Rubén Figueroa

Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments

We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this article, and we show with an example that similar results with the classical definitions are false. We also introduce an example showing that the problems considered need not have the least (or the greatest) solution between given lower and upper solutions, but we can prove that they do have minimal and maximal solutions in the usual set-theoretic sense. Sufficient conditions for the existence of lower and upper solutions, with some examples of application, are provided too.

Second-order functional differential equations with past, present and future dependence☆

Abstract In this paper we obtain a result on existence of extremal quasisolutions and solutions for second-order functional differential equations with functional boundary conditions and past, present and future dependence. In order to do that, we develop a generalized monotone method in presence of lower and upper solutions. Some examples of application are provided too.

A Schauder-type theorem for discontinuous operators with applications to second-order BVPs

We prove a new fixed point theorem of Schauder type, which applies to discontinuous operators in noncompact domains. In order to do so, we present a modification of a recent Schauder-type theorem of Pouso. We apply our result to second-order boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory.

Discontinuous functional differential equations with delayed or advanced arguments

We provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced. The boundary condition is allowed to be discontinuous and to depend functionally on the unknown solution.

A version of Krasnosel'skiĭ's compression-expansion fixed point theorem in cones for discontinuous operators with applications

We introduce a new fixed point theorem of Krasnoselskiĭ type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and discontinuous nonlinearities.