Positive solutions of fourth-order problems with dependence on all derivatives in nonlinearity under Stieltjes integral boundary conditions

In this article, we investigate the existence of positive solutions to fourth-order problems with dependence on all derivatives in nonlinearities subject to the Stieltjes integral boundary conditions $$\begin{aligned}& \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u'(0)+\beta _{1}[u]=0, \qquad u''(0)+\beta _{2}[u]=0, \qquad u(1)=\beta _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ and $$\begin{aligned}& \textstyle\begin{cases} -u^{(4)}(t)=g(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=\alpha _{1}[u], \qquad u'(0)=\alpha _{2}[u], \qquad u''(0)=\alpha _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ where $f: [0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{-}^{3}\to \mathbb{R}_{+}$ , $g: [0,1]\times \mathbb{R}^{3}_{+}\to \mathbb{R}_{+}$ are continuous and $\beta _{i}[u]$ , $\alpha _{i}[u]$ ( $i=1,2,3$ ) are linear functionals involving Stieltjes integrals of signed measures. Some growth conditions are posed on nonlinearities f, g, meanwhile the spectral radii of corresponding linear operators are restricted, which means the superlinear or sublinear conditions. On the cones in $C^{3}[0,1]$ we apply the theory of fixed point index, the existence of positive solutions is obtained. We also give some examples under mixed multi-point and integral boundary conditions with sign-changing coefficients.