A two-point boundary value problem for third order asymptotically linear systems

Abstract

We consider a third order system ${\boldsymbol x}'''={\boldsymbol f}({\boldsymbol x})$ with the two-point boundary conditions ${\boldsymbol x}(0)={\boldsymbol 0}$, ${\boldsymbol x}'(0)={\boldsymbol 0}$, ${\boldsymbol x}(1)={\boldsymbol 0}$, where ${\boldsymbol f}({\boldsymbol 0})={\boldsymbol 0}$ and the vector field ${\boldsymbol f}\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ is asymptotically linear with the derivative at infinity ${\boldsymbol f}'(\infty)$. We introduce an asymptotically linear vector field ${\boldsymbol \phi}$ such that its singular points (zeros) are in a one-to-one correspondence with the solutions of the boundary value problem. Using the vector field rotation theory, we prove that under the non-resonance conditions for the linearized problems at zero and infinity the indices of ${\boldsymbol \phi}$ at zero and infinity can be expressed in the terms of the eigenvalues of the matrices ${\boldsymbol f}'({\boldsymbol 0})$ and ${\boldsymbol f}'(\infty)$, respectively. This proof constitutes an essential part of our article. If these indices are different, then standard arguments of the vector field rotation theory ensure the existence of at least one nontrivial solution to the boundary value problem. At the end of the article we consider the consequences for the scalar case.

Keywords

• boundary value problem
• spectrum
• asymptotically linear vector field
• index of isolated singular point