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Multivalent guiding functions in forced oscillation problems. (English) Zbl 0840.34038

This article deals with the \(T\)-periodic problem for the system of differential equations \({d\rho/ dt}= R(t, \rho, \varphi, \xi)\), \({d\varphi/ dt}= \phi(t, \rho, \varphi, \xi)\), \({d\xi/ dt}= g(t, \rho, \varphi, \xi)\) \((0< \rho< \infty\), \(- \infty< \varphi< \infty\), \(\xi\in \mathbb{R}^{N- 2})\) with continuous functions \(R\), \(\phi\) and \(g\) which are periodic with respect to \(T\) and \(\varphi\), respectively with periods \(T\) and \(2\pi\). The main result is the existence of at least one \(T\)-periodic solution in terms of a multivalent vector guiding function.
The latter is a pair of smooth functions \(W_1(\xi)\) and \(W_2(\rho, \varphi)\) satisfying some conditions including \((\text{grad } W_1(\xi), g(t, \rho, \varphi, \xi)) < 0\) \((|\xi|\geq r_0)\), \(\alpha< R(t, \rho, \varphi, \xi) {\partial\over \partial\rho} W_2(\rho, \varphi)+ \phi(t, \rho, \varphi, \xi) {\partial\over \partial\varphi} W_2(\rho, \varphi) < \beta\) \((- \infty< \varphi< \infty\), \(\rho\geq \rho_0\), \(|\xi|\leq r)\), \({2\pi(k- 1)\over \alpha}< T< {2\pi k\over \beta}\) for some \(\alpha\), \(\beta\), \(\rho_0\), \(r_0\), \(r> 0\). Three examples are presented.
Reviewer: P.Zabreiko (Minsk)

MSC:

34C25 Periodic solutions to ordinary differential equations
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References:

[1] Mawhin, J., (Topological Degree methods in Nonlinear Boundary Value Problems, Reg. Conf. Ser. in Math., Vol. 40 (1979), American Mathematical Society: American Mathematical Society New York) · Zbl 0414.34025
[2] Krasnosel’skii, M. A.; Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis (1984), Springer: Springer RI · Zbl 0546.47030
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