Abstract
In this chapter, we are concerned with the initial value problem:
where E is a real Banach space and f : [0, T] × E → E has a decomposition f = g + h with g and h Carathéodory functions satisfying respectively, a compactness and Lipschitz assumptions. Our results rely on Krasnoselskii fixed point theorem for contraction plus compact mappings and don’t use homotopy arguments. It is worth remarking here that the periodic problem could also be discussed in this setting (we leave this as an exercise). Our main existence principle is obtained in section 16.2. This result will be used in section 16.3 to obtain more applicable existence results. More precisely, in section 16.3, we give existence theorems of Wintner type. Also in section 16.3, existence theorems are obtained under an assumption which is equivalent to an assumption of existence of upper and lower solutions to (1.1) in the scalar case. No growth condition is assumed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of noncompactness and condensing operators, Birkhäuser, Basel, 1992.
K. Deimling, Ordinary differential equations in Banach spaces, Springer Verlag, New York/Berlin, 1977.
N. Dunford and J.T. Schwartz, Linear operators part.I: General theory, Interscience, New York, 1967.
M. Frigon and J.W. Lee, Existence principles for Carathéodory differential equations in Banach spaces, Topological Methods in Nonlinear Analysis, 1, (1993), 91–106.
M. Frigon and D. O’Regan, Existence results for initial value problems in Banach spaces, Diff. Eqns. and Dynamical Systems, 2 (1994), 41–48.
A. Granas, R.B. Guenther and J.W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential systems J. Math. pures et appl., 70 (1991), 153–196.
M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspehi Mat. Nauk., 10 (1955), 123–127.
V. Lakshmikanthan and S. Leela, Nonlinear differential equations in abstract spaces, Pergamon, NewYork, 1981.
J.W. Lee and D. O’Regan, Existence results for differential equations in Banach spaces, Commentat. Math. Univ. Carol., 34 (1993), 239–251.
J.W. Lee and D. O’Regan, Topological transversality. Applications to initial-value problems, Ann. Polonici Math., 48 (1988), 247–252.
R.H. Martin, Nonlinear operators and differential equations in Banach spaces, Wiley, New York, 1976.
D.R,. Smart, Fixed point theorems, Cambridge University Press, 1980.
A. Wintner, The nonlocal existence problems for ordinary differential equations, Am. J. Math., 67 (1945), 227–284.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
O’Regan, D. (1997). Differential equations in abstract spaces. In: Existence Theory for Nonlinear Ordinary Differential Equations. Mathematics and Its Applications, vol 398. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1517-1_16
Download citation
DOI: https://doi.org/10.1007/978-94-017-1517-1_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4835-6
Online ISBN: 978-94-017-1517-1
eBook Packages: Springer Book Archive