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Locally lipschitzian guiding function method for ODEs. (English) Zbl 0935.34007
The author treats the existence of periodic solutions to the problem \[ x'(t) = f(t,x(t)), \quad x(0)=x(T), \tag{*} \] where \(f:[0,T] \times \mathbb{R}^n \to \mathbb{R}^n\) is a Carathéodory function with integrably bound growth. Under the assumption that \(f\) has a locally Lipschitzian guiding function \(V\) with Ind\((V) \neq 0\) it is proved that problem (*) admits at least one solution. Moreover, it is shown that every coercive direct potential has Ind\((V)=1\), and so it can be used as a guiding function in the above theorem.

MSC:
34A60 Ordinary differential inclusions
34C25 Periodic solutions to ordinary differential equations
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[1] J. Andres, L. Górniewicz, M. Lewicka, Partially dissipative periodic processes, Topology in Nonlinear Analysis, vol. 35, Banach Center Publications, 1996, pp. 109-118. · Zbl 0845.34026
[2] Aubin, J.P.; Cellina, A., Differential inclusions, (1982), Springer New York
[3] R. Bader, W. Kryszewski, Fixed point index for compositions of set – valued maps with proximally∞ - connected values on arbitrary ANR’s, Set Valued Anal. 2 (3) (1994) 459-480. · Zbl 0846.55001
[4] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley-Interscience New York · Zbl 0727.90045
[5] L. Górniewicz, Topological approach to differential inclusions, in: Proc. Conf. on Topological Methods in Differential Equations and Inclusions, Université de Montreal, 1994, NATO ASI Series C, Kluwer, Dordrecht, pp. 129-190. · Zbl 0834.34022
[6] L. Górniewicz, S. Plaskacz, Periodic solutions of differential inclusions in\(R\^{}\{n\}\), Boll. UMI 7-A (1993) 409-420. · Zbl 0798.34018
[7] M.A. Krasnosel’skii, The Operator of Translation Along the Trajectories of Differential Equations, Nauka, Moscow, 1966 (in Russian) [English translation: American Math. Soc., Translations of Math. Monographs, vol. 19, Providence, RI, 1968.]
[8] J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations, in: Proc. Conf. on Topological Methods in Differential Equations and Inclusions, Université de Montreal, 1994, NATO ASI Series C, Kluwer, Dordrecht, pp. 291-375. · Zbl 0834.34047
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