Existence problems for homoclinic solutions

Abstract

The problem $\dot{x}=f\left(t,x\right),x\left(-\infty \right)=x\left(+\infty \right)$, where $x\left(\pm \infty \right):={lim}_{t\to \pm \infty }x\left(t\right)\in {ℝ}^n$, is considered. Some existence results for this problem are established using the fixed point method and topological degree theory.