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.