Extremal solutions of systems of measure differential equations and applications in the study of Stieltjes differential problems

TL;DR

This paper investigates extremal solutions for measure differential equations using lower and upper solutions, and applies these results to Stieltjes differential problems, including a bacteria population model.

Contribution

It introduces methods to find extremal solutions for quasimonotone measure differential systems and applies these to unify and solve Stieltjes differential equations.

Findings

Existence of greatest and least solutions established.

Results applied to a bacteria population model.

Unified approach to discrete, continuous, and impulsive systems.

Abstract

We use lower and upper solutions to investigate the existence of the greatest and the least solutions for quasimonotone systems of measure differential equations. The established results are then used to study the solvability of Stieltjes differential equations; a recent unification of discrete, continuous and impulsive systems. The applicability of our results is illustrated in a simple model for bacteria population.