Research Topics in Analysis

✍ Scribed by Shouchuan Hu , Nikolaos S. Papageorgiou

• Publisher: Birkhäuser
• Year: 2024
• Tongue: English
• Leaves: 731
• Series: Birkhäuser Advanced Texts Basler Lehrbücher
• Edition: 1
• Category: Library

✦ Synopsis

This book, the second of two volumes, presents significant applications for understanding modern analysis. It empowers young researchers with key techniques and applications to explore various subfields of this broad subject and introduces relevant frameworks for immediate deployment.

The applications list begins with Degree Theory, a useful tool for studying nonlinear equations. Chapter 2 deals with Fixed Point Theory, and Chapter 3 introduces Critical Point Theory. Chapter 4 presents the main spectral properties of linear, nonlinear, anisotropic, and double-phase differential operators. Chapter 5 covers semilinear and nonlinear elliptic equations with different boundary conditions, while Chapter 6 addresses dynamic systems monitored by ordinary and partial differential equations. Chapter 7 delves into optimal control problems, and Chapter 8 discusses some economic models, providing a brief presentation of Game Theory and Nash equilibrium.

By offering a clear and comprehensive overview of modern analysis tools and applications, this work can greatly benefit mature graduate students seeking research topics, as well as experienced researchers interested in this vast and rich field of mathematics.

✦ Table of Contents

Preface
Note on Notations
Contents
1 Degree Theory
Introduction
1.1 Brouwer Degree
1.2 Leray-Schauder Degree
1.3 Degree for Maps of Monotone Type
1.4 Degree for Multifunctions
Remarks
2 Fixed Point Theory
Introduction
2.1 Metric Fixed Point Theory
2.2 Topological Fixed Point Theory
2.3 Order Fixed Point Theory
2.4 Fixed Points for Multifunctions
Remarks
3 Critical Point Theory
Introduction
3.1 Compactness Conditions-Pseudogradients
3.2 Deformation Theorem
3.3 Minimax Theorems
3.4 Multiple Critical Points
3.5 Critical Points Under Constraints
3.6 Critical Groups
Remarks
4 Spectra of Differential Operators
Introduction
4.1 Scalar and Vector Operators
4.2 Linear Operators
4.3 Nonlinear Operators
4.4 Double Phase and Anisotropic Operators
Remarks
5 Elliptic Boundary Value Problems
Introduction
5.1 Maximum, Antimaximum and Comparison Principles
5.2 Nonlinear Elliptic Equations
5.3 Singular Problems
5.4 Problems with Convection
5.5 Anisotropic and Double Phase Problems
Remarks
6 Evolution Equations
Introduction
6.1 Initial Value and Periodic Problems
6.2 Evolution Inclusions with Nonlocal Conditions
6.3 Structure of the Solution Set
6.4 Continuous Dependence Results
Remarks
7 Calculus Of Variations
Introduction
7.1 Calculus of Variations
7.2 Optimal Control-Relaxation
7.3 Sensitivity Analysis
Remarks
8 Mathematical Economics and Game Theory
Introduction
8.1 Equilibrium Theory
8.2 Dynamic Models
8.3 Nash Equilibrium
Remarks
References
Further Reading
Index

✦ Subjects

Degree Theory, Fixed Point Theory, Critical Point Theory, Spectra of Differential Operators, Elliptic Boundary Value Problems, Evolution Equations, Calculus Of Variations, Mathematical Economics, Game Theory