In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{cc}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)\hfill & 0\<x\<\pi ,t\in ℝ,i\in \mathbb{Z}\left(TC\right)\hfill \\ {\varphi }^i(0,t)={\varphi }^i(\pi ,t)=0\hfill & \forall t,i.\hfill \end{array}\right.$$We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^i\forall i\in \mathbb{Z}$ and some $N\in \mathbb{N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{c}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)0<x<\pi ,t\in \mathrm{I}\mathrm{R},\mathrm{i}\in \mathrm{Z}\mathrm{Z}\left(\mathrm{TC}\right){\varphi }^{\mathrm{i}}(0,\mathrm{t})={\varphi }^{\mathrm{i}}(\pi ,\mathrm{t})=0\forall \mathrm{t},\mathrm{i}.\hfill \end{array}\right.$$ We consider the case of “closed chains" i.e.${\varphi }^{i+N}={\varphi }^i\forall i\in ZZ$ and some $N\in IN$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system $$\left\{\begin{array}{c}\ddot{u}\left(t\right)=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0.\hfill \end{array}\right.$$ Some new existence theorems are obtained by the least action principle.

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...