Boundary value problems for first order systems on the half-line
Keywords
Ordinary differential equation, half-line, Sobolev space, boundary and initial value problem, Fredholm operator, a-priori boundsAbstract
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.Downloads
Published
2005-03-01
How to Cite
1.
RABIER, Patric J. and STUART, Charles A. Boundary value problems for first order systems on the half-line. Topological Methods in Nonlinear Analysis. Online. 1 March 2005. Vol. 25, no. 1, pp. 101 - 131. [Accessed 27 April 2024].
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