Abstract
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 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.
Citation
Patric J. Rabier. Charles A. Stuart. "Boundary value problems for first order systems on the half-line." Topol. Methods Nonlinear Anal. 25 (1) 101 - 133, 2005.
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