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Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign. (English) Zbl 0894.34036

The author studies the system x ¨=W ' (z,t)=0, with x(0)=x(T), x ˙(0)=x ˙(T), where W(x,t)=A(t)x,x+b(t)V(x). Here, b is a continuous T-periodic function, VC 2 ( N ×) has superquadratic behavior, A is a continuous T-periodic symmetric matrix function, with A(t)ξ,ξ>0, for all ξ N , and |ξ|=1.

The author shows that if lim x0 V(x)/|x 2 |=0, and 0 T b(t)dt>0 then the system has a T-periodic solution. With some additional assumptions including 0 T b(t)dt<0 (reversing the sign) it also follows that a T-periodic solution exists. Using a variational approach, she proves that the corresponding Lagrangian action integral

1 2 0 T |x ˙ 2 |dt-1 2 0 T [A(t)x,x+b(t)V(x)]dt

has a critical point in an appropriately chosen finite subspace of the original Hilbert space H 1 .

34C25Periodic solutions of ODE