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.