Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-29T15:30:13.283Z Has data issue: false hasContentIssue false

On existence theorems for differential equations in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Józef Banaś
Affiliation:
Institute of Mathematics and Physics, Technical University, 35–084 Rzeszów, Poznańska 2, Poland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we show that a number of existence theorems for the Cauchy problem of ordinary differential equations in Banach spaces are only apparent generalizations of the previous ones.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Ambrosetti, A., “Un teorema di esistenza per le equazioni differenziali negli spazi di Banach”, Rend. Sem. Mat. Univ. Padova 39 (1967), 249360.Google Scholar
[2]Banaś, Józef and Goebel, Kazimierz, Measures of noncompactness in Banach spaces (Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, New York, 1980).Google Scholar
[3]Banaś, Józef, Hajnosz, Andrzej and Wedrychowicz, Stanislaw, “Relations among various criteria of uniqueness for ordinary differential equations”, Comment. Math. Univ. Carolin. 22 (1981), 5970.Google Scholar
[4]Bompiani, E., “Un teorema di confronto ed un teorema di unicita per lequazione differenziale y′ = f(x, y)“, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 1 (1925), 298302.Google Scholar
[5]Coddington, E. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
[6]Deimling, Klaus, Ordinary differential equations in Banach spaces (Lecture Notes in Mathematics, 596. Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[7]Deimling, Klaus, “On existence and uniqueness for Cauchy's problem in infinite dimensional Banach spaces”, Proc. Colloq. Math. Soc. Janos Bolyai Differential Equations 15 (1975), 131370.Google Scholar
[8]Goebel, Kazimierz and Rzymowski, Witold, “An existence theorem for the equations x′ = f(t, x) in Banach space”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 367370.Google Scholar
[9]Kuratowski, K., Topology I, new edition, revised and augmented (translated by Jaworowski, J.. Academic Press, New York and London; Państwowe Wydawnictwo Naukowe, Warsaw; 1966).Google Scholar
[10]Lakshmikantham, V. and Leela, S., Nonlinear differential equations in abstract spaces (Pergamon, New York, 1981).Google Scholar
[11]Martin, R.H., Nonlinear operators and differential equations in Banach spaces (John Wiley and Sons, New York and London, 1976).Google Scholar
[12]Olech, Czeslaw, “Remarks concerning criteria for uniqueness of solutions of ordinary differential equations”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 661666.Google Scholar
[13]Sadovskii, B.N., “Limit compact and condensing mappings”, Uspehi Mat. Nauk 27 (1972), 81116.Google Scholar
[14]Szarski, Jacek, Differential inequalities (Monografie Matematyczne, 43. PWN – Polish Scientific Publishers, Warszawa, 1965).Google Scholar
[15]Szufla, Stanislaw, “Measure of noncompactness and ordinary differential equations in Banach spaces”, Bull. Acad. Felon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 831835.Google Scholar
[16]Walter, Wolfgang, Differential and integral inequalities (Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar