| Summary: | In this article, we study the structure of global solutions to the
boundary-value problem
$$\displaylines{
-x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr
\Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
\Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr
x(0)=x(1)=0,
}$$
where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$,
$\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$,
$\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$,
and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$
are continuous. By a comparison principle and spectral properties
of the corresponding linear equations, we prove the existence of
solutions by using Rabinowitz-type global bifurcation
theorems, and obtain results on the behavior of positive solutions
for large $\lambda$ when $f(x)=x^{p+1}$.
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