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Periodic solutions of Hamiltonian systems with superquadratic potential. (English) Zbl 0632.34036
The authors consider the Hamiltonian H(t,p.q), $H(t,p.q)=\sum^{n}_{i,j=1}a_{ij}(t,q)p_ ip_ j+\sum^{n}_{i=1}b_ i(t,q)p_ i+V(t,q),$ with suitable growth conditions on the coefficients. The important ones are (i) V(t,q) grows more than $$| q|^{\alpha}$$ $$(\alpha >2)$$ at infinity, (ii) some plausible growth rates of $$a_{ij}(t,q)$$ with $$q_ j$$ in addition to $$\sum_{i,j}a_{ij}(t,p,q)p_ ip_ j>\nu (q)| p|^ 2$$ $$(\nu (q)>0)$$, and (iii) relative growths of $$| a_{ij}|$$, $$b^ 2_ j$$ and $$| \partial b_ j/\partial q_ kq_ k|^ 2$$ with respect to V(t,q) as $$q\to \infty$$. It is shown that, in addition to the above, (1) if $$\partial H/\partial t=0$$, the system has infinitely many nonconstant T-periodic solutions for every prescribed periodic T, and (2) if H(t,p.q) is T-periodic in t and satisfies suitable bounds on the second order differential coefficients with respect to p,q, then the system has at least a nonconstant T-periodic solution. In order to establish these results, the authors first demonstrate an abstract theorem on critical points for strongly indefinite functionals. As a prelude to this, they recall some notions on the critical point theory and the index theory.
Reviewer: N.D.Sengupta

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 70H05 Hamilton’s equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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