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2011 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Fixed Point Theory in Ordered Sets and Applications

From Differential and Integral Equations to Game Theory

verfasst von: Siegfried Carl, Seppo Heikkilä

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph provides a unified and comprehensive treatment of an order-theoretic fixed point theory in partially ordered sets and its various useful interactions with topological structures. The material progresses systematically, by presenting the preliminaries before moving to more advanced topics. In the treatment of the applications a wide range of mathematical theories and methods from nonlinear analysis and integration theory are applied; an outline of which has been given an appendix chapter to make the book self-contained. Graduate students and researchers in nonlinear analysis, pure and applied mathematics, game theory and mathematical economics will find this book useful.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
In this introductory chapter we give a short account of the contents of the book and discuss simple notions and examples of the fixed point theory to be developed and applied to more involved applications in later chapters. As an introduction to the fixed point theory and its applications let us recall two fixed point theorems on a nonempty closed and bounded subset P of ℝm, one purely topological (Brouwer’s fixed point theorem) and one order-theoretically based. A point \(x\ \epsilon\ P\) is called a fixed point of a function \(G : P \rightarrow P \ \text {if}\ x = Gx.\) We assume that ℝm is equipped with Euclidean metric.
Siegfried Carl, Seppo Heikkilä
2. Fundamental Order-Theoretic Principles
Abstract
In this chapter we use the Chain Generating Recursion Principle formulated in the Introduction to develop generalized iteration methods and to prove existence and comparison results for operator equations and inclusions in partially ordered sets. Algorithms are designed to solve concrete problems by appropriately constructed Maple programs.
Siegfried Carl, Seppo Heikkilä
3. Multi-Valued Variational Inequalities
Abstract
In this chapter we provide existence, comparison, and extremality results for multi-valued elliptic and parabolic variational inequalities that will be used in subsequent chapters about discontinuously perturbed problems of this kind. The subject of this chapter is not only a prerequisite for the following chapters, but also is of independent interest. Our presentation is based on and includes results recently obtained in [38, 39, 66, 69, 70, 72], which partly generalize related results of [62] on this topic. Moreover, the theory about multi-valued elliptic and parabolic variational inequalities allows us to treat a wide range of nonsmooth elliptic and parabolic problems in a unified way.
Siegfried Carl, Seppo Heikkilä
4. Discontinuous Multi-Valued Elliptic Problems
Abstract
In this chapter we study various multi-valued elliptic boundary value problems involving discontinuous and nonlocal nonlinearities. The basic tools in dealing with these kinds of problems are on the one hand the existence and comparison results of Sect. 3.2 and on the other hand the abstract fixed point results provided in Chap. 2.
Siegfried Carl, Seppo Heikkilä
5. Discontinuous Multi-Valued Evolutionary Problems
Abstract
In this chapter we consider multi-valued evolutionary problems involving discontinuous data. Abstract fixed point results developed in Chap. 2 and the theory on multi-valued parabolic variational inequalities provided in Chap. 3 are the main tools used in the treatment of such kind of problems.
Siegfried Carl, Seppo Heikkilä
6. Banach-Valued Ordinary Differential Equations
Abstract
The main purpose of this chapter is to derive well-posedness, extremality, and comparison results for solutions of discontinuous ordinary differential equations in Banach spaces. A novel feature is that functions in considered differential equations are allowed to be Henstock–Lebesgue (HL) integrable with respect to the independent variable. HL integrability can be replaced also by Bochner integrability, although it is assumed explicitly only in the last section.
Siegfried Carl, Seppo Heikkilä
7. Banach-Valued Integral Equations
Abstract
In this chapter we derive existence, uniqueness, extremality, and comparison results for solutions of discontinuous nonlinear integral equations in various spaces of functions with values in an ordered Banach space E
Siegfried Carl, Seppo Heikkilä
8. Game Theory
Abstract
The main goal of this chapter is to study the existence of extremal Nash equilibria for normal-form games. Our interest is focused on games with strategic complementarities, which means roughly speaking that the best responses of players are increasing in actions of the other players. Properties, known as ‘increasing differences,’ ‘(quasi)supermodular,’ and ‘single crossing property,’ are used to formalize, or even to define strategic complementarities, cf. [9, 12, 74, 75, 146, 179, 218]. In the last section of this chapter we consider the existence of winning strategies in a pursuit and evasion game. In addition we obtain new fixed point results for set-valued and single-valued mappings.
Siegfried Carl, Seppo Heikkilä
9. Appendix
Abstract
In this chapter we provide basic facts of the theory of operators of monotone type, as well as the calculus of Clarke’s generalized gradient as it is used in Chaps. 3, 4, and 5. The focus, however, is on the basic analysis of vectorvalued, HL integrable functions used in the theory of differential and integral equations and inclusions presented in Chaps. 6–7, which represents in itself a new development that is of interest in its own. With the tools provided by that theory we are able to convert the problems under consideration into operator equations and inclusions that then can be solved by means of the results derived in Chap. 2, see Chaps. 6–7. The application of the order-theoretic results of Chap. 2 to the problems studied in Chaps. 6–7 requires a detailed analysis about the existence of supremums and infimums of chains, as well as the existence of order centers of sets in ordered function spaces, which is provided in Sect. 9.2.
Siegfried Carl, Seppo Heikkilä
Backmatter
Metadaten
Titel
Fixed Point Theory in Ordered Sets and Applications
verfasst von
Siegfried Carl
Seppo Heikkilä
Copyright-Jahr
2011
Verlag
Springer New York
Electronic ISBN
978-1-4419-7585-0
Print ISBN
978-1-4419-7584-3
DOI
https://doi.org/10.1007/978-1-4419-7585-0