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

Abstract

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

On fourth order retarded equations with functional boundary conditions: A unified approach

“…Fourth order ODEs with nonlocal and/or nonlinear boundary terms have been studied in the past, also in view of applications to mechanical systems, see for example the recent papers [4,7,14,21,29] and the references therein. Functional differential equations, in the case of fourth and higher order, have also been considered in the past, we mention here the manuscripts [2,16].…”