Minimax Theory and Applications Edited by Biagio Ricceri Department ofMathematics, University ofCatania Catania, Italy

and

Stephen Simons Department ofMathematics, University of California at Santa Barbara, Santa Barbara, California, U.S.A.

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KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Contents Preface

Nonlinear Two Functions Minimax Theorems Cao-Zong Cheng and Bor-Luh hin

xi

1

1. Introduction 2. Nonlinear Minimax Theorems 3. Two Functions Minimax Theorems of Type A/Type B 4. Two Functions Minimax Theorems of Mixed Type References

1 5 10 18 19

Weakly Upward-Downward Minimax Theorem Cao-Zong Cheng, Bor-Luh Lin and Feng-Shuo Yu

21

References

28

A Two-Function Minimax Theorem Antonia Chinni

29

1. Introduction 2. The Main Result 3. Remarks and Examples Related to Theorem 2.2 References

29 30 31 33

Generalized Fixed-Points and Systems of Minimax Inequalities Paul Deguire

35

1. Introduction 2. Applications References

35 36 40

vi

CONTENTS

A Minimax Inequality for Marginally Semicontinuous Functions Gabriele H. Greco and Maria Pia Moschen References On Variational Minimax Problems under Relaxed Coercivity Assumptions Joachim Gwinner 1. Introduction 2. Some Preliminary Remarks 3. A Unilateral Boundary Value Problem and its Variational Mimimax Formulation 4. The Semicoercive Case 5. Lagrangian Minimax Problems References

41 50

53 53 55 57 60 64 69

A Topological Investigation of the Finite Intersection Property Charles D. Horvath

71

1. Introduction 2. The Finite Intersection Property 3. Topological Spaces with a Convexity Structure 4. Conclusion References

71 74 81 88 89

Minimax Results and Randomization for Certain Stochastic Games Albrecht Irle

91

1. Introduction 2. Randomization of Stopping Times 3. Compact Embedding and Equivalence of Randomization 4. Minimax Results in Discrete Time 5. A Minimax Result in Continuous Time References

91 93 95 98 99 103

Intersection Theorems, Minimax Theorems and Abstract Connectedness Jürgen Kindler

105

1. Introduction 2. Abstract Continuity

105 107

CONTENTS

vii

3. Abstract Connectedness 4. Intersection Theorems 5. Minimax Theorems References K-K-M-S Type Theorems in Infinite Dimensional Spaces Hidetoshi Komiya

108 110 113 120 .

121

1. Introduction 2. Selection of Base Spaces and Preliminaries 3. Balanced Families 4. K-K-M-S Type Theorems in Infinite Dimensional Spaces 5. Application to Game Theory 6. Extensions of K-K-M-S Theorem References

121 122 123 127 130 132 134

Hahn-Banach Theorems for Convex Functions Marc Lassonde

135

1. Separation of Convex Functions 2. Continuity of Convex Functions References

137 140 144

Two Functions Generalization of Horvath's Minimax Theorem Bor-Luh Lin and Feng-Shuo Yu

147

References

156

Some Remarks on a Minimax Formulation of a Variational Inequality 157 Giandomenico Mastroeni 1. Saddle Point Conditions and Variational Inequalities 2. Applications to the Classical Variational Inequality 3. Connections with Complementarity Problems 4. Vector Variational Inequalities 5. Further Developments References

157 159 161 162 164 166

Network Analysis Michael M. Neumann and Maria Victoria Velasco

167

1. Introduction 2. From Finite to Infinite Networks

167 167

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CONTENTS

3. Tools from Functional Analysis 4. Existence of Flows 5. Existence of Potentials 6. Symmetrie, Antisymmetric and Net Flows 7. Marginal Problems 8. Concluding Remarks References On a Topological Minimax Theorem and its Applications Biagio Ricceri 1. 2. 3. 4.

170 173 178 180 185 186 188 . 191

Introduction Preliminaries Proof of Theorem 1.1 An Application of Theorem 1.1 to the Problem inf x / = inf 9x / 5. A Variational Property of Integral Functionals References

191 193 196

Three Lectures on Minimax and Monotonicity Stephen Simons

217

0. Introduction 1. Multifunctions and Monotonicity 2. A Convexification of E x E* and the Three Affine Maps 3. Monotone Subsets and their "Pictures" 4. For Reflexive Spaces Only 5. The Convex Function Determined by a Multifunction 6. Surrounding Sets and the Dom-Dom Lemma 7. The "Dom-Dom Constraint Qualification" 8. A "Sum Theorem" for Reflexive Spaces References

217 219 221 222 224 227 228 234 236 239

198 203 216

Fan's Existence Theorem for Inequalities Concerning Convex Functions and its Applications 241 Wataru Takahashi 1. Introduction 2. Generalization of Fan's System Theorem 3. Basic Results in Functional Analysis 4. Applications References

241 242 248 252 259

CONTENTS An Algorithim for the Multi-Access Channel Problem Peng-Jung Wan, Ding-Zu Du and Panos M. Pardalos

ix ....

261

1. Introduction 2. The Algorithm 3. Analysis 4. Conclusion References

262 262 265 269 269

Author Index

271

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