The authors consider the second order boundary value problem (1) \(-u'' = f(t,u)\), \(0<t<1\), \(\alpha u(0) - \beta u'(0) = 0\), \(\gamma u(1) + \delta u'(1) = 0\), where \(f\) is continuous and \(f(t,u) \geq 0\) for \(t \in[0,1]\) and \(u \geq 0\), \(\alpha, \beta, \gamma, \delta \geq 0\) and \(\alpha \beta + \alpha \gamma + \alpha \delta>0\). They prove the existence of two positive solutions of (1) provided \(f(t,u)\) is superlinear at one end (zero or infinitely) and sublinear at the other. It is shown that these results also imply the existence of multiple positive radial solutions of certain semilinear elliptic boundary value problems. The proofs are based on the fixed point arguments.