On a general variational framework for existence and uniqueness in Differential Equations

TL;DR

This paper introduces a versatile variational framework based on the contraction mapping principle, applicable to various existence problems in differential equations, including ODEs, PDEs, hyperbolic problems, and Navier-Stokes systems.

Contribution

It develops a general variational approach that broadens the applicability of existence results in differential equations beyond traditional methods.

Findings

Applicable to initial-value and Cauchy problems for ODEs

Extends to non-linear, monotone PDEs and hyperbolic problems

Includes steady Navier-Stokes systems as examples

Abstract

Starting from the classic contraction mapping principle, we establish a general, flexible, variational setting that turns out to be applicable to many situations of existence in Differential Equations. We show its potentiality with some selected examples including initial-value, Cauchy problems for ODEs; non-linear, monotone PDEs; linear and non-linear hyperbolic problems; and steady Navier-Stokes systems.