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Global bifurcation and multiple results for Sturm-Liouville problems. (English) Zbl 1223.34030
The following nonlinear Sturm-Liouville boundary value problem
$\begin{cases} -(p(t)u'(t))' + q(t)u(t) = \lambda a(t)f(u(t)), \quad 0<t<1, \\ \alpha_1 u(0) + \beta_1 u'(0) = 0, \quad \alpha_2 u(1) + \beta_2 u'(1) = 0\end{cases}$
is considered, where $$f$$: $${\mathbb R} \to {\mathbb R}$$ is a continuous function and there exists $$f_0$$, $$f_\infty \in (0,\infty)$$ such that
$f_0 = \lim_{|x|\to0}\, \frac{f(x)}{x}, \quad f_\infty = \lim_{|x|\to\infty} \frac{f(x)}{x}.$
A global bifurcation result is obtained, and then the existence of solutions having exactly $$k-1$$ zeros in $$(0,1)$$ is derived, where $$k \in {\mathbb N}$$.

##### MSC:
 34B24 Sturm-Liouville theory 34C23 Bifurcation theory for ordinary differential equations
##### Keywords:
global bifurcation; Sturm-Liouville problems
Full Text:
##### References:
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