An abstract Gronwall inequality on a Banach lattice

TL;DR

This paper develops an abstract Gronwall inequality within Banach lattices using spectral bounds, leading to new results on uniqueness, continuous dependence, and connections to the maximum principle for semilinear problems.

Contribution

It introduces an abstract Gronwall inequality based on spectral bounds in Banach lattices, extending classical results to a more general setting.

Findings

Establishes a spectral bound-based Gronwall inequality in Banach lattices.

Proves uniqueness and continuous dependence for semilinear problems.

Explores the relationship between the inequality and the maximum principle.

Abstract

An abstract version of the celebrated inequality is described by means of the spectral bound of an operator defined on a Banach lattice. As a consequence, uniqueness and continuous dependence results for the general semilinear problem $Lu=N(u)$ are established and a connection with the maximum principle is explored.