Generalizations of the Lax-Milgram theorem

TL;DR

This paper extends the classical Lax-Milgram theorem to nonlinear cases and provides conditions for representing bounded linear functionals, with applications to singular differential equations.

Contribution

It introduces linear and nonlinear generalizations of the Lax-Milgram theorem with new sufficient conditions and demonstrates applications to differential equations.

Findings

Established conditions for representing bounded linear functionals.

Provided two applications to singular differential equations.

Abstract

We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations.