Non-homogeneous BVPs for second-order symmetric Hamiltonian systems

Abstract

By making use of Bolle’s method, we show that the following problem has infinitely many solutions: x¨+V′(x)=0,x(0)cosα−x˙(0)sinα=x0,x(1)cosβ−x˙(1)sinβ=x1,\begin{array}{rcl}\ddot{x}+{V}^{^{\prime} }\left(x)& =& 0,\\ x\left(0)\cos \alpha -\dot{x}\left(0)\sin \alpha & =& {x}_{0},\\ x\left(1)\cos \beta -\dot{x}\left(1)\sin \beta & =& {x}_{1},\end{array} where α,β∈(0,π)x0,x1∈Rn\alpha ,\beta \in \left(0,\pi ){x}_{0},{x}_{1}\in {{\rm{R}}}^{n} are given and V∈C2(Rn,R)V\in {C}^{2}\left({{\rm{R}}}^{n},{\rm{R}}) is even and is super-quadratic at infinity.

Keywords

• sturm-liouville
• hamiltonian system
• non-symmetric
• 34b15
• 34b24
• 34l11