Morse Theory and Existence of Periodic Solutions of Elliptic Type

Abstract

In this paper we study the problem of finding periodic solutions of elliptic type for Hamiltonian systems of the following form

$$ \left\{ {\begin{array}{*{20}{c}} {\dot x = J(Ax + N'(t,x))}\\ {x(0) = x(T)\quad with\quad T > 0;} \end{array}} \right. $$

(H)

whereJ is the matrix

$$ J \equiv \left( {\begin{array}{*{20}{c}} 0&{{I_n}}\\ { - {I_n}}&0 \end{array}} \right) $$

andI _n is the identity of Rⁿ,

$$\begin{array}{*{20}{c}} {A \equiv \left( {\begin{array}{*{20}{c}} {{A_n}}&0\\ 0&{{A_n}} \end{array}} \right),}\\ {{A_n} \equiv \left( {\begin{array}{*{20}{c}} {{a_1}}\\ 0 \end{array} \ddots \begin{array}{*{20}{c}} 0\\ {{a_n}} \end{array}} \right)} \end{array}$$

${a_j} \in R,{a_j} \ne 0,j = 1, \ldots ,n,N \in {C^2}(R \times {R^{2n}},R)$ denotes the partial gradient with respect to the second variable.