An abstract Gronwall inequality on a Banach lattice

TL;DR

This work formulates an abstract Gronwall-type inequality in real Banach lattices via the spectral bound $\rho_K$ of a positive operator $K$ and a right-inverse framework for a linear operator $L$. It proves that if $s>\rho_K$ and $sz\le Kz$, then $z\le 0$, which leads to a maximum principle for problems of the form $Lx=N(x)$ with $\mathcal{P}x=x_0$. The authors provide rapid resolvent-based proofs, extend the theory to finite dimensions and non-Fredholm elliptic operators (demonstrated via a strong maximum principle for $-\Delta$), and establish uniqueness and continuous dependence under a lattice-Lipschitz assumption on $N$. The results unify and sharpen classical Gronwall-type inequalities, connect to maximum principles, and offer a versatile framework for semilinear problems in Banach lattices with potential broad applications.

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.

Theorems & Definitions (20)

• Lemma 1.1
• Lemma 1.2
• Theorem 1.3
• Corollary 1.1
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3