Resolvent and Gronwall inequalities and fixed points of evolution operators

TL;DR

This paper develops new resolvent and Gronwall inequalities for functions of multiple variables, leading to a generalized fixed point theorem applicable to operators on topological spaces.

Contribution

It introduces kernels and resolvents on preordered sets and extends Banach's fixed point theorem to broader topological contexts.

Findings

Established sharp resolvent inequalities

Derived Gronwall inequalities for multivariable functions

Proved a generalized fixed point theorem

Abstract

We introduce kernels and resolvents on preordered sets and derive sharp resolvent inequalities that entail Gronwall inequalities for functions of several variables. In this way, we can prove a fixed point result for operators on topological spaces that extends Banach's fixed point theorem and allows for a wide range of applications.