×

On forced dynamical systems with a singularity of repulsive type. (English) Zbl 0708.34041

The existence of periodic solutions of the system \(-\ddot x=\nabla F(x)+h(t),\) where F is a potential with a singularity in zero of repulsive type (like for instance, the electrostatic potential between two charges of the same sign), satisfying also the condition: lim F(x)\(=\infty\) as \(x\to 0\), is discussed. There are given results representing conditions on h sufficient for the existence and nonexistence of T-periodic solutions. These results seem to be important since the collection of all of them permits us to established conditions being “not far away” from certain, almost full, characterization of discussed systems, from that point of view.
Reviewer: A.Pelczar

MSC:

37-XX Dynamical systems and ergodic theory
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. func. analysis, 14, 349-381, (1973) · Zbl 0273.49063
[2] Coti Zelati, V., Dynamical systems with effective-like potential, Nonlinear analysis, 12, 209-222, (1988) · Zbl 0648.34050
[3] Gordon, W., Conservative dynamical systems involving strong forces, Trans. AMS, 113-135, (1975) · Zbl 0276.58005
[4] Lazer, A.; Solimini, S., Nontrivial solutions of operator equations and Morse indices of critical points of min – max type, Nonlinear analysis, 12, 761-775, (1988) · Zbl 0667.47036
[5] Lazer, A.; Solimini, S., On periodic solutions of nonlinear differential equations with singularity, Proc. AMS, 2, 109-114, (1988) · Zbl 0616.34033
[6] Rabinowitz, P.H., Some minimax theorems and applications to nonlinear partial differential equations, (), 161-177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.