Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 118, pp. 1-21.

Title: Existence and multiplicity of heteroclinic solutions for a
       non-autonomous boundary eigenvalue problem

Authors: Luisa Malaguti (Univ. of Modena and Reggio Emilia, Italy)
         Cristina Marcelli (Polytechnic Univ. of Marche,  Italy)

Abstract: 
 In this paper we  investigate the boundary eigenvalue problem
 $$\displaylines{
 x''-\beta(c,t,x)x'+g(t,x)=0 \cr
 x(-\infty)=0, \quad x(+\infty)=1
 }$$
 depending on the real parameter $c$. We take $\beta$ continuous
 and positive and assume that $g$ is bounded and becomes active
 and positive only when $x$ exceeds a threshold value
 $\theta \in ]0,1[$. At the point $\theta$ we allow $g(t, \cdot)$
 to have a jump. Additional monotonicity properties are required,
 when needed. Our main discussion deals with the non-autonomous case.
 In this context we prove the existence of a continuum of values $c$
 for which this problem is solvable and we estimate the interval
 of such admissible values. In the autonomous case, we show its
 solvability for at most one  $c^*$. In the special case when
 $\beta$ reduces to $c+h(x)$ with $h$ continuous, we also give a
 non-existence result, for any real  $c$.  Our methods combine
 comparison-type arguments, both for first and second order
 dynamics, with a shooting technique. Some applications  of the
 obtained results are included.

Submitted Submitted April 15, 2003. Published November 28, 2003.
Math Subject Classifications: 34B40, 34B18, 34C37.
Key Words: Boundary eigenvalue problems; positive bounded solutions;
           shooting method