A rapidly convergent iteration method and non-linear partial differential equations - I

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1. 1 J. Schauder, Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung zweiter Order in beliebiger Anzahl von unabhängigen Veränderlichen. Fund. Math.24, 1935, pp. 213-246. Zbl0011.35202JFM61.0541.02
2. 2 K.O. Friedrichs, Symmetric Hyperbolic Linear Differential Equation8. Comm. Pure Appl. Math.7, 1954, pp. 345-392. Zbl0059.08902MR62932
3. 3 J. Nash, The Imbedding of Riemannian Manifolds, Ann. Math.63, 1956, pp. 20-63. Zbl0070.38603MR75639
4. 4 J. Schwartz, On Nash's 'Implicit Function Theorem, Comm Pure Appl. Math, 13, 1960, pp. 509-530. Zbl0178.51002MR114144
5. 4' J. Mosfr, A New Technique for the Construction of Solutions of Nonlinear Differential Equations. Proc. Nat. Acad. Sciences, Vol. 47, No. 11, pp. 1824-1831, 1961. Zbl0104.30503MR132859
6. 4" J. Moser, On Invariant Curves of Area-Preserving Mappings of an Annulus. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl.IIa, No. 1, 1962, pp. 1-20. Zbl0107.29301MR147741
7. 5 C.L. Siegel, Iteration of Analytic Functions. Ann. of Math.43, 1942, pp. 607-612. Zbl0061.14904MR7044
8. 5' C.L. Siegel, Über die Normalform Analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss., Göttingen, Math. Phys. Kl, IIa, 1952, pp. 21-30. Zbl0047.32901MR57407
9. 6 A.N. Kolmogorov, Doklad. Nauk, USSR98, 1954, p. 527-530. Zbl0056.31502
10. 7 A.N. Kolmogorov, General Theory of Dynamical Systems and Classical Mechanics. Proc. of Intl. Congress of Math., Amsterdam, 1954, (Amsterdam: Erven P. Nordhoff, 1957), Vol. 1, pp. 315-333. Zbl0095.17103
11. 8 V.I. Arnold, «Small divisors » I, On mappings of a circle onto itself, Isvest. Akad. Nauk, ser. mat., 25, No. 1, 1961, pp. 21-86. MR140699
12. 9 V.I. Arnold, Small Divisor and Stability Problems in Classical ana Celestial Mechanics. Uspekhi Mat. Nauk USSR, Vol. 18, Ser. 6 (119), 1963, pp. 81-192. MR170705
13. 10 V.I. Arnold, Proof of A N. Kolmogorov's Theorem on the Preservation of Quasi-Periodic Motions under Small Perturbations of the Hamiltonian. Uspekhi Mat. N. USSR, Vol. 18, Ser. 5 (113), 1963, pp 13-40. Zbl0129.16606MR163025
14. 11 K.O. Friedrichs, Symmetric Positive Linear Differential Equations, Comm. Pure Appl. Math.11, 1458, pp. 333-418. Zbl0083.31802MR100718
15. 12 N.N. Bogolioubov, and Y.A. Mitropolski, The Method of Integral Manifolds in Nonliriear Mechanics, Contrib. to Differential Equations, Vol. II, 1963, pp. 123-196. Zbl0137.28406
16. 13 S.P. Diliberto, Perturbation Theorems for Periodic Surfaces. I, Rend. del Circ. Mat. Palermo (2), 9, 1961, pp. 265-299II, Rend. del Circ. Mat. Palermo (2), 10, 1962, pp. 111-161. Zbl0109.31401MR142853
17. 13' S.P. Diliberto, Pertubation Theory of Invariant Surfaces I-IV. Mimeographed ONR Reports, Berkeley, 1956/57. 
18. 14 Kyner, W.T., Invariant manifolds, Rend. del Circolo Matematico Palermo, Ser. II, Vol. 9, 1961, pp. 98-110. Zbl0104.06303MR149038
19. 14' J.K. Hale, Integral Manifolds of Perturbed Differential Systems, Annals of Math., 73, 1961, pp. 496-531. Zbl0163.32804MR123786
20. 15 I. Kupka, Stabilité des varietés invariants d'un champ de vecteurs voui- les petites perturbations. Compt. Rend. Acad. Sc. Paris, 258, Group 1, 1964, pp. 4197-4200. Zbl0117.05403MR162036
21. 16 L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, Ser. 3, 13 (1959), pp. 116-162. Zbl0088.07601MR109940
22. 17 Gagliardo, E., Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8, 1959, pp. 24-51. Zbl0199.44701MR109295
23. 18 J.J. Kohn and L. Nirenberg, Non-coercive boundary value problems, to appear in Comm. Pure Appl. Math. (See under « elliptic regalarization »). Zbl0125.33302MR181815
24. 19 W.T. Kyner, A fixed point theorem. Contributions to the Theory on Nonlinear Oscillation, Vol. 3, Princeton University Press, Annals of Math. Studies36, 1956, pp. 197-205. Zbl0073.33501MR81472
25. 20 W.T. Kyner, Small Periodic Perturbations of an Autonomous System of Vector Equations. Contributions to the Theory of Nonlinear Oscillations Vol. 4, Princeton, 1958. Zbl0082.30103MR101946
26. 21 R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, NYU Report, IMM-NYLT 333, Oct. 1964, to be published in Comm. Pure Appl. Math. 
27. 22 C.L. Siegel, Vorlesungen über, Himmelsmechanik, Springer, 1956. Zbl0070.45403MR80009
28. 23 H. Cremer, über die Häufigkeit der Nichtzentren, Math. Ann.115, pp. 607-611, 1942. Zbl61.0331.03JFM64.0289.02
29. 24 N. Levinson, Transformation of an analytic function of several variables to a canonical form. Duke Math. Journal28, 1961, pp. 345-353. (See Theorem 2). Zbl0100.29002MR145101
30. 25 N.I. Achieser, Vorlesungen über Approximationstheorie, Akd. Verlag. Berlin1953, translated from Russian, Chapter V. Zbl0052.29002MR61692

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