# Estimate of the validity interval for the Antimaximum Principle and application to a non-cooperative system

## TL;DR

This paper investigates the conditions under which the Antimaximum Principle holds for non-cooperative systems near a principal eigenvalue, providing estimates, applications, and counterexamples to clarify its validity range.

## Contribution

It offers precise estimates of the Antimaximum Principle's validity interval for systems and demonstrates its application and limitations through examples and counterexamples.

## Key findings

- Validity interval for the Antimaximum Principle is precisely estimated.
- The principle applies only within a restricted positive cone.
- Counterexamples show the necessity of the hypotheses.

## Abstract

We are concerned with the sign of the solutions of non-cooperative systems when the parameter varies near a principal eigenvalue of the system. With this aim we give precise estimates of the validity interval for the Antimaximum Principle for an equation and an example. We apply these results to a non-cooperative system. Finally a counterexample shows that our hypotheses are necessary. The Maximum Principle remains true only for a restricted positive cone.