Global structure of positive solutions for third-order semipositone integral boundary value problems

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition

$$\begin{array}{rl}& y^{‴}+\beta y^{″}+\alpha y^{\prime }+\lambda f(t,y)=0,t\in (0,1),\\ & y(0)=y^{\prime }(0)=0,y(1)=\chi {\int }_0^{1}y(s)ds,\end{array}$$

where $\alpha \in (0,\mathrm{\infty })$ and $\beta \in (-\mathrm{\infty },\mathrm{\infty })$ are two constants, $\lambda ,\chi$ are two positive parameters, and $f\in C([0,1]\times [0,\mathrm{\infty }),\mathbb{R})$ with $f(t,0)<0$. Our analysis mainly relied on the bifurcation theory.