On the discontinuous second-order deviated Dirichlet problem with non-monotone conditions

TL;DR

This paper establishes the existence of extremal solutions for a second-order Dirichlet problem with deviation arguments, allowing for nonlinearities that are not necessarily continuous or monotone, using a generalized monotone method.

Contribution

It introduces a novel approach to handle non-monotone, discontinuous nonlinearities in second-order Dirichlet problems with deviation arguments.

Findings

Existence of extremal solutions under non-monotone conditions

Application of a generalized monotone method with lower and upper solutions

Extension of solution theory to discontinuous nonlinearities

Abstract

We provide a new result on the existence of extremal solutions for second-order Dirichlet problems with deviation argument. As a novelty in this work, the nonlinearity need not be continuous or monotone. In order to obtain this new result, we use a generalized monotone method coupled with lower and upper solutions.