Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 135, pp. 1-15.
Title: Boundary-value problems for nonautonomous nonlinear systems
on the half-line
Author: Jason R. Morris (State Univ. of New York, Brockport, NY, USA)
Abstract:
A method is presented for proving the existence of solutions for
boundary-value problems on the half line. The problems under study
are nonlinear, nonautonomous systems of ODEs with the possibility
of some prescribed value at $t=0$ and with the condition that
solutions decay to zero as $t$ grows large. The method relies
upon a topological degree for proper Fredholm maps.
Specific conditions are given to ensure that the boundary-value
problem corresponds to a functional equation that involves an
operator with the required smoothness, properness, and Fredholm
properties (including a calculable Fredholm index).
When the Fredholm index is zero and the solutions are bounded
a priori, then a solution exists. The method is applied
to obtain new existence results for systems of the form
$\dot{v}+g(t,w)=f_1(t)$ and $\dot{w}+h(t,v)=f_2(t)$.
Submitted October 4, 2011. Published October 17, 2011.
Math Subject Classifications: 34B40, 34B15, 34D09, 46E15, 47H11, 47N20.
Key Words: Ordinary differential equation; half-line; infinite interval;
boundary and initial value problem; Fredholm operator;
degree theory; exponential dichotomy; properness; a priori bounds.