Competing Operators and Their Applications to Boundary Value Problems / by Marek Galewski, Dumitru Motreanu.

Format:

Book

Author/Creator:

Galewski, Marek.

Series:

SpringerBriefs in Mathematics, 2191-8201

Language:

English

Subjects (All):

Operator theory.

Approximation theory.

Operator Theory.

Approximations and Expansions.

Local Subjects:

Operator Theory.

Approximations and Expansions.

Physical Description:

1 online resource (172 pages)

Edition:

1st ed. 2026.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2026.

Summary:

This book addresses problems driven by differential operators that lack monotonicity. The authors’ methods rely on coercivity and continuity, allowing for the construction of an approximative scheme whose convergence is induced by coercivity. This observation leads to a new type of solution, which is precisely a limit of finite-dimensional approximation schemes and leads to the weak solution, provided that the operator driving the equation is at least pseudomonotone. This new type of solution is called a generalized solution. To systematically treat its existence, the authors introduce an abstract existence tool that serves as a counterpart to the Browder-Minty Theorem in the non-variational case and the Weierstrass-Tonelli Theorem if the problem is potential. Thus, the authors utilize many already developed techniques, suitably modified due to the absence of the monotonicity assumption. The authors obtain three abstract results, also in the non-smooth case, which they apply to nonlinear boundary value problems. In their applications, they also deal with problems depending on an unbounded weight, which forces them to implement a suitable truncation technique. The book includes an extended chapter covering analysis on abstract tools from the theory of monotone operators and minimization techniques, supplied with proofs and comments that allow for a better understanding of the authors’ approach towards generalized solutions. It includes necessary background on Sobolev spaces, introduces the non-variational generalized solution, and investigates the existence of solutions for variational problems and inclusions.

Contents:

Introduction

Background from function spaces

A resume on existence methods

Generalized solutions for non-potential problems

Generalized solutions - variational problems

Generalized solutions for inclusions

Index

References.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-032-15445-6

9783032154453

OCLC:

1579290358