Journal of Inequalities and Applications
Volume 6 (2001), Issue 5, Pages 483-506
doi:10.1155/S1025583401000303
Bifurcation of solutions of nonlinear Sturm–Liouville problems
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, Gdańsk 80-952, Poland
Received 15 July 1999; Revised 1 March 2000
Copyright © 2001 Jacek Gulgowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A global bifurcation theorem for the following nonlinear Sturm–Liouville problem is given
{u″(t)=−h(λ,t,u(t),u′(t)),
a.e. on (0,1)u(0)cosη−u′(0)sinη=0
(∗)u(1)cosζ+u′(1)sinζ=0
with η,ζ∈[0,π2].
Moreover we give various versions of existence theorems for boundary value problems
{u″(t)=−g(t,u(t),u′(t)),
a.e. on (0,1)u(0)cosη−u′(0)sinη=0
(∗∗)u(1)cosζ+u′(1)sinζ=0.The main idea of these proofs is studying properties of an unbounded connected subset of the set of all nontrivial solutions of the nonlinear spectral problem (∗), associated with the boundary value problem (∗∗), in such a way that h(1,⋅,⋅,⋅)=g.