Positive Solutions of a Third-Order Boundary Value Problem with Full Nonlinearity

This paper is concerned with the existence of positive solutions of the third-order boundary value problem with full nonlinearity

$$\begin{aligned} \left\{ \begin{array}{lll} u'''(t)&{}=f(t,u(t),u'(t),u''(t)),\quad t\in 0,1],\\ u(0)&{}=u'(1)=u''(1)=0, \end{array}\right. \end{aligned}$$

where \(f:0,1]\times \mathbb {R}^+\times \mathbb {R}^+\times \mathbb {R}^-\rightarrow \mathbb {R}^+\) is continuous. Under some inequality conditions on f as |(x, y, z)| small or large enough, the existence results of positive solution are obtained. These inequality conditions allow that f(t, x, y, z) may be superlinear, sublinear or asymptotically linear on x, y and z as \(|(x,y,z)|\rightarrow 0\) and \(|(x,y,z)|\rightarrow \infty \). For the superlinear case as \(|(x,y,z)|\rightarrow \infty \), a Nagumo-type growth condition is presented to restrict the growth of f on y and z. Our discussion is based on the fixed point index theory in cones.