Existence of minimal and maximal solutions to first--order differential equations with state--dependent deviated arguments

TL;DR

This paper establishes new existence results for solutions to first-order differential equations with state-dependent deviated arguments, including delay differential equations, using novel definitions of lower and upper solutions.

Contribution

It introduces new definitions of lower and upper solutions for differential equations with deviating arguments and proves existence of minimal and maximal solutions under these conditions.

Findings

Existence of minimal and maximal solutions is proven.

Classical definitions of lower and upper solutions may be insufficient.

Examples demonstrate the applicability of the new framework.

Abstract

We prove some new results on existence of solutions to first--order ordinary differential equations with deviating 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 paper, 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.