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

DOI: http://dx.doi.org/10.12775/TMNA.2005.005

#### 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 {\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.

$(0,\infty)$, 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.

#### Keywords

Ordinary differential equation; half-line; Sobolev space; boundary and initial value problem; Fredholm operator; a-priori bounds

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