In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem
$\left\{ \begin{array}{l} - u''(x) = \lambda f(u),\;\;\;\;0 < x < 1,\\u(0) = 0,\\\frac{{u(1)}}{{u(1) + 1}}u'(1) + \left[ {1 - \frac{{u(1)}}{{u(1) + 1}}} \right]u(1) = 0,\end{array} \right.$
where $λ>0$ is a bifurcation parameter, $f(u)>0$ for $u>0$. We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for $f(u) = e^{u},\ f(u) = a^{u}(a>0),\ f(u) = u^{p}(p>0),\ f(u) = e^{u}-1,\ f(u) = a^{u}-1(a>1)$ and $f(u) = (1+u)^{p}(p>0)$. Our methods are based on a detailed analysis of time maps.
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