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Continuity moduli criteria for ODE’s in a Banach space. (English) Zbl 0531.34046
The main result of this paper is a modulus of continuity criterion for the convergence of successive approximations to the solution of the Cauchy problem \(x'(t)=f(t,x(t))\), 0\(\leq t\leq T\), \(x(0)=z\) in infinite- dimensional Banach spaces. This new criterion seems to contain all previously known criteria of its type. Many examples and related results are also presented.
Reviewer: Simeon Reich

MSC:
34G20 Nonlinear differential equations in abstract spaces
47J25 Iterative procedures involving nonlinear operators
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[1] Brauer, F, Some results on uniqueness and successive approximations, Canad. J. math., 11, 527-533, (1959) · Zbl 0142.34305
[2] Brauer, F, A note on uniqueness and convergence of successive approximations, Canad. math. bull., 2, 5-8, (1959) · Zbl 0084.07801
[3] Brauer, F; Sternberg, S, Local uniqueness, existence in the large, and the convergence of successive approximations, Amer. J. math., 80, 421-430, (1958) · Zbl 0082.06801
[4] Brauer, F; Sternberg, S, Errata to our paper “local uniqueness etc.”, Amer. J. math., 81, 797, (1959) · Zbl 0114.28603
[5] Deimling, K, Ordinary differential equations in Banach spaces, (), 106, 1977
[6] Deimling, K, On approximate solutions of differential equations in Banach spaces, Math. ann., 212, 79-88, (1974) · Zbl 0281.34056
[7] Evans, J.W; Feroe, J.A, Successive approximations and the general uniqueness theorem, Amer. J. math., 96, 505-510, (1974) · Zbl 0308.34005
[8] Krasnoselskii, M.A; Krein, S.G, On a class of uniqueness theorems for the equation y′ = f(x, y), Uspekhi mat. nauk., 11, 209-213, (1956)
[9] Kooi, O, The method of successive approximations and a uniqueness theorem of Krasnoselskii and Krein in the theory of differential equations, Nederl. akad. wetensch. indag. math., 20, 322-327, (1958) · Zbl 0084.28402
[10] Lakshmikantham, V, Existence and comparison results for differential equations, (), 459-473
[11] Lakshmikantham, V; Mitchell, A.R, On the redundancy of monotony assumption, J. math. phys. sci., 10, 219-230, (1976) · Zbl 0355.34002
[12] Luxemburg, W.A.J, On the convergence of successive approximations in the theory of ordinary differential equations, Canad. math. bull., 1, 9-20, (1958) · Zbl 0081.07802
[13] Luxemburg, W.A.J, On the convergence of successive approximations in the theory of ordinary differential equations, II, Nederl. akad. wetensch. indag. math., 20, 540-546, (1958) · Zbl 0084.07703
[14] Luxemburg, W.A.J, On the convergence of successive approximations in the theory of ordinary differential equations, III, Nieuw. arch. wisk., 6, 93-98, (1958) · Zbl 0085.30201
[15] Olech, C, Remarks concerning criteria for uniqueness of solutions of ordinary differential equations, Bull. acad. polon. sci. Sér. sci. math. astronom. phys., 8, 661-666, (1960) · Zbl 0173.33701
[16] Olech, C, A connection between two certain methods of successive approximations in differential equations, Ann. polon. math., 11, 237-245, (1962) · Zbl 0106.05701
[17] Olech, C; Pliś, A, Monotonicity assumption in uniqueness criteria for differential equations, (), 43-58 · Zbl 0153.10902
[18] Rosenblatt, A, Über die existenz von integralen gewöhnlicher differentialgleichungen, Ark. mat. astro. fys., 5, 1-4, (1909) · JFM 39.0372.01
[19] Vidossich, G, Global convergence of successive approximations, J. math. anal. appl., 45, 285-292, (1974) · Zbl 0321.34054
[20] Vidossich, G, Most of the successive approximations do converge, J. math. anal. appl., 45, 127-131, (1974) · Zbl 0346.34002
[21] Walter, W, Bemerkungen zu verschiedenen eindeutigkeitskriterien für gewöhnliche differentialgleichungen, Math. Z., 84, 222-227, (1964) · Zbl 0125.04401
[22] Walter, W, Differential- und integral-ungleichungen, (1964), Springer-Verlag New York/Berlin · Zbl 0119.12205
[23] De Blasi, F.S; Myjak, J, Generic properties of differential equations in a Banach space, Bull. acad. polon. sci. Sér. sci. math. astronom. phys., 26, 395-400, (1978) · Zbl 0393.34034
[24] Derrick, W; Janos, L, A global existence and uniqueness theorem for ordinary differential equations, Canad. math. bull., 19, 105-107, (1976) · Zbl 0344.34002
[25] Kamke, E, Differentialgleichungen reeller funktionen, (), 139-141
[26] Kikooze, M.S, On the question of uniqueness of a solution of the Cauchy problem and the convergence of successive approximations, Differential’nye uravnenija, 2, 1553-1560, (1966) · Zbl 0146.11302
[27] Kisielewicz, M, On the non-convergence of successive approximations in the theory of ordinary differential equations, (), 267-270
[28] Lipschitz, R, Sur la possibilité d’intégrer complètement un système donné d’équations différentielles, Bull. sci. math. astro., 10, 149-159, (1876) · JFM 08.0177.01
[29] Müller, M, Über die eindeutigkeit der integrale eines systems gewöhnlicher differentialgleichungen und der konergenz einer gattung von verfahren zur approximation dieser integrale, Sitzungsber. heidelb. akad. wiss. math.-natur. kl., 9, (1927) · JFM 53.0400.03
[30] Nevanlinna, R, Über die methode der sukzessiven approximationen, Ann. acad. sci. fenn. ser. A, 291, (1960) · Zbl 0094.10603
[31] Pelczar, A, The method of successive approximations, Wiadom. mat., 20, 2, 80-84, (1976)
[32] Picard, E, Sur LES méthodes d’approximations successives dans la théorie des équations différentielles, (), 353-367, Paris · JFM 29.0312.01
[33] Rao, M.Ramamohana, Some problems on general uniqueness and successive approximations, (), 205-212 · Zbl 0196.34801
[34] Rao, M.Ramamohana, The local uniqueness and successive approximations, Bul. inst. politehn. iasi (N.S.), 9, No. 13, 13-18, (1963)
[35] Santoro, P, Convergence delle approximazioni successive per le equazioni differentiziali di ordine n, Riv. mat. univ. parma, 2, 301-312, (1961), (2) · Zbl 0113.29203
[36] Wazewski, T, Sur la convergence des approximations successives pour LES équations différentielles ordinaraires au cas de l’espace de Banach, Ann. polon. math., 16, 231-235, (1965) · Zbl 0154.39803
[37] Winter, A, On the convergence of successive approximations, Amer. J. math., 68, 13-19, (1946)
[38] Ziebur, A.D, Uniqueness and the convergence of successive approximations, (), 899-903 · Zbl 0108.27504
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