Morse Theory and Existence of Periodic Solutions of Elliptic Type

• B. D’Onofrio
• I. Ekeland
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

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.

Keywords

Periodic Solution Hamiltonian System Unit Circle Hermitian Form Morse Theory

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