Sixth Mississippi State Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 15 (2007), pp. 239-249. 

Title: On the exact multiplicity of solutions for boundary-value 
       problems via computing the direction of bifurcations

Authors: Joaquin Rivera (Univ. of Iowa, Iowa City, USA) 
         Yi Li  (Univ. of Iowa, Iowa City, USA) 
 
Abstract: 
We consider positive solutions of the Dirichlet problem 
$$\displaylines{
 u''(x)+\lambda f(u(x))=0\quad\hbox{in }(-1,1), \cr
 u(-1)=u(1)=0.
}$$
depending on a positive parameter $\lambda $. We use two formulas derived in 
[18] to compute all solutions $u$ where a turn may occur and to
compute the direction of the turn. As an application, we consider quintic a
polynomial $f(u)$ with positive and distinct roots. For such quintic
polynomials we conjecture the exact mutiplicity structure of positive
solutions and present computer assisted proofs of such exact bifurcation
diagrams for various distributions of the real roots. The limiting behavior
of the solutions on these bifurcation branches as $\lambda \to \infty $ and
their stabilities are also investigated.

Published February 28, 2007.
Math Subject Classifications: 34B15.
Key Words: Bifurcation points; direction of the turn; multiplicity of solutions.