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

Kisyński, J.; Pelczar, A.

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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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J. Kisyński A. Pelczar

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Rozprawy Matematyczne tom/nr w serii: 76 wydano: 1970

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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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Instytut Matematyczny Polskiej Akademii Nauk

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Warszawa

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Rozprawy Matematyczne tom/nr w serii: 76

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74

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Opis fizyczny

Dissertationes Mathematicae, Tom LXXVI

Daty

wydano

1970

Twórcy

autor

J. Kisyński

autor

A. Pelczar

Bibliografia

[1] A. Alexiewicz and W. Orlicz, Some remarks on the existence and uniqueness of solutions of the hyperbolic equation $\frac{∂^2z}{∂x∂y} = f(x, y, z, \frac{∂z}{∂x}, \frac{∂z}{∂y}$, Studia Math. 15 (1956), p. 201-215.
[2] G. Arnese, Sull'approssimazione, col metodo di Tonelli, delle soluzioni del problema di Darboux per I'equazione s = f(x, y, z, p, q), Ricerche di Matematica 11 (1962), p. 61-75.
[3] G. Arnese, Sul problema di Darboux in ipotesi di Carathéodory, ibidem 12 (1963), p. 13-31.
[4] A. Bielecki, Une remarque sur l'application de la méthode de Banach-Caccioppoli-Tikhonov dans la théorie de l'équation s = f(x, y, z, p, q), Bull. Acad. Polon. Sci, Cl. III, 4 (1956), p. 265-268.
[5] A. Bielecki and J. Kisyński, Sur un problème de Mlle Szmydt relatif à l'équation $\frac{∂^2}{∂x∂y} = f(x, y, z, \frac{∂z}{∂x}, \frac{∂z}{∂y}$, ibidem 6 (1958), p. 321-325.
[6] F. Brauer, Some results on uniqueness and successive approximations, Canad. J. Math. 11 (1969), p. 527-533.
[7] T. Bruno, Sull'unicità e l'approssimazione dette soluzioni del problema di Darboux, Rend. dell'Accademia di Scienze Fisiche e Matematiche della Società Nazionale di Scienze, Lettere ed Arti in Napoli, ser. 4, vol. 33 (1966), p. 50-63.
[8] T. Bruno, Sull covergenza delle approssimazioni successive per il problema di Darboux, ibidem, ser. 4, vol. 34 (1967), p. 29-43.
[9] F. Cafiero, Su due teoremi di confronto relativi ad un'eqazione differenziale ordinaria del primo ordine, Bolletino delia Unione Matematica Italiana, 3 (1948), p. 124-128.
[10] C. Carathéodory, Vorlesungen über reelle Funktionen, Leipzig-Berlin 1927.
[11] C. Ciliberto, Sull'approssimazione delle soluzioni del problema di Darboux per l'equazione s = f(x, y, z, p, q), Ricerche di Matematica 10 (1961), p. 106-138.
[12] C. Ciliberto, Teoremi di Confronto e di unicità per soluzioni del problema di Darboux, ibidem 10 (1961), p. 214-242.
[13] C. Ciliberto, Sul problema di Darboux, Rend. Acc. Lincei (VIII), 30 (1961), p. 460-466.
[14] C. Ciliberto, Il problema di Darboux per un'equazione di tipo iperbolico in due variabili Ricerche di Matematica 4 (1955), p. 15-29.
[15] C. Ciliberto, Sul problema di Darboux per l'equazione s = f(x, y, z, p, q), Rend. dell’Accademia di Scienze Fisiche e Matematiche della Società Nazionale di Scienze, Lettere et Arti in Napoli, Serie 4, 22 (1955), p. 1-5.
[16] C. Ciliberto, Sul alcuni problemi relaiivi ad una equazione di tipo iperbolico in due variabili, Bull. U.M.I. (3), 11 (1956), p. 383-393.
[17] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, New York-Toronto-London 1955.
[18] J. Conlan, The Cauchy problem and the mixed boundary value problem for non-linear hyperbolic partial differential equation in two independent variables, Arch. Rational Mech. and Analysis 3 (1969), p. 355-380.
[19] R. Conti, Sul problema di Darboux per l'equazione $z_{xy}=f(x, y, z, z_x, z_y)$, Annali dell'Univ. di Ferrara, (N.S.), Ser. VII, Sc. Mat. 2 (1953), p. 129-140.
[20] R. Conti, Sull'equazione in integrodifferenziale di Darboux-Picard, Le Matematiche 13 (1958), p. 30-39.
[21] J. B. Diaz, On analogue of the Euler-Cauchy polygon method for the numerical solution of $u_xy = f(x, y, u, u_x, u_y)$, Arch. Rational Mechanics and Analysis 1 (1958),p. 367-390.
[22] J. B. Diaz and W. Walter, On uniqueness theorems for ordinary differential equations and for partial differential equations of hyperbolic type, Trans. Amer. Math. Soc. 96 (1960), p. 90-100.
[23] N. Fedele, Sull'approssimazione, col metodo di Tonelli, delle soluzioni di certi problemi relativi ad equazioni non lineari di tipo iperbolico, Rend. Acc. Scienze Fis. e Mat., Napoli 31 (1964).
[24] N. Fedele, Sull'integrale superiore e quello inferiore di alcuni problemi relativi ad equazioni non lineari di tipo iperbolico, in ipotesi di Carathéodory, Le Matematiche 21 (1966), p. 167-197.
[25]) C. Foias, G. Gussi and V. Poenaru, Une méthode directe dans l'étude du problème de Cauchy pour les équations aux dérivées partielles, hyperboliques, du second ordre, à deux variables, Revue de Mathématiques pures et appliqués 1 (1956), p. 61-98.
[26] K. Goebel, On strong contractions, Bull Acad. Polon. Sci., Sér. sci. math., astr. et phys. 15 (1967), p. 309-312.
[27] F. Guglielmino, Sul problema di Darboux, Ricerche di Matematioa 8 (1909), p. 180-196.
[28] F. Guglielmino, Sull'esistenza delle soluzioni dei problemi relativi alle equazioni non lineari di tipo iperbolico in due variabili, Le Matematiche 14 (1959), p. 67-80,
[29] F. Guglielmino, Sul problema di Nicoletti per le equazione alle derivate parziali, ibidem 13 (1958), p. 40-60.
[30] F. Guglielmino, Sul problema di Goursat, Ricerche di Matematica 19 (1960), p. 91-105.
[31] Ph. Hartman and A. Wintner, On hyperbolic partial differential equations, Amer. J. Math. 74 (1952), p. 834-864.
[32] M. Hukuhara, Le problème de Darboux pour l'équation s = f(x, y, z, p, q), Annali di Matematica pura ed applicata (IV), 51 (1960), p. 39-54.
[33] M. Hukuhara and T. Satō, Theory of differential equations (in japanese), Modern Math. Series, 11 (1957), Kyōritsu Shuppan Co Ltd, Chapter 6. (Chapter 6 is translated in french by T. Satō — mimeograph).
[34] E. Kamke, Differenlialgleichungen reeller Funktionen, Leipzig 1950.
[35] J. Kisyńki, Application de la méthode des approximations successives dans la théorie de l'équation $∂^2z/∂x∂y = f(x, y, z, ∂z/∂x, ∂z/∂y)$,, Ann. Univ. M. Curie-Skłodowska, Sectio A, 14 (6) (1960), p. 66-86.
[36] J. Kisyńki, Solutions généralisées du problème de Cauchy-Darboux pour l'équation $∂^2z/∂x∂y = f(x, y, z, ∂z/∂x, ∂z/∂y)$, ibidem 14 (6) (1960), p. 87-109.
[37] J. Kisyńki, Sur l'existence des solutions de l’équation $∂^2z/∂x∂y = f/(x, y, z, ∂z/∂x, ∂z/∂y)$, ibidem 15 (7) (1961), p. 85-95.
[38] J. Kisyńki, On second order hyperbolic equations with two independent variables, Coll. Math, 22 (1970), p. 135-151.
[39] J. Kisyński and W. Tym, Sur la convergence des approximations successives pour l'équation $∂^2z/∂x∂y = f/(x, y, z, ∂z/∂x, ∂z/∂y)$, Ann. Univ. M. Curie-Skłodowska, Sectio A, 16 (9) (1962), p. 107-121.
[40] A. Lasota, Sur une généralisation d'un problème de Z. Szmydt concernant l'équation $u_{xv} = f(x, y, u, u_x, u_v), Bull. Acad. Polon. Sci, Cl. III, 5 (1957), p. 15-18.
[41] A. Lasota, Sur un nouveau problème aux limites relatif à l'équation de la corde vibrante, ibidem 5 (1957), p. 843-846.
[42] P. Leehey, On the existence of not necessarily unique solutions of classical hyperbolic boundary value problems for non-linear partial differential equations in two independent variables, Ph. D. Thesis, Brown University, June 1950.
[43] W. Mlak, Note on maximal solutions of differential equations, Contributions to Diff. Equations 1 (1962), p. 461-465.
(44) R. H. Moore, On approximate solutions of non-linear hyperbolic partial differential equations, Archive for Rational Mechanics and Analysis 6 (1960), p. 75-88.

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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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