A topological method for bounded solutions of nonautonomous ordinary differential equations

Ward, James R.

doi:10.1090/S0002-9947-1992-1066450-8

A topological method for bounded solutions of nonautonomous ordinary differential equations
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by James R. Ward

Trans. Amer. Math. Soc. 333 (1992), 709-720

DOI: https://doi.org/10.1090/S0002-9947-1992-1066450-8

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Abstract:

The existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied. This is done by associating the given equation to nonlinear autonomous ones by means of a family of skew-product flows related by homotopy. The existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain Conley index associated with the autonomous differential equations. The existence of nontrivial bounded solutions is also considered. The differential equations studied are perturbations of homogeneous ones.

References

Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
È. Muhamadiev, On the theory of periodic solutions of systems of ordinary differential equations, Dokl. Akad. Nauk SSSR 194 (1970), 510–513 (Russian). MR 0274873
Margaret C. Memory and James R. Ward Jr., The Conley index and the method of averaging, J. Math. Anal. Appl. 158 (1991), no. 2, 509–518. MR 1117579, DOI 10.1016/0022-247X(91)90253-V
Krzysztof P. Rybakowski, The homotopy index and partial differential equations, Universitext, Springer-Verlag, Berlin, 1987. MR 910097, DOI 10.1007/978-3-642-72833-4
George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
James R. Ward Jr., Conley index and nonautonomous ordinary differential equations, Results Math. 14 (1988), no. 1-2, 191–209. MR 956013, DOI 10.1007/BF03323225
James Robert Ward Jr., Averaging, homotopy, and bounded solutions of ordinary differential equations, Differential Integral Equations 3 (1990), no. 6, 1093–1100. MR 1073059