Boundary value problems for first order systems on the half-line

We prove existence theorems for first order boundary value problems on $(0,\infty)$, of the form $\dot{u}+F(\cdot,u)=f$, $Pu(0)=\xi$, where the function $F=F(t,u)$ has a $t$-independent limit $F^{\infty}(u)$ at infinity and $P$ is a given projection. The right-hand side $f$ is in $L^{p} ((0,\infty),{\mathbb R}^{N})$ and the solutions $u$ are sought in $W^{1,p}((0,\infty),{\mathbb R}^{N})$, so that they tend to $0$ at infinity. By using a degree for Fredholm mappings of index zero, we reduce the existence question to finding {\it a priori} bounds for the solutions. Nevertheless, when the right-hand side has exponential decay, our existence results are valid even when the governing operator is not Fredholm