Positive Solutions for Some Semi-positone Problems with Nonlinear Boundary Conditions via Bifurcation Theory

Abstract

Bifurcation theory is used to prove the existence of positive solutions of some classes of semi-positone problems with nonlinear boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} -u''=\lambda f(t, u), \qquad t\in (0,1),\\ u(0)=0, \quad u'(1)+c(u(1))u(1)=0,\\ \end{array}\right. } \end{aligned}$$

where $c:[0, \infty )\rightarrow [0, \infty )$ is continuous, $f:[0, \infty )\rightarrow \mathbb {R}$ is continuous and $f(t,0)<0$ for $t\in [0,1]$.