On positive solutions for some second-order three-point boundary value problems with convection term

In this paper, a fixed point theorem in a cone and some inequalities of the associated Green’s function are applied to obtain the existence of positive solutions of second-order three-point boundary value problem with dependence on the first-order derivative $$\begin{aligned}& x''(t) + f\bigl(t, x(t), x'(t)\bigr) =0, \quad 0< t< 1, \\& x(0) =0, \qquad x(1) =\mu x(\eta ), \end{aligned}$$ where \(f: [0, 1] \times [0, \infty ) \times R \rightarrow [0, \infty )\) is a continuous function, \(\mu >0\), \(\eta \in (0, 1)\), \(\mu \eta <1\). The interesting point is that the nonlinear term is dependent on the convection term.