Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory

Marc Notes: Includes bibliographical references and index. Review Quotes: From the reviews: The book provides a self-contained exposition of an order-theoretic fixed point theory in partially ordered sets together with its interactions with topological structures. The book is well written and provides a very detailed study of theoretic topics along with many illustrative examples and applications of various types covering an immense range of problems. the book is very well designed for studying nonlinear problems via a fixed point approach. Both graduate students and mathematicians will find this book attractive, useful and highly instructive. (Wojciech Kryszewski, Mathematical Reviews, Issue 2012 c) Brief Description: This unified, comprehensive treatment of an order-theoretic fixed point theory in partially ordered sets shows its various useful interactions with topological structures. Covering the preliminaries before moving to more advanced topics makes it approachable. Jacket Description/Back: 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. It begins with a discussion of some simple examples of the order-theoretic fixed point results along with simple applications from each of the diverse fields. The fixed point theory is then developed and preliminary results on multi-valued variational inequalities regarding the topological and order-theoretical structure of solution sets are covered. This is followed by more advanced material which demonstrates the power of the developed fixed point theory. 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 in 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.""Table of Contents: Preface -- 1. Introduction -- 2. Fundamental Order-Theoretic Principles -- 2.1. Recursions and Iterations in Posets -- 2.2. Fixed Point Results in Posets -- 2.2.1. Fixed Points for Set-Valued Functions -- 2.2.2. Fixed Points for Single-Valued Functions -- 2.2.3. Comparison and Existence Results -- 2.2.4. Algorithmic Methods -- 2.3. Solvability of Operator Equations and Inclusions -- 2.3.1. Inclusion Problems -- 2.3.2. Single-Valued Problems -- 2.4. Special Cases -- 2.4.1. Fixed Point Results in Ordered Topological Spaces -- 2.4.2. Equations and Inclusions in Ordered Normed Spaces -- 2.5. Fixed Point Results for Maximalizing Functions -- 2.5.1. Preliminaries -- 2.5.2. Main Results -- 2.5.3. Examples and Remarks -- 2.6. Notes and Comments -- 3. Multi-Valued Variational Inequalities -- 3.1. Introductory Example -- 3.2. Multi-Valued Elliptic Variational Inequalities -- 3.2.1. The Sub-Supersolution Method -- 3.2.2. Directedness of Solution Set -- 3.2.3. Extremal Solutions -- 3.2.4. Equivalence to Variational-Hemivariational Inequality -- 3.3. Multi-Valued Parabolic Variational Inequalities -- 3.3.1. Notion of Sub-Supersolution -- 3.3.2. Multi-Valued Parabolic Equation -- 3.3.3. Parabolic Variational Inequality -- 3.4. Notes and Comments -- 4. Discontinuous Multi-Valued Elliptic Problems -- 4.1. Nonlocal and Discontinuous Elliptic Inclusions -- 4.1.1. Hypotheses, Main Result, and Preliminaries -- 4.1.2. Proof of Theorem 4.1 -- 4.1.3. Extremal Solutions -- 4.1.4. Application: Difference of Clarke's Gradient and Subdifferential -- 4.2. State-Dependent Clarke's Gradient Inclusion -- 4.2.1. Statement of the Problem -- 4.2.2. Notions, Hypotheses, and Preliminaries -- 4.2.3. Existence and Comparison Result -- 4.2.4. Application: Multiplicity Results -- 4.3. Discontinuous Elliptic Problems via Fixed Points for Multifunctions -- 4.3.1. Abstract Fixed Point Theorems for Multi-Functions -- 4.3.2. Discontinuous Elliptic Functional Equations -- 4.3.3. Implicit Discontinuous Elliptic Functional Equations -- 4.4. Notes and Comments -- 5. Discontinuous Multi-Valued Evolutionary Problems -- 5.1. Discontinuous Parabolic Inclusions with Clarke's Gradient -- 5.2. Implicit Functional Evolution Equations -- 5.2.1. Preliminaries -- 5.2.2. Main Result -- 5.2.3. Generalization and Special Cases -- 5.2.4. Application -- 5.3. Notes and Comments -- 6. Banach-Valued Ordinary Differential Equations -- 6.1. Cauchy Problems -- 6.1.1. Preliminaries -- 6.1.2. A Uniqueness Theorem of Nagumo Type -- 6.1.3. Existence Results -- 6.1.4. Existence and Uniqueness Results -- 6.1.5. Dependence on the Initial Value -- 6.1.6. Well-Posedness of a Semilinear Cauchy Problem -- 6.2. Nonlocal Semilinear Differential Equations -- 6.2.1. Existence and Comparison Results -- 6.2.2. Applications to Multipoint Initial Value Problems -- 6.3. Higher Order Differential Equations -- 6.3.1. Well-Posedness Results -- 6.3.2. Semilinear Problem -- 6.3.3. Extremal Solutions -- 6.4. Singular Differential Equations -- 6.4.1. First Order Explicit Initial Value Problems -- 6.4.2. First Order Implicit Initial Value Problems -- 6.4.3. Second Order Initial Value Problems -- 6.4.4. Second Order Boundary Value Problems -- 6.5. Functional Differential Equations Containing Bochner Integrable Functions -- 6.5.1. Hypotheses and Preliminaries -- 6.5.2. Existence and Comparison Results -- 6.6. Notes and Comments -- 7. Banach-Valued Integral Equations -- 7.1. Integral Equations in HL-Spaces -- 7.1.1. Fredholm Integral Equations -- 7.1.2. Volterra Integral Equations -- 7.1.3. Application to Impulsive IVP -- 7.1.4. A Volterra Equation Containing HL Integrable Functions -- 7.2. Integral Equations in Lp-Spaces -- 7.2.1. Preliminaries -- 7.2.2. Urysohn Equations -- 7.2.3. Fredholm Integral Equations -- 7.2.4. Volterra Integral Equations -- 7.2.5. Application to Impulsive IVP -- 7.3. Evolution Equations -- 7.3.1. Well-Posedness Results -- 7.3.2. Existence and Uniqueness Result -- 7.3.3. Continuous Dependence on x0 -- 7.3.4. Special Cases -- 7.3.5. Application to a Cauchy Problem -- 7.3.6. Extremal Solutions of Evolution Equations -- 7.3.7. Evolution Equations Containing Bochner Integrable Functions -- 7.3.8. Application -- 7.4. Notes and Comments -- 8. Game Theory -- 8.1. Pure Nash Equilibria for Finite Simple Normal-Form Games -- 8.1.1. Preliminaries -- 8.1.2. Existence and Comparison Results -- 8.1.3. An Application to a Pricing Game -- 8.2. Pure and Mixed Nash Equilibria for Finite Normal-Form Games -- 8.2.1. Preliminaries -- 8.2.2. Existence Result for the Greatest Nash Equilibrium -- 8.2.3. Comparison Result for Utilities -- 8.2.4. Dual Results -- 8.2.5. Applications to Finite Supermodular Games -- 8.2.6. Application to a Multiproduct Pricing Game -- 8.3. Pure Nash Equilibria for Normal-Form Games -- 8.3.1. Extreme Value Results -- 8.3.2. Smallest and Greatest Pure Nash Equilibria -- 8.3.3. Special Cases -- 8.3.4. Applications to a Multiproduct Pricing Game -- 8.3.5. Minimal and Maximal Pure Nash Equilibria -- 8.4. Pure and Mixed Nash Equilibria of Normal-Form Games -- 8.4.1. Definitions and Auxiliary Results -- 8.4.2. Existence and Comparison Results -- 8.4.3. Applications to Supermodular Games -- 8.5. Undominated and Weakly Dominating Strategies and Weakly Dominating Pure Nash Equilibria for Normal-Form Games -- 8.5.1. Existence of Undominated Strategies -- 8.5.2. Existence of Weakly Dominating Strategies and Pure Nash Equilibria -- 8.5.3. Examples -- 8.6. Pursuit and Evasion Game -- 8.6.1. Preliminaries -- 8.6.2. Winning Strategy -- 8.6.3. Applications and Special Cases -- 8.7. Notes and Comments -- 9. Appendix -- 9.1. Analysis of Vector-Valued Functions -- 9.1.1. -Measurability and -Integrability of Banach-Valued Functions -- 9.1.2. HL Integrability -- 9.1.3. Integrals of Derivatives of Vector-Valued Functions -- 9.1.4. Convergence Theorems for HL Integrable Functions -- 9.1.5. Ordered Normed Spaces of HL Integrable Functions -- 9.2. Chains in Ordered Function Spaces -- 9.2.1. Chains in Lp-Spaces -- 9.2.2. Chains of Locally Bochner Integrable Functions -- 9.2.3. Chains of HL Integrable and Locally HL Integrable Functions -- 9.2.4. Chains of Continuous Functions -- 9.2.5. Chains of Random Variables -- 9.2.6. Properties of Order Intervals and Balls in Ordered Function Spaces -- 9.3. Sobolev Spaces -- 9.3.1. Definition of Sobolev Spaces -- 9.3.2. Chain Rule and Lattice Structure -- 9.4. Operators of Monotone Type -- 9.4.1. Main Theorem on Pseudomonotone Operators -- 9.4.2. Leray-Lions Operators -- 9.4.3. Multi-Valued Pseudomonotone Operators -- 9.5. First Order Evolution Equations -- 9.5.1. Evolution Triple and Generalized Derivative -- 9.5.2. Existence Results for Evolution Equations -- 9.6. Calculus of Clarke's Generalized Gradient -- List of Symbols -- References -- Index. Review Quotes: From the reviews: The book provides a self-contained exposition of an order-theoretic fixed point theory in partially ordered sets together with its interactions with topological structures. The book is well written and provides a very detailed study of theoretic topics along with many illustrative examples and applications of various types covering an immense range of problems. the book is very well designed for studying nonlinear problems via a fixed point approach. Both graduate students and mathematicians will find this book attractive, useful and highly instructive. (Wojciech Kryszewski, Mathematical Reviews, Issue 2012 c)"

Contributor Bio:  Heikkila, Seppo Heikkila is an Associate Professor of Mathematics in the Faculty of Sciences and a Docent of Applied Mathematics in the Faculty of Technology at the University of Oulu, Finland.