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Periodic solutions of a class of abstract nonlinear equations of the second order. (English) Zbl 0574.34024
The aim of this paper is to prove the existence of weak periodic solutions of the abstract differential equation (1) $$F(u)=\phi (u)+h$$, where $$F(u)\equiv u''+\psi (u')+{\mathcal A}u$$, $$u'=du/dt$$, $$\psi$$ and $$\phi$$ are nonlinear mappings of a Hilbert space H into itself with linear growth and $${\mathcal A}$$ is a linear elliptic operator from $$V\subset H$$ into $$V^*$$. The results obtained here are applied to the jumping- nonlinearity problem for ordinary and partial differential equations.
##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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