Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound third-order derivative term, and theory fixed-point index used suitable open sets obtain existence positive solutions. The have quadratic growth term. Previous results literature are not applicable our case, as shown by examples.