSIVE APPROXIMATIONS IN COMPACT SETS 1.1. Notation and definitions................................................................................................................................................. 17 1.2. Uniqueness...................................................................................................................................................................... 18 1.3. Existence and successive approximations................................................................................................................ 19 1.4. Existence without uniqueness...................................................................................................................................... 22 1.5. Some generalizations of the results from 1.2-1.4..................................................................................................... 23 1.6. Some supplementary remarks..................................................................................................................................... 24 Chapter II. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS 2.1. Notation and definitions................................................................................................................................................. 27 2.2. Union of solutions........................................................................................................................................................... 28 2.3. Global uniqueness.......................................................................................................................................................... 29 2.4. Definition of the condition (W) and some remarks................................................................................................... 31 2.6. Local existence of solutions.......................................................................................................................................... 32 2.6. Lemmas............................................................................................................................................................................ 33 2.7. Limits of solutions on the boundary............................................................................................................................. 36 2.8. Prolongations................................................................................................................................................................... 38 2.9. Global existence under the assumptions on uniqueness...................................................................................... 39 2.10. Global existence without uniqueness....................................................................................................................... 41 2.11. Global existence without uniqueness by the method of A. Bielecki, T. Dłotko and M. Kuczma...................... 43 2.12. Existence of solutions under the assumptions (Y) and (Ỹ)................................................................................... 46 2.13. Local convergence of successive approximations under the assumptions (V)............................................... 47 2.14. Remarks on some generalizations........................................................................................................................... 49 Chapter III. CONTINUOUS DEPENDENCE OF SOLUTIONS ON GIVEN FUNCTIONS 3.1. Continuous dependence on $λ, ψ, {φ^a}$............................................ 51 3.2. Continuous dependence on ƒ....................................................... 52

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Comparison of solutions and successive approximations in the theory of the equation $∂^2z/∂x∂y = f(x, y, z, ∂z/∂x, ∂z/∂y)$

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Kisyński J., Pelczar A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction........................................................................................................................................................................................................... 5 I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^1'*(Δ_{a,b};E)$ AND $L^{1,*}_1(Δ_{a,b};E)$......................... 7 1. Basic function classes ................................................................................................................................................................................... 7 2. The Cauchy-Darboux problem ...................................................................................................................................................................... 12 II. Comparison of solutions ............................................................................................................................................................................... 18 3. The growth estimations.................................................................................................................................................................................. 18 4. Maximal solutions............................................................................................................................................................................................ 26 5. A theorem on extension of inequalities........................................................................................................................................................ 28 6. Effective estimation in the case $M_1$, (b)................................................................................................................................................. 30 III. COMPARATIVE CRITERIA OF EXISTENCE AND UNIQUENESS OP SOLUTIONS OF THE CAUCHY-DARBOUX PROBLEM...................................................................................................................................................................................... 35 7. Basic classes of comparative functions...................................................................................................................................................... 35 8. Existence and uniqueness of solutions of the Cauchy-Darboux problem............................................................................................ 42 9. Remarks on the continuous dependence of solutions on boundary data and on the second member........................................ 47 10. Examples......................................................................................................................................................................................................... 49 BIBLIOGRAPHICAL REMARKS.......................................................................................................................................................................... 66 BIBLIOGRAPHY..................................................................................................................................................................................................... 68 INDEX OF SYMBOLS............................................................................................................................................................................................ 74

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Semistability of motions and regular dependence of limit sets on points in general semi-sytems

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Pelczar A.

Annales Polonici Mathematici

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1983

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tom 42

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nr 1

263-282

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On a modification of the method of Euler polygons for the ordinary differential equation

48%

Pelczar A.

Annales Polonici Mathematici

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1964

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tom 15

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nr 2

195-202

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On invariant points of monotone transforamtions in partially ordered spaces

48%

Pelczar A.

Annales Polonici Mathematici

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1965

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tom 17

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nr 1

49-53

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Some qualitative problems in the theory of second order partial differential equations

48%

Pelczar A.

Annales Polonici Mathematici

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1990-1991

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tom 52

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nr 1

1-17

7

On the invariant points of a transformation

36%

Pelczar A.

Annales Polonici Mathematici

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1961-1962

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tom 11

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nr 2

199-202

Ograniczanie wyników

5 Annales Polonici Mathematici

7 Pelczar A.

1 Kisyński J.

1 1990

1 1983

1 1973

1 1970

1 1965

1 1964

1 1961

Some functional differential equations

Comparison of solutions and successive approximations in the theory of the equation $∂^2z/∂x∂y = f(x, y, z, ∂z/∂x, ∂z/∂y)$

Semistability of motions and regular dependence of limit sets on points in general semi-sytems

On a modification of the method of Euler polygons for the ordinary differential equation

On invariant points of monotone transforamtions in partially ordered spaces

Some qualitative problems in the theory of second order partial differential equations

On the invariant points of a transformation