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.

We show the existence of a nontrivial solution to the semilinear elliptic equation $-\Delta u+u=b\left(x\right){\left|u\right|}^{p-2}u,u>0,u\in H_0^1\left({ℝ}_{+}^N\right)$ under some suitable conditions.

An inverse problem, the determination of the shape and a convective coefficient on a part of the boundary from partial measurements of the solution, is studied using 2-person optimal control techniques.