Topological Methods in Nonlinear Analysis

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

Patric J. Rabier and Charles A. Stuart

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 25, Number 1 (2005), 101-133.

Dates
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1466705097

Mathematical Reviews number (MathSciNet)
MR2133394

Zentralblatt MATH identifier
1090.34014

Citation

Rabier, Patric J.; Stuart, Charles A. Boundary value problems for first order systems on the half-line. Topol. Methods Nonlinear Anal. 25 (2005), no. 1, 101--133. https://projecteuclid.org/euclid.tmna/1466705097


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