Abstract
In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fučik spectra.
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REFERENCES
V. V. Filippov, “On homological properties of the sets of solutions to ordinary differential equations,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 188 (1997), no. 6, 139–160.
V. V. Filippov, “A mixture of Leray–Schauder and Poincaré–Andronov methods in the problem on periodical solutions to ordinary differential equations,” Differentsial'nye Uravneniya [Differential Equations], 35 (1999), no. 12, 1709–1711.
V. V. Filippov, “Remarks on periodic solutions of ordinary differential equations,” J. Dynamical and Control Systems, 6 (2000), no. 3, 431–451.
J.-P. Gossez and P. Omari, “Nonresonance with respect to the Fučik spectrum for periodic solutions of second order ordinary differential equations,” Nonlinear Anal. Th. Methods Appl., 14 (1990), 1079–1104.
J.-P. Gossez and P. Omari, “Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance,” J. of Differential Equations, 94 (1991), no. 1, 67–82.
P. Habets, P. Omari, and F. Zanolin, “Nonresonance conditions on the potential ith respect to the Fučik spectrum for the boundary value problem,” Rocky Mountain J. Math., 25 (1995), no. 4, 1305–1340.
J. Mawhin, “Continuation theorems and periodic solutions of ordinary differential equations,”in: Topological Methods in Differential Equations and Inclusions, Kluwer Academic Publishers, Dordrecht–Boston–London, 1995, pp. 291–376.
S. R. Gabdrakhmanov, “Spaces of solutions and the Fučik spectrum,” Differentsial'nye Uravneniya [Differential Equations], 39 no. 3, 298–298.
V. V. Filippov, “Topological structure of spaces of solutions to ordinary differential equations,” Uspekhi Mat. Nauk [Russian Math. Surveys], 48 (1993), no. 1, 103–154.
V. V. Filippov, Spaces of Solutions to Ordinary Differential Equations [in Russian ], Moskov. Gos. Univ., Mosco, 1993.
V. V. Filippov, “Basic topological structures of the theory of ordinary differential equations,” in: Topology in Nonlinear Analysis, vol. 35, Banach Center Publications, Warsaw, 1996, pp. 171–192.
V. V. Filippov, Basic Topological Structures of Ordinary Differential Equations, KluwerAcademic Publishers, Dordrecht–Boston–London, 1998.
V. V. Filippov, “On differential inclusions of second order,” Fundamental'naya i Prikladnaya Matematika [Fundamental and Applied Mathematics], 3 (1997), no. 2, 587–623.
V. V. Filippov, “On the existence of periodic solutions,” Mat. Zametki [Math. Notes], 61 (1997), no. 5, 769–784.
V. V. Filippov, “On the uniqueness theorem for solutions of the Cauchy problem for ordinary differential equation,” Differentsial'nye Uravneniya [Differential Equations], 30 (1994), no. 6, 1005–1009.
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Filippov, V.V. On Fučik Spectra and Periodic Solutions. Mathematical Notes 73, 859–870 (2003). https://doi.org/10.1023/A:1024062116222
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DOI: https://doi.org/10.1023/A:1024062116222