Nonlinear and Convex Analysis: Proceedings in Honor of Ky Fan 0824777778, 9780824777777

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Nonlinear and

Convex Analysis

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft

Zuhair Nashed

Rutgers University New Brunswick. New Jersey

Unfrersity of Del.aware Newark. Delaware

CHAIRMEN OF THE EDITORIAL BOARD

S. Kobayashi

Edwin Hewitt

University of California, Berkeley Berkeley, California

University of Washington Seattle, Washington

EDITORIAL BOARD M. S. Baouendi Purdue University

Donald Passman University of Wisconsin-Madison

Jack K. Hale Brown University

Fred S. Roberts Rutgers University

Marvin l'v!arcus University of California, Santa Barbara W. S. Massey Yale University Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and University of Rochester Anil Nerode Cornell University

Gian-Carlo Rota Massachusetts Institute of Technology David Russell University of Wisconsin-Madison Jane Cronin Scanlon Rutgers University Walter Schempp Universitii t Siegen

Mark Teply University of Wisconsin-Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS 1. N. .lacohson, Lxccptional Lie AlgL·bras 2. L. -A'. U1ulahl and F Poulsen, Thin Sets in llannonic Analysis

3. I. Sa take. Classification Theory of Semi-Simple Algebraic Croups 4. F Hir:ehruch, W D. Newmann, and S.S. Koh, Dillerentiable Manifolds and Quadratic Forms (out of print) 5. I. Cizal'c/. Riemannian Symmetric Spaces of Rank 01\L' (out of print) 6. R. B. Burckcl, Characterization of C(X) Among Its Subalgcbras 7. B. R. McDonald..-1. R. Magid, and K. C Smith, Ring Theorv: Proceedings of the Okbhoma ConfrrenL-c 8. Y.-T. Siu, Techniques of Fxtension on Analytic Objects 9. S. R. Caradus, W. /:. Pfajji•11hcrger. and B. Yood, Calkin Algebras and Algebras ol Operators on Banach SpaL"L'S 10. /:". 0. Roxin. P.-T. Uu. and R. L. Stcmherg. Ditlen:ntial Can1csand Contrnl lheor, 11 . .l'vl. Orzech and C. S111all. The Brauer Group of Commutative Rings 12. S. Thomeier, Topology ,ind Its Applications 13 . ./. ,"1. Lopez and K. A. Ross. Sidon Sets 14. W. W. ComjcJrl and S. ,Vcgrepontis, Continuous P,eudometncs 15. K. McKennon and .I. iv!. Roher/son, Locally Convex Spaces 16. M. Canncli and S. Malin. Representations of the Rotation and Lorentz Groups: An Introduction 17. G. B. Seligman, Rational Methods in Lie Algebras 18. D. G. de Figueiredo. Functional Analysis: Proccedint:s of 1hc Brazilian l\lathematical Society Symposium 19. L. Cesari, R. Kannan, and .I. D. Sclzuur, Nonlinear I unctional Analysis and Differential Equations: Proceedings of the Michigan State University Conference 211 . ./. ./. Schaffer, Geometry of Spheres in Normed Spaces 21. K. Yano and M. Kon, Anti-Invariant Subnianifolds 22. W. V. Vasco11cf'ios, The Rings of Dimension Two 23. R. J-:. Chandler, Hausdorff Compactifications 24. S. P. Franklin and B. VS. Thomas, Topology: Proceedings of the Memphis State University Conference 25. S. K. Jain, Ring Theory: Proceedings the Ohio University Conference 26. B. R. McDonald and R. A. Morris. Ring Theory II: Proceedings of the Second OkLihom,1 Conference 27. R. B. Mura and A. Rhe11Hulla. Orderablc (;roups 28 . ./. R. Graef: Stability of Dynamical Systems: Thl'Ory and Application, 29. lf..C. Wang, Homo~eneous Branch Algebras 30. F. 0. Roxin, P.-T. /,i11, and R. /,. Sternhcrg. Differential Games and Control The,m II 31. R. D. Porter. Introduction to Fibre Bundles 32. M. Altman, Contrac·tors and Contractor Direction, Theory and Applications 33 . .I. S. Colan. Decomposition and Dimension in .\1odule Categories 34. c;_ Fairweather, 1-inite flemL'llt Galcrkin Methods for Di!lerL'nti,il 1-.quatlllns 35 . .I D. Sall_\', Numbers of Cennators of Ideals in Lllcal Rings 36. S S. Miller, Comple~ Analysis: Proceedings of the S.U.:\i.Y. Brnckport ConlcrenCL' 37. R. Gordon, Representation Theory of Algebra,: ProcL·edings ,,f the Phil:tdclphia ConferenLT 38. J'vl. Goto and F !)_ Grosshans. Scmisnnpk Lie Al~d,ra, 39. A. I. Arruda. X. CA. da Costa, and R. Ch1ia1111i. .\Li1hematical 1.o~ic: Procccdin).!s of the 1-'irst Brazilian Conference

or

40. F. Van 0ystaeyen, Ring Theory: Proceedings of the 1977 Antwerp Conference 41. F. Van 0ystaeyen and A. Verschoren. Reflectors and Localization: Application to Sheaf Theory 42. M. Satyanarayana, Positively Ordered Semigroups 43. D. L. Russell, Mathematics of finite-Dimensional Control Systems 44. P.-T. Liu and E. Roxin, Differential Games and Control Theory III: Proceedings of the Third Kingston Conference, Part A 45. A. Geramita and J. Seberry. Orthogonal Designs: Quadratic Forms and Hadamard Matrices 46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 4 7. P.-T. Liu and J. G. Sutinen. Control Theory in Mathematical Economics: Proceedings of the Third Kingston Conference, Part B 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van 0ystaeyen, Ring Theory: Proceedings of the 1978 Antwerp Conference 52. B. Kedem, Binary Time Series 53. J. Barros-Neto and R. A. Artino. Hypoelliptic Boundary-Value Problems 54. R. L. Sternberg, A. J. Kalinowski. and J. S. Papadakis. Nonlinear Partial Differential Equations in Engineering and Applied Science 55. B. R. McDonald, Ring Theory and Algebra III: Proceedings of the Third Oklahoma Conference 56. J. S. Golan, Structure Sheaves over a Noncommutative Ring 57. T. V. Narayana, J. G. Williams, and R. M. Mathsen. Combinatorics, Representation Theory and Statistical Methods in Groups: YOUNG DAY Proceedings 58. T. A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. 0. A. Nielson, Direct Integral Theory 62. J. E Smith, G. 0. Kenny. and R. N. Ball, Ordered Groups: Proceedings of the Boise State Conference 63. J. Cronin, Mathematics of Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. T. G. McLaughlin, Regressive Sets and the Theory of !sols 67. T. L. Herdman, S. M. Rankin, III, and H. W. Stech. Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra: Analytic Methods 69. W. G. McKay and J. Patera. Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz, C. Praeger, and R. Su/Iii-an, Algebraic Structures and Applications Proceedings of the First Western Australian Conference on Algebra 75. L. Bican, T. Kepka, and P. Nemec, Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry: Proceedings of the Second University of Oklahoma Conference 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C Yang, Factorization Theory of Meromorphic l'unctions 79. 0. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry. Nonlinear Analysis and Applications 81. K. B. Hannsgen, T. L. Herdman, H. W. Stech. and R. L. Wheeler, Volterra and Functional Differential Equations

82. N. L. Johnson, M. J. Kallaher, and C. T. Long, Finite Geometries: Proceedings of a Conference in Honor of T. G. Ostrom 83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra: Proceedings of the Trento Conference 85. A. V. Fiacco, Mathematical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty, W. Oettli, and J. Stoer, Optimization: Theory and Algorithms 87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups 88. M Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istriifescu, Strict Convexity and Complex Strict Convexity: Theory and Applications 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations 91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L. P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L. Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M Rassias and T. M Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightbourne, III, and S. M. Rankin, III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A. Baker and L. M. Batten, Finite Geometries 104. J. W. Brewer, J. W. Bunce, and F. S. Van Vleck, Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology: Manifolds, Varieties, and Knots 106. D. W. Kueker, E. G. K. Lopez-Escobar, and C. H. Smith, Mathematical Logic and Theoretical Computer Science 107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis: Proceedings in Honor of Ky Fan Other Volumes in Preparation

Nonlinear and Convex Analysis PROCEEDINGS IN HONOR OF KY FAN Edited by

BOR,LUH LIN 'The University of Iowa Iowa City, Iowa

STEPHEN SIMONS University of California Santa Barbara, California

~ CRC Press v

Taylor & Francis Group Boca Raton London New York

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Preface

Professor Ky Fan retired at the end of the 1984-85 academic year, and the Department of Mathematics of the University of California at Santa Barbara held a conference in his honor June 23-26, 1985. This volume contains expanded versions of the talks given at the conference, as well as papers contributed by others, many of whom would have liked to attend the conference but were unable to come. Before writing about the conference, I will give a brief history of Professor Fan's career, although it is not really possible to do it justice in a few lines. Professor Fan obtained a B.S. from the National Peking University in 1936 and a D.Sc. Math from the University of Paris in 1941.

He was a mem-

ber of the Institute for Advanced Study at Princeton from 1945 to 1947, and held professorial positions at the University of Notre Dame, Wayne State University, and Northwestern University before coming to the University of California at Santa Barbara in 1965. He has been a visiting professor at the University of Texas at Austin, Hamburg University in Germany, and the University of Paris IX in France. Elected a member of the Academics Sinica in 1964, he served as director of the Institute of Mathematics at the Academics Sinica for two terms from 1978 to 1984.

He has been on many editorial boards, including that

of the Journal of Mathematical Analysis and Applications since its founding in 1960. A student and collaborator of M. Frechet, Professor Fan was also influenced by J. von Neumann and H. Weyl.

He made fundamental contributions

to operator and matrix theory, convex analysis and inequalities, linear and nonlinear programming, topology and fixed-point theory, and topological iii

iv

Preface

groups.

Many of his results have become part of the basic literature in

these fields.

Furthermore, his work in fixed-point theory has found wide

application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations, and has had many consequences in nonlinear functional analysis. Professor Fan has trained twenty-two Ph.D. students, many of whom hold positions at prestigious universities both in the United States and abroad.

Seven of them were at the conference:

John Cantwell, Michael

Geraghty, Donald Hartig, Charles llimmelberg, Ronald Knill, Bor-Luh Lin, Michael Powell, and Raimond Struble. In all, sixty-three mathematicians from three continents came to the University of California at Santa Barbara for the conference.

The one hour

talks were given by Professors Jean-Pierre Aubin of the University of Paris, Felix Browder of the University of Chicago, Zhang Gong-Qing (K. C. Chang) of Peking University, Andrzej Granas of the University of Montreal, Paul Ilalmos of the University of Santa Clara, and Shizuo Kakutani of Yale University.

Professor Samuel Karlin of Stanford University was also scheduled

to give a one hour talk, but was prevented from coming by i 11 ness.

Twenty-

s even shorter talks were also given, many in areas pioneered by Professor Fan. On the evening of June 25, those attending the conference joined members of the Department of Mathematics for a banquet 1n honor of Professor Fan.

The banqucteers were rather startled when one of the spcakers---a

student of Professor Fan who prefers to remain nameless-started to undress.

To the relief (and amusement) of everyone it was merely to demon-

strate that he was wearing a Ky Fan T-shirt. I spoke to many of the participants in the conference, trying to find out more about Professor Fan's career before he came to Santa Barbara. They confirmed unanimously my own feelings; they spoke of his high standards in mathematics, not only for himself, hut also for his colleagues and students.

But they also spoke of his fine qualities as a human being and

his generosity. Although Professor Fan has now retired from his teaching duties, he remains mathematically active, and will undoubtedly continue to do so for many years to come. Thanks are due to Marvin Marcus, Associate Vice Chancellor of Research and Academic Development, and David Sprecher, Provost of the College of Letters and Science at the University of California at Santa Barbara, whose

Preface

V

offices provided financial support for the conference; to James Robertson, Chairman of the Department of Mathematics at the University of California at Santa Barbara and all his staff, who worked so hard to make the conference a success; to all those who participated - in many cases coming a great distance and braving an unpleasant airline strike to get to Santa Barbara; to all those who could not participate in the conference but sent their good wishes; and to Professor Earl Taft and Professor Zuhair Nashed and Marcel Dekker, Inc., for including the Proceedings in their Lecture Notes in Pure and Applied Mathematics Series.

Stephen Simons

Contents

Preface

iii ix

Contributors

xiii

Conference Participants

xv

List of Publications of Ky Fan Smooth and Heavy Viable Solutions to Control Problems

Jean-Pierre Aubin

On Simplified Proofs of Theorems of von Neumann, Heinz, and Ky Fan, and Their Extended Versions

Gong-ning Chen

Vandermonde Determinant and Lagrange Interpolation in Rs Charles K. Chui and M. J. Lai

Applications of Nonstandard Theory of Locally Convex Spaces Carl L. Devito

Local Invertibility of Set-Valued Maps

Halina Frankowska

Some Minimax Theorems Without Convexity

Andrzej Granas and Fon-Che Liu

Weak Compactness and the Minimax Equality

Chung-Wei Ha

Nonlinear Volterra Equations with Positive Kernels

Norimichi Hirano

Some Results on Multivalued Mappings and Inequalities Without Convexity Charles D. Horvath

Strong Equilibria

1

15

23 37

47 61 77

83

99 107

Tatsuro Ichiishi

vii

viii

Contents

A Variational Principle ,\ppl i cat ion to the \onl inear Complementarity Problem George Isac and M.

Thera

On a Best Approximation Theorem Hidetoshi Kamiya

A Vector-Minimization Problem in a Stochastic ContinuousTime n-Person Game Hang-Chin Lai and Kensukc Tanaka

Calculation of the A-Function for Several Classes of \ormed Linear Spaces

Robert H. Lohman and Thaddeus J. Shura

Existence of Positive Eigenvectors and Fixed Points for A-Proper Type Naps 1n Cones

W. V. Petryshyn

On the Nethod of Successive Approximations for Nonexpansive Mappings Simeon Reich and I. Shafrir

147

151

167

1 75

19 3

Quasilinear Ellipticity on the \-Torus Victor L. Shapiro

A Local ~linimax 'li1eorern \\'i thout Compactness

Shi Shu-zhong and Chang Kung-ching (Zhang Gong-qing)

203

211

Covering Theorems of Convex Sets Related to Fixed-Point Theorems

Mau-Hsiang Shih and Kok-Keong Tan

Shapley Selections anJ Covering Theorems of Simplexes

Mau-Hsiang Shih and Kok-Keong Tan

235

245

Cer:erali zatim;s of Convex Suprcmi zation Duality Ivan Singer

On the Asymptotic Behavior of i\lmost-Drbits of Commutative Semigroups in Banach Spaces fvataru Takahashi and Jong Yeoul Park

On a Factorization of Operators Through a Subspace of c 0 Yau-Chuen Wong

Trace Formula for Almost Lie Algebra of Operators and Cyclic One-Cocycles

253

271

295

Daoxing Xia

299

Index

309

Contributors

JEAN-PIERRE AUBIN Mathematical Research Center, University of ParisDauphine, Paris, France CHANG KUNG-CHING (ZHANG GONG-QING) Department of Mathematics, Peking University, Beijing, People's Republic of China GONG-NING CHEN Department of Mathematics, Beijing Normal University, Beijing, People's Republic of China CHARLES K. CHUI Department of Mathematics, Texas A&M University, College Station, Texas CARLL. DeVITO* Tucson, Arizona

Department of Mathematics, University of Arizona,

HALINA FRANKOWSKA Mathematical Research Center, University of ParisDauphine, Paris, France ANDRZEJ GRANAS Department of Mathematics, University of Montreal, Montreal, Quebec, Canada CHUNG-WEI HA Department of Mathematics, ]l;ational Tsing Hua University, llsinchu, Taiwan, Republic of China NO RIMI CHI HIRANO Yokohama, Japan

Department of Ma thematics, Yokohama National University,

CHARLES D. HORVATH Canada

Champlain Regional College, St. Lambert, Quebec,

TATSURO ICHIISHI** Iowa City, Iowa

Department of Economics, The University of Iowa,

GEORGE ISAC Department of ~!athematics, Royal Military College, St.-Jean, Quebec, Canada *Current affiliation: **Current affiliation:

Naval Postgraduate School, Monterey, California Ohio State University, Columbus, Ohio

ix

x HIDETOSHI KOMIYA

Contributors College of Commerce, Nihon University, Tokyo, Japan

HANG-CHIN LAI Institute of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China M. J. LAI Department of Mathematics, Texas A&M University, College Station, Texas FON-CHE LIU Institute of Mathematics, Academia Sinica, Taipei, Taiwan, Republic of China ROBERT H. LOHMAN Department of Mathematics and Sciences, Kent State University, Kent, Ohio JONG YEOUL PARK Department of Mathematics, Busan National University, Busan, Republic of Korea W. V. PETRYSHYN Department of Mathematics, Rutgers University, New Brunswick, New Jersey SIMEON REICH* Department of Mathematics, The Technion-Israel Institute of Technology, Haifa, Israel I. SHAFRIR Department of Mathematics, The Technion-Israel Institute of Technology, Haifa, Israel VICTOR L. SHAPIRO, Department of Mathematics and Computer Science, University of California, Riverside, Riverside, California SHI SHU-ZHONG** Paris, France

Mathematical Research Center, University of Paris-Dauphine,

MAU-HSIANG SHIH Department of Mathematics, Chung Yuan University, ChungLi, Taiwan, Republic of China THADDEUS J. SHURA Department of Mathematics and Sciences, Kent State University at Salem, Salem, Ohio IVAN SINGER Department of Mathematics, National Institute for Scientific and Technical Creation, Bucharest, Romania WATARU TAKAHASHI Department of Information Science, Tokyo Institute of Technology, Tokyo, Japan KOK-KEONG TAN Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada KENSlJKE TANAKA Japan

Department of Mathematics, Niigata University, Niigata,

*Current affiliation: California

The University of Southern California, Los Angeles,

**Current affiliation: China

Nankai University, Tianjin, People's Republic of

xi

Contributors

M. TI-IERA France

Department of Mathematics, University of Limoges, Limoges,

YAU-CHUEN WONG Department of Mathematics, The Chinese University of Hong.Kong, Hong Kong DAOXING XIA Tennessee

Department of Mathematics, Vanderbilt University, Nashville,

Conference Participants*

Archbold, R. (Aberdeen, Scotland) Aubin, J.-P. (Paris, France) Browder, F.E. (Chicago, IL) Beer, G.A. (Los Angeles, CA) Bruck, R.E. (Los Angeles, CA)

Kottman, C .A. (Springfield, VA) Laetsch, T. (Tucson, AZ) Lai, H.-C. (Taiwan, China) Lai, M. (College Station, TX) Lange, L.H. (San Jose, CA)

Border, K.C. (Pasadena, CA) Cantwell, J.C. (St. Louis, MO) Chang, K.C. (Beijing, China) Chu, II. (College Park, MD) Chuan, J.-C. (Taiwan, China)

Lapidus, M. (Iowa City, IA) Lassonde, M. (Clermont, France) Lau, A.T. (Edmonton, Alberta) Lee, T.Y. (University, AL) Lin, B.-L. (Iowa City, IA)

Chui, C. K. (College Station, TX) Crandall, M.G. (Madison, WI) Deprima, C.R. (Pasadena, CA) Donoghue, W. F. (Irvine, CA) Fournier, G. (Sherbrooke, Quebec)

Liu, F.-C. (Taiwan, China) Liu, S.-C. (Taiwan, China) McLinden, L. (Urbana, IL) Mikusinksi, P. (Orlando, FL) Namioka, I. (Seattle, WA)

Frankowska, H. (Paris, France) Geraghty, M.A. (Iowa City, IA) Granas, A. (Montreal, Quebec) Greim, P. (Charleston, SC) Gretsky, N. (Riverside, CA)

Ng, K.F. (Hong Kong) Pak, H. (Detroit, MI) Petryshyn, W. V. (New Brunswick, NJ) Phelps, R. (Seattle, WA) Powell, M.H. (Santa Cruz, CA)

Halmos, P.R. (Santa Clara, CA) Hartig, D.G. (San Luis Obispo, CA) Hilario, R. (Chino, CA) Himmelberg, C.J. (Lawrence, KS) Hirano, N. (Yokahama, Japan)

Reich, S. (Los Angeles, CA) Schechter, M. (Irvine, CA) Shapiro, V.L. (Riverside, CA) Shekhtman, B. (Riverside, CA) Shih, H.-H. (Taiwan, China)

Hoffman, H. (Los Angeles, CA) Horvath, C. (Montreal, Quebec) Hu, T. (Taiwan, China) Ichiishi, T. (Iowa City, IA) Kakutani, S. (New Haven, CT)

Struble, R. (Rayleigh, NC) Swaminathan, S. (Halifax, Nova Scotia) Takahashi, W. (Tokyo, Japan) Tan, K.-K. (Halifax, Nova Scotia) Thera, M. (Limoges, France)

Kalisch, G.K. (Irvine, CA) Knill, R.J. (New Orleans, LA) Kamiya, H. (TokyQ, Japan) Komuro, N. (Hokkaido, Japan)

Wang, J.L.-M. (University, AL) Wong, Y. C. (!long Kong) Wu, S.Y. (Iowa City, IA) Xia, D. (Nashville, TN)

*Other than those from The University of California at Santa Barbara xiii

List of Publications of Ky Fan 1.

Sur une representation des fonctions abstraites continues, C.R. Acad. Sci. Paris 210(1940), 429-431.

2.

Surles types homogenes de dimensions, C.R. Acad. Sci. Paris 211 (1940), 175-177.

3.

Espaces quasi-reguliers, quasi-normaux et quasi-distancies, C.R. Acad. Sci. Paris 211(1940), 348-351. Caracterisation topologique des arcs simples dans les espaces accessibles de M. Frechet, C.R. Acad. Sci. Paris 212(1949), 1024-1026.

4. 5.

Surles ensembles possedant la propriete des quatre points, C.R. Acad. Sci. Paris 213(1941), 518-520.

6.

Surles ensembles monotones-connexes, les ensembles filiformes et les ensembles possedant la propriete des quatre points, Bull. Soc. Royale Sci. Liege 10(1941), 625-642.

7.

Sur le theoreme d'existence des equations differentielles dans !'analyse generale, Bull. Sci. Math. 65(1941), 253-264.

8.

Sur quelques notions fondamentales de !'analyse generale, J. Math. Pures et Appl. 21(1942), 289-368.

9.

Expose sur le calcul symbolique de Heaviside, Revue Scientifique 80 (1942), 147-163.

10.

Sur le comportement asymptotique des solutions d'equations lineaires aux differences finies du second ordre, Bull. Soc. Math. France 70(1942), 76-96.

11.

Les fonctions asymptotiquement presque-periodiques d'une variable entiere et leur application a l'etude de !'iteration des transformations continues, Math. Zeitschr. 48(1943), 685-711.

12.

Une propriete asymptotique des solutions de certaines equations Lineaires aux differences finies, C.R. Acad. Sci. Paris 216(1943), 169-171.

13.

Quelques proprietes caracteristiques des ensembles possedant la propriete des quatre points et des ensembles filiformes, C.R. Acad. Sci. Paris 216(1943), 553-555.

14.

Nouvelles definitions des ensembles possedant la propriete des quatre points et des ensembles filiformes, Bull. Sci. Math. 67 (1943), 187-202.

15.

Entfernung zweier zufalligen Grossen und die Konvergenz nach Wahrscheinlichkeit, Math. Zeitschr. 49(1943/44), 681-683.

16.

Sur !'extension de la formule generale d'interpolation de M. Borel aux fonctions aleatoires, C.R. Acad. Sci. Paris 218(1944), 260-262.

17.

Apropos de la definition de connexion de Cantor, Bull. Sci. Math. 68(1944), 111-116.

18.

Un theoreme general sur les probabilites associees a un systeme d'evenements dependants, C.R. Acad. Sci. Paris 218(1944), 380-382.

19.

Une definition descriptive de l'integrale stochastique, C.R. Acad. Sci. Paris 218(1944), 953-955.

20.

Sur !'approximation et !'integration des fonctions aleatoires, Bull. Soc. Math. France 72(1944), 97-117.

21.

Le prolongement des fonctionnelles continues sur un espace semixv

xvi

Publications List of Ky Fan ordonne, Revue Scientifique 82(1944), 131-139.

22.

Conditions d'existence de suites illimitees d'evenements correspondant a certaines probabilites donnees, Revue Scientifique 82 (1944), 235-240.

23.

Generalisations du theoreme de M. Khintchine sur la validite de la loi des grands nombres pour les suites stationnaires de variables aleatoires, C.R. Acad. Sci. Paris 220(1945), 102-104.

24.

Remarques sur un theoreme de M. Khintchine, Bull. Sci. Math. 69 (1945), 81-92.

25.

Two mean theorems in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 31 (1945), 417-421.

26.

(with M. Frechet) Introduction a la topologie combinatoire, I. Initiation, Vuibert, Paris (1946).

Spanish translation: Introduccion a la topologfa combinatoria (traducida por D.A.H. Nogues), Editorial Universitaria de Buenos Aires (la. edicion, 1959; 2a. edicion, 1961; 3a edicion, 1967). English translation: Initiation to combinatorial topology (translated by H.W. Eves), Prindle, Weber & Schmidt, Boston, LondonSydney (1967). 27.

On positive definite sequences, Ann. of Math. 47(1946), 593-607.

28.

(with S. Bochner) Distributive order-preserving operations in partially ordered vector sets, Ann. of Math. 48(1947), 168-179.

29.

On a theorem of Weyl concerning eigenvalues of linear transformations, I, Proc. Nat. Acad. Sci. U.S.A. 35(1949), 652-655.

30.

On a theorem of Weyl concerning eigenvalues of linear transformations, II, Proc. Nat. Acad. Sci. U.S.A. 36(1950), 31-35.

31.

Partially ordered additive groups of continuous functions, Ann. of Math. 51(1950), 409-427. --

32.

Les fonctions definies-positives et les fonctions completement monotones (Memorial des Sci. Math., Fasc. 114), Gauthier-Villars, Paris (1950).

33.

(with A. Appert) Espaces topologiques intermediaires (Actualites Sci. et Industr., Fasc. 1121), Hermann, Paris (1951).

34.

Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760-766.

35.

Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38(1952), 121-126.

36.

Note on a theorem of Banach, Math. Zeitschr. 55(1952), 308-309.

37.

A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. 56(1952), 431-437.

38.

(with N. Gottesman) On compactifications of Freudenthal and Wallman, Proc. Kon. Nederl. Akad. Wetensch. Amsterdam, Ser. A, 55 (1952), 504-510. Also in: Indag. Math. 14 (1952), 504-510.

39.

Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39(1953), 42-47.

40.

Some remarks on commutators of matrices, Archiv. der Math. 5 (1954), 102-107. (with R.A. Struble) Continuity in terms of connectedness, Proc. Kon.Nederl. Akad ..Wetensch. Amsterdam, Ser. A, 57(1954), 161-164. Also in: Indag. Math. 16(1954), 161-164.

41.

Publications List of Ky Fan

xvii

42.

(with A.J. Hoffman) Lower bounds for the rank and location of the eigenvalues of a matrix, Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, 117-130. (National Bureau of Standards Applied Math. Series, vol. 39, Washington 1954.)

43.

Inequalities for eigenvalues of !Iermitian matrices, Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, 131-139. (National Bureau of Standards Applied Math. Series, vol. 39, Washington 1954.)

44.

(with G.G. Lorentz) An integral inequality, Amer. Math. Monthly 61(1954), 626-631.

45.

(with J. Todd) A determinantal inequality, J. London Math. Soc. 30 (1955), 58-64.

46.

(with A.J. Hoffman) Some metric inequalities in the space of matrices, Proc. Amer. Math. Soc. 6(1955), 111-116.

47.

(with O. Taussky and J. Todd) Discrete analogs of inequalities of Wortinger, Monatsh. Math. 59(1955), 73-90.

48.

Some inequalities concerning positive-definite Hermitian matrices, Proc. Cambridge Philos. Soc. 51(1955), 414-421.

49.

(with O. Taussky and J. Todd) An algebraic proof of the isoperimetric inequality for polygons, J. Washington Acad. Sci. 45(1955), 339-342.

50.

(with I. Glicksberg) Fully convex normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 41(1955), 947-953.

51.

A comparison theorem for eigenvalues of normal matrices, Pacific J . Ma th . 5 ( 19 5 5 ) , 911- 913 .

52.

On systems of linear inequalities, Linear Inequalities and Related Systems, 99-156 (Annals of Math. Studies, No. 38, Princeton Univ. Press, 1956). Also in Russian translation, Moscow, 1959.

53.

(with G. Pall) Imbedding conditions for Hermitian and normal matrices, Canad. J. Math. 9(1957), 298-304.

54.

(with P. Davis) Complete sequences and approximations in normed linear spaces, Duke Math. J. 24(1957), 183-192.

55.

(with I. Glicksberg and A.J. Iloffman) Systems of inequalities involving convex functions, Proc. Amer. Math. Soc. 8(1957), 617-622.

56.

Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations, Math. Zeitschr. 68 (1957), 205-216.

57.

Topological proofs for certain theorems on matrices with non-negative elements, Monatsh. Math. 62(1958), 219-237.

58.

Note on circular disks containing the eigenvalues of a matrix, Duke Math. J. 25(1958), 441-445.

59.

Linear inequalities and closure properties in normed linear spaces, Seminar on Analytic Functions, Vol. 2, 202-212 (Institute for Advanced Study, Princeton 1958).

60.

(with I. Glicksburg) Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25(1958), 553-568.

61.

On the equilibrium value of a system of convex and concave functions, Math. Zeitschr. 70(1958), 271-280.

62.

Convex sets and their applications, Argonne National Laboratory, Argonne (19 59) .

xviii

Publications List of Ky Fan

63.

(with A,S. Householder) A note concerning positive matrices and M-matrices, Monatsh. Math. 63(1959), 265-270.

64.

Note on M-matrices, Quarterly J. Math. (1960), 43-49.

65.

Combinatorial properties of certain simplicial and cubical vertex maps, Archiv der Math. 11(1960), 368-377.

66.

A generalization of Tychonoff's fixed point theorem, Math. Ann. 142(1961), 305-310.

67.

On the Krein-Milman theorem, Proceedings of Symposia in Pure Mathematics, Vol. 7, Convexity, 211-219 · (Amer. Math. Soc., Providence 1963).

68.

(with R, Bellman) On systems of linear inequalities in Hermitian matrix variables, Proceedings of Symposia in Pure Mathematics, Vol. 7, Convexity, 1-11 (Amer. Math. Soc., Providence 1963).

69.

Invariant subspaces of certain linear operators, Bull. Amer, Math. Soc. 69(1963), 773-777.

70.

Invariant cross-sections and invariant linear subspaces, Israel J. Math. 2(1964), 19-26.

(Oxford Second Ser.) 11

Russian Translation: Matematika: Period, Sb. Perevodov Inostran. Statei. (Mathematics: Periodical Collection of Translations of Foreign Articles) 13 (6), Moscow, 1969. MR 40 #7057, 71.

Inequalities for M-matrices, Proc. Kon. Nederl. Akad. Wetensch. Amsterdam, Ser. A, 67(1964), 602-610. Also in:

Indag. Math. 26(1964), 602-610.

72.

Sur un theoreme minimax, C.R. Acad. Sci. Paris 259(1964), 39253928.

73.

A generalization of the Alaoglu-Bourbaki theorem and its applications, Math. Zeitschr. 88(1965), 48-60.

74.

Invariant subspaces for a semigroup of linear operators, Proc. Kon. Nederl. Akad. Wetensch. Amsterdam, Ser A. 68(1965), 447-451, Also in:

Indag. Math. 27(1965), 447-451.

Russian Translation: Matematika: Period. Sb. Perevodov Inostran. Sta tei. (Mathematics: Periodical Collection of Translations of Foreign Articles) 13 (6), Moscow, 1969. MR 40 #7057. 75.

Applications Of a theorem concerning sets with convex sections, Math. Ann. 163(1966), 189-203. Also in: Contributions to Functional Analysis Heidelberg-New York, 1966).

(Springer, Berlin-

76.

Some matrix inequalities, Abhandl. Math. Seminar Univ. Hamburg 29 (1966), 185-196.

77.

Sets with convex sections, Proceedings of the Colloquium on Convexity, Copenhagen, 1965, 72-77 (K¢benhavns Univ. Mat. Inst., 1967.

78.

Subadditive functions on a distributive lattice and an extension of Szasz's inequality, J. Math. Anal. Appl. 18(1967), 262-268.

79.

Inequalities for the sum of two M-matrices, Inequalities, Proceedings of a Symposium, 105-117 (Academic Press 1967).

80.

Simplicial maps from an orientable n-pseudomanifold onto sm with the octahedral triangulation, J. Combinatorial Theory 2(1967), 588-602.

Publications List of Ky Fan

xix

81.

An inequality for subadditive functions on a distributive lattice, with application to determinantal inequalities, Linear Algebra and ~ - 1 (1968), 33-38.

82.

On infinite systems of linear inequalities, J. Math. Anal. Appl. 21(1968), 475-478.

83.

A covering property of simplexes, Math. Scandinavica 22(1968), 17-20.

84.

Asymptotic cones and duality of linear relations, J. Approximation Theory 2(1969), 152-159. Also in: Inequalities II, Proceedings of the Second Symposium on Inequalities, 179-186 (Academic Press, 1970).

85.

Extensions of two fixed point theorems of F. E. Browder, Math. Zeitschr. 112(1969), 234-240. Abstract in:

Set-valued mappings, selections and topological

properties of 2x. Proceedings of the Conference held at SUNY at Buffalo, May 1969, edited by W. M. Fleischman (Lecture Notes in Mathematics, Vol. 171, Springer, 1970), 12-16. 86.

On a theorem of Pontryagin, Studies and Essays presented to Yu-Why Chen, Mathematics Research Center, National Taiwan Univ. (1970), 197-200.

87.

A combinatorial property of pseudomanifolds and convering properties of simplexes, J. Math. Anal. Appl. 31(1970), 68-80.

88.

On local connectedness of locally compact Abelian groups, Math. Ann . 18 7 ( 19 7 o) , 114 -116 .

89.

Simplicial maps of pseudomanifolds, Annals of the New York Academy of Sciences 175 (Art. 1), (1970), International Conference on Combinatorial Mathematics, 125-130.

90.

Combinatorial properties of simplicial maps and convex sets, Proceedings of the Twelfth Biennial Seminar of the Canadian Mathematical Congress: Time Series and Stochastic Processes, Convexity and Combinatorics (Canadian Math. Congress, 1970) 231-241.

91.

On the singular values of compact operators, J. London Math. Soc. (2) 3 (1971), 187-189.

92.

A minimax inequality and applications, Inequalities III, Proceedings of the Third Symposium on Inequalities, 103-113 (Academic Press, 1972).

93.

Covering properties of convex sets and fixed point theorems in topological vector spaces, Symposium on Infinite Dimensional Topology (Annals of Math. Studies, No. 69, Princeton Univ. Press, 1972), 79-92.

94.

Generalized Cayley transforms and strictly dissipative matrices, Linear Algebre a n d ~ - 5(1972), 155-172.

95.

Fixed-point theorems in functional analysis (Supplementary manual for recording of a lecture), Audio Recordings of Mathematical Lectures, No. 46, Amer. Math. Soc. (1972), 7 pages.

96.

On Dilworth's coding theorem, Math. Zeitschr. 127(1972), 92-94.

97.

On real matrices with positive definite symmetric component, Linear and Multilinear Algebra 1(1973), 1-4.

98.

Sums of eigenvalues of strictly J-positive compact operators, J. Math. Anal. Appl. 42(1973), 431-437.

xx 99.

Publications List of Ky Fan On similarity of operators, Advances in Math. 10(1973), 395-400.

100.

On strictly dissipative matrices, Linear Algebra and Appl. 9 (1974), 223-241.

101.

Two applications of a consistency theorem for systems of linear inequalities, Linear Algebra and Appl. 11(1975), 171-180.

102.

Orbits of semi-groups of contractions and groups of isometries, Abh. Math. Sem. Univ. Hamburg 45(1976), 245-250.

103.

Extension of invariant linear functionals, Proc. Amer. Math. Soc. 66 (1977), 23-29.

104.

Analytic functions of a proper contraction, Math. Zeitschr. 160 (1978), 275-290.

105.

Distortion of univalent functions, J. Math. Anal. Appl. 66(1978), 626-631.

106.

Julia's lemma for operators, Math. Annalen 239(1979), 241-245.

107.

(with T. Ando) Pick-Julia theorems for operators, Math. Zeitschr. 168 (1979), 23-34.

108.

Fixed-point and related Game Theory and Related Game Theory and Related 1978; Managing Editors: Holland, 1979, 151-156.

109.

Schwarz's lemma for operators on Hilbert space (lecture presented at the Romanian-American Seminar on Operator Theory and Applications, 20-24 March 1978), Analele stiintifice ale Universitatii Al. I. Cuza Iasi, Supliment la tomul 25, s. I a 1979, 103-106.

110.

Harnack's inequalities for operators, General Inequalities, Oberwolfach; edited by E.F. Beckenbach), Birkhauser Verlag, BaselBoston-Stuttgart, 1980; 333-339.

111.

A further generalization of Shapley's generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Game Theory and Mathematical Economics (edited by 0. Moeschlin and D. Pallaschke), North-Holland, 1981, 275-279.

112.

Evenly distributed subsets of Sn and a combinatorial application, Pacific J. Math. 98 (1982), 323-325.

113.

Iteration of analytic functions of operators, Math. Zeitschr. 179 (1982), 293-298.

114.

Iteration of analytic functions of operators. II, Linear and Multilinear Algebra 12(1983), 295-304.

115.

Normalizable operators, Linear Algebra and Appl. 52/53(1983), 253-263.

116.

Some properties of convex sets related to fixed point theorems, Ma th . Ann. 2 6 6 ( 19 8 4 ) , 519- 5 3 7 .

117.

An identity for symmetric bilinear forms, Linear Algebra and Appl. 65 (1985), 273-279.

118.

The angular derivative of an operator-valued analytic function, Pacific J. Math. 121 (1986), 67-72.

theorems for non-compact convex sets, Topics (Proceedings of the Seminar on Topics, Bonn/Hagen, 26-29 September, 0. Moeschlin and D. Pallaschke), North-

Nonlinear and Convex Analysis

and Heavy Viable Solutions to Control Problems JEAN-PIERRE AUBIN Mathematical Research Center, University of ParisDauphine, Paris, France

ABSTRACT We introduce the concept of viability domain of a set-valued map, which we study and use for providing the existence of smooth solutions to differential inclusions. We then define and study the concept of heavy viable trajectories of a controlled system with feedbacks.

Viable trajectories

are trajectories satisfying at each instant given constraints on the state.

The controls regulating viable trajectories evolve

according a set-valued feedback map.

Heavy viable trajectories

are the ones which are associated to the controls in the feedback map whose velocity has at each instant the minimal norm.

We con-

struct the differential equation governing the evolution of the controls associated to heavy viable trajectories and we state their existence. DEDICATION I would have liked to find an original way to dedicate this lecture to Professor Ky Fan, but I did not see any better solution than to simply confess that it is both an honor and a pleasure to have been invited to this conference held in his honor. I have been deeply influenced by the theorems discovered and proved by Professor Ky Fan, and, in particular, by his 1968 famous inequality.

Let me just repeat what I tell my students 1

DOI: 10.1201/9781003420040-1

Aubin

2

when I begin to teach the Ky Fan inequality.

I tell them a lot

of stories, how the young Ky Fan came to Paris in 1939 for one year with only a metro map, how he had to survive during the darkest years of the history of my country, how he met Frechet and worked with him, etc.

But most important, I choose the

Ky Fan inequality as the best illustration of the concept of "labor value" of a theorem. Indeed, most of the theorems of nonlinear functional analysis are equivalent to the Brouwer fixed point theorem.

But when

we prove that statement (A) is equivalent to statement (B), there is always one implication, say "A implies B", that is more difficult to prove than the other one. ment (B)

We then can say that state-

"incorporates" more labor value than statement (A).

An empirical law shows that the more labor value a theorem incorporates, the more useful it is.

And my point is that among

all the theorems equivalent to the Brouwer fixed point theorem I know, the Ky Fan inequality is one which is the most valuable.

1.

VIABLE SOLUTIONS TO A CONTROL PROBLEM

IRn, U: X + X be a set-valued map with closed graph Let X and f : Graph U-->- X be a continuous map. We consider the control problem with feedbacks

( 1. 1)

i)

x' (t) = f(x(t),u(t)}

ii)

for almost all t _:: _ 0,

iii)

u (t) E u(x (t))

x(0) = x 0 given in Dom U

Instead of selecting a solution x( •) to (1) which minimizes a given functional, as in optimal control theory( 1 ),we are only selecting solutions which are viable in the sense that, given a closed subset KC X ( 1 • 2)

\It > 0

,

X ( t)

E K

A first issue is to provide necessary and sufficient con-

Smooth and Heavy Viable Solutions to Control Problems

3

ditions linking the dynamics of the system (described by f and U) and the constraints bearing on the system (described by the closed subset K) such that the viability property

vx 0 EK , there exists a solution to ( 1)

( 1 • 3)

viable in K

holds true.

This allows us to describe the evolution of the

viable controls u(•),

(the controls which govern viable solutions)

A second issue is to provide conditions for having smooth viable solutions to a control problem, in the sense that the viable control function is absolutely continuous instead of being simply measurable. A third issue is to give a mathematical description of the "heavy viable solutions" of the control system which we observe in the evolution of large systems arising in biology and economic and social sciences.

Such large systems keep the same control

whenever they can and change them only when the viability is at stakes, and do that as slowly as possible.

In other words, heavy

viable solutions are governed by those controls who minimize at each instant the norm of the velocity of the viable controls. In the case when f(x,u)

= u, system (1) reduces to the differen-

tial inclusion x' (t) E U(x (t)), x (0)

=

x0 :

heavy (viable) solu-

tions to this system minimize at each instant the norm of the acceleration of viable solutions; in other words, they evolve with maximal inertia. Hence the name heavy viable solutions (or inert viable solutions). 0,

and

such that

sup Re £ 1 (co h (Ue)) s c - t

.e 1 (h (x)),

< c s Re

r = 1.

a contradiction, so that (3) holds when

k

the function into

A.

In the general case of

k(y) = h(ry)

defined by

is holomorphic

(3) now follows directly from the result just proved

and from the convexity of balls. co u

Obviously,

of

(3) whether

A

is hermitian and continuous, we have

bra pseudonorm on (3).

if not, by a separation

c

is bounded in

e

A,

A

(4)

is a simple consequence

or not.

If the involution for

([3], [8]) that

and is continuous on

A,

whence

p

is an alge-

(5)

is valid by

For the last part of the theorem, as before, it will be enough to r = 1.

verify the result in the case

But this follows directly from

the mean value property for vector-valued holomorphic functions [5, p. 99]:

(2n) -1J2n

h(x) = h(f(O)) = for each (6).

x

in

A

( 7)

0

!xl 0 < 1

with

and

p(x) < 1,

where

f

is as in

That completes the proof of the theorem. We note that there exists an alternative proof of Theorem 1

REMARK.

by deriving it from the formula COROLLARY 1. Let

Suppose that

A

(7). has hermitian and continuous involution.

be a domain containing

G

rt

for some

0 < r s 1,

r,

and define

A1 as in Theorem 1 and holomorphic function on

A0 = [x EA: o(x) c OJ. If k is a complex 0, then (3)-(5) of Theorem 1 are valid with

D

h(x)

replaced by k(x)

where

r

Proof. function

A0

and

defined by

k(x)

= (2ni) -1 ~k(A) (Ae-x) -1 dA, n

( 8)

is any contour that surrounds It is well known [10] that k

defined by so that

(8) k : Ao

Ar, into A, virtue of the properties of whenever

by the function

O

~

rt.

A0

o(x)

in

is open in

0. A

and that the

is a continuously differentiable mapping of -➔

A

is holomorphic by definition.

In

p given above, we have A0 ~ r co Ue Corollary 1 now follows from Theorem 1.

3. SIMPLIFIED DISCUSSION OF THEOREMS I, II, III, AND THEIR EXTENDED VERSIONS In this section we shall introduce the use of the results of the pre-

Proofs of Theorems of von Neumann, Heinz, and Ky Fan

19

ceding section in the simplified proofs of Theorems I, II, III, and their extended versions of Tao [11].

A result (see [6, p. 109] and

[10, p. 309]) of fundamental importance for us is the following LEMMA I.

Let

E

normal operator

be the spectral resolution of the identity for a V

in

B(H),

and suppose that

O

morphic function on a domain 0

< r s 1.

Then

g(rV)

Jcc l V) 0

containing

g

is a complex holoin

o(rV)

C

for some

r,

g(rt)dE(t),

( 9)

where the integral exists with respect to the norm topology in

B(11),

and fig(rV)li = sup[ Jg(r,\.)

i: ,\.

( 10)

E CT\V)j.

We note that the involution on

B(Il),

the Hilbert space adjoint, is

clearly hermitian and continuous, and that [T le B(Il): l:T

B (II) l

Proof of Theorem I. since

o(V)

s

[Tl

CT

s p(T) =

iiTL

for each

V

so that

l}.

Observe that

O ~

lies on the unit circle.

E

~

o(V)

By Corollary l

with

in

U

r = l

and

v(V)

for

by (10),

i: u (T) ,i

s sup[ u (V)

V E

uJ

l} s 1,

s sup[ lu(,\.) I

as desired. Proof of Theorem II. each

V

D,

in

0

s;•i

(Re v(V)w,w) for all

w

"

. 2

ii WI! C,

is in

•

in E

w,w Since

where E (U) = (E(u)w,w). w,w is a positive measure on o(V) Rev(,\.) ~

t.

It is well known [10]

0

~

0

whose total variation

is continuous on the compact set

tor each

w

in

I!,

i.e.,

Re v(V)

~

o(V) 0

by

The result now follows directly from (3) of Corollary 1.

Proof of Theorem I I I . exist real

v(v)*

Re v(ciO)dEw,w(O)

II,

(Re v(V)w,w)

definition.

and since

i Re v ( e , ) dE ( 0 ) ,

J,

and therefore

that each

l

r

we have

r2,1

Re v(V)

By (9), with

r,

T

If

0 < r < 1,

is a proper contraction of and a contraction

;\

T

of

B (H),

B(H)

there

such that

s sup[ I h (r,\.) I : i ,\. I = 1] < l. Thus, as before, By /! h (T) i applying a similar argument to that used in the preceding theorem, we

T

=

rT.

can obtain

Re k (T) > 0

for each

11

T ii < 1.

It is natural to expect that the proofs of the extended versions of Theorems I, II, and III, given by Tao [11], should be also greatly

20

Chen

simplified by the same approach mentioned above. case.

For notations, let

the set of all A

A

A

0.

Recall [11] that

A

and ~(T)

A

g(A)

is normal in

denotes a

B(H) 0 =[TE B(H): cr(T) c OJ,

function

~

C

and denote by

B(H)-valued holomorphic functions

g(cr)g(A) = g(A)g(cr) main

be a domain in

These are indeed the

on

O

B(H)

A

g

on O for all

NH{O)

such that A,cr in

B{H)-valued function, with dothat arise from a

B(H)-valued

by the formula

A

g(T) where

r

(11)

is as in (1). A A

A

THEOREM IV. Suppose that u,v are in NH(O) and h is in NH (t.), each of which commutes with T in B(H). Then Theorems I, II, and III A v, A hold J.'f u, v, and h therein are replaced by u, and h/,, respectively. Proof.

Consider only the extended result with respect to Theorem I.

In a similar way one can verify the others.

By a slight extension of

Theorem 10.38 in [10], we can assert that the function by (11) is a holomorphic mapping of Lemma I

B(H) 0

into

B(H).

~(T)

defined

Instead of

we use an analogous lemma, that is Lemma 6.2 of [11].

Thus,

as before, we are able to conclude that the extended result is valid. We conclude with the remark that each of Theorems 3.3, 3.4 (2)-(4), and 3.5 of [11] may be shown to be valid by means of the same methods, and is merely an immediate consequence of Theorem 1 and Corollary 1 together with either Lemma I or Lemma 6.2 of [11]. REFERENCES 1. N. Dunford and J.T. Schwartz, "Linear Operators," Interscience, New York, pt. I, 1958; pt. II, 1963. 2. Ky Fan, Analysis functions of a proper contraction, Math. Z. 160 (1978), 275-290. 3. L. Harris, Banach algebras with involution and Mobius transformations, J. Functional Analysis 11(1972), 1-16. 4. E. Heinz, Ein v. Neumannscher Satz uber beschrankte Operatoren im Hilbertschen Raum, Nachr. Akad. Wiss. Gottingen Math.-Phys. KI.II (19 5 2) , 5- 6. 5. E. Hille and R.S. Phillips, "Functional Analysis and Semigroups," Rv. Ed., Amer. Math. Soc., Providence, R.I. 1957. 6. B. Sz.-Nagy and C. Foias, "Harmonic Analysis of Operators on a Hilbert Space," North-Holland, Amsterdam, 1970. 7. J. Von Neumann, Eine Spektraltheorie fur allgemeine Operatoren eines unitaren Raumes, Math. Nachr. 4(1950/1951), 258-281. 8. V. Ptak, On the spectral radius in Banach algebras with involution, Bull. London Math. Soc. 2(1970), 327-334.

Proofs of Theorems of von Neumann, Heinz, and Ky Fan 9. F. Riesz and B. Sz.-Nagy, "Functional Analysis," Fredrick Ungar, New York, 1955. 10.

w.

Rudin, "Functional Analysis," McGraw-Hill, New York, 1973.

11. Tao Zhiguang, Analytic operator functions, J. Math. Anal. Appl. 103(1984), 293-320.

21

Determinant and Lagrange Interpolation in Rs CHARLES K. CIIUI Department of Mathematics, Texas A&M University, College Station, Texas HANG-CHIN LAI Institute of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

The problem of multivariate polynomial interpolation is very old. Among the papers published during the last decade, we only include [1-16] in the References. Let N~ lem can be stated as follows: study the location of nodes (or sample points) in

R•

such that for every data {f; : i

= 1, ... , Nn, p(w) =

= (n;•). The prob{ x; : i = 1, ... , N n

there is a unique polynomial

L aiwi

lilSn

with total degree n which interpolates the given data at the nodes, namely: p(x;)

=

1, ... , N~. Here and throughout, we use the usual multivariate notation wi

w~•,

j1

+ ... + is

= w{

1 •••

f;, i

=

Iii =

(j1 , ... , is E Z+), etc. Of course, the problem is equivalent to the study of the

nonsingularity of the square matrix

= [x{JT, Iii :S: n, is the i th column of the matrix. While the determinant of this matrix in the cases = l is the wellknown Vandermonde determinant, which is always nonzero for arbitrary

where 0 F,Eint R(T,F,)

( 5)

The purpose of this section is to provide a sufficient condition for (5) when F, is an equilibrium of F, i.e. OEF(F,). We shall apply results of Section 1. The set of solutions ST(F,) is closed in w111 (O,T) whenever Graph (F) is closed in JR.n

Consider the continuous linear operator A from the Banach space w111 (O,T) into the finite dimensional space JR.n x JR

n_

defined by A(x)

1 1

x(T) for all xEW '

Theorem 1.1. states then that if x

0

(0,T)

denotes the constant tra-

jectory x 0 (•) = F, and {w(T) :wECST(F,)(x 0 lation (5) holds true.

)}

= ]Rn then the re-

Let B denote the closed unit ball in JR.n.

We say that a

set-valued map Fis Lipschitzian (in the Hausdorff metric) on an open neighborhood V of F, if for a constant L> 0 x,yE V F ( x) C F ( y) + L llx-y II B

and all

Frankowska

52

Thanks to this property we can compute a subset of c 8 T(s) (x 0 ) : Theorem 2.1. Assume that F has closed graph and is Lipschitzian around the equilibriums- Then every solution of the differential inclusion

(6)

I

w' ( t) E CF ( s, 0) w (t)

a.e. in [ 0, T]

w (0) = 0

belongs to c 8 T(s) (x 0

).

•

The proof of the last result is based on a Filippov Theorem (1967]. We say that the inclusion (6) is aontrottabte if its reachable set at some time T > 0 is equal to the whole space. Theorems 1.1. and 2.1. together imply Theorem 2.2. Assume that F has closed graph and is Lipschitzian around the equilibriums- The inclusion (3) is locally controllable around s if the inclusion (6) is controllable.• Remark. Actually the idea of the proof of Theorem 1.1. allows to prove a stronger result: We denote by co F(~) the closed convex hull of the set F(~). Theorem 2.3. Assume that F has closed graph and is Lipschitzian around the equilibrium S• The inclusion (3) is locally controllable around s if the inclusion

(7)

~ w'

~

E

w(O)

cl [CF ( s , 0) w + CcoF ( ~ ) ( 0) ] = 0

is controllable. • The proof requires a very careful calculation of variations of solutions (see Frankowska (1984]). A necessary condition for the controllability of the inclusions (6), (7) is Dom CF ( s , 0) : = { w E lRn : CF ( s , 0) w 'f' j} = lRn

Local Invertibility of Set-Valued Maps

53

Whenever it holds true the right-hand sides of (6),

(7)

are set-valued maps whose graphs are closed convex cones. Such maps, called "closed convex processes", are set-valued analogues of linear operators.

The controllability of such

differential inclusions is the issues of the next section. Before, we provide the following Using Theorem 2.3 one can obtain a classical result

Example.

on local controllability of control system (4) without assuming too much regularity.

Let Ube a compact in lRmand let

f : lRn x u-+ lRn be a continuous function. (E;,u)ElRnxU, f(E;,u)

=

Assume that for some

0 and for some S>O, L>O and all uEU;

x,y Es+ SB

~ llf(x,u) - f(y,u) 11.:_Lllx-yll

?~; (• ,u) Theorem 2.4.

is continuous on

s + SB

If the sublinearized differential inclusion

) w' E

~!

? w(O) =

(E;,u)w+Cco f(E;,U) (0) 0

is controllable, then the system (4) is locally controllable around t;. 3.

Controllability of Convex Processes A convex process A from lRn to itself is a set-valued map

satisfying Vx, yEDom A,\,µ> 0 ,

\A(x) +µA(y) C A(\x+µy)

or, equivalently, a set-valued map whose graph is a convex cone.

Convex processes are the set-valued analogues of linear

operators.

We shall say that a convex process is closed if its

graph is closed and that it is strict if its domain is the whole space.

Convex processes were introduced and studied in

Frankowska

54

Rockafellar [1967], [1984]).

[1970],

[1974)

(see also Aubin-Ekeland

We associate with a strict closed convex process A

the Cauchy problem for the differential inclusion

(8)

{

x' (t) EA(x(t))

a.e.

x(O) = 0

We say that the differential inclusion (8) is controllable if the reachable set R:={x(t):xEw 111 (0,t) is a solution of (8), t>O} is equal to the whole space JRn. A particular case of (8) is a linear control system

( 9)

f x'

- Fx+GU

lx(O~

=

uEU

0

where U is an m-dimensional space and FE L (JRn, JRn), GEL (JRm, JRn) are linear operators. We observe that the reachable set R(T,O) of (8) at time T is convex.

Since OE A(O) the family {R(T,0) }T>O is increasing.

Moreover, R = U R(T,0). Hence (8) is controllable if and only 'I'>O if it is controllable at some time T>O, i.e.3 T>O such that R(T,O) = JRn a)

The rank condition Let A be a strict closed convex process.

Set A1 (0)

A(O)

and for all integer i > 2 set Ai (0) = A(Ai- 1 (0)) Theorem 3.1. if and only if

The differential inclusion (8) is controllable

Local Invertibility of Set-Valued Maps

55

•

In the case of system (9) for all x E m.n Ax

Fx + Im G.

Thus (-A)

m

(0) = Im G + F (Im G) + ... +F

m-1

(Im G)

The Cayley-Hamilton theorem implies then the Kalman rank condition for the controllability of the linear system (9):

r k [ G, FG, . . F

n-1

G] = n

Theorem 3.1. is a consequence of the following "Eigenvalue" criterion for controllability b) We say that a subspace P of m.n is invariant by a strict closed convex process A if A (P) c P. A real number A is called an eigenvalue of A if Im(A-\I) f f

m.n, where I denotes the identity operator.

Theorem 3.2.

The differential inclusion (8)

is controllable if

and only if A has neither proper invariant subspace nor eigen-

•

values.

It is more convenient to write the above criterion in a "a:Aal forn1":

c)

"Eigenvector" criterion for controllability The convex processes can be transposed as linear operators.

Let A be a convex process; we define its transpose A* by pEA*(q)

¢>

V(x,y) EGraph A,

<

It can easily be shown that\ is an eigenvalue of A if and only if for some q E Im (A-,\I)

1

, q f

0

AgEA*q We call such a vector q f is equivalent then to

O an eigenvector of A*.

Theorem 3.2

56

Frankowska

The differential inclusion (8) is controllable if

Theorem 3.3.

and only if A* has neither proper invariant subspace nor eigenvectors.

A

The proof of Theorem 3.3 is based on a separation theorem and the KY-FAN coincidence theorem [1972].

(See Aubin-

Frankowska-Olech [1985] ). Examples:

a)

Let F be a linear operator from :m.n to its elf, L

be a closed convex cone of controls and A be the strict closed convex process defined by A(x)

:=Fx+L

Then its transpose is equal to A* (q)

I

= F:q ,,

When L = { 0}, i.e. , when A= F, we deduce that A* = F*, so that transposition of co11'.;ex processes is a legitimate extension of transposition of linear operators. Consider the control system

{X '

( 1 0)

= Ax+u,

uEL

lx(O)=O

Corollary 3.4. The following conditions a"e equivalent. a)

the system (10) is controllable

b)

For

SOffi,;

m +1 L + F(L)+ ••• +Fm(L)=

L-F(L)+ ..• +(-1)mFm(L) = :m.n (see Korobov (1980]).

c)

F has neither proper invariant subspace containing L nor eigenvalue A satisfying Im(F-AI) + L-1 :m.n.

d) e)

F* has neither proper invariant suspace contained in . . L. + L+ nor eigenvector in n-1 the subspace spanned by L, F(L) , ..• ,F (L) is equal to JRn and F* has no eigenvector in L+ (see Brammer (1972])

A

Local Invertibility of Set-Valued Maps

b)

57

Consider the control system with feedback in m.2 :

( 11 )

+y +U +

X

1

=

y

I

= -x +w

XV

XU

U, W E U

= [ 0 , 1]

j +1 vEV(x)=}- 1

X?O

x

either

0

(a)

there is x0

(b)

there is y0 E Y such that g(x,y 0 ) 2).. - c for all

E

X such that

f(x 0 ,y)

s)..

for all

y

£

Y ,

or Proof. Sup f(x,y)

YEY

>)..

(i ) ,

=>

hence

(ii )

If

(ii) (a)

Inf Sup f(x,y)

2)..

x EX

does not hold, then for each x EX ,

XEX YEY

which implies by (i) that

This research was supported in part by a grant from the National Research Council of Canada. 61

DOI: 10.1201/9781003420040-6

62

Granas and Liu

Sup Inf g(x,y) cc>...

Thus there is

y EY X EX

g ( x ,y O ) cc >.. -

for al l

£

(ii)=> (i)

Let

x EX . y

=

y 0 c Y such that So ( i i ) ( b)

Inf g(x,y 0 ) cc>.. -

X ,X

£,

i.e.,

ho l d s .

Inf Sup f(x ,y) EX y EY

We may assume that

y

> - oo.

If

X

y < +

00

we apply ( i i ) with

,

(ii)(b)

holds, i.e.,there is

x EX ,

or

>..

=y

-

y 0 EY

such that

Sup Inf g ( x ,Y) cc y - 2 £

Sup Inf g(x ,y) cc y .

Thus {i) holds.

does not hold, hence there is

If

y = +

y 0 e Y such that

does not hold for this

g(x,y 0 ) cc.\.-£ =y-2£

But since

y eY x eX

y eY X eX

Since (ii)(a)

£

for all

0 is arbitrary, we have

E >

oo,

>..,

then for any

g(x,y 0 ) cc>.. -

>.. E R , (i i )(a)

for all

£

x EX

Thus Sup Inf g(x,y) cc A X EX

y eY

If we let

>..

+

+

oo,

then

complete.

{l .2)

Lemma.

Let

Sup Inf g{x,y) yeY xEX

f ,g : X

x

+

00

•

E

•

Hence (i) holds.

The proof is

Y + R be two numerical functions and suppose

that one of the following conditions holds :

(*)

X

is a compact topological space and

X for each

(**)

+

f{x,y)

is lower semi-continuous on

y

+

q(x,y)

is upper semi-continuous on

ye Y.

Y is a compact topological space and Y for each

x

x c X .

Then the following two statements are equivalent (i)

Inf Sup f(x,y) s Sup Inf g(x,y) xeX ycY yeY xeX

(ii)

For each

>.. e R , either

(a)

there is

such that

f(x 0 ,y) s >.. -

for all

y

(b)

there is

y 0 e Y such that

g(x,y 0 ) cc>..

for all

x EX .

E

y ,

or

Some Minimax Theorems Without Convexity

Proof.

Suppose that(*) holds.

straightforward manner from (l. l).

63

The implication (II)

To show that (i)

=>

=>

(i)

(ii) , let

follows in a \ E R be given

and assume that (ii)(b)

does not hold. By Lemma (l.l), for each n = 1,2, ... , we have Inf Sup f(x,y) ,; A +nl- . But, since x + Sup f(x ,y) is lower semi continuous yEY XEX YEY l on X , there is xn E X such that Sup f(xn,y) = Inf Sup f(x,y) ,; A + n . From XEX yEY YEY this letting n + oo , and using the compactness of X and the lower-semicontinuity on

X of

Thus

we get a point X + Sup f(x,y) ' YEY (ii)(a) holds and we have shown that

XO E X such that Sup f(x 0 ,y) ,; \ . YEY ( i ) => ( i i ) The proof of our asser-

tion in the case (**) being similar is omitted. Now we are able to formulate the main result of this section representing a generalization of a known theorem of Ky Fan Ll J. (l.3)

Theorem.

Let

X,Y

be two compact topological spaces and

f,g

Xx Y • R two real-valued functions such that

(i)

x

+

f(x,y)

is lower-sernicontinuous on

X for each

y

E

Y

(ii)

y

+

g(x ,y)

is upper-semi continuous on

Y for each x

E

X.

Then the following conditions are equivalent:

A. exists

For any two finite sets

XO E X and

For each

for all

,;

,; n

X,

{yl , ... ,ylll}

and

l ,; k ,; m

x0 EX

1c

for all

y

E

Y

~ 1c

for all

X E

X

y0 E Y such that g(x,y0 )

C.

l

\ER, either there is f ( x0 ,Y) ,;

or there is

C

Yo E y such that

f(xo,yk) ,; g(xi ,yo) B.

{xo,· .. ,xn}

The following minimax inequality holds

such that

C

y

'

there

64

Granas and Liu

Min Sup f(x,y) s Max Inf q(x,y) . XEX YEY YEY XEX Proof.

We al ready know from Lemma (1.2) that B and

The implication C

=>

C

are equivalent.

A being obvious, it remains only to prove that

A implies

Assume that A is verified ; we are now going to prove that B holds. A

ER be given.

B Let

Define

L(y) = {x E X

f(x,y)

,s A}

for each y E Y

R(x) = {y E Y

q(x,y)

~ A}

for each x EX

n L(y)"' ¢ YEY are closed subsets of

To show that B holds

is equivalent to show that either

X and Y or n R(x) "' ¢ • Since the sets L(y) and R(x) XEX respectively. it is sufficient to show that either {L(y)}yEY or {R(x)}xEX has the finite intersection property.

y does not have the finite inYE m tersection property: there are y1 , .•. ,Ym E Y such that n L(yk') = ¢ , i.e., k=l

(*)

Max f(x,y.) 1 l,sk~

Let now {x 1 , .•. ,xn}

>

A

Suppose that {L(y)}

for each x EX

be any finite subset of X . By (i) there exist x0 EX and

y0 E Y such that

or :5

:5

n •

From(*) we have then

Thus y0

n

€

n R(xi) . Hence {R(x)} XE X has the finite intersection property and i =l

the proof is complete. 2.

A Minimax Inequality of the von Neumann type. In this section, we prove a minimax theorem which represents a further gene-

ralization of a minimax theorem of Ky Fan ll J under "convexity conditions" milder than in our previous note [4].

Some Minimax Theorems Without Convexity

65

In what follows given positive integer t , we let [tJ = {i E N I s i s t} t-1 t t and denote by ti = {(x 1 , ... ,xt) ER I x1 , ... ,xt.: 0 , i~l xi = l} the standard simplex in Rt (2.1)

We introduce first some terminology. Definition.

function on X

Let X be a set and F

=

{f} a family of real-valued

We shall say that X is finitely-convex with respect to

F

provided for any f 1 ,f2 , ... ,fn E F and any x1 ,x 2 EX there is a point xd such that

for all

i E [nJ ; X is called finitely-concave with respect to

above definition the symbol (2.2)

Definition.

"s"

is replaced by

11

2:"

for

in the

•

Let X,Y be sets and f : Xx Y + R a real-valued

function ; consider the family {fy}yEY of functions f(x,y)

F , if

fy : X + R where fy{x) =

x EX . We shall say that X is finitely f-convex (resp. f-concave)

if X is finitely convex (resp. concave) with respect to the family {fy}yEY Similar definitions can also be given for

Y , by considering the family

We can now state and prove the main result of this section : (2.3) Theorem.

Suppose that X and

Y are two non-empty compact spaces

and that f,s,t,g : X x Y + R are four real-valued functions such that (i )

f(x,y) s s(x,y) s t(x,y) s g(x,y)

(ii)

x

(iii)

X is finitely s-convex ; i.e., for any x1 ,x 2 • X and y1 , ... ,Yn E Y there

+

f{x,y)

for all

( X ,y)

€

X

X

y ;

is lower semicontinuous on X for each y E Y ;

is x EX such that s(x,yi) s

½[s{x 1 ,yi)

~ s(x 2 ,yi)J for all

= l ,2, ... ,n ;

(iv)

Y is finitely t-concave ; i.e., for any y1 ,y 2 E Y and x1 , ... ,xm EX there exist y E Y such that t(xj,y).:} [t(xj,yl) ~ t(xj,y 2)J for all j = l ,2, ... ,m ;

(v)

y

+

g(x,y)

is upper semicontinuous on

Y for each x EX .

66

Granas and Liu

Then Min Sup f(x,y) s Max Inf g(x,y) . X Y Y X Proof.

Let

E >

be given.

0

To establish our assertion, it is clearly

sufficient to show that Min Sup f(x,y) s Max Inf{g(x,y)} X

Y

Y

X

+ E

= Max{Inf g(x,y) Y

+ E} •

X

Hence, in view of Theorem (l.3), we need to show only that for any two finite sets x1 , ••• ,xn "- X and y1 , ••• ,Ym "- Y there exist x0

"-

X and

y0

"-

Y such that

(l )

"- [nJ,

for all

k "- [mJ

.

by

For this purpose define a function n

m

L L

G(a,B)

i =l k= l

G

L:in-l

x

L:im-l -->R

aiBk s(xi ,yk)

for a = (a1 , ... ,an) "- lln- l ,

By the von Neumann Minimax

Theorem in Rn , there exist - ) "- lln- l a- = (-a 1 , ... ,an

and

such that (2)

G(a,B) s G(a.S)

for all

(a,B) "- L:in-l

From the definition of G and (2)

x

L:im-l .

we get

(3)

Next we choose

a= (;1 , ...

,;n) "- lln-l

and

B=

(s1 , ... ,Bi

11 )

rationals as their components such that (4)

(5)

n

I

i =l m

I

k=l

ai s (xi ,yk) s Sk t(xi ,Yi)

s

n ai s(xi ,yk) + f.2 i =l

I

m

I

k=l

E

Bk t(xi ,yk) + 2

for all

k d[mJ

for all

"- [n J

with dyadic

Some Minimax Theorems Without Convexity

Now, because exist

x0

and

EX

X is finitely s-convex and y0

s(xo,yk)

(7)

t(xi,y 0 )"" i

E

I

$

i =l Ill

I

k= l

In I and

Y is finitely t-concave, there

Y such that

E

n

(6)

Let

67

~ ai s(xi,yk)

for a 11

Bi t(xi,yk)

for a11

k

E

Un J be given.

k

E 1111 )

c Ln J .

Then using the assumption (i) of the

Theorem and (4), (5,), (6), and (7) we get (8)

(9)

Finally, from (3), (8), and (9) we conclude that

Since

i c [nJ and

kc [mJ were fixed arbitrarily, this implies (1), and the proof

of the theorem is complete. As an immediate consequence we obtain the following result established in our Note [4] :

(2.4)

Theorem.

Let

X,Y

be two compact spaces and let

f,s,t,g : X xy

be four real-valued functions satisfying conditions (i), (ii) and (v) of Theorem (2.3).

Assume furthermore that

(iii)*

For any

xc

x 1 ,x 2 , ... ,xn c X and

X such that for a 11 s(x ,Yl

n

'° I

i =l

(a 1 ,a. 2 , ... ,an) c lln-l

y c Y

ai s(xi ,_vl

there is a point

➔R

68

Granas and Liu

For any y1 ,y 2 , ... ,Ym y

€

Y such that all t(x,y) ~

€

Y and

x

€

m

I

j =l

there is a point

X

t(x,yi)

i3J•

Then Min Sup f(x,y) X

Proof.

Y

~

Max Inf g(x,y) . Y

X

It is enough to observe that (iii)* (resp. (iv)*)

implies the condi-

tion (iii) (resp. (iv)) of Theorem (2.3) . By taking in (2.3)

f = s = t = g we obtain among special ca~es of Theorem

(2.3) the following result (2.5) Theorem.

Let X,Y be two non-empty compact spaces and f

Xx Y ... R

be a real-valued function such that : (i)

x ... f(x,y)

is lower semi-continuous on

X for each y

€

Y

(ii)

y ... f(x,y)

is upper semi-continuous on

Y for each x

€

X

(iii)

For any x1 ,x 2 f(x,yi) ~

(iv)

€

X and y1 ,y 2 , ... ,Yn

½[f(x 1 ,yi)

+ f(x 2 ,yi)J

Y there is x such that

€

for all

For any y1 ,y 2 EX and any x1 ,x 2 , ... ,xm EX

i

E

LnJ ;

there is y

€

Y such that

The the following minimax equality holds Min Max f(x,y) = Max Min f(x,y) . X

Y

Y

X

As a special case we have the following (2.6) Corollary (Nikaido- von Neumann).: Let X and

Y be two compact

convex subsets in linear topological spaces and f : Xx Y... R a real-valued function such that

Some Minimax Theorems Without Convexity

x ... f{x,y)

(a)

(b) y ... f(x,y)

69

is lower semi-continuous and convex on X for each y is· upper semi-continuous and concave on

E

Y;

for each x EX.

Y

Thenthe following minimax equality holds Min Max f{x,y) = Max Min f(x,y)

X Y

Y X

A theorem concerning systems of inequalities.

3.

In this section we establish a further generalization of a theorem of Ky Fan [2] concerning systems of inequalities ; our result extends three previous generali-

zations given in C3J,[4J and [5J. X be a non-empty set and

Let

G =

F = {f},

{g}

two families of real-

valued functions defined on X ; we write F s G provided for each f g

€

G such that f(x) s g(x)

for all

€

F there is

x EX

We need the following Definition. A family

(3.1)

any f 1 ,f 2

for all

€

F there is an

f

E

F =

{f}

is weakly F-concave on X provided for

F such that

x e X . *) We may now formulate our second main result Theorem.

(3.2)

Let X be a compact space and

families of real-valued functions defined on

*)

We recall that a family

F = {f}

~

-~ aifi(x)

l

=l

Ky Fan [lJ.

for all

x

E

be three non-empty

X satisfying :

is called F-concave on X provided for any

f 1 ,f2 ,... ,fn e F and a = (a l ' a 2 '· · · •0 n ) f{x)

F,G,H

E

~n-1 0

there is f

E

F

such that

X . This notion was first formulated by

70

Granas and Liu

2)

Each function in

F is lower-semicontinuous on

3)

X is finitely convex with respect to

4)

The family

H is weakly

X

G;

F-concave.

Then (I)

(II)

A ER , the following alternative holds :

For each (a)

There is

h EH

such that

h(x)

(b)

There is

x0 E X such that

> A

for all

x EX ;

f(x 0 ) s: A for all

f

E

F.

Min Sup f(x) ,; Sup Inf h(x) XEX fEF hEH XEX Proof.

Since (II) follows e:isily from (I), we only need to prove (I). A ER

To show (I), let

be given ;

we assume that the condition (I)(a)

is not verified and proceed to show that (I)(b) holds. n and each

For each positive integer

f E F , we let Sn (f) = {x E X I f(x) s: 1' + ~}

Clearly each

Sn(f)

is closed in

first that for each fixed intersection property. gl , ... ,g 111

E

G

and

X and therefore compact.

n , the family of compact sets

Let

f 1 , ... ,f111

h1 , ... ,h111

E I{

F(x ,E;)

G(x ,E;)

H(x,E;) = for

m

I

F , G, H

E;ifi (x)

'

.I E;igi (x) l =1

'

i =l

F be given.

for all

=

Since

x EX

and

Xx 6m-l ➔ R defined

has the finite

F s: G ,; H , there are

i E CmJ. by

Ill

m

.I i=,;.h.(x) l =1 l l

,

x E X and We are going to apply Theorem (2.3) with

g

{Sn(f)}fEf

such that

fi(x) s g(x) s hi(x) Consider now the functions

E

We are going to show

H.

We observe first that

Y

6m-l ,

f

F, s

= t =G

and

71

Some Minimax Theorems Without Convexity

i' ,

F(,,()

(*)

E,

G(x ,E,)

➔

E, ➔ H(s ,E,)

is lower semi-continuous on X for each t, m- l is concave on 6 for each X E X ; m- l is upper semi-continuous on 6 for each

Furthermore since (by the assumption) given

x1 ,x 2 c X there is

x

E

X

X

6m- l

E

X E

X •

is finitely convex with respect to

G ,

such that E [m

J ,

and this implies

for all

E,

E

L\m-l .

Thus

X is finitely G-convex.

the hypotheses of Theorem (2. 3) and

g

=

H.

are verified with

Y

= lim- l , f

=

F ,

s

=t =G

Consequently by Theorem (2.3) we get

(**)

Min Max X 6m-l Let

Because of this fact and(*),

n

E

F(x ,E,) s Max 6m- l

N be a given integer.

Inf H(x ,E,) X

From (**), by applying Theorem ( l. 3), we

have the following alternative, either there is

(a)

such that

., ,\ +

ln

for all

t, EL\nl-1'

or there is

(b)

H(x ,r;) :, ,\ + -l n

such that

for all

X E

X •

By adapting arguments of S. Simons [5J and those of our earlier Note [3J we are going to show that (b) is not true. holds.

By restri ctinq

we may assume that

E,

Suppose to the contrary that the condition (b)

the following set-up to some face of L\m-l is in the interior of

L\

m-1

•

Define

if necessary,

h : L\ m-1 -, R u

{ - ro}

by k(E,) = Inf H(x,E,) XEX

Since E, ,

H(x,E,) ;,, F(x,E,) k

and

x

➔

,

F(x,E,)

for

E,

E

L\m-l .

is lower semi-continuous on

is real-valued and is bounded from gel0\'1 on

L\m-l .

X for each

Furthermore, since

k

Granas and Liu

72

m-1

is concave on the convex

t:,

,

k is continuous at

and k is continuous at 1; , there is a point n

=

Because

1; •

(n 1 , .•. ,11,i) with dyadic raand

tional coordinates n1 ,n2 , ... ,11,i such that n E Int t:,m-l since the family H h(x)

m 2

I

n1hi(x) = H(x,n)

i =l

2

k(n)

Finally,

> >..

x EX . This contradicts our assumption that (I)(a)

for all

> >.. •

is weakly F-concave, there is h E H such that

{h}

a:

k(n)

is not verified.

Thus

the condition (b) does not hold and hence in view of (a) there is x0 EX such that F(xo,1;)

s A

nl

T

for all

I;

m-1

t:,

€

•

This means that

XO

€

family {Sn{f)}fEF has the finite intersection property.

m n s {f.) ' i.e.' the i =l n ,

A = n S {f) "' f EF 11 n is compact and An~ An+l) that n An"' ..

f E F.

for all

n =l

Thus (I}{b)

Next it is clear Obviously, if holds and the proof

As an immediate consequence of (3.2) we obtain the following result Let X be a compact space, Y an arbitrary set and

(3.3) Theorem.

f,g,h : Xx Y ➔ R three real-valued functions verifying l.

f (x ,y) s g (x ,Y) s h( x ,y)

2.

x

➔

f{x,y)

for a11

(x ,y)

€

Xx Y;

is lower semi-continuous on X for each y E Y .

Assume furthermore that one of the following conditions (A), (B), C)

is

satisfied : Al..

For any x1 ,x 2 E X and y1 ,y2 , ... ,Yn E Y there is x E X such that l [g{x, ,Yi) + g(x2,Yi) J for all s 2 g ( x,yi)

(A)

A4.

For any y1 ,y 2 E Y there is h( x,y)

2

y€

i1 [h(x,y 1 ) + h(x,y 2 )J

i

E (n

€

X.

Y such that for a11

x

J •

Some Minimax Theorems Without Convexity

83.

there is

For any x1 ,x 2 , ... ,xm E X and

x E )(

such that

g(x,y) 84.

(B)

73

m

.l

s

l

=1

aig(xi ,y)

For any y1 ,y 2 , ... ,Yn

E

for all

y

€

Y

there is

Y and

y E Y such that h(x ,y)

(C)

{"

n 2:

l

j =1

S}(x ,Y)

for all

x EX •

X is convex

C4.

X-+

C5.

The family {hy}yEY

g(x,y)

is convex on

for each y E y •

X

is F-concave.

Then the following two assertions hold : (I)

For any

A

ER either (a)

x0 EX

there is

f(x 0 ,y)

such that

s A

for all

y EY

or

(s) there is y0 E Y such that h(x,y 0 ) (II)

A

>

x EX .

for all

Inf Sup f(x,y) s Sup Inf h(x,y) X

y

y

X

By taking in Theorem (3.3),

f

=

g

=

h we obtain as a special case the fol-

lowing result, which should be compared with Theorem (2.5) : (3.4) Theorem. f : Xx

Y-+

for each y 1.

Let

X be a compact space,

R a function such that x-+ f(x,y) E

Y.

Y an arbitrary set and

is lower semi-continuous on

X

Assume that the following two additional conditions are verified there is

x- EX

such that

74

Granas and Liu

2.

there is

y-

E

Y such that

f(x,y);:, [f{x,y 1 ) +f(x,y 2 )J

for all

x EX.

Under the above hypotheses we have : (I )

For any

"

€

R either

there is

(a)

XO

E

f ( X ,y) 0

X such that

"

$

or there is

( 13)

(II)

Inf Sup f(x ,y) X

€

y

f(x,y 0 )

>

Yo

for a11

y

€

y

such that for a11

"

X E

X•

Sup Inf f ( x ,y) . y X

y

As a special case we get the following

(3.5) Corollary. (Ky Fan-tnkaido-K!,eser).

Let

X be a compact convex space,

Y an arbitrary set and f : Xx Y ➔ R a real-valued function such that x ➔ f(x,y) is lower-semicontinuous on X for each y E Y . Assume that one of the following additional hypotheses is verified : 1.

( Ky Fan).

The family

2.

(Nikaido)

Y is a convex subset of a vector space and

Y for each 3.

(Kneser)

{f} y y, y

is F-concave. y

➔

f(x,y)

is concave on

x EX

Y is a vector space and

y

+

f(x,y)

Then Inf Sup f(x ,y) X

y

Sup Inf f(x,y) . y

X

is affine on

Y for each

x EX •

Some Minimax Theorems Without Convexity

75

References 1.

2.

Ky Fan, Minimax Theorems, Proc. Nat. Acad. Sci:, U.S.A., 39 (1953), 42-47. , Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations, Math. Z. 68 (1957), 205-217.

3.

A. Granas & F.C. Liu, Remark on a theorem of Ky Fan concerning systems of inequalities, Bull. Inst. Math. Acad. Sinica 11 (1983), 639-643.

4.

_________ , Theoremes du minimax sans convexite, C.R. Acad. Sci. Paris, t. 300, no ll (1985), 347-350.

5.

S. Simons, Remark on a remark of Granas and Liu concerning systems of inequalities, Bull. Inst. Math. Acad. Sinica (to appear).

Compactness and the Minimax Equality CHUNG-WEI HA Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

Let E be a real locally convex Hausdorff topological vector space, E' be the topological dual of E and let X be a bounded subset of E. (We suppose throughout that X and Y are nonempty.) We denote by cr(E,E') and t(E,E'), respectively, the weak topology and the Mackey topology on E induced by E'. It is easy to show (see, e.g., [SJ, Remark 6) that if Xis cr(E,E')-compact, then, for any subset Y of E', I]

inf sup inf S sup inf sup t E ,jT (X) y E Y XE t E $(Y) XE X yE I]

(1)

where 3-" (X) and ,y; (Y) denote the families of nonempty finite subsets of X and Y, respectively. Moreover, if Xis convex and cr(E,E')-compact, then for any convex subset Y of E', inf sup ( y ,x> < y EY XE X

sup inf XE X y E y

(2)

(2) follows immediately from (1) and the fact that for any convex sets

X C E and y CE I ' we always have I]

and

inf E $ (Y) sup

t E $(X)

sup XEX

inf YE 11

inf yEY

sup XE t

< Y,x>

= inf yE y

sup < Y,x> XEX

(3)

( y ,x) = sup inf ( y ,x) XE X yE y

(4)

(see, e.g., [SJ, Lemma 11). The relation (2) is the minimax equality referred to in the title; obviously (2) with the inequality sign reversed is always true. In this paper, we shall show that each of the properties (1) and (2) in a way characterizes the weak compactness of 77

DOI: 10.1201/9781003420040-7

78

Ha

X. Our main results are Theorems 1 and 2 below. As applications, we obtain directly from Theorem 2 DeBranges' s characterization [ 1] of the t(E' ,£)-open convex sets in E' and a theorem of Fan [2] generalizing the Alaoglu-Bourbaki theorem. If E is assumed to be t(E,E')-complete, then Theorem 2 becomes a result of Simons [5]. In the following we shall assume the notations given above. The reader is referred to [4] for the terminology not herein defined. Theorem 1. Let X be a bounded set in E. If (1) holds for any convex set Y in E' for which the left-hand side of (1) is positive, then Xis relatively a(E,E')-compact in E. Proof: Suppose that X is not relatively a(E,E' )-compact in E, then the a(E'*,E')-closure of X in the algebraic dual E'* of E' contains an element$ which is not in E. For any elements x 1 , ... ,xm in X, there exists y E E' satisfying $(y) > 1 but I< y,xi>I < 1/3 for all 1 ~ i ~ m. Let y = {y EE 1 :

Then Y is convex in E'

t

sup E S(X)

'

$(y)

~

l}

(5)

and

inf yE y

sup xEt

I< y ,x >I

~

1/3.

On the other hand, since$ is in the a(E'*,E')-closure of X, for any elements y 1 , ... ,Yn in Y, there exists x E X satisfying I$ (y j )- < y j ,x > < 1/3 and so < y j ,x > > 2/3 for all 1 ~ j ~ n. Thus,

I

inf r)ES(Y)

sup xEX

inf y Er)

< y,x>

~

2/3.

(6)

Hence (1) does not hold for the convex set Y defined in (5), which by (6) makes the left-hand side of (1) positive. This completes the proof. Theorem 2. Let X be i bounded convex set in E. lf (2) holds for any convex set Y in E' for which the left-hand side of (2) is positive, then Xis relatively a(E,E')-compact in E. Clearly Theorem 2 follows from Theorem 1, ( 3) and (4). Now we shall use Theorem 2 to prove the following result of DeBranges ([1], Theorem 4):

Weak Compactness and the Minimax Equality

79

set- in Theorem 3. Let Ube a convex E'. U is t(E',E)-open if and only if A n u = fJ for any convex set H in E' such that H n u = fJ' - -where A denotes the a(E' ,£)-closure of H in E'. Since the a(E'E)-closure and the t(E' ,£)-closure of a Proof: convex set Hin E' are the same, the condition is clearly necessary. Conversely, by a translation we may assume that the origin O of E' belongs to U. It suffices to prove that O is in the t(E' ,£)-interior of U. Let X be the polar U0 of U in E, that is, X = {x E E : (u,x>

~

1

for

u E U}

then X is bounded, closed, convex and contains O. (X is bounded because U is radial round all of its points.) We shall show that Xis a(E,E')-compact. To this end, let Y be a convex set in E' and a> 0 be a real number such that a < inf sup yEY XEX

(7)

By dividing both sides of (7) by a, we may assume that a= 1. Then (7) implies that Y n U = {J. By Zorn's lemma, there exists a maximal convex set H in E' containing Y but disjoint from U. Clearly H is a(E' ,E)-closed. It follows from the separation theorem that there exists an element x E E and a real number ~ such that ( y,x>

~

~

(8)

>

for all y E Y and u E U. Since OE U, Thus, (8) implies that x E X and so 1

~

sup

XE

X

~

> 0 and so we can assume

~

= 1.

(9)

inf < y ,x > yE Y

By Theorem 2, X is a(E,E' )-compact. Now if y ff U, then by applying the preceding argument to {y} in place of Y, we obtain an element x EX such that< y,x> ~ 1. Hence {u

which shows completed.

that

E

E'

O is

u ,X ) in

< 1

for

X E

X}

C

the t(E' ,£)-interior of U.

U The proof is

80

Ha

Next we shall show that the following result of Fan ([2], Theorem 1) is a consequence of Theorem 2. Theorem 4.

Let X be a bounded set in E.

U = {u

E

u ,x >

E'

~

1

for

x E X}

of X in E' has ~ nonempty t(E' ,E)- interior, cr(E,E')-compact in E.

(10)

then X is relatively

Proof: Since X and the closed convex hull of X u {0} have the same polar in E', we may assume that Xis closed, convex and contains 0. To apply Theorem 2 as in the proof of Theorem 3, let Y be a convex set in E' such that 1 < inf sup ( y,x) yEY xEX

(11)

Then (11) implies that Y n U = {J. Since U has a nonempty t(E' ,£)interior, by the Hahn-Banach theorem there exist a nonzero element x EE and a real number ~ such that

~

~

~

(u,x>

(12)

for all y E Y and u E U. Since O E U, ~ ~ 0. Suppose first that = 0. By the bipolar theorem ( [4], p.126) x E X. But then X also contains all the positive multiples of x. Since x 1 0, this contradicts the boundedness of X. Hence ~ > 0 and so we may assume that ~ = 1. Then again x EX and so (12) implies (9). This completes the proof. ~

As pointed out in [2], the t(E,E')-interior of the set U in (10) does not necessarily contain the origin of E', and so Theorem 4 strictly generalizes the Alaoglu-Bourbaki theorem. Our proofs show a dual relationship between Theorems 3 and 4. Theorem 5. Let X be a closed bounded set in E andlet -the - closed convex hull C of X in E be t (E ,E' )- complete. Then the following statements are equivalent: (a) (b)

Xis cr(E,E')-compact; If y is~ real number and Z is the convex hull of an equi-

Weak Compactness and the Minimax Equality

81

continuous sequence {y.} in E' such that J

inf ZEZ

sup ( z-u,x XEX

(13)

for any a(E' ,E)- cluster point u of the sequence {yj}, then y ~ O; (c) li a, ~ are real numbers, {yj} is an equicontinuous sequence in E' and {xi} is~ sequence in X such that (yj,xi>

~

a

for

j

( y. ,x. )

~

~

for

j > i

J

Then

1

~

i

l

(14)

; (d) The inequality (1) holds for any equicontinuous (convex) set Yin E' (for which the left-hand side of (1) is positive). a~~

Proof: Suppose that Xis a(E,E')-compact. (b) is a property of the vector space C(X) of real-valued continuous functions on X. For given a a(E' ,£)-cluster point u of the sequence {yj} in E', there exists a subsequence of {yj} which converges pointwise on X to u (see [4], p.185). Without disturbing (13) we may assume that {yj} itself converges pointwise on X to u. Since {yj} is uniformly bounded in C(X), from Lebesgue's bounded convergence theorem, {yj} converges to u in the weak topology of C(X). Since the closures of Zin C(X) in the weak and the uniform topology coincide, u can be arbitrarily approximated uniformly on X by elements in Z. This shows that y ~ 0. To prove that (b) implies (c), let Z be the convex hull of the sequence {yjl given in (c). Then (14) implies that (13) holds for any a(E' ,£)cluster point u of {yjl, where y = a - ~- Hence a~~- Now we assume (c). Let Y be an equicontinuous set in E', for which we suppose that the left-hand side of (1) is not -oo; the right-hand side of (1) is not +oo. Let a' ~ be real numbers such that a < ~

>

rt E !7(Y) XE X

inf y E rt

( y ,x)

(15)

sup inf E !7(X) yEY

sup xEt

( y ,x)

(16)

inf

f;

We choose arbitrarily an x 1 E X such that < Yi ,x 1 > exists y 2 E Y such that Y1,··•,Yn-l E Y have been

sup

element y 1 EY, then by (15) there exists > a. For the element x 1 EX by (16) there < ~ for all 1 ~ i ~ n-1, and again by (15) there exists XnE X such that< yj,xn> > a for all 1 ~ j ~ n. Continuing in this way we obtain an equicontinuous sequence {y.} in E' J and a sequence {xi} in X satisfying (14). Hence a~ ~- This proves (d). In proving (d) implies (a), we shall make use of the assumption that the closed convex hull C of Xis t(E,E')-complete. Suppose that X is not a(E,E')-compact, then the closure of X in the algebraic dual E'* of E' contains an element$ which is not in the t(E,E')-completion of E. By Grothendieck' s completion theorem ( see [ 4] , p .148), there exists a balanced convex equicontinuous set U in E' such that the restriction of $ to U is not a(E' ,£)-continuous at OE U. After suitably scaling U, the proof of Theorem 1 goes through for the set Y={yEU:

$(y)~l}

The proof is completed. It is clear that if a bounded set X in E satisfies Theorem 5(b), so does also its closed convex hull C in E. Thus, we obtain a new proof of Krein' s theorem ( see [ 4] , p .189), which says that if the closed convex hull C of a a(E,E' )-compact set X in E is l(E,E' )complete, then C is also a(E,E')-compact. It is noteworthy that Theorem 5(b) was used by Pryce (3) as a starting point in his proof of James's theorem on weakly compact sets (see also [5]). Finally we remark that, if X is assumed to be convex, then Theorem 5(d) can be replaced by the condition that (d') The relation (2) holds for any equicontinuous convex set Y in E' (for which the left-hand side of (2) is positive). Thus, we obtain Simons's result Theorem 15 in [5).

References 1.

L. DeBranges, Vectorial topology, J. Math. Anal. Appl., 69(1979), 443-454.

2.

K. Fan, A generalization of the Alaoglu-Bourbaki theorem and its applications, Math. Z., 88(1965), 48-60.

3.

J. D. Pryce, Weak compactness in locally convex spaces, Proc. Amer. Math. Soc., 11(1966), 148-155.

4.

H. H. Schaefer, Topological vector spaces, Springer-Verlag, New York-Heidelberg-Berlin, 1970.

5.

S. Simons, Maximinimax, minimax, and antiminimax theorems and a result of R. C. James, Pacific J. Math., 40(1972), 709-718.

Volterra Equations with Positive Kernels NORIMICHI HIRANO Yokohama, Japan

1•

Department of Mathematics, Yokohama National University,

Our purpose in this paper is to consider

Introduction.

the existence of solutions to the nonlinear Volterra equation u(t)

( 1. 1 )

+

f

t O a(t-s)Au(s) ds

f ( t),

0 < t _:_ T,

where O < T _:_ oo, a(t) is a real valued function and A is a nonlinear operator from one space to another.

In case where A is

a maximal monotone operator on a real Hilbert space H,

the

existence of solutions of (1 .1) has been studied by several authors(cf.

(1 ],[6]). The setting in which A is a maximal monotone

operator from a real Banach space V c H to its dual V'

has also

been studied by Barbu[1 ], Crandall et al.[5], and Kiffe and Stecher[9].

In [7],

the author considered the case where A is a

pseudo-monotone operator from H into itself and gave existence theorems for the equation (1 .1 ).

From the point of view of

applications to the case where A is a differential operator, this assumption on A is restrictive.

In this paper, we extend the

result in [7] to the case where A is a pseudo-monotone operator from a real Hilbert space V CH into its dual V'. Assuming that a(t) is a kernel of positive type, we show the existence of solutions of (1.1)

2

Statement of

for f

EL

2

(0,T;V).

the main result.

Throughout this paper, V

will denote a real Hilbert space, densely and continuously imbedded in a real Hilbert space H. Identifying H with its own 83

DOI: 10.1201/9781003420040-8

Hirano

84

dual H',

v CH cv'.

we have

Let (x,y) be the pairing between

an element x EV' and y EV. If x, y EH, then (x,y) is the ordinary inner product in H.

II • II , I • I

By

II • II*,

and

we denote '\,

'\,

'.\,

the norms of V, Hand V', respectively. We denote by V, H, v' • spaces L 2 (O,T,V), L 2 (O,T;H) and

L 2 (O,T;V'), respectively. We

denote by the pairing between~• and u, v EH,

~- Then for each

is the inner product in~-

and~• are again

the

l•I

denoted by 11•11,

'\,

The norms of~.~

and 11•11*' respectivel-y.

-v

-If"

the cl-osed convex

For a subset D of V, we denote by co D and co D

hull of D with respect to the topology of~ and~, respectively. A nonlinear mapping A from V into V' is said to be monotone if for all yi E

Axi, i = 1,2.

A is said

to be maximal monotone if it has no proper monotone extension, A nonlinear (single-valued) mapping from V into V' is said to be pseudo-monotone (*)

if

A satisfies the following conition:

If a sequence {un}C V satisfies that un

in V

and

(Au, u - v)

lim sup (Aun' un - u)

~

n+co

~

lim inf (Aun' un - v) n+co

u(weak convergence)

+

O, it follows that for all v EV,

and Aun

converges weakly to Au in V'. In the following,

we will assume that A is a pseudo-monotone

operator from V into V' with domain D(A) = V.

In addition,

we

will assume that A satisfies the conditions

II ull>

( 2. 1 )

II Aull*

( 2. 2)

c 2 11ull2 ~ c3 + (Au, u)

~

c, ( 1 +

for u EV, for

U

E

v,

where c, , c2 and c 3 are positive constants. 1 For each a(t) EL (O,T),

L

a

denotes the linear continuous

Nonlinear Volterra Equations with Positive Kernels

85

operator defined by

f

(Laf)(t) = for each f £ L 2 (O,T). =

I

T

t

for O

:a(t-s)f(s) ds

t

~

~

T,

Then the adjoint operator La* is given by for O

a(s-t)f(s) ds

t

~

~

T.

We state the assumptions for the kernel function a(t). (i)

a(t) £ L 2 (0,T) and is of positive type on [O,T], i.e., for

each f £ L2 (O,T),

f:

(ii)

La*

f(t)

f:

a(t-s)f(s) ds dt

* is injective, i.e., Laf

Remark.

~

0,

=O

for O

means f

~

t

~

T.

= O.

L and L* are bounded linear on a a are positive on these spaces if a(t)

the operator L*

and ~•. Also L and a a satisfies the condition (i). Sufficient conditions for a(t) to satisfy (i) are investigated in (11] and (12]. (ii)

if a ( 0)

~

0 and a' ( t)

£

1

a(t) satisfies

L ( O, T) •

We now state our main result: Theorem.

Let A:V

V' be a pseudo~monotone operator satisfying (2.1) and (2,2), Let a(t) £ t 2 (0,T) be a function satisfying (i) and (ii). Then for each f £ L 2 (o,T;V), (1 .1) has a solution in L 2 (0,T;V). ~

Hirano

86

Remark.

Since a monotone hemicontinuous operator is

pseudo-monotone, our result is an extension of the results for monotone operators (cf.

[ 1, 9 J).

It is well known that the sum of

monotone hemicontinuous operator and completely continuous operator is pseudo-monotone. Then Corollary. Let A:V

➔

we have

Suppose that Vis compactly imbedded in H.

V' be a monotone single-valued hemicontinuous operator

satisfying (2.1) and B:H

+

H be a continuous operator satisfying

(2.1 ). Suppose that A+ B satisfies (2.2) with A replaced by A+ B. Let a(t) E L 2 (0,T) be a function satisfying (i) and (ii). Then for each f EL 2 (O,T;V), the equation

f t0a(t-s)(Au(s)

u(t) +

( 2. 3)

+ Bu(s)) ds

f ( t),

0 < t

< T,

has a solution in L 2 (0,T;V).

3.

Proof of Theorem.

We first state a well known result

which is crucial for our argument. Proposition A(cf. [3), [4)). of~-

Let T :~

+

~•

Let K be a closed convex subset

be a monotone operator and T 0 :K ➔

pseudo-monotone operator.

~•

such that (g + Tou, v - u) > 0

for all g E Tv,

v EK.

The following lemma is also known and easy to verify. Lemma A. sup Then

u E 11'_

Let u be an element of~•. (U

1

V)

be

Then there exists an element u of K

VE

'\;

V,

!vi

~

Suppose that
0 A-Ix EB,

Hence, every closed convex cone with a base is necessarily pointed. Proposition2.l

(V. Klee[31])

Let K be a closed convex pointed cone in equivalent:

E. The following are

The Nonlinear Complementarity Problem

129

(1)

K

is locally (weakly) compact;

(2)

K

admits a (weakly) compact base.

When

K

is not pointed, an useful property initially remarked by

V•. Klee [31] ( see al so B. Anger and J. Lembcke [ I J) and pointed out to the authors by J.M. Borwein [8]) will be a key tool for what follows. Proposition 2.2.

Let

K be a convex closed cone in

(1)

K

is locally (weakly) compact;

(2)

K

is the direct sum of a locally (weakly) compact convex S with a finite dimensional subspace L ,.

pointed subcone (3)

there exists a continuous sublinear functional

B := {x EK : g(x)=J}

exists

c

Proof

E. The following are equivalent

>

0

is (weakly) compact, generates

such that

(1)=(2)

g (x) ~ c

11

x

for each

As easily seen, the set

which is finite dimensional since if we denote by

11

K

Since for each we deduce that S+L = K

K. Furthermore, there

is a subspace

is locally (weakly) compact. Hence, L

L

and by

P (resp. Q)

(resp. onto M), the set

is a locally (weakly) compact convex subcone of S n ( -S) = (Kn ( -K)) n M

such that

x E K.

L :=Kn (-K)

M a topological co~plement of

the continuous linear projection onto

g

K

S :=KnM

which is pointed since

LnM={O}. x EK,

Q(x) = x-P(x) E K-L cK+K cK

Q(K) c Kn M = S, and therefore

and

Q(x) EM,

Kc P(K) +Sc L+S c K. Hence,

as desired. (2)=> (3) Since

S+L = K

and

S

is a locally (weakly) compact

convex cone, the standard Klee's result, provides a compact convex base B for s. If e * EE * separates B from the origin, then X* :=e * o Q EK * ·= {x * EE * :x*(x) ~o for all x E K} and x * (x) > 0 for each XEK\{O}. For any equivalent norm in g(x) :=

L, the functional

II P(x) II

g

given by

+x* (x)

is sublinear and continuous. As easily seen, B := {x EK: g(x)=l} K

and is (weakly) compact. Hence, B

such that

g(x) ~cllxll

for all

xEK.

is bounded and there exists

generates c

>

0

Isac and Thera

130

(3)= (I)

Let

a< I

and set

U := {x EE: g(x) l' a}. Then

convex closed neighbourhood of the origin such that (weakly) compact. Hence

Un K is (weakly) compact and therefore

locally (weakly) compact, as desired,

that generates the topology on iEI

which is K is

D

In the locally convex setting, He say that uniformly positive if for a given g(x) >,cpi(x), for all

U is a

Un Kc [0,a] B

g : E ->JR

is K-

saturated family of semi-norms

E, there exists and each

c >0

{p.}. I l.

such that

l.E

xEK, xfO,

We observe that if we only require the existence of a sublinear continuous functional, which is K-uniformly positive, we only get a convex open neighbourhood

U of the origin such that

U n K is bounded. Then we

K is locally bounded [22]. In fact we have :

say that

Proposition 2.3

Let

K

be a cone in

z,Jhich is supposed to be locally convex.

E

The following are equivalent (1)

There exists a sublinear functional

g

which is K-uniformly

positive and continuous; (2)

K is locally bounded.

Proof : We only have to prove (2)=>(1). Choose a neighbourhood origin, which is closed, convex and circled. given by

pU(x) := inf{t > 0 : t

Furthermore, if with

therefore i.e.,

pU(x) =0 -I

x = ti (ti x)

x EU}

for some

and

ti> 0, lim ti= 0

-I

I

Then, the gauge

U of the

Pu

of

U

is sublinear and continuous, x EK\ {0}

t: x EU. Since

we can select a net -I

(x.). I l.

l.E

K is a cone, ti x EU n K and

belongs to the asymptotic cone

T (UnK) 00

of

U n K,

xe:{de:E: '3(ti)id->0, ti>0, -3(di)iEI' die:E, d=lim tidi}. Since

u n K is bounded, T ,,(UnK) 0

contradiction. Hence, PU

reduces to

{0}, so that

x =0

and we get a

is a convenient sublinear mapping.

REMARK: We observe that in Proposition 2,2, whenever convex locally (weakly) compact cone, L

reduces to

Klee's result and its well-known consequence: E*

D

K is a pointed {0}

and we recapture

admits a K-uniformly

continuous linear functional which is necessarily strictly positive.

131

The Nonlinear Complementarity Problem 3

A VARIATIONAL PRINCIPLE.

Theorem 3. I

Let

K

E

c

that satisfy: is positively homogeneous of order

nals defined on each x

E

f

K, x

0

be functio-

and

p

for

T 1 (x) > 0

T 2 (x)

---:,:0. II XII p

lim sup llx 11 ➔ + 00 XEK

(?,)

T1, T2

K

TI

(1)

be a cone in a named space and let

is closed, locally (weakly) compact and convex, and T 1 is (weakly) lower semi-continuous, then {x EE: T 1 (x)-T 2 (x) :s >..} is (weakly) If

K

>.. ER.

relatively compact for all Since

Proof :

K

is a closed locally (weakly) compact convex cone, by and a sublinear continuous mapping

c >0

Proposition 2.2(3) we get

g

such that

~

g(x)

cl/pll xii

x EK.

for all

Note LA:= {xEK :T 1 (x)-T 2 (x) :,:>..}, the >..-level set of T 1-T 2 • Since g is weakly continuous, it suffices to show that sup g(x) is finite. X XELA n If not, pick xn ELA such that g(xn) ~ n. Then y :=-(--) belongs to g xn n -1 g ({!}) nK which is (weakly) compact. Thus, there exists a subsequence with

(xnk), which tends (weakly) t o y

Ilk

)

----::: - - - - : s - - - [g(x

Ilk

)JP

[g(x

so that, T 1 (y)

Thus, T 1 (y):,:O,

yEK

A+T 2 (x~)

A+T 2 (x~)

T 1 (x~)

Tl(y

g(y)=l. Hence,

and

:sf

Ilk

cllx

)JP

~

llp

T2 (x~)

lim inf k -->-+oo

----:so II x

Ilk

11P

;;(y)=I, which is impossible.

□

Corollary 3. 2

Let

T

be a p-positively homogeneous functional defined on a cone E, which satisfies

K given in a named space

T(x)>O

forxEK,x/0.

Isac and Thera

132

If K is closed,locally (weakly) compact and convex and Tis (weakly) lower semi-continuous,then each level set of Tis (weakly) compact. REMARK:

A similar result was proved by Janos [28 :Thm. 1.2] for a positive-

ly continuous functional. The next result, extends the Weierstrass variational principle. Theorem 3.3

Under the assumptions of Proposition 3,1, if T1-T 2 is (weakly) lower semi-continuous then T1-T 2 achieves its minimum on K and the set of minimal points of T1-T2 is (weakly) compact. Proof :

Every level set of

T 1-T 2

is (weakly) relatively compact, hence

(weakly) compact, since (weakly) closed by (weak) lower semi-continuity of T 1-T 2 . Therefore,

m :=inf {T 1 (x)-T 2 (x)} X€K

tion property applied to the family

is finite and the finite intersec-

(FE)E>O

given by

X€

provides an element

FE :={xEK :T 1 (x)-T 2 (x) :;:m+d satisfies :

min {T 1 (x)-T 2 (x)} XEK

n

that

F

E>O E

□

Corollary 3.4

Under the assumptions of Corollary 3.2, every p-positively homogeneous functional T that satisfies T(x)

achieves its minimum on (weakly) compact, Proof

Set

T2 =0

>

0

for x

€

K, x -IO ,

K and moreover the set of minimal points of

in the preceding Theorem.

T is

□

See also J.P. Penot [43] for a more general criterion using the concept of asymptotically compact set, REMARK :

We notice that assumption (2) of Theorem 2.1 applies with

for a quasi-bounded mapping i.e, a mapping such that

p

>

1,

The Nonlinear Complementarity Problem

133

l..U20J

q(f) := lim sup

llxll->-+oo llxll

< +oo •

XEK

As observed in [ 21], this occurs whenever D00 f EE *

that is, whenever we may find

I f(x)-D f(x) I 00

lim

II

is asymptotically linear,

f

such that

ll->-+

X

XEK

II XII

00

= 0

The reader interested by these concepts should consult for instance

t 21],

[ 33].

4

APPLICATION TO THE NONLINEAR COMPLEMENTARITY PROBLEM. Let

T map a closed convex cone

Kc E

nonlinear complementarity problem relative to

into

E*

T and

and consider the

K

N,C.P. (T,K) : Find XE K such that T(x) EK* and = Since

K is a cone, as easily observed, N.C.P, (T,K)

to the variational inequality V.I(T,K) V.I(T,K): Find XEK, such that

o.

is equivalent

defined by ~O for each yEK.

which can also be rewritten as, V,I(T,K) : Find x EK where

31/J

K

-T(x)E 31/JK(x),

stands for the convex subdifferential of the indicator

(x)

function of

such that

K defined by, 1/J

K

(x)

=0

if

X E

K

and

1/J

K

(x) = +

00 ,

else.

The complementarity problem was studied in infinite dimensional spacesin[2]

[4]

[7]

[24]

[25]

[26]

[29]

andvariationaline-

qualities for potential operators in [44]. Let

K be a closed convex set of a Banach space

that a continuous operator P(E) =K

and

P(x) =x

P

for each

defined on

E

E. We recall

is a projection onto

K if

x EK.

We obtain the existence of continuous projections using different approaches. When

K is a closed convex cone, the following holds true:

Isac and Thera

134

Theorem 4 .1

Let a > O,

[34]

K be a closed convex cone in a Banach space

there exists a projection

P

0:

onto

K

such that for all

llx-Po:(x)II:: (J+o:)d(x,K)

where as usual

xEE

d(x,K) := inf llx-yjj yEK

We recall that a norm

11

satisfies the property t>,O

E. For each

or else

defined on

11

is strictly convex if it

E

implies

llx+yll = llxll + llyll

x =t y

for some

y=O.

The corresponding normed space is then called strictly convex. The class of strictly convex Banach spaces introduced by Clarkson [13] includes all Hilbert spaces, as well all

Lp((t,1,µ)

for

1

O, bl, b2 ~

y

x

is a convex combination of

x E ext(B), x =b 1 =b 2

x,yEext(B), with

b1

and the claim is proved, In par-

x11yfO, the relations

O a Galer>kin appr>oximation for>

T : K->E *

(a) if conver>gent to

y,

be a closed convex cone.

Kc E

(K) n nE ]N

of K. Suppose that

the following assumption ar>e satisfied (xn)

then

is weakly conver>gent to

x

and if

(yn)

is stY'Ongly

138

Isac and Thera

lim sup ~ n-++oo

(b) Then,

1,,J

(xn)

solves

'.t>

X

is sequentially weakly lower semi-continuous,

xl->

solves

N,C.P, (T,Kn), each weak sequential Zimit point of N.C.P,(T,K). Moreover, if K is locally compact or E is

n

reflexive, it suffices that Proof :

x=

be bounded.

(xn)

Let

x

be a weak sequential limit point of

(w) - lim

x

• Since

k➔+oo

Ilk

solves

Letting

X

Ilk convexity of

:= p

Ilk

N,C.P. (T ,Kn),

(x)

and

lim inf = 0

I

x +(I --)x and using the ~Ilk Ilk~ we have z EK • Furthermore, since X solves Ilk Ilk ~ also solves V,I(T,K ) so that, z

Ilk

:= -

n

Hence, >, 0 k

and therefore, since

k

assumption (a) yields Let

~

N.C.P.(T,K

(xn), that is,

y EK

>,O. Hence

(;;-n) k

norm-converges to

x,

=O.

be arbitrary and set

solves

N.C.P.(T,K obtain,

) , we have >,O, and using again assumption (a), we Ilk Ilk Ilk >, 0 for every y E K. Hence T(x) E K* and x so 1ves

N,C.P(T,K), as

desired. □

REMARK

Assumption (a) is fulfilled if

(I)

continuous. In this case, if norm-converges to

(xn)

and

(T(x ))

REMARK

n

(2) :

n

x

and

(yn)

- n n n

is strongly bounded, we have If

is sequentially weakly

y, since

- n

T

weakly converges to

T

lim

n-++oo

n

n

=

is completely continuous, i.e, T maps weakly

convergent sequences to

x

into strongly convergent sequences to

T(x),

as do most of the operators used in the deformation equations of Elasticity theory, then

xt->

tion 4.4 is trivial.

is sequentially weakly continuous and Proposi-

The Nonlinear Complementarity Problem REMARK

i.e. for each sequence zero i_n

(resp. in

E

has the Dunford-Pettis property (D.P.P in short),

E

Suppose

(3)

139

{x } CE (resp. {x*} c E*) n n * E ) then lim X * (X ) = Q • n oo n n

that tends weakly to

It is well-known that for each compact set D.P.P, as well each If E

E

T, C(T)

is Dunford-Pettis then for each operator E*

equipped with the weak topology into

xl-->

admits the

for each probability measure.

1 1 (µ)

T

continuous from

equipped with the weak topology

is weakly continuous and Proposition 4.4 is again trivial.

Lemma 4.5. -

Let

T be the Gateaux derivative of a convex functional

following implication holds true:

'

lim sup 11 X 11 ➔ +oo Proof

11

::=

X 11 p

0

=

Because of the convexity of

¢(x)-¢(0)

. llm sup 11 X 11 ➔ +oo

¢ we have

¢'(x)(x)

::=

1. The

from which we derive the desired result.

O

Theorem 4.6. -

Let Let

K be a closed locally compact convex cone in a Banach space.

T :=T 1-T 2

where

Ti

(i=l,2)

maps K into

and suppose the

E*

following assumptions hold true (1)

If

y,

converges to (2)

then

n

n

oo

n

is sequentially weakly upper semicontinuous; is the Gateaux derivative of

T.

1.

¢ 1 (x) >O

for each

is convex and Then, N.C.P(T,K) Let

1.

{Kn}

is lower

p,lower semicontinuous

XEK\{O};

lim sup

1

::=

X I I Hoo 11 X 11 p+ XE:K 11

Proof :

and

¢.

is positively homogeneous of order

¢l

and satisfies

norm-

is sequentially weakly lower semicontinuous while

semicontinuous; ( 4)

and if

x

lim sup :;;

x t->

x l-> ( 3)

is weakly convergent to

{xn}

O.

is solvable. be a Galerkin approximation of

conditions (3), (4) and (5), Theorem 3.3 applied to

K. Using Lemma 4.5, provides

X

n

EK

n

Isac and Thera

140

such that



on X is both upper

is a continuous sublinear DOI: 10.1201/9781003420040-12

Komiya

148

functional on E and A is a continuous multi-valued mapping of X into E. Theorem 1 Let N be a normed vector space, X a compact convex subset of N1 and pa continuous sublinear functional on N. Let F be a continuous multi-valued mapping of X into N with nonempty values Fx such that for any u, v £ Fx 1 there exists w £ Fx with p(w - (u + v)/2) s O. Then there exist x £ X and f £ N' such that max f(X) f s ¢, f(x) inf p(Fx).

inf f(Fx)

Proof The dual space N' of N will have the weak* topology. If. Y = (f £ N': f s ¢}, then Y is compact, Define a multi-valued mapping A of X into Y by Ax

{f £ Y: inf f(Fx) = inf (Fx)}.

Ax is nonempty by [4, Basic Theorem) and it is easy to see Ax is convex. Let {(xa,fa)} be a net in the graph Gr(A) of A converging to (x,f) £ X x Y. Take a point x I of Fx. Since the functional y ~ inf ¢ (Fy - x 1 ) is upper semicontinuous, for any£> O, inf (Fxa - x') < £ eventually. Hence there exist x~ £ Fxa with (x~ - x 1 ) < £ eventually. Then we have f

a

(X 1 a

)

1 .: f a (x') a - ¢(x a

-

x')

Hence we have f(x') ~ inf (Fx), since the functional y i-+ inf (Fy) is lower semicontinuous, and hence inf f(Fx) ~ inf (Fx). The reverse inequality is easy, hence we have (x,f) £ Gr(A). We have proved that Gr(A) is closed. Since Y is compact, A is upper semicontinuous and has closed values. Define a multi-valued mapping B of Y into X by Bf

{x £ X: f(x)

max f(X)}.

It is easily seen that Bf is nonempty and convex. If {(f ,x )} is a net a a

On a Best Approximation Theorem

in Gr(B) converging to (f,x) for any x 1 £ X,

£

149

Y x X, then

f(x)

f(x 1 ) = lim a f a (x') :a lim a f a (x) = f(x). a Therefore we have (f,x) £ Gr(B). We have proved that Gr(B) is closed. Since Xis compact, Bis upper semicontinuous and has closed values. By the coincidence theorem, there exist x £ X and f £ Y such that f £ Ax and x £ Bf. These x and fare the desired ones. Remark (1) The conditions for the multi-valued mapping F of Theorem 1 are satisfied if Fis a continuous multi-valued mapping of X into N with nonempty convex values. (2) We obtain [6,Theorem 5,1] by taking Fx = Px - Qx. We see from Theorem 1 that the assumption that P and Q have closed values in [6, Theorem 5.1] is unnecessary. (3) Theorem 1 holds even if the normed vector space N is replaced by a Macky space, but it is not known whether Theorem 1 holds or not when N is replaced by a general locally convex topological vector space.

3, A normed vector space N with a vector order.: is called an ordered normed vector space. The functional¢ on N defined by cj>(x) = inf {JJyJJ: y.: x} is called the canonical half-norm for the given vector order (cf. [1]). The functional¢ is continuous and sublinear. A point f of N1 is said to be poisitve if f(x) .: 0 for all x.: O, and we write f.: 0 if f is positive. When the positive cone is closed, x ~ 0 if and only if cj>(x) = O, and f s cjl if and only if f.: 0 and jfj s 1. Now we have the following thorem. Theorem 2 Let N be an ordered normed vector space with closed positive cone and X be a compact convex subset of N. Let F be a continuous multi-valued mapping of X into N with nonempty values Fx such that for any u, v £ Fx, there exists w £ Fx with w s (u + v)/2. Then there exist x £ X and f £ N1 such that f.: O,

JJfJJ :a 1,

and inf f(Fx)

f(x) = max f(X) inf cj>(Fx).

It is known that if N is a normed vector lattice (cf. [5]), then

150

Komiya

the canonical half-norm is the norm of the positive part, that 1s, l(x) = ~x+~. Hence we have the following corollary of Theorem 2. Corollary 3 Let L be a normed vector lattice and convex subset of L. Let F be a continuous multi-valued L with nonempty values Fx such that for any u, v € Fx, w E Fx with w ~ (u + v)/2. Then there exist x EX and f 2: 0, II fll s 1 , f ( x) = max f ( X) , inf f(Fx)

inf {lly+II: y

€

X be a compact mapping of X into there exists f EN' such that

Fx}.

REFERENCES [1] Arendt W., Chernoff P.R. and Kato T., 11 A Generalization of Dissipa:tivity and Positive semigroup 11 , J. Optim. Theory, vol. 8, pp. 167180, 1983. [2] Fan K., 11 Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces", Proc. Nat. Acad. Sci. U.S.A., vol. 38, pp. 121.,.125, 1952. [3) Kakutani S., 11 A Generalization of Brouwer's Fixed Point Theorem", Duke Math. J., vol. 8, pp. 457-459, 1941. [4] K~nig H., 11 0n Certain Applications of the Hahn-Banach and Minimax Theorems", Arch. Math., vol. 21, pp. 583-591, 1970. [5] Schaefer H.H., "Banach Lattices and Positive Operators'', Berlin/ Heidelberg/New York: Springer, 1974. [6] Simons S., 11 An Existence Theorem for Quasiconcave Functions with Applications", an expanded version of the talk given at AMS meeting in Eugene, Oregon in August 1984.

Vector-Minimization Problem in a Stochastic Continuous-Time n-Person Game HANG-CHIN LAI Institute of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China KENSUKE TANAKA Japan

1.

Department of Mathematics, Niigata University, Niigata,

Introduction The authors investigated a cooperative n-person game with

a discount factor in the case of discrete time countable state space.

( See Lai

and Tanaka

space [10]

).

and We

study the game system with a side-payment in case of the tinuous time space.

a now con-

In this case we would apply the Kolmogorov

forward differential equation to settle the transitionprobabilities which plays an important role in our game system.

From

the transition probability, we can define a one parameter contraction semigroup of operators on a space of all bounded real valued functions on the countable state space so that it has a fixed point.

Then it will reduce to an optimal

multistrategy

on which the collective total expected multiloss will 151

make

a

DOI: 10.1201/9781003420040-13

152

Lai and Tanaka

collective stability in the dynamic game system for a side-payment.

This vector valued optimization problem in game theory

is pretty different from usual multiobjectivemathematical problems which one could refer to Yu [16], Tanino and Sav,aragi [15], Lai and Ho [11] and their references.

The reasons

come

from

that the total expected multi loss is depending on each component which is steered by each player upon his chosen in the strategy space.

The cooperative discounted Markov

game

considered

in this paper has many connection with the noncooperative cases which are developed by the authors in [7],

[8],

[9]

We organize this paper into five sections.

and

[1~.

Section 2

is

the formulation of our game system which will be discussed

in

this paper.

In section 3, we give some necessary

assumptions

and the preliminaries on this game system in. which we introduce the Kolomogorov forward differential equation which wU 1 derive a one paprmeter semigroup of operators on the space of bounded real valued functions of the state space

S.

Section

4

will

give a cone dominated structure in which we would describe optimal multi-strategy existed in the game so that

an

the

total

expected multiloss makes a collective stability under a

side-

payment, and finally we summarized the main results of our present game system that under some conditions, there is a payment

side-

forced a vultistratevy to be optimum, and the related

properties, for example, a convex

multiplier, lower

support

function and super-differential are discussed in the context.

A Minimization Problem in a Stochastic n-Person Game

2,

153

Formulation of an E_-person discounted dynamic game Consider a game system as the following Cl,

2n+3

objects:

) r1, r 2 ,···,rn ,ex.

(2.1)

Here (1)

S = {1, 2,

•··,}

is a countable set, namely,

the

state

space; (2)

Ai

is the action space of player

Ai

is a compact metric space, i

We assume that each

i. E

:t-T

= {1,

2,

• · •, n}

is

the player set; (3)

q = q(• Is, a)

is a bounded function, namely a transition

rate function, for any

-a

where (4)

i r ( s, a)

= (a

1

'

a

(s, a) 2

'

... '

a

an)

X

E

A, A

n -

IT

i=1

s

is a real valued function on

loss rate function of player (5)

Es

X

Ai. A,

namely

a

i EN.

is a positive number, as a discount factor.

In order to know

easily our game system,

we

give some

interpretation as follows. For any time

t

E

[O,oo),

all players observe the state of

the game process and classify it to one possible state which is determined by a stochastic process Then all players choose their actions

{X(t)l

at FA

st ES

at time t.

under some pro-

such that each player t After this the game process moves to a new state st, according bability

JJs

to the transition function Markov process.

q(st' Jst, at)

associated

with

a

154

Lai and Tanaka

A strategy [O, oo)

1T

i

of the player

i

N

£

chosen at time

is described as a function which maps the information

game process from past history up to present state space carried by

i th player.

dent of the time

t

Especially if

n

i =

of

t

£

the

jnto

the

ni

is indepen-

( t)

action

and depends only on the present state, the strategy

is said to be stationary. In this case, for any t £ (0' 00)' there is mapping µ i = µ i : s ... P(Ai), that is, µ i £ [P(Ai)]s, for simplicity,we t write P(Ai) instead of [P(Ai)]S as the stationary strategy space of the player

i

N.

£

Here

P(Ai)

on the Borel measurable space field of the metric space

Ai

is the set of all probability measures (Ai, B(Ai)) for each

i

B(Ai)

where £

N.

Since

is the Borel Ai

is assumed

C(A i ), the space of all real-valued continuous func-

to be compact so

tions on A\ is a separable Banach space in the supremum norm topology. It is know that space of M(Ai).

P(Ai)

C(Ai).

is a weak* closed unit sphere in

M(Ai), the dual

Hence P(Ai) is a metrizable compact convex subset of

(Cf. Lai and Tanaka [8]).

Throughout this

paper,

we

assume

that each player uses only the stationary strategy. In this game systems, as the multistrategy players, the i th tial state

i

1, 2,

s

£

ii

is

chosen by all

player's total expected discounted loss under an iniS

n.

is defined by

We denote a vector (2.3)

and ask whether there exists an optimal mul tistrategy 1/J(ii)(s)

µ*

such

that

makes a collective stability rather than the individual opti-

mal strategies of players under some convex dominated structure. 1 2 n n this paper, we will use a side-payment d = (d, d, ·· · ,d) £ 1R

In with

A Minimization Problem in a Stochastic n-Person Game

li~lldi\ = 1

155

to make a collective loss function

w(µ)(s)> =

+ Q(µ*, o )u(µ*)(s)}

min {-.-Function for Several Classes of Normed Linear Spaces ROBERT H. LOHMAN Department of Mathematics and Sciences, Kent State University, Kent, Ohio THADDEUS J. SHURA Department of Mathematics and Sciences, Kent State University at Salem, Salem, Ohio

Let X be a normed space and let x E BX.

A triple (e, y, :\.) is said to be

amenable to x in case e E ext(¾:), y E BX , 0 < :\.

~

1 and x = :\.e + (1-:\.)y,

In this

case, :>i.(x) is defined by :>i.(x) = sup(:\.: (e, y, :>i.) is amenable to x}. Xis said to have the A-property if each x E ¾: admits an amenable triple, ally, if X has the :\.-property, then the :\.-function:\.: ¾:

~

Inform-

[O, 17 gives a measure

of the "largest" positive weight assigned to an extreme point e in the different representations x = :\.e + (1 - :>i.)y as x ranges over the closed unit ball BX of X, If X has the :\.-property, define :>i.(X)

inf {:\.(x): x E ¾:l,

Xis said to have the

uniform :\.-property in case :\.(X) > 0, These ideas were introduced and studied in [2],

In particular, spaces with

the :\.-property or uniform :\.-property were shown to have geometric features which may be of potential use in applications,

For example, if X has the :\.-property,

then every real-valued convex function on¾: which attains its maximum value must attain this value at a member of ext(¾:),

Consequently, if Xis a Banach

space with the :\.-property, then BX is the closed convex hull of its set of extreme points (Theorem 3.3 of [2]).

A much stronger conclusion holds if X has the uniform

:\.-property (Theorem 3.1 of [2]).

Namely, if O < :\. < :\.(X), then for each x E BX

there is a sequence (ek) in ext(BX) such that for every n, we have n llx - 6 :\.(1 - :>i.)k-le II ~ (1 - :>i.)n. k=l k Evidence is given in [2] that neither the :\.-property nor uniform :\.-property is It is shown that many spaces of the type t 1(X), t 00 (X), c(X), CX(T) possess

rare,

one of these properties and formulas for the :\.-functions of these spaces are obtained,

Also, every finite-dimensional normed space X possesses the uniform :\.-property and, for each x E Bx, we have :>i.(x) ~ (1 + n)- 1 , where n is the dimension of X considered as a real vector space, Since normed spaces with the :\.-property or uniform :\.-property satisfy strong geometric conditions, it is of interest to expand the catalog of spaces with one 167

DOI: 10.1201/9781003420040-14

Lohman and Shura

168

of these properties,

That is the purpose of this note, We first consider t -direct co sums, noting that parts of the following result were obtained independently by

David Trautman, Let (X ) be a sequence of normed spaces and let X = ($ !) X ) , • n n~ co Assume that each space Xn has the >..-property and let >..n denote its >..-function, (a) X has the >..-property if and only if there is a finite subset P0 of the set P of positive integers such that inf [>..n(Xn): n E I"P 0 } > 0, In this case, Theorem 1,

x = (xn ) E BX' then >..(x) = inf>.. n n(xn ) • (b) X has the uniform >..-property i f and only if inf >.. (X ) > 0, n n n 11 11 Proof. ( a) ~ Write B = R_ • Let x = (x ) E BX and note that we have n -xn n a= inf>.. (x) > 0, If O .. (x ) - s < >.. ~ 1, such that x = >.. e + (1 - >.. )y • n n n nn n n nn nn Since O < a - s ~ >.. (x) - s, Proposition 1,2 of [2] guarantees that we can write if

n

n

x = (a - s)e + (1 - ~ - s)z, where z EB. Thus, x = (a - e)e + (1 - a - e)z, n n n n n where e = (en) E ext(Bx), z = (zn) E Bx• Since a - £ > O, X has the >--property and >..(x) i!: a - s. Letting e tend to O yields >..(x) i!: a. On the other hand, assume x = >..e + (1 - >..)y, where e =(en) E ext(Bx), y = (yn) E BX and O < >.. ~ 1, For each n, we have e E ext(B ), y EB and x = >..e + (1 - >..)y. Therefore,>..~>.. (x) n n n n n n n nn for all n, This implies >.. ~ a and, taking the supremum over all such >.., shows that >..(x) ~ a, Assume that inf [>..n(Xn): n E P\P O } = 0 for each finite subset PO of P. Then there is an increasing sequence (n.) in P and vectors z EB such that -1 K Ilk nk >.. (z ) < k for all k, If we define (x) E BX by x = z , k E P, and x = 0 Ilk Ilk n Ilk Ilk n otherwise, then there is no triple amenable to x, Consequently, X fails to have the >..-property, 11

=>11

(b)

away from

11

« 11

Follows from (a), noting that a i!: inf>.. (X ) and hence is bounded n n n

o, independent of x.

Fix m E P and let x E B • Define x = (x ) E BX by x = x if n = m and m n n x = 0 otherwise, Given O < s < >..(x), we can write x = >..e + (1 - >..)y, where n = (e) n E ext(R_), -x y = (y) n E BX and O < >..(x)-s ;!;>..~ 1, Since em E ext(B m), ym EB m and x = >..e + (1 - >..)y, we have>.. (x) i!: Ai!: >..(x) - 8 i!: >..(X) - s, Since g is m m m arbitrary, we obtain>.. (x) l!: >..(X), Therefore, inf>.. (X) l!: >..(x) > 0, m n n n Remarks 2, If each space X is strictly convex, then>.. (x ) = (1 + llx II )/2 n n n n and the formula for >..(x) given in part (a) of Theorem 1 coincides with that given 11

=>11

e

in [2] (see Remark 1,14), It is also clear that Theorem 1 holds for tco-direct sums of arbitrary families of normed spaces, Next, we consider t 1 -direct sums of normed spaces, The situation is more complicated than for tco-direct sums and we obtain upper and lower estimates for >..(x). A precise formula for >..(x) is not yet known,

Calculation of the A-Function for Normed Linear Spaces

169

Let (Xn) be a sequence of normed spaces and let X = (13:1 6 X ) • n t1

Theorem 3,

Assume that each space Xn has the :>..-property and let An denote its :>..-function. Then X has the :>..-property and if x = (xn) E

¾,

f

x

O, we have ~(x)

:§

:>..(x)

\(x),

:§

where -:>..(x) = (1 - !!xii + 2max (A n (x n /!Ix n!I )!Ix n II : x n

and

f

0) )/2

f(x) = min ( sup:>.. (x ), (1- !!xii+ 2max!!x !J)/2), n n n n Proof.

Since !Ix II ... 0, the sequence (A (x /llx II )!Ix 11: x ,/ 0) is either finite n n n n n n

or is a null sequence.

In either case, there is a positive integer N such that

:>..i~/ll~ll lll~ll = max p..n(xn/11xn11 lllxn11 : xn Let O < e < :>..N(~/ll~lllll~II.

f

0).

There is a triple (eN, ~' :>..N) amenable to ~/llxNIJ

such that :>..N(~/JJ~JJ) - e < AN may assume that 1 - :>..Nll~II

f

0.

:§

By making AN slightly smaller, if necessary, we

1.

Since ~/11~11 = :>..NeN + (1 - :>..N)~, we obtain

~ = :>..N!l~\JeN + (1 - :>..Nll~!!)vN' where vN = [(1 - :>..N)l!~l!/1 - :>..NJJ~!J]~.

Define

e = (en) E ext(BX) and y = (yn) EX by

e

n

=

l

o,

n '/- N

eN'

n = N

[2/(1 + llxll - 2:>..N!l~li)]~, y

n

=

n

2(1 - :>..N!l~[J)vN + (IJx/1 - 1 )~ 1 + JJx\\ - 2:>..Nl\~Jl

and let A = (1 - !Jx!J + 2:>..N\l~JJ)/2.

N

n = N

'

I t is clear that x = :>..e + (1 - :>..)y.

f

Moreover,

a routine computation shows IIYil

:§

2 (!1xll - 11~11) + 2(1 - :>..Nll~II )JlvNII + 1 - llxll 1 + \\xi[ - 2ANJ\xJ\

:§

2(llx!I - llxJI) + 2(1 - :>..N)l!~ll + 1 - 11:,ql 1 + llx\\ - 2:>..Nll~II

= 1.

Since 1 ~A= [l - !Ix/I+ 2:>..N(~/l!xJll!l~IIJ/2 + [AN - Ai~/llxJl)Jll~\I

~ ~(x) -

e!!xJI,

we obtain :>..(x) ~ ~(x) - e!lxJI.

Letting e tend to zero shows :>..(x)

.

-

:§

:>..(x), which

establishes the first inequality and the fact that X has the :>..-property, That :>..(x)

:§

(1 - llx!J + 2max\1xn11)/2 is established in the same manner as in the

proof of Theorem 1.1 of [2].

In addition, if e

=

(en) E ext(¾), y

and O < :>. ~ 1 are such that x = :>..e + (1 - :>..)y, we have e Since !!Y JI :§ 1 and x = :>..e + (1 - :>..)y , we have A m mm m follows that :>. ( x) :§ I"( x) •

:§

m

=

(yn) E BX

E ext(R_ ) for some m.

:>. (x ) mm

-xm

:§

sup>.. (x ) • nnn

It

170

Lohman and Shura

Theorem 3 easily extends to t 1 -direct sums of arbitrary families of normed spaces. Also, if each Xn is strictly convex and xn f o, then we have Remarks 4.

:>..n(xn/11xn11) = 1 and :>..n(xn) = (1 + llxnl\)/2.

~ ( x)

=

i (x)

Hence, in this case, - 11 xi I

= :>,. ( x) = ( 1

2maxl Ixnl I)/ 2 ,

>

which agrees with the formula given in [2] (see Remark 1.12). space X of Theorem 3 never has the uniform :>..-property. positive integer and choose~ E Xk such that

11~\I

= 1, 1

Observe that the

To see this, let m be a c'.e

k c'.e m.

Then the unit

vector x = ( xJ /m, ••• , x /m, O, O, ... ) of X satisfies :>..(x) c'.e i(x) c'.e 1/m and so .

:>..(X) = O.

m

Finally, if one of the spaces Xm fails to have the :>..-property, there is

a unit vector x EX

m

which fails to admit an amenable triple.

in the proof of Theorem 1, it is easy to check that triple in the space X of Theorem 3.

x fails

If~ is defined as

to admit an amenable

Therefore, X fails to have the :>..-property.

We next consider Lorentz sequence spaces of the type d(a, 1).

Before proceed-

ing to our results, we review some notation and known facts about this class of spaces.

For our purposes, let a

sequence with a 1 in c 0 such that

(an) E c 0\t 1 be a positive, strictly decreasing

= 1. The space d(a, 1) consists of all real sequences x = (xn)

IxTI ( n ) Ia n

sup 0

< "' ,

where the supremum is taken over all permutations

TI

of the set of positive integers.

If llxll is taken to be this supremum, then d(a, 1) is a Banach space.

There is con-

siderable literature on the Lorentz sequence spaces d(a, 1.) and, more generally, on the Lorentz sequence spaces d(a, p).

The interested reader should refer to refer-

ences [1], [3]-[6]. If x = (xn) E d(a, 1) and x M1 (x)

f

O, write

= llxt,

F1 (x)

= (n : lxnl = M1 (x)}.

If cF (x) denotes the characteristic function of F1 (x), define 1

M2 (x) = l\x - xcF 1 (x)I\"',

F2 (x)

= (n: \xn\ = M2 (x)}

and write M3(x) = llx - xcF 1 (x) - xcF 2 (x)t' etc. Then M1 (x)>O, ~(x)IO and if

~(x)>O, then ~(x) > ~+ 1 (x). Also, if M/xl > o, ~(x) > 0 and j f k, then Fj(x) and Fk(x) are disjoint. We let N(x) = (k ~(x) - ~+ 1 (x) > O} and, if k E N(x), we define

~(x) = card(

k

U

i=l

F.(x)), l

0, then we can write llxll =

0

kEN(x)

II xii

as

~(x)[sk(x) - sk_ 1 (x)].

Calculation of the \-Function for Normed Linear Spaces

171

The extreme points of Bd( a, l) are precisely the vectors of the form k

(I;an) n=l

e=

where jl < ... < ik, .:n (see

[3]

and

-J

k

(I;.:nei ), n=l n

:!: land (ei) is the standard unit vector basis of d(a, 1)

[7]).

Also, observe that if b = ( b ) is a sequence of ± l's, then the mapping n Tb:d(a, 1) - d(a, 1), defined by Tb((x )) =(bx), is a linear isometry of d(a, 1) n n n onto d(a, 1). In addition, any rearrangement of coordinates is a linear isometry

1)

of d(a,

1).

onto d(a,

Y is

Finally, note that if T:X -

a linear isometry of the

normed space X onto a normed space Y and x E BX' then x admits an .amenable triple if and only if T(x) admits an amenable triple. Theorem~.

Let x

=

f

(xn) E Bd(a, l)' x A( x) ~

In this case, \(x) = \(T(x)). Then

0,

sup [ Ic~ ( x) - I•~+ 1 ( x)] sk ( x) • kEJlf(x)

Consequently, the space d(a, 1) has the A-property, Proof. For each n, choose b = + l such that bx = Ix I. n n n n ing Tb(x), where b = (bn), we may assume xn ~ 0 for all n, If k

E N(x),

let e

E

ext(Bd(a, l)) be defined by

e

= (

sk ( x) )

and let A = [ M ( x) - M 1 ( x) ] s ( x) . K k+ _ k First, if A< l, define y

=

-l

k

6 e) i=l nEF. (x) n

( I;

l

There are two cases to consider,

(yn) E d(a, 1) by

- A)' y

=

n

[H ( x) k

-

~

~

k

r/. U

n - M,

. ( x) K+J

1 - \

l f l

Then, by consider-

J

F. (x)

i=l

l

n E Yi(x) for some i E (1, ... , k)

'

k - l, then (x) - [t~(x) - l\+ 1 (x)J ~ //1 0 and xn = 0 for n ~ N + 1. If x E ext(Bd(a, l)), then A(x) = 1 and, since the nonzero values of x are the constant M1 (x) and M2 (x)

= O,

n1 (x)

we obtain

1 =

llxll = M1 (x)(

6 an)

n=l

Thus, we may assume x ~ ext(BX). Suppose that x = Ae + (1 - A)y, where O NJ

L

(N+l, N+2, ••• )'-.(il' ... ,

\l-

In what follows, we will denote the nth coordinate of a sequence z by either zn or z(n).

Then, noting that the nth coordinate of the sequence x = Ae +

zero if n ~ N + 1 and e(n)

=

0 if n ~ (i 1 , ... , ik), we have 1 = llxll =

N

6 x a

n=l n n

(1 - A)y is

Calculation of the A-Function for Normed Linear Spaces 6

nEI +

[>.. c

E

ck(

+ (1 - :>..)y. ]a. + 6 (1 - :>..)y a n n l n l n nEJ

[:>..e(i ) + (1 - :>..)y. ]a. + E [:>..e(n) + (1 - :>..)y ]a n in in nEL n n

nEK

= :>. ck(

Since

E:

k n

.,

a. ) + :>. ck( I: E: a. nEI n in nEK n in

E E:

k

+ ( 1 - :>..)

E

y a n=l n n

.,

CD

a. n=l n in EE:

1 and I; y a ~ E IYnlan ~ n=l n n n=l

~

ck(

( 1) and

k

E

JlyjJ ~ 1, it follows that

1

E: a. n=l n in

6 y a = 1. n=l n n

(2) From ( 1), we obtain for 1

173

~

n

~

k.

k

E E: n ai

n=l

k

E

n=l

n

an' which forces

E:

CD

n

= 1 for 1

~

n

~

k and i

n

= n

a,

CIO

E y a < E I y I a , forcing E y a < 1. n=l n n n=l n n n=l n n

If yn < 0 for some n, then

Therefore, (2) implies yn ~ 0 for all n. This means that e = ck(

(3)

x

(4)

x

k

E

n=l

n n

en) and

=:>..c

~

N.

n

= ( l - :>. )y ,

(3)

l~n!k n ~ k + 1.

n

If N < k, then N + 1 ~ k and Therefore, we have k

+(1-:>..)y,

k

implies ~+l ~:>..ck>

o,

which is impossible.

By (3) and (4), y is of the form

y = ( xl - :>..ck, ••• , xk - :>..ck, 1 - :>. 1 - X

xk+1, ••• ,

r---:---x-

~ ,

r---:---x-

0, 0, • • • ) •

Then 1 = IIYII N

[n~/nan - ¾¾ -xk+lak+l + ¾+lak + xk¾+1

k 1 - :>..c (Ea ) + :>..c a - Ac a ](1 - A)k n=l n k k k k+l [l - A + (Ac

k

+

:x_

K+l

-

:x_

)(a. - a

K

K

K+l

)](1 - A)-l

Since¾ - ¾+l > O, it must be the case that Ack+ ¾+l - ¾ ;e

•

o.

Therefore, we

Lohman and Shura

174

have A;,;;

("k -

xk+l)ck-l"

But A> 0 implies xk > xk+l"

Hence, there is an integer

j such that M.(x) = xk, M. J(x) = xk 1 and n.(x) = k (i.e., s.(x) = c -l). J J+ + J J k yields A( x) ;,;;

rhis

max [ M. ( x) - M. 1 ( x) ] s . ( x) . jEN(x) J J+ J

An appeal to Theorem 5 proves the asserted equality for A( x). To prove that d(a, 1) fails to have the uniform A-property, let m = 2k be an even positive integer.

X

Define the unit vector x by

=

m m [ 6 (m - n + l)a ]-1[ L (-m_-_n_+_l)e ]. n=l m n n=l m n

From the first part of Theorem 6, we have m

m

\(x) = ![ 0 (-m_-_n_+_l)a ]- 1 ( 6 a ) m n=l m n n=l n m m m ( 6a )/[0a + 6(m- n)a] n= 1 n n= 1 n n= 1 n m

k

m

;,;; ( 0 a ) /[ 0 a + k( 6 a ) ] n=l n n=l n n=l n m m m ;,;; (0a )/[Ea+ (~)6an] n=l n n=l n n=l = 4/(4 +

m).

As a consequence, \(d(a, 1)) = O, completing the proof. REFERENCES 1.

Altshuler,

z.,

Casazza, P. G. and Lin, B. L., On symmetric basic sequences in

Lorentz sequence spaces, Israel 2.

J.

Math., 1'._5(1973), 140-155.

Aron, R. M. and Lohman, R. H., A geometric function determined by extreme points of the unit ball of a normed space, (preprint).

3.

Calder, J. R. and Hill, J.B., A collection of sequence spaces, Trans. Amer.

4.

Casazza, P. G. and Lin, B. L., On symmetric basic sequences in Lorentz sequence

5.

Casazza, P. G. and Lin, B. L., On Lorentz sequence spaces, Bull. Inst. Math.

6.

Casazza, P. G. and Lin, B. L., Some geometric properties of Lorentz sequence

7.

Davis, W. J., Positive bases in Banach spaces, Rev. Roum. Math. Pures Et Appl.,

Math. Soc., 152(1970), 107-118. spaces. II, Israel J. Math., 17(1974), 191-218. Acad. Sinica, 2(1974), 233-240. spaces, Rocky Mount. J. Math., 7(1977), 683-698. 16(1971), 487-492.

of Positive Eigenvectors and Fixed Points for A-Proper Type Maps in Cones W. V. PETRYSHYN Department of Mathematics, Rutgers University, New Brunswick, New Jersey

0.

Introduction

Let

X

be a real Banach space, K

projectionally complete scheme for and

D

X

0

a bounded neighborhood of

boundary and

DK

the closure of

a cone in with

ln

DK

r

X,

= {Xn,Pn} a for each n EN+

Pn(K)CK

with

X

relative to

The purpose of this paper is two-fold.

the

DK=

K.

First, using the approach

of [14, 23] and the index theory for A-proper vector fields developed by Fitzpatrick and Petryshyn in [8], in Section 1 we give a complete proof of Theorem 1, which is an improved version of Theorem 3 stated without proof in [23] and which asserts that if such that

T11

and some fixed and Ci):

I-T-µF y

A e (O,d/6]

>

0

T, F:DK + K

is A-proper w.r.t. with

such that

r

6 = inf{il Fxll: x e clDK} > dy!CT, where

T = 0, then clearly

be A-proper for each

assumption that

qI-F: DK+ X

µ

e

a = a(K) e [1/2 ,l] K

is the

and

IK(O,DK)

=

(0,1/y]

{l} 'le {O}, d

=

=

b

is equivalent to the

is A-proper for each

Py-compact in the sense of [17, 18].

x e cl DK

6 > by/a, and the requirement

sup{[lxll: x E 3DK} > O, (i) reduces to I-µF

E [0,1/y]

provided

d = sup{ II x-Txll : x e 3DK}.

that

µ

IK(T ,DK) 'le { D}, then there exist x-Tx = \Fx

"quasinormality" constant associated with Note that if

are bounded maps

for each

q

~

y, i.e., F

is

In this case, Theorem 1 yields a

new result, i.e., Corollary 1, which establishes the existence of an eigenvalue such that·

n > 6/b

Fx = nx.

and the corresponding eigenvector Since every k-ball-contractive map

Py-compact for any fixed exist

n

>

o/b

and

x E 3DK

of

F:DK + K

F is

y > k, it follows from Corollary 1 that there

x E 3DK

such that

6 = inf{//F,x/1: x e 3DK} > bk/a.

:x = nx

provided

The latter result was proved by Massabo

and Stuart in [14] for the case when

F:DK + K

is k-set-contractive and

Supported by NSF Grant MCSS0-3002 175

DOI: 10.1201/9781003420040-15

176

K

Petryshyn

is a normal cone with

being the constant of normality.

0

For the

extension of some eigenvalue results in [12, 9, 14] to multivalue k-setcontractive maps see [23]. Let us add that when

o

> 0

(i.e., when

= I-T-µF

T

)J

is A-proper for each

µ > 0

and

y = 0), then Theorem 1 and its Corollary 1 (when

T = 0) were first proved by Fitzpatrick and Petryshyn in [8] by a method which cannot be extended to the more general class of maps studied in In particular, Corollary 1 contains Theorem 5.5 of Krasno-

this paper.

selskii [12] when

F

is completely continuous.

Using Theorem 1 and the fixed point index properties in [8], it is shown in Theorem 2 in Section 2 that if D1 and D2 are bounded neigh-2 borhoods of O in X, T, F, C: DK+ K are bounded mappings such that I-T-µF

is A-proper for each

µ e [0,1/y], condition Ci) holds on

aD~,

then under very general boundary conditions (see (Cl) and (C2)) the map -2 -1 T has a fixed point x 0 e DK\DK. In case Tµ = I-T-µF is A-proper for each µ > 0 and o > O, Theorem 2 was proved by the author in [24] by a method which cannot be extended to maps treated in this paper. special cases of Theorem 2, when k-ball-contractive, or

T

and/or

F

Various

are completely continuous,

T

is P 1 -compact, will be considered. cular, it will be shown that, for suitable choices of C and

In partiF, Theorem

2 contains, on the one hand, fixed point theorems which are of ''coneexpansion and cone-compression" type in the sense of Krasnoselskii [12] for various classes of maps

T

and, on the other hand, it contains fixed

point theorems for T when and/or T00 at oo along K

has a Frechet derivative T T at 0 with T 0 and/or T00 being of type

in the sense of Amann [l]. or L~(X) authors will be given in Section 2.

1.

0

L~(X) The exact references to various

Basic eigenvalue results

We first introduce some definitions and state those results which will be needed in the sequel.

Let

X

be a real Banach space, { Xn} C X

a sequence of increasing finite dimensional subspaces, and for each in

let

N+

Pn(x)

+

x

in

P n

convergence in

be a linear projection of for each

X

X

and set

x

in

X.

X

We use when

onto

"+"

Xn

n

such that

to denote the strong

a = supjj Pn II n

•

The following class of A-proper (and, in particular, Py-compact) mappings introduced and studied by the author in [18, 19] (see [20] for the survey) proved to be quite useful since, on the one hand, this class includes completely continuous, Py -compact and ball-condensing vector

Positive Eigenvectors and Fixed Points for Maps in Cones

177

fields, strongly accretive maps and their perturbations by k-ballcontractive operators and others and, on the other hand, the notion of the A-proper mapping proved to be also useful in the constructive solvability of operator equations; furthermore, the theory of A-proper maps can be applied directly to the solvability of differential equations when the latter are formulated as operator equations with operators acting between two different spaces. (1)

Definition.

Let

D

be any subset of

X.

A-proper w.r.t. continuous for each

n

{x

a subsequence

if

=

and

T:D C:: X

X

+

is said to be

nk

}

q > 0

X

and i f

{x n. [ x n.

n.

) +

g

for some

X

D

such that

J

and

E

y > 0, the map

-

=

y

if

q

Dn. }

E

J

J

is A-proper for each

qI-F

y > 0

J

For a fixed

g.

N+

E

T n. (x

sequence such that

Tx

A map

iff

F:D

J

in

g X

nk

X

+

dominating

+

is

n

is any bounded

X, then there exist k

as

X

+

and

00

is said to be p y-compact y

(i.e., q > y

if

0) .

The notion of an A-proper mapping evolved from the· concept of a P-compact map introduced by the author in [17] to obtain constructive fixed point theorems and surjectivity theorems for monotone type maps. In terms of Definition 1 is A-proper for each

q

, we say that >

D, (i.e., F

use, we recall that, for a bounded set B.

J

is a ball in

with

X

diam(Q.) < d} J

-

with radius

r}

F:D

+

is P-compact if

X

is P0 -compact).

n

U B. ,

Q, S ( Q) = inf { r > 0: Q C and

qI-F

For subsequent j =1 J

a(Q) = inf{d > 0: QC

n

U

Q. j=l J

are ball-measure and set-measure if noncompactness

A continuous map F: DC X + X is said to be k-ball-contractive Ck-set-contractive) if S(F(Q)) 2 kS(Q) (a(F(Q)) 2 ka(Q)) for each bounded set QC D and some k > O. Clear-

of

Q, respectively.

ly, if

F:D

+

X

is completely continuous, then

(and 0-set-contractive).

If

D

F

semicontractive in the sense of Browder [3], then contractive for some

k

E

(0,1)

said to be ball-condensing if that

S(Q) # 0.

is

(see [21,26]). S(F(Q)) < S(Q)

For various properties of

□ -ball-contractive

F:D

is a open set and F

A map

for each

S(Q)

and

+

is strictly

X

is k-ballF:D

X

+

QCD a(Q)

is such

and examples

of the above classes of mappings see [6, 16, 25]. It is known that A-properness is invariant under compact perturbaIn some cases we can say more to indicate the generality of

tions.

178

Petryshyn

A-proper maps. and

T:X

w.r.t. when

F:D

➔

X

is k-ball-contractive

is c-accretive and continuous, then Tµ _ T+µF is A-propel for each µ e (-ck- 1 ,ck- 1 ). If c =a= 1, the same holds

r1

F

Thus, for example, if

X

➔

is ball-condensing and

is k-ball-contractive, then

µ e [-1,1].

F

In particular, if

F:D

➔

y > k.

is P -compact for any fixed y

X

For

other examples of A-proper maps see [4, 6, 15, 20, 27] and the more recent papers by the author, Browder, Fitzpatrick, Milojevi~, Webb, Fitzpatrick-Petryshyn, Petryshyn-Yu, Massabo-Nistri, Xianling, Dupuis, Toland, and others. Since the fixed point index for A-proper vector fields introduced in [8] will play an essential role in what follows, we introduce this notion here and state those of its properties which we shall use.

KCX

Let

be a closed and convex set and suppose that Let

for each DK= D ()Ki 0 DK

and let

relative to

w.r.t.

ra

D

K.

and

DK

x-Tx

n > No' where

and

oDK

and

Suppose

0

~

for

x e aDK.

DK

is such that

P

(K)

CK

n such that

I-T

is A-proper

Then the A-properness of

such that

x-Tn(x) i O

relative to

n

X

the closure and the boundary of

D = D () xn, DK K = K () X n n' n

is the boundary of

n

aDK

T:DK ➔ K

N0 e N+

implies the existence of and

be a bounded and open set in

n

= Dnn Kn' T n

K n

I-T

x e aDK

for

n

p TJn DK

Following in the

spirit of the multivalued topological degree defined by Browder and Petryshyn in [ 4], the following notion was introduced in [ 8 J .

(2) Definition. Let X e ;)DK. denoted by

X, r

K, DK and T:DK K be as above and X i Tx for a' fixed 12.oint index of T on D with respect to K, Then the --➔

Z' = NU {-N} U {±

IK(T,DK)' is a subset of

00 }

determined

as follows:

(i) {n.}CN+

J

An integer such that

me IK(T,DK) iK

nj

(ii) such that Here

(Tn. ,DK J nj

if there is an infinite sequence . + ) = m f or each J e N .

if there is an infinite sequence

lim iK (Tn. ,DK j nj J nj "iK" n

= ± 00 •

{n.}

J

C N+

is the finite dimensional fixed point index defined

via the Brouwer degree.

Note that the observation immediately preceding

Definition 2 implies that

IK(T,DK) ¥ 0.

In what follows we need the following properties: (Pl)

If

IK(T,DK)

t {O}, then

T

has a fixed point in

DK.

n

Positive Eigenvectors and Fixed Points for Maps in Cones ( :, 2 )

I£

T(x)

(F3)

If

F:[

x-F(x,t) e K t

x e DK

for

I"'

(,L)

i~ either tion:

anc

t

(I-F(T,•),DK)

U C0

J :::>

U 3D 2 K' then

3DlK

(T,

or

e [O,l], r(t,x)

~

O

is constant in

t

e [O,l].

Cl U J 2 , D1 () D 2

D

,

IK(:,DK) C lK)

and

is an A-proper homotopy such that

,l]xl\

then

[O,l],

E

x e DK

for

J

~

179

IK(T,

0

x e 3DK

£or

and if

We sa"; that

:KC:,D 2 K)

F:[O,l]xQ

Q C X, then

anc if

r)

➔

{t

X n.

is an A-proper homotopy

} C

[ O , l] , { x

"

r, ·,

:" (

g

➔

) = g. We rec al J trEt t

ax+b:;

If

E

K

a,L e

for

K

+

[c],oo)

is a cone, we wri~e

we have

I xii

In case

T

,'.:_

ell YI!

i

e~Ji:alent to the

some

ye ~

C)

(C,J]

ex

I ·II Ii ·II

_

c:

X

llx+-,,11

r-1)_

re

e:i.ger.',

K

E

I7 is eas

to

I x+ i

i.ndeper1dent of

Y/lxl]

>

x




a=

y

y

(Q)

[l,oo)

for

[23] _

µ

in the Banach e

(0,1), or

a!'e EOt norma: b ·,+ Ll ' C (Q)

and

( Q)

~s a bo~~ded domain.

C

2:t v.'a2 sf'.:.._.-:i,h'r. b\·

and that

K.

€

~onriegative furictioris

:he same cones in

t

l

CT
0 for each ye~

normality constant

o = o(K)

by

If one sets

o(y) =

and defines the quasio

o(K) = sup{o(y): ye K}, then it was

shown in [5] that for any cone in

X

one has the estimate

1/2 < o < 1

( 1. 3)

and in general the constant

1/2

in (1.3) is the best possible.

In view of the above results, our first theorem in this paper is the following improvement of Theorem 3 stated in [23] without proof. Theorem 1. scheme in

X

Let

K

Pn(K) CK

with

bounded neighborhood of such that

I-T:DK

be a cone in

+

X

X,

{Xn,Pn}

a projection

+

n e N , and let

for each

O e X.

ra =

Suppose

T,F:DK

ra

is A-proper w.r.t.

+

K

DC X

be a

are bounded maps

and the following condi-

tions hold: (Hl)

IK(T,DK) # {O}.

CH2)

There is a constant

is A-proper w.r.t. µ > 0

if (H3)

y

=

o.

r

y

for each

~

0

such that

6 = inf{II Fxll: x e aDK} > dd, where

quasinormality constant associated with Then there are

x e aDK

and

K

Tµ = I-T-µF:DK

if

µ e (0,1/y]

and

A e (O,d/6]

y > 0

+

X

and for each

a e [1/2,1]

is the

d = sup{llx-Txll :x e aDK}. such that

x-Tx = AFx,

i.e., (1. 2) holds. Proof.

We shall first consider the case when

y > 0.

Now, choose

Positive Eigenvectors and Fixed Points for Maps in Cones E

( 0 , o)

E

0 > yd/(o-E).

such that

II x+yll and

X

0

in

K,

11

= II tO/t)x+y)II

x+tyll

and

y = YE E K such that o(y)+E > o(K) and for all X in X. Hence, for all t > 0

> ( O-E l II xii

> o(y)II xii

o(y)

Then, by definition of

0

a = o ( K) , there exists

181

> t ( o - El

= tll n yod = ~

a> 1, in contradiction to

It is obvious that

H(t,x) ¥ 0

We claim that

0, i.e., x -Tx o o

(H3), d ..::_ II x 0 -Tx 0 II

Tn

be arbitrary and consider the map H(t,x) = Tn(x)-tmy.

If not, then there exist

H(t 0 ,x 0

T (x ) n o

Tn = I-T-nF, then

Let

given by

is an A-proper homotopy. that

and

d

and

Thus ( 0, l]

and

Then the preceding discussion, the choice of

x 0 -Tx 0 = nFx 0 +t 0 my.

E > 0, and condition (H3) imply that d > llx 0 -Tx 0 II

= llnFx 0 +t 0 my!! > (o-Elnllrx 0 II > Co-Elna

> (o-E)n•yd/(o-E)

which is impossible.

=d

since

Thus, H(t,x)

= 1/y,

n ~

0

x e aDK

for

t e [0,l].

and

Therefore, by the homotopy property (P3), IK(T+nF,DK) = IK(T+nF+my,DK).

me N+

Now, the above equality holds for any implies that

IK(T+nF,DK) = {0}.

then for each

m

in

(I-T-nF)(xm) = my. T

and

F

N+

we could choose

Since

and this, we claim,

Indeed, if this were not the case,

y ¥ 0

and

xm e DK

such that

(I-T-nF)(DK)

is bounded because

are bounded maps, we have a contradiction when

m

is large.

Consequently, IK(T+nF,DK) = {0}. Consider now the mapping x-Tx-tnFx

for

t

boundedness of

E [0,l]

F

that

H0 :[0,l] DK ➔ X

and H0

x E DK.

given by

is an A-proper homotopy.

IK(T,DK) ~ {0}, while we have just proven that Hence there exist some

t e (0,1)

i.e., x = Tx+AFx

x

In case

o

y

> dE/(o-E),

o(y) > o-E

and

=

with

E dDK

0, we let

n

=

A= tn

1/E

and choose

As before, there exists llx+yll ::_ (o-Elllxll

such that

and

Ee (0,o) 0

x

H0 (t,x) = 0,

(O,d/8].

y = yE € K

for all

argument is the same as in the case when

€

By (Hl) we have

IK(T+nF,DK) = {0}

x e aDK

and

H0 (t,x) =

It follows from (H2) and the

in

such that

such that K.

The rest of the

y > 0. Q .E .D.

We note that Theorem 1 includes a number of special cases.

In what

182

Petryshyn

follows we only state here two new special cases which extend some known results. Corollary 1. be fixed and let

=

6

Let F:DK

inf{[[ Fxl[ :x

E

➔

clDK} > by/ CT , b

Then there exist Proof. when

T

=

K, CT ' r and D be as in Theorem 1. Let y -> 0 K be a bounded and P -compact map such that y

x E clDK

=

and

sup{[lxl[

E

X

A E (0,b/6]

such that

Corollary l follows from ~heorem l when

0, then

=

IK(O,DK)

{l}-/ {O}

( 1. 4)

cl DK}.

x = AFX.

T = 0.

by (P2), i.e.,

Indeed,

(Hl) holds.

The hypothesis (H2) reduces in this case to the condition that is Py-compact, while (H3) reduces to (1.4).

F:DK ➔ K

Consequently, Corollary l

follows from Theorem 1. Remark 1.

When

y = 0, ~heorem land Corollary 1 were proved in

[8] by a method which cannot be extended to the more general

maps when 0

y > 0.

Since a completely continuous map

F:DK

class of K

➔

is

0 -compact, Theorem Vl.l in [12] is a special case of Corollary 1 when

the Banach space

r

has a projectionally complete scheme

X

~oreover, since every k-ball-contractive map

F

= {Xn,Pn}.

can be easily shown to

y > k when r = r 1 by using the argument of Webb, it follows that Corollary l in [23] and, in particular, Theorem

be ~-compact for any fixed l

in [14]

(for

w.r.t.

x = AF

Corollary 2.

and

r1 .

provided Let

Suppose that

1

::.s } -compact and

\ e ( 0,b / 6]

8 > b/CT.

K, CT T: DK

Let

IK(T,D) f. {O}.

k > 0

fork-ball-contractive

r 1 , Corollary l shows that there exist

such that

=

normal) follow from Corollary l

Furthermore, since a ball-condensing map

maps.

r

K

and ➔

F:

D

be as in Theorem 1 and let k 0 e [0,1)

is k 0 -ball-contractive with

K ➔

K

beak-ball-contractive map with

and such that

dk 0 = inf{IIFx[I: x e clDK} > CT(l-k), 0

where

d = sup{[lx-Tx[I: x

A e (O,d/6]

Proof.

such that

E

clDK}.

Then there exist

x e clDK

and

x-Tx = \Fx.

Note that when we take

y

such t:iat

y

>

k

1-k

and 0

8 > dy/CT, then to deduce Corollary 2 from Theorem 1, it suffices to

show that

Tµ = I-T-µF

But, for any such

is A-proper w.r.--'.::".

µ, the map

~+µ?

is

r1

for each

µ

E

(0,1/y].

(k 0 +k/y)-ball-contractive with

Positive Eigenvectors and Fixed Points for Maps in Cones

183

k 0 +k/y < k 0 +k(l-k 0 )/k = 1.

is

Hence, by the results of Webb [27], Tµ

A-proper.

Q.E.D.

Let us note that for some applications, and from the practical point of view, the following version of Theorem 1 may prove to be useful since we don't have to compute

in case

d = inf{llx-Txl/: x E clDK}

K

is a normal cone. Theorem l'.

If

K

is a normal cone in

Theorem 1 remains valid with

A E (D,Tb/o]

X, then the conclusion of if condition (H3) is replaced

by (H3') and

o = inf{/IFxll: x E clDK} > by/o, where

o = llT

with

T > 1

such that

b = sup{l/xll: x e clDK} if

II ul/ .'.:_ T/1 vii

u,v EK

and

D < u < v.

Proof.

The proof of Theorem l' is similar to that of Theorem 1.

It suffices to show that if we let with

E > 0

such that

H:[D,l]xDK ➔ X and

x e clDK.

by

=

o > bE/o

=

n bET

1/y if

if y

H(t,x) = Tn(x)-tmy, then

=

y > D 0)

H(t,x) 1 0

H(t,x) 7 D, then

If we can show that

and this together with (Hl) yields the conclusion. H(t,x) 1 0 when

for

y > D

there are

x 0 e clDK

x 0 -Tx 0 -nFx 0 -t 0 my 11•11

t E [0,1]

and

(the case when

=

and 0.

semimonotone on

t

0

X

D

for

1/E

t E [0,1]

IK(T+nf,DK) = {O} To show that

is handled similarly).

E [D,l] x0

=

n

x E clDK, we first consider the case

y

Then

(and

and define

=

such that

H(t 0 ,x 0

Tx 0 +nfx 0 +t 0 my > nfx 0

and (H3') holds, we have

)

Suppose that

=

and thus, since

b > Jlx 0 /I

> onl1Fx 0 I

.:_

ono > on•by/o = b, a contradiction. Q .E.D.

2.

Existence of positive fixed points

Using Theorem 1 and the properties of the fixed point index stated in Section 1, we are now in position to extend Theorem 2 for cones proved in [24] so as to obtain the following new theorem concerning the existence of positive fixed points. Theorem 2.

Let

K

be a cone in

scheme as in Theorem 1, D1 with

D1 C D2 , and let

is A-proper w.r.t. ( Cl)

r.

-2 T:DK

D2

and ➔

K

X,

r

= {Xn,Pn}

a projectional

bounded neighborhoods of

be a bounded map such that

D

in

-2 I-T:DK

X ➔

Assume also that the following conditions hold:

There is a bounded map

-1

C:DK

➔

K and

such that

X

184

Petryshyn

the restriction for

X €

for

X €

C

n

Pn C

of

to

µ > 1 and n > N furthermore, X t- tT (x)+Cl-t)C ( X) aD 1 o' K ' n n n aD 1 t € [ 0, 1] and n > N 0 K ' n

(C2)

-2 F:DK

There is a bounded map

(H2) and (H3) of Theorem 1 and A E

is continuous and

K

➔

x-Tx t- Afx

which satisfies conditions 2 for all X E aDK and

(O,d/8]. Then there exists

x0

-2 -1 DK\DK

such that

x 0 = T(x 0

The same assertion is true if we assume that condition (Cl) holds on D2 while E

).

K

1

(C2) holds on

DK.

Proof. First, we may assume without loss of generality that T 1 2 has no fixed points on aDK and on aDK. Second, we will only provide a detailed proof for the case where y > 0 in (H2) since the case when

y = O

is handled in a similar way.

Now, for each tC ( x)

for

n

n > N0

t E [0,1]

define a map

,

and

of condition (Cl) that

x E

Dl

K

n

x-Mn(t,x) t- 0

-1

Mn:[O,l]xDK ➔ Kn by Mn(t,x) = n It follows from the first part

for

t

E

[0,1], x

1

oDK

E

and

n

the properties of the finite dimensional fixed point for all

= { l}

n > N . 0

part in condition (Cl), the map by

n > N0 t

E

all

= tTn(x)+(l-t)Cn(x) and

[O,l]

n > N0 -1

I-T:DK

➔

X

X

€

1

aDK. n

➔

for

x t- Gn(t,x)

for

by Definition 2 since x-Tx t- 0

for

in virtue of condition (C2), Theorem 1 (with Indeed, if

given for each

n

) = iK (Cn,D~) = {l} n n

and, therefore,

IK(T,D~) = {O}.

K

is such that

Hence

is A-proper and

Now, in view of the second

IK(T,D~)

1 x E aDK.

On the other hand,

DK= D~) implies that

were not equal to

{O}, then

since the first part of condition (C2) implies (H2) and (H3) of Theorem

1, it would follow from that theorem that there exist A E

(O,d/o]

tion in (C2). Now, let

x E aD~

and

such that Thus Q

=

x-Tx = Afx, in contradiction to the second assump2 IK(T,DK) = {O}.

D2 \n 1

has no fixed points on

and observe that oQK.

aQK

=

aD~ U aD~

and that

Thus, by the additivity property (P4),

T

Positive Eigenvectors and Fixed Points for Maps in Cones

since

is the singleton.

that

{-1}.

-2,-1

such that

DK DK

It follows from the last equality

Hence, by property (Pl), there exists

= T(x 0

x0

185 E QK =

x0

).

To pr'ove the second part of Theorem 2, note that when condition 2 DK then, by the same arguments as above, we see that Similarly, when (C2) holds on D1 then, in view of IK(T,D~) = {l}. 1 K 1 Theorem 1 with DK= DK, we see that IK(T,DK) = {O}. The property (P4) (Cl) holds on

implies in this case that

{l}

ceding case, there exists

x0

and so again, as in the pre-

such that

QK

E

x

T(

0

x ). 0

Q.E.D.

For the purpose of applications we state Theorem 2 in the following practically useful form which, as we will indicate below, unifies and extends a number of results obtained earlier by this and other authors for special classes of maps by using various methods. Theorem 2'.

Let

K

and

be as in Theorem 2, let

T

r 1 ,r 2 e (0, 00 )

J\(

with

0 ,r) ➔ K r = max{r 1 ,r 2 }, let BK(O,r) = B(O,r)() K and let T: be a bounded map such that I-T:BK(O,r) ➔ X is A-proper. Suppose also 2 that (Cl) and (C2) of Theorem 2 hold with D1 B( □ ,r 1 ) and D = B ( 0 , r 2 ) . Then there exist X e K such that min{r 1 ,r 2 } ~ ~x 0 j ~ max{r 1 ,r 2 }

and

x

0

= T(x ).

0

0

Special cases. the maps

T, F

We shall now show that, for suitable choices of

and

point theorems:

C, Theorems' 2 and 2' contain two classes of fixed

(A)

Variations of the cone-expansion and the cone-

compression theorems of the type established in [12] when

T

is compact

and Nussbaum, Potter, Reich, Hahn, Fitzpatrick-Petryshyn, Milojevic and others when T is k-ball-contractive (see [2, 22]) for exact references. ( B) The fixed point theorems when T:K ➔ K has a Frechet derivative

T0

at

0

are of type (A)

and/or

L~(X)

T00

or

at

Li(X)

along

such that

K

T

in the sense of [l].

and/or

0

Cone-expansion and cone-compression type theorems.

X E aD 1 K

and

X

II h[[

n

fc tTx

> dy/o

defining

\ft

E

for and

-2 F:DK

[O,l] t

➔

K

by

n 2:_ N0 ", holds when and

[ 0 'l]

E

x-Tx

\f

fc

11h

for

Fx = h

while (H2) also holds since

X

1

e oDK.

X

c oD 2

for

X

I-T-µF

K

and

-2 T: DK

Choosing /\ E

➔

is P 1 -compact

K h

0

€

K

such that

( 0' d/[[ h[[], and then

-2 DK we see that (H3) holds, is A-proper for each ).1 > 0 E

C=0

"x fc tP n Tx

in (Cl), then condition (Cl), which in this case reduces to for

When

T

186

Petryshyn

because

F

is compact.

Thus, in this case we deduce from Theorem 2'

the earlier results in Goncharov [10] when and of Hamilton [11] when

T

D2 = B(0,r 2 ), and of Milojevic [15] when 2 and

T:BK(0,r) + K is P-compact D1 = B(0,r 1 ) and

is P 1 -compact with D1

D2

and

are as in Theorem

is singlevalued. Theorem 2 in [28] is also a special case of Theorem 2.

T

An immediate consequence of Theorem 2' is the following extension of Corollary 1 in [24] which is a new result. Corollary 3. with

(Dl) x

E

for

3BK(0,r 1 )

(D2) with

+

K

There is a bounded ball-condensing map

Cx t µx

that for

Let r = r 1 and T:BK(0,r) Suppose further that:

k 0 e (0,1).

x e 3BK(0,r 1 ) and and t e [0,l].

µ > 1, and

be k 0 -ball-contractive C:BK(0,r 1 )

There is a bounded k-ball-contractive map

k > 0

such that

x-Tx t AFX

for

x

0

F:BK(0,r 2 )

dk o = inf{II Fxll: x e 3BK(0,r 2 )} > 0(1-k)

x e 3BK(0,r2)

Then there is

+

such

+

K

and

0

A e (0,d/o].

and

K

x t tCx+Cl-t)Tx

e K

Proof.

To deduce Corollary 3 from Theorem 2' it suffices to show ' (Dl) (Cl) , and that implies that the map Tµ = I-T-µF lS A-proper w.r.t. k and rl for each µ e (0,1/y], where y lS chosen such that y > 1-k 0

o > dy/0.

The latter fact has been shown to be the case in our proof

of Corollary 2, while the fact that (Dl) implies (Cl) follows in the standard way from the fact that

C

is ball-condensing and

Cx t µx

for x e 3BK(0,r 1 ) and µ > 1 and that F(t,•) = tC+(l-t)T is also ball-condensing (in the sense that S(F([0,l]xA)) < S(A) for each A

E

BK(0,r)

with

SCA) t 0)

and

x t Cx + (1-t)Tx

for

t e [0,1]

x e 3KB(0,r 2 ).

When all

A> 0

F

and

Q.E.D. is compact (i.e., k = 0

in (D2)) and

x-Tx t AFx

for

and

x e 3BK(0,r 2 ), Corollary 3 was first proved by the author in [22] for a general Banach space X where it was also shown that in this case it extends and unifies the corresponding results of Krasnoselskii, Gustafson-Schmitt, Turner, Gatica-Smith, Edmunds-PotterStuart, Nussbaum, Potter, Fitzpatrick-Petryshyn, Amann, Milojevic, Petryshyn and others.

mentioned authors.

See [22] for the exact references of the above

187

Positive Eigenvectors and Fixed Points for Maps in Cones ~onzero fixed points of differentiable maps.

( B)

In order to

deduce from Theorem 2 the existence of nonzero fixed points for compact maps, which are along

0

rechet differentiable either at

K, we recall firs~ some definit~ons.

1

0

er at

0

We emit some details except

for one lemma which is necessary bGt net proved in [24].

A map

T:K

X

➔

is said to be (Freche~) differentiable at

if there exists a map

K

( C , h)

T(O)+c'

is said to be

The ma? e ~(X)

It

h

1,;i th

~ 00

are u~iq~ely determined. T~,(K) CK

and

K

I

ti I )

00

( 2 .1)

➔ 0•

if there is

K

II hll

as

(2•2)

➔

is a tctal cone, then the maps

C

ancl

It ~s easy to

'l (K) CK

II hll

as

along

K 0 (

at

the derivati__\/es o~

,

in

(h ) =

not hard to show that i"

co

o(llhll)

w(O,h) =

sy~pto!icaJly linear along

such that for each

T(h)

anC

1,Jith

0

h e K

such that for each

e L(X)

along

K

ee that if

respectively,

T:K

➔

K, then

,,;henever they exist.

Before we appJ.y Theorc~ 2'

t.o ob~a~~ the exister1ce of nonzero

0 1 -ccnpact maps T:K ➔ K satis~ying either (2.1) or .2), we will need ~he fol:owing s le fact (see [12]): If u,v e K,

fixed points for u 1-

C

} C

a

ex::sts

II

()

(J,oo)

S'Jcr_ that

a

lS

3·J.cf'. ~:-1at

e

¢ : 1

}

for all

e K

\1-A

and this implies that

then

n

;\

i~

co::_r,c

BC::,,·)

n

in

X.

Then

K

such that ;\h

A~ f

for all

BK(G,~) = 8(0,r)

K, the closure

~K(C,r)

K, ,:,nc:; the bo:nc'a:n·

of

3BK(O,r)

n

K

BK(C,r) of

3(=,llnK.

( 0, r)

the ~ollcw1ng lemma which is related 1:1

r,

~errirr.a 1.

If

[

= ].

(',r)

_,_

co~tinuc~s 3nd A-proper w.r.t. to any closed subset

rcst_:-ict

( :,1 ) }C

M

of

clos e

S\;c'i

tl:at

G(

)

➔

g

as

188

Petryshyn

k .-,.

for some

oo

all

g

in

n, Pn(x) .-,. x

that for each

K.

Since

for each and

k

x

in

ck= 1/k

= 1, Pn

Pn(K) CK, IIPnll K, and

G

C

Pn+l

for

is continuous, it follows n(k) EN+

there exists

n(k) > k

with

such that ( 2. 3)

for sufficiently large

and

k

JI Txk-Twn(k)IJ ~ Ek .-,. 0

r1

and our properties of the scheme Pn(k)Gwn(k) .-,. g

k .-,.

as

= {Xn,Pn}

wn(j) .-,. x

x . .-,. x J

as

j .-,.

as

j .-,. and

00

k .-,.

This

00

imply that

whence on account of the A-properness of

00 ,

it follows that there exist a subsequence that

as

x EM

wn(j)

Gx = g.

and

00

since

M

and

x E BK(0,r)

G such

This and (2.3) imply that is closed.

This proves Lemma 1.

We now are in position to deduce from Theorem 2 the following corol-

laries for differentiable maps. Assuming (Cl) of Theorem 2, our first corollary in this section shows how (C2) of Theorem 2 is verified when

T0

+ L (X).

or

T00

is of type

1

Corollary 4.

Let

is A-proper w.r.t.

T:K .-,. K

be a bounded map such that

I-T:K .-,. X

and assume also that:

fl

> 0' N e N+ and a bounded map 0 such that (Cl) of Theorem 2 holds for Dl = BK(0,r). K (al)

There exist

r

Suppose further that either (bl) or (dl) holds, where T(0) = 0, T

(bl) that

(I-T 0 )jK T

T0

has the derivative

is A-proper w.r.t.

r1

and at

has the derivative

oo

T0

at E

along +

is A-proper w.r.t.

TOOELl(X).

Then, in either case, T

has a fixed point in

+

0

along

K

such

L1 (X) K

such that

0

K.

Since (al) holds, to deduce Corollary 4 from Theorem 2, it

Proof.

suffices to show that condition (C2) of Theorem 2 is implies by either (bl) or (dl) for a suitable ball B = D2 . Since the arguments are almost identical in both cases, we prove these implications simultaneously for T

satisfying (bl) and (dl).

First note that (bl) and (dl) can be stated

as: (dB) where

B

iK

=

T(h) = TB(h) + QB(h) 0

or

B

= oo

and

is A-proper and

Now, since

TB

E

1

TB

TB E

with E

+

L(X)

IIOB(h)JI

= o(ilhll) .-,. B

is such that

(h E K)'

TB(K) ~ K,

L 1 (X).

L+(X), there are

AB> 1

and

hB

€

0

K

such that

Positive Eigenvectors and Fixed Points for Maps in Cones

TB(hB) = ABhS. r 00 > r

We claim that we can choose F:BK(o,r 6 )

and define a map

K

+

rS > 0

by

189

with

F 6 (x) ~ hB

r

0

< r

and

for

x e BK(o,r 6 ) such that (C2) of Theorem 2 holds with y = 0 in (H2) and (H3). Indeed since FS:BK(O,rS) + K is compact, it follows that I-T-µF:BK(o,r 6 )

+

X

y = O.

holds for

is A-proper w.r.t.

for each

µ

>O, i.e.,

inf{JIFxll: x e aBK(o,r 6 )} = lih 6 11

6

Since

r1

( H 2)

we

> 0

r 6 > 0 chosen 0. Indeed, if above is such x-Tx ~ AFx for x e aBK(O,rS) and A> this were not the case, then there would exist sequences {rs} n ' see that (H3) holds when

OB} C n

or

CO 00 ) '

and

S = oo, AS > 0 n

{ K6 } C

n

and

Finally, we show that

y = 0.

11

K

x~II

such that

= rS n

rS n

+

for each

S

as

n

with

N

E

+

S = 0

and

It follows from last equality and (d 6 ) that

Since

llzSII n

as

1

is bounded.

llxSII n

+

S, it follows that

Hence we may assume that

as

{zB} and n. J with ns > 0 (I-TS)(z 6 ) = nSh s

rl, it follows from Lemma 1 that there is a subsequence ZS n. J because

ZS e aBK(O,l)

such that

since

ZS

➔

and

Since

lows from the last equality, by induction, that zB-nS(l+AS+A~+ ... +A~)(hS) e K and

for each

t'+

' .

in

n

Since

ns > 0

AS> 1, the last relation is impossible by the remark preceding

Corollary 4.

Thus,

(C2) holds and so Corollary 4 follows from Theorem 2

provided that (al) and either (bl) or (dl) hold.

Q.E.D. Remark 2.

Under the stronger assumption that

compact and the restrictions of

T

0

and

Corollary 4 was first obtained in [24].

+ K is P 1 are also P 1 -compact, For the proof given in [24] to

T00

to

T:K

K

be correct it seems that one should use Lemma 1 instead of Proposition l.lC from [20]. In our next corollary, assuming (C2) of Theorem 2, we show how (Cl) of that theorem is verified when T00 1K

T:K

is also~ -compact and either

+

T0

K or

is P 1 -compact, T 0 \K T00

lies in

L~(X).

or

Petryshyn

190

Corollary 5.

Suppose

T:K

K

+

is a bounded P 1 -compact map such

that: (a2)

There exist

r > 0

and a bounded map

F:BK(O,r)

K

+

such

that condition (C2) of Theorem 2 holds. Suppose further that either (b2) or (d2) holds, where: (b2)

T(O) = 0

such that

T 0 1K

( d 2)

and

T

has the derivative

is P 1 -compact and

has the derivative

T

P1-compact and

T e L~(X). 00

Proof.

T

T 0 e L~(X). T00

at

along

00

Then, in either case,

at

0

0

along

K such that

K

T j K is 00

0

T has a fixed point in K.

To deduce Corollary 5 from Theorem 2, it suffices to show

that (Cl) of Theorem 2 is implied by either (b2) or (d2) for a suitable B = D1 .

ball

As before, the proof that either (b2) or (d2) implies (Cl) will be carried simultaneously. derivative at TS(x) i µx x-TSx i O

S

for for

NS e N+

and

n ~ NS.

Choose

mS/2jxj

for

all

along µ ~ 1

=

0

K.

Sine;

and

x e K.

or

S

=

00

and let

TS

be the

TS e L~(X), it follows that

with

3BK(O,r 8 ).

Now, since

TS\K

II x-P nT S (x)II ~ msll xii

such that

rs> 0 E

t e [O,l], all

ro < r

and

roo > r

for

x e Kn

such that

and

IITx-Tsxll
NS

we have

is P 1 -compact and TB e L~(X), it follows from PnTS(x) i µx for all x € aBK(O,rs), all n ~ NB and

Moreover, since Lemma 1 that -

S

is P 1 -compact and x e aKB(O,l), it follows from Lemma 1 that there exist

ms > 0 x

Let

TslK

+

NB c N . Thus, (Cl) holds with C = TB and so Corollary S follows from Theorem 2 provided that (a2) and either (b2) or (d2) hold. Q.E.D.

some

Remark 3.

When

y = 0

in condition (a2), Corollary 5 was proved

by the author in [24]. An immediate consequence of Corollaries 4 and 5 and Remarks 2 and 3 is the following special case established in [24]. K

is a bounded and ? 1 -compact map

and the derivatives

Too at 0 and at 00 are P 1 -compact. ~ 00 ,K K provided one of the following

Corollary 6. with along

T(O) = 0 K Then

Suppose

are such that T

T:K

➔

To\K and has a fixed point in

T

0

and

Positive Eigenvectors and Fixed Points for Maps in Cones

191

conditions hold: (a)

T0

E

Li(X)

and

Too

E

L~(X)

(b)

T

E

L~(X)

and

T

E

Li (X).

0

0

Corollary 6 contains the corresponding results of [12] when T completely continuous and of [l], [7] and [22] when T is k-ballcontractive with

k


0

(4)

is also satisfied. Our first two results (Theorems 5 and 8) are valid in every Banach space.

On the other hand, our next two results

are valid only in special Banach spaces.

(Theorems 11 and 14)

As a matter of fact, in order

to formulate Theorem 14 we introduce a new geometric property of (infinite-dimensional) Banach spaces.

We conclude with a consequence of

Theorem 8 which involves the fixed point property for nonexpansive mapings (Theorem 15). Our first result will be preceded by several lemmas. Lemma 1.

(y

l

is defined by (2) and xt E D(Jt), then n-1 (1+1/t)nlJtxt-ynj s _II (1+(1-c.)/t) !Jtxt-Yol i=0 l. n-1 • n-1 + (1/t) -~ ci(l+l/t)l.lxt-Tyil -~ (1+(1-ck)/t) i-0 k=i+l for all n .e 1.

Proof.

If

Since

n

(l+t}Jtxt

(1+1/t) n+l IJtxt - Yn+l I

s (1+1/t) (1-cn) (1+1/tlnlJtxt-ynl + cn(l+l/t)n(l/t) (l+t) IJtxt-TYnl s (1+1/t) (1-cn) (1+1/t)n!Jtxt-ynl + cn(l+l/t)n(l/t) (lxt-TYnl+tlJtxt-ynll (1+(1-c n )/t) (1+1/t)nlJtxt-y n I+ c n (1+1/t)n(l/t) lxt-Ty n

I-

The result now follows by induction. Although the next lemma is well known, we include a proof for completeness. (znJ are defined by (2), then n-1 ly 0 -ynl s 2ly 0 -z 0 l + (i~ol lz 0 -Tz 0 1

Lemma 2.

for all

If

(ynJ

n .e 1.

and

Successive Approximations for Nonexpansive Mappings Proof.

We first note that

195

lzn+ 1 -Tzn+ll :s: lzn+l-Tznl + !Tzn-Tzn+ll

,, I (1-c n ) z n +c n Tz n -Tz n I + I z n -z n +l I [lz -Tz IJ is decreasing. Hence n n

I z n -Tz n I,

so that the sequence

jz 0 -zn+ll:s:iz 0 -znl +cnlzn-Tznl,, lz 0 -znl +cnlz 0 -Tz 0 j. Since jyo-Yn+ll $ IYo-zol + lzo-zn+ll + lzn+l-yn+ll +jz 0 -zn+ll' the result now follows by induction.

$

2jyo-zol

The range condition (3) is satisfied if and only if there exists a sequence

[(ti,xt_JJ

such that

J.

ti ➔ °', xt. E D(Jt.l J.

(5)

lxt_l/ti ➔ 0.

and

J.

J.

Our next lemma is also known [8, Proposition 3], but we present a different and simpler proof. main

D(A)

and range

,;; lx 1 -x 2 +r(y 1 -y 2 ) clear that

I

A= I-T

Recall that a subset

R(A)

for all

A

of

is said to be accretive if [xi,yi] EA,

i = 1,2,

is accretive whenever

T

and

X

x

X

with do-

lx 1 -x 2 1

r > 0.

is nonexpansive.

It is

The

analog of condition (3) is therefore lim inf d(0,R(I+tA))/t = 0.

(6)

t ➔"'

Lemma 3.

Let

X

be a Banach space,

that satisfies (6), and

( (ti,xt_JJ

A c X xX

an accreti ve opera tor

a sequence satisfying (5).

Then

J.

Proof.

Let

d= d(0,R(A)).

Since

(xt-Jtxt)/t

belongs to

is clear that

0 < s < t

On the other hand, for

and

[x,y] EA

jJs(x+s~-Js( (s/t)xt+(l-(s/t) )Jtxt) ,;; lx+sy-{(s/t)xt+(l-(s/t))Jtxt)

we have

I

I

,. (1-(s/t)) jx-Jtxtl + (s/t) lx-xtl + sly!. Hence and lim sup !Jt x i ➔~

Therefore

i ti

l/t. ,, IYI· J.

li~ sup jJt xt l/t. ,;; d i ➔ ro

i

i

J.

and the result follows.

Finally, we note the following simple fact.

R(A),

it

Reich and Shafrir

196

Lemma 4.

For each

n

~

1,

lim t(l-

t➔m

n-1

n-1

I1 (1-c./(l+t)))

i=0

L

i=0

1

C .•

1

We are now in a position to establish our first result. Theorem 5. T: D

X

➔

(ynJ

Let

D

be a closed subset of a Banach space

a nonexpansive mapping which satisfies (3).

X

and

If the sequence

is defined by (2), then n-1 lim IY I/( L c.) = d(0,R(I-T)). n➔ m n i=0 1 n-1

L'. c. by a n and d ( 0 , R ( I -T) ) by d. On the one i=0 1 hand, Lemma 2 shows that lim sup iY I/a :!. d. On the other hand, n n

Proof.

Denote

Lemma 1 shows that

IY 0 -ynl

n➔ °'

IYo-Jtxtl - !Jtxt-ynl ~ n-1 n-1 n-1 (1- I1 (l-c./(l+t)l)!Jtxt-Yol- (1/(l+t)) L c.lxt-Ty.l I1 (1-ck/(l+t)) 1 1 k=i+l i=0 i=0 1

for all that

n

!y 0 -y

~

l

I~ n

~

and

xt E D(Jt).

ad n

for all

Therefore Lemmas 3 and 4 now imply

n ~ 1,

and the result follows.

In order to establish our second result we need two more lemmas. Lemma 6. If {en} is bounded away from defined by (2), then for all k ~ 1,

Proof.

Fix

{!y n -Ty n IJ

k

~

1

0

and

1,

and

{yn}

is

and denote the limit of the decreasing sequence

by

L. Since n+k-1 L'. (yJ.+1-yJ.l Yn+k - Yn = j=n

it is clear that

lim sup IYn+k-yn I/ ( n➔ m

n+k-1

L

j=n

C.) J

n+k-1

L

j=n

:!.

c. (Ty. -y.) , J J J

L.

Now we note that by [4, Proposition l] IYn+k-yn! ~ ITYn+k-Tyn!

~ I Ty n+ k -yn I - ITyn -yn I ~

n+k-1

n+k-1 1 (1-cJ.)- (ITYn+k-Yn+kl-!Tyn-yn!)+( _L CJ.) jTyn-yn'· J=n J=n

.rr

Assuming, as we do, that the sequence and l we conclude that n+k-1 lim inf IYn+k-yn[/( _L cj) ~ L. n➔ °' J=n

{c

n

J

is bounded away from

0

Successive Approximations for Nonexpansive Mappings

197

Hence the result. Lemma 7.

[c n J

If

is bounded away from

O

and

1,

and

defined by (2), then the limit of the decreasing sequence is independent of the initial point Proof.

Let

limits of tively.

{z J

y0 •

be another sequence defined by (2), and denote the

n

{!y -Ty IJ and (lz -Tz IJ n n n n We first note that

I !Yn+k-yn!

Since the sequence

s,

(en}

n.

L(z 0 )

and

respec-

n+k-1

( L

j=n

C . ) -

J

1.

0,

is bounded away from

find an integer

I I y + k -y I I n n for all

L(y 0 )

- lzn+k-znl I "" 1Yn+k- 2 n+kl + [yn-znl "" 2 lyn-znl "- 2!y 0 -z 0

positive

by

Iz

k

we can, given a

such that

n+k-1 + k -z I/ ( C . ) I < s n n j=n J

z:

By Lemma 6 we can now find, for this

k,

an integer

N

such that

and

n+k-1 I lz +k-z I/( Z: c.) -L(z 0 ) I< n n j=n J

for all .:

n ~ N.

!L(y 0 )-L(z 0 ) I< 3s,

Thus

is arbitrary.

Theorem 8. T: D

➔

X

Let

i.

L(y 0 )

and

be a closed subset of a Banach space

D

a nonexpansive mapping which satisfies

bounded away from

0

(2) , then for all

k

lim [y +l-y l/c n n n

n ➔ 0>

L(z 0 ) because

and ~

1,

and the sequence

1,

~~:

( 3) •

(yn}

X

and

If {c J is n is defined by

n+k-1 !Yn+k-yn[/( j~n cj)

n-1 lim IY [/( Z: c.) = d(O,R(I-T)). n➔.,, n j=O J Proof.

Since

lim IYn+l-ynj/cn

is independent of the initial point

n ➔ ro

y0

by Lemma 7, it must equal

d(O,R(I-T)).

The result now follows by

combining Theorem 5 with Lemma 6.

We do not know if this result remains valid

when condition (3) is not

assumed.

{c

It is indeed true if the sequence

Theorem 2.1]. (3)

n

J

is constant [1,

In this case the limit of Theorem 5 also exists even if

is not satisfied [8, Lemma 1], but it is not always equal to

d ( 0, R ( I-T) ) .

198

Reich and Shafrir

We now turn our attention to special Banach spaces.

We begin with

another lemma. Lemma 9.

Let

T: D

a nonexpansive mapping which satisfies (3).

by

➔

an

X

and

D

be a closed subset of a Banach space

d(0,R(I-T))

by

d.

If the sequence

Ex*

(2), then there is a functional z 2 ((y 0 -yn)/an,z) ;;,: d for all n.

Proof.

with

let zt belong to { (ti,xt_l} satisfy (5). Let

For

of a subnet of (yO-yn,zt)

J.

lzl = d

such that

J((y 0 -Jtxt)/t),

xt E D(Jt)

the sequence

X and n-l Denote L c. i=0 1 {yn} is defined by

z

and let

be a weak-star limit

Since

{zt_J. J.

(yO-Jtxt,zt) + (Jtxt-yn,zt) ;;,: lztl(lyo-Jtxtl- !Jtxt-yn!l,

we can use Lemmas 1, 3 and 4 to conclude that for all

n.

This inequality and Theorem 5 now show that clear that

lzl ~ d,

lzl ~ d.

Since it is

the proof is complete.

Recall that the norm of a Banach space ferentiable if for each

x

lim ( I x+ty I - Ix I ) / t t""0 is attained uniformly for

X

is said to be Frechet dif-

in its unit sphere

U = fx EX: Ix! = 1}, (7)

y

in

U.

We shall then write that

X

is

(F) •

Our next lemma is known (cf. Lemma 10.

x*

[2]).

is (F) i f and only i f every sequence

which there exists

w E

x*

with

lwl = 1

such that

{x n J

C

X

for

lim lxnl n"""'

lim (xn,w) converges. n"""' Theorem 11. Let D be a closed subset of a Banach space X and T: D _, X a nonexpansive mapping which satisfies ( 3) • Let the sen-1 quence {ynJ be defined by (2) and denote 1: c. by a n If x* i=0 J. has a Frechet differentiable norm, then v = strong limy /a exists, n➔"' n n and -v is the unique point of least norm in cl (R (I-T)). Proof.

Let the sequence

f (ti,xt_lJ J.

the functional obtained in Lemma 9. u = strong

lim Jt.xt_/ti i """' J. J.

satisfy (5), and let

z

Ex*

It is known [8, Theorem 3] that

exists, and that

-u

is the unique point

be

Successive Approximations for Nonexpansive Mappings

199

of least norm in cl(R(I-T)). Moreover, the proof of [8, Theorem 3] shows that z E J(-u). Combining Theorem 5, Lemma 9 and Lemma 10 we see that Since u

=

v

X

v.

strong

limy /a also exists, and that z E J(-v). n n is certainly strictly convex, [8, Lemma 5] now shows that n ➔"'

This completes the proof.

exists when X is lim y /a n n assumed to be only reflexive and strictly convex. These results improve upon Theorem 1 and Corollary 2 of [3] because we show that the weak range condition (3) is sufficient for the conclusions to hold.

A similar argument shows that the weak

n➔°'

(This condition, introduced in [8], is, in fact, also necessary.) Since X is uniformly convex if and only if the norm of x* is uniformly Frechct differentiable, they also improve upon previous results of the first author [11, 12]. We continue with a convergence result in the setting of Theorem 8.

To

this end, we introduce a new geometric property of (infinite-dimensional) Banach spaces. Recall that to each functional w in the unit sphere of x* there corresponds a face F of the unit sphere U of X,

namely all those

x EU

the norm of a Banach space

tiable (LUF) if for each face formly for all y in U and

x*

whenever

(x,w) = 1.

for which X

We shall say that

is locally uniformly F~echct differenF x

of in

is uniformly convex or

U the limit (7) is attained uniF. It is clear that X is (LUF) X

is (F) with compact faces.

In

analogy with Lemma 10, we also have the following characterization. Lemma 12.

x*

is (LUF) if and only if every sequence [x J c X for n lxl = 1 and a sequence [w} c J(x) n converges.

which there exists x EX with such that limlx I = lim(x ,w) n ➔~

n

n ➔ oo

n

n

We also note in passing that if convex, then x* is (LUF). Lemma 13.

If

x*

X

is reflexive and locally uniformly

is (LUF), then every accretive operator

that satisfies (6) has the following property: in

R(A)

Proof.

for which Denote

least norm in

limlb

n➔ °'

d(0,R(A)) cl(R(A)).

n

I =

d(0,R(A))

by

d

and let

converges. -u

A c Xx X

every sequence

[b n J

be the unique point of

The proof of [8, Theorem 4] and Lemma 3 show

that to each b there corresponds a point z in J(-u) such that n n lim (b , z ) = d2, (bn,zn) :.,; d2. Since lim i b I = d, we have, in fact, 11 .. °' n n n➔ °' n and the conclusion follows by Lemma 12. This lemma improves upon Lemma 3.2 in [11] where

X

was assumed to be

200

Reich and Shafrir

uniformly convex and

A

is assumed to satisfy the stronger range con-

dition corresponding to (4).

It can be shown that the converse of

Lemma 13 is also valid. We can now establish our convergence result.

Such a result has been

known so far only in uniformly convex spaces [11, Theorem 3.7(c)].

It

sharpens the conclusion of Theorem 11. Theorem 14. T: D

➔

X

Let

D

be a closed subset of a Banach space

a nonexpansive mapping which satisfies (3).

X

and

Assume that

(en} is bounded away from O and 1, and let the sequence [yn} be defined by (2). If x* is (LUF), then the strong lim(y -Ty) exists n➔ m n n and coincides with the unique point of least norm in cl(R(I-T)). Proof.

[y n -Ty n JC R(I-T)

Since

limly n -Ty n I

and

limly +l-y l/c n n n by Theorem 8, the result follows by Lemma 13.

= d(O,R(I-T))

x*

It can be shown that if 14 is no longer valid.

n➔m

n➔m

is not (F), then the conclusion of Theorem

We do not know, however, if this differentia-

bility condition is sufficient for Theorem 14 to hold. We shall say that the norm of a Banach space

X

with a unit sphere

is locally uniformly Gateaux differentiable (LUG) if for each and each face x

in

F.

F

of

U

y Eu

the limit (7) is attained uniformly for all

x*

It can be shown that if, in the setting of Theorem 14,

is (LUG), then the weak

U

lim(y -Ty) exists. We expect to present a n n more complete discussion of the differentiability properties (LUF) and n➔ m

(LUG), as well as their consequences, elsewhere. We t~rn now to another consequence of Theorem 8. Recall that a closed convex subset C of a Banach space has the fixed point property for nonexpansive mappings (FPP for short) if every nonexpansive

T: C

➔

C

has a fixed point [6, 7, 9]. Theorem 15. and

T: C

away from

Let C

➔

O

C

be a closed convex subset of a Banach space

a nonexpansive mapping. and

1,

Assume that

If each bounded closed convex subset of fixed point free if and only if Proof.

[yn]

and let the sequence limly

n➔ m

n

I

X

X

[c] is bounded n be defined by (2).

has the FPP, then

T

is

m

(yn} of (yn} is bounded, we k let R = lim sup!y 0 -y I and B = [x EC: limk➔ ~uplx-ynkl $ R}. It is k➔ m nk ,_ clear that B is a nonempty, bounded closed convex subset of C. Since

Assuming that a subsequence

lim(y

n➔ m

nk

-Ty

nk

)

=

0

by Theorem 8, it is also invariant under

T.

Successive Approximations for Nonexpansive Mappings

201

Hence the result. It can also be shown that if, in the setting of Theorem 15, uniformly convex and (F), and lim yn

n-->co

T

X

is

has a fixed point, then the weak

exists and is a fixed point of

T.

In this case, however, a

better result is already known [10, Theorem 2]. Finally, we note that Theorems 5, 8 and 15 can be shown to carry over to these self-mappings of the Hilbert ball which are nonexpansive with resepct to the hyperbolic metric [5].

These results extend several

theorems in [13] and provide an affirmative answer to the question raised at the end of that paper.

It is expected that a complete dis-

cussion, as well as related results on implicit iterations, will be presented elsewhere. REFERENCES 1.

J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. 4(1978), 1-9.

2.

K. Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25(1958), 553-568.

3.

T. Fujihira, Asymptotic behavior of nonexpansive mappings and some geometric properties in Banach spaces, Tokyo J. Math. 7(1984), 119-128.

4.

K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Contemporary Math. 21(1983), 115-123.

5.

K. Goebel and S. Reich, "Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings," Marcel Dekker, New York and Basel, 1984.

6.

W. A. Kirk, Fixed point theory for nonexpansive mappings, Lecture Notes in Math., Vol. 886, Springer, Berlin and New York, 1981, pp. 484-505.

7.

W. A. Kirk, Fixed point theory for nonexpansive mappings II, Contemporary Math. 18(1983), 121-140.

8.

A. T. Plant and S. Reich, The asymptotics of nonexpansive iterations, J. Functional Anal. 54(1983), 308-319. s. Reich, The fixed point property for nonexpansive mappings, I, II, Arner. Math. Monthly 83(1976), 266-268; 87(1980), 292-294.

9. 10.

S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67(1979), 274-276.

11.

S. Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, J. Math. Anal. Appl. 79(1981), 113-126.

12.

S. Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators II, Mathematics Research Center Report #2198, 1981; J. Math. Anal. Appl. 87(1982), 134-146.

13.

S. Reich, Averaged mappings in the Hilbert ball, J. Math. Anal. Appl. 109(1985), 199-206.

Quasilinear Ellipticity on the N-Torus VICTOR L. SHAPIRO, Department of Mathematics and Computer Science, University of California, Riverside, Riverside, California

1.

Introduction.

In this paper, we intend to establish one new result and

extend two previous results [3, Thins. 1 & 2], for second order quasilinear elliptic partial differential equations defined on the N-torus.

The

main point of the extension is that now the leading order coefficients

aii(x,u) are allowed to have also a linear growth condition with respect to u, i.e., laii(x, u)I ~ a(x) + 11lul where a(x) E L 2 (0) and f7 is a positive constant. Here, O={x:-1r 0 such that lii(x,z)l::;a(x)+11lzl for every zER and a.e. xEO, i,i=l, ... ,N. (Q-3) Q is symmetric; that is, aii(x,z) = aii(x,z) for every z ER and a.e. x E 0, i,J· = 1, ... , N. 203

204

Shapiro

(Q-4) Q is uniformly elliptic almost everywhere in O; that is, there is a constant r,0 > 0 such that

e

et>.

for every z ER, a.e. x E O, and every E RN Clel 2 =er+···+ (Q-6) There is a nonnegat,i.ve :function b(x) E L 2 (0) and a positive constants tJ1 and tJ2 such that

:for every pERN, zER, and a.e. zEO, j=l, ... ,N. (Q-6) For every u E W 1 •2 (0), the vector b(z,u,Du) = [b1(z,u,Du),

... ,bN{z,u,Du)] is weakly solenoidal, i.e.,

f0

bi(x,u,Du)D;v(x)dx = 0 :for

every u and v E W 1•2 (0) where the summation convention is used.

(Q-7) If {un}~=l is a sequence of functions in L2 (0) which tend strongly

to u E L2 {0) and {wn}~=l is a sequence of vector-valued :functions which

tend weakly tow E [L2 {0)]N, then {b(z,un,wn)}~=l tends weakly to b(z,u,w)

E [L 2 {0)]N, i.e., with bi(z,u,w) b{z,u,w), then lim j=l, ... ,N.

n--oo

f0

= bi(x,u,w1, .. ,,wn) f0

bi(x,un,wn)vdx =

the j-th component of

bi(x,u,w)vdx V v E L 2 {0) and

By strong convergence in (Q-7), we mean convergence in norm. In this paper, we intend to establish the following new result.

Assume (Q-1) - (Q-7) and let sufficient condition that the equation THEOREM 1.

Qu=

f

E L 2 (0). Then a neceuary and

f

{1.2)

has a distribution solution u E W 1 •2 ( 0) is that fO f dx = 0. If Qu = -~u where ~ is the Laplacian, the above theorem is well known. It is also well known if bi = 0, j = 1, ... , N and aii does not depend on u and is smooth as a :function of z.

(See [2, p. 674] .)

However

in the case considered here where the aii(x,u) are unbounded in:,; and possibly grow linearly in u, the result obtained in Theorem 1 is new. To be quite explicit, what we mean by u E W 1 •2 (0) being a distribution solution of Qu = f is the following:

(1.3)

205

Quasilinear Ellipticity on the N-Torus

for every ¢> E 0 00 (0), where the summation convention is used for i,j

l, ... ,N. 2.

We first dea.l with the sufficiency condition of

Proof of Theorem 1.

the theorem a.nd establish this pa.rt via. a. Ga.lerkin argument. to accomplish this, we observe tha.t there is a. sequence va.lued functions in 0 (a.) (b)

00

In order

{1Pk}k=l

of rea.1-

(0) with the following properties:

= (2,r)-N/2; J0 1Pk1Ptdx = Dkt

1/11

Also, given

,p E 0

00

where Dkt is the Kronecker-5 k, l

= l, 2, ....

(2.1)

(0) a.nd t > 0, :3 constants c1, ... , Cn such tha.t

lt/J(x) - Cq,Pq(x)I < t uniformly for x E O a.nd j in (2. 2) with q

=

= l, ... , n.

ID;,f;(x) - cqDjt/,q(x)I < t

a.nd

= 1, ... , N

(2.2)

where the summation convention is used

Here, a.nd in the sequel throughout the rest of the pa.per, we sha.11 use the notation

=

(v, w)o for v a.nd w in L 2 (0).

lo

v(x)w(x)dx

In ca.se v and w a.re a.lso in W 112 (0), we sha.11 set

(v, w)i

= (v, w)o + (D;v, D;w)

where the summation convention is used for j set

(2.3)

llvllo = (v, v)g'

and

=

1, ... , N.

(2.4) Also we sha.11

llvlli = (v, v)f.

(2.5)

H~ sha.11 designate the Hilbert space spanned by {1Pk}r'= 2 using the Thus w E H~ means :l{wn}~=l such tha.t llw - wnll1 -+ 0 a.s

W 112 (0)-norm.

n-+ oo where ea.ch Wn is a. finite linear combination of elements of the

sequence

,p2,,f;3, ... ,,pk,··· .

lows tha.t v E

W 112 (0)

As a. consequence of (2.1) and (2.2), it fol-

ca.n be written uniquely in the form V

= C11/}1 + W.

where w EH~ a.nd c1 is a. constant.

(2.6)

Also, it is clear tha.t

(2.7)

206

Shapiro

Furthermore, it is easy to see from (2.1)(a), (2.4), (2.6), and (2.7) and elementary Fourier analysis that

llwll~ $

(Diw, Diw}o

(2.8)

V wEH~

where the summation convention is used for j = 1, ... , N. The first lemma we state is the following. LEMMA 1.

Let f( x) be the function in L 2 ( 0) given in the statement of Theorem

1, and assume that Q satisfies (Q-1) - (Q-6). ~

2, there is a function un

= 72'¢;2 + · · · + "f~"Pn

Then if n is a given positive integer such that

(ii(·, un)Diun, Di'I/Jk}o + (bi(·, Un, Dun)Diun, "Pk}o + (un, "Pk}on- 1 = (1/Jk,J)o

(2.9)

fork= 2, .. . ,n where the summation convention is used for i,j = 1, ... ,N and ,; are constants q = 2, ... , n. The proof of this lemma is essentially the same as the proof given in [3, Lemma 1], and the reader should have no difficulty in filling in the details. Next for n fixed and~ 2, we multiply both sides of (2.9) by 'k for k = 2, ... , n and sum on k. Observing that Di(un) 2 = 2un D;un and using (Q-6), we see that (aii(.,un)D;un,Diun)o

+ (un,un)on-1 = (un,/}o.

We conclude from (Q-4) and this last fact that

(2.10) where the summation convention is used for j = l, ... ,N. Since un is in 1. H-, it follows from (2.8) that llunllo$(D;un,D;un)g. We therefore obtain from (2.4), (2.6), and (2.10) that for n = 2,3, ....

{2.11)

From this last fact, it follows that there is a subsequence of {un}:= 2 • which for ease of notation we take to be the. full sequence, with the following properties:

3u E H- such that lim un(x) = u(x)

n-+oo

lim

n- 00

[ wD;undx

lo

= [

lo

lim llun - ullo = O;

(2.12)

for a. e. x E O;

(2.13)

n-+oo

wD;udx V w E L 2 (0),j

= 1, ... , N.

(2.14)

Quasilinear Ellipticity on the N-Torus

207

Using (2.11) - (2.14), in conjunction with (Q-6) - (Q-7), it follows exactly as in the proof [3, Thm. 1] that

fork= 2,3, .... Next, we fix i and j and set (2.16) and observe from (2.13) and (Q-1) that for a.e. zEO.

(2.17)

With a(z) and~ as in (Q-2), we set On(z) = [2a(z} + ~lunl + ~(u}] 2 and observe from (2.13) that lim On(z) = 4[a(z) +~u(z)]2 a.e. inn. n-oo we have that for a.e. zEO and n=2,3, ...

(2.18} Also

(2.19)

Since a(z) E L 2 (0}, it follows from (2 .12) that lim

n - 00

f

}0

On(z)dz = 4

2 Jof [a(z) + ~juj dz.

(2.20)

We invoke [2, Thm. 16, p. 89] with hn replacing fn and conclude from (2.16) - (2.20) that lim J0 hn(z)dz = 0. Now i and j were fixed but arbitrary; n-oo so we record this last fact, using (2.16), as follows: (2.21) for i,j= 1, ... ,N. But then using (2.21) and proceeding exactly as in the proof of [3, Thm. 1], we see that (2.22) fork= 2,3, ...

208

Shapiro

Next, we see from (2.11) that lim {un,ifak)on- 1 = 0 fork= 2,3, .... n-+oo We consequently obtain from (2.9) on passing to the limit as n--+ oo and using (2.16) and (2.22) that

(ii(·, u)D;u, D;ifak)o + {b'°(-, u, Du)D;u, 1Pk)o = {ifak,J)o

(2.23)

for k = 2, 3,... . Now using (Q-6) and (2 .1) (a), we see that the lefthand side of (2.23) is zero when k = 1. Likewise from the hypothesis of the theorem, we see the right-hand side of (2.23) is zero when k = 1. We conclude that (2.23) also holds fork= 1. But then it follows from (2.23) and the uniform convergence aspect of (2.2) that

for all JR 1

Suppose that

of Banach spaces.

Ex F

is a

c 1 -function de-

of two closed convex sets

We say that

f

E

and

satisfies the twist Palais Smale

condition (T.P.S), if { (xn,yn)} c Ex F,

any sequence (T.P.S.)

= Sup f(x ,y)

f(x ,y) n n

yEF

n

is bounded and

n

inf ( f' (x , y ) , -11 h II) hEE-x x n n h n

along which

-;>

a. .: O,

possesses a

convergent subsequence. Remark 1.1. F

F

is a compact set, then any function

satisfies (G.P.S.).

tion

f

defined on

Remark 1.2. g

If

If

F

If

Ex

E

F,

and

F

satisfies (G.P.S.).

If

defined on

are compact sets, then any func-

satisfies (T.P.S.).

is a Banach space, and if

satisfies P.S. over

g

E

Ex F,

and then

F f

g

satisfies P.S., then

are Banach spaces, and if

f

satisfies (T.P.S.).

The main result of this paper is the following Theorem A. let

f E c1 (1)

Let

E, F

be two closed convex sets of Banach spaces, and

(EX F,lR 1 )

V XE E,

be a function satisfying

y I--> -f(x,y)

is bounded below and quasi-convex,

and satisfies (G.P.S.), (2) (3)

3 y0 E F

Then there exists (1)

(2)

such that

xi--> f(x,y 0 )

(T.P.S.) holds for (x,y)

is bounded below,

f.

E E

X

F

such that

min max f (x,y), = max t(x,y) xEE yEF yEF V h E E-x. (f~ (x,y) ,h) ~ 0

f (x,y)

Moreover, if

X 1------;l>

f (x,y)

there is a neighborhood

is locally pseudo-convex at

U(x)

of

x

such that

x,

i.e.,

A Local Minimax Theorem Without Compactness Y x E U (x) ~ f (x,y) ;:: f (x,y)

(f' (x,y) ,x-x) ;:: 0 X

then

(x,y)

213

is a saddle point of

f,

V x E U (x)

i.e.,

f (x,y) :,; f (x,y) :,; f (x,y) V x E U(x),

Vy E F.

Remark 1.3.

It is well known that the crucial point in proving von

Neumann-Sion Minimax Theorem, is to prove the equality: min max f(x,y) = max min f(x,y) xEE yEF yEF xEE which depends heavily on the convexity hypothesis of the functions

Vy E F.

x I--> f(x,y) point

of

view

of

However, our approach, which starts from the

critical point

theory,

concerns merely

the

local

behavior near the "saddle point," so that the convexity hypothesis is not necessary. The proof of Theorem A is based upon the following two fundamental principles: (Ekeland's Variational Princiele [ 7]) • Let E be a complete metric Suppose G:E -a> JRl u [ +a, J be 1. s. c. with G(x) F +"' + that XO E E and € > 0 are given such that

space, and let

G(x 0 ) :,; inf G(x) + €. xEE Then

V k > O,

there exists

x€k EE

such that

G(x 8 k) :;;G(x 0 ), 1 d(x0,x€k):,; k,

G(x) > G(x€k) - ked(x,x~k) where

V x /

xek'

m. 1

is the metric of E. + This principle is very useful in many problems. d : E X E -,:,

It has been appli-

ed to give a new proof of the Mountain Pass Lemma (cf. [2] and [12]), and will be applied time to time in this paper. (Loe-sided Minimax Theorem).

be two convex sets of Hausdorff linear topological spaces, and let f: Ex F -,:, :JR 1 . Suppose that (1)

Let

E, F

V x E E,y I--> f(x,y)

is u.s.c. and quasi-concave,

(2)

Y y E F,x I--> f(x,y)

is 1.s.c. and quasi-convex,

(3)

3 xC E E

f(x 0 ,y) :::

A]

and

is compact.

Then there exists

y E F

A. < inf sup f (x,y)

xEE yEF

such that

such that the set

(y E F

I

214

Shi and Chang

inf sup f(x,y) = sup inf f(x,y). xEE yEF yEF xEE

inf f(x,y) xEE

The reader is referred to [2], [3] and [4]. The paper is organized as follows: given in §2.

The proof of Theorem A will be

Some extensions, which need weaker smooth conditions on

the function will be discussed in §3, in which the locally Lipschitzian f are used to replace the c 1 -condition.

condition for the function

§4 deals with applications, in which a variational inequality problem, an elliptic system and an existence theorem of nonlinear programming are studied.

2.

Proof of Theorem A. The proof is divided into several steps.

In order to make it easy

to understand, we start with a slightly abstract version. Theorem 2.1. and let

Let E, F be two closed convex sets of Banach spaces, f E c1 (Ex F, JR 1 ). Suppose that

(1)

V x E E,y I--> f(x,y)

(2)

3 y 0 E F,

(3) (4)

is bounded above and quasi-concave,

such that

(T.P.S.) holds for

V x EE,

x

f(x,y 0 )

is bounded below,

the set

M(x) = {y E F

J

is compact and nonempty, (5) V x EE, V h E E-x, there exist y n E M(x+tnh) , mulate point y E M(x). Then there exists a pair (1)

f (x,y)

(2)

(f' (x,y) ,h)

=

1-->

f, f(x,y)

sup f (x,y)] yEF

V positive sequence

n = 1, 2, . . .

such that

(x,y) E E X F such that = min max f(x,y), xEE yEF v h E E-x. 0

max f(x,y) yEF

X

Proof of Theorem 2.1.

;::

Let G(x)

sup f(x,y). yEF

It is easily seen from the assumptions (1) and (2) that is l.s.c. and bounded below. Principle,

v

€

with tn I 0, has an accu-

> 0,

According to the Ekeland's Variational

3 x 8 EE G(xe)

G: E -~ m. 1

such that ~

inf G(x) + e, xEE

(2 .1)

215

A Local Minimax Theorem Without Compactness For each

h E E-xe,

we have G(x +th)-G(x) e t e .: -eilhll-

G+I (x ;h) ~ lim e l-->+O Since for any positive sequence

[tn]

with

tn" 0,

( 2. 2)

we have

~ lim tl [G(x e +t n h)-G(x e )], n-->"" n

G+' (x ;h) e

~ 1 im tl [ f ( x e + t n h, y n ) - f ( x e , y n ) ] n-->"" n lim (f' (x +t e h,y ) ,h) x e n n n for some

en E (0,1),

provided by the assumption (4), with

yn E

M(xe +tnh). According to the assumption (5), we may so choose {y] 1 . 1: that [y ] has an accumulate point yn E M(xe). The C -continuity of n f implies that ( 2. 3)

Combining (2.2) with (2.3), it follows 'th E E-x .

(2. 4)

€

P.: : (E-xe) x M (xe) -;;:,. lR 1

Let us define a function

by

(f' (x ,y) ,h) + ellhll x e

Pe (h,y) We shall verify that (1) (2) ( 3)

V y E M(xe), h V h E E-x e, y

is continuous and convex,

1--->

Pe (h,y)

is continuous and quasi-concave,

M(xe)

y 1--7 f(x,y).

function

P .: (h,y)

is compact and convex.

M(xe)

The convexity of tion

1--->

It remains to prove the quasi-concavity of the

y 1--> Pe (h,y);

In fact,

follows from the quasi-concavity of the funcbecause all other items are easily seen.

V y 1 ,y 2 E M(xe),

and

Vt.. E [O,l],

(f~(xe, (1-t..)y 1 +t..y 2 ) ,h) 1 lim t[f(xs:+th, (1-t..)y 1 +t..y 2 )-f(xe' (l-t..)y 1 +t..y 2 )]

uo

.: lim min[½[f(xe+th,y 1 )-G(x€)], t[f(xe+th,y 2 )-G(xs:)]]

uo

min[ (f~ (xs: ,y 1 ) ,h), (f~ (x,y 2 ) ,h)], i.e., the function Now

Pe

y

t-->

PE:(h,y)

is quasi-concave.

satisfies all conditions of the Lop-sided Minimax Theorem,

we conclude that

j

ys: E M(xs:)

such that

216

Shi and Chang inf hEE-x

P E:

E:

(h, y ) = inf E: hEE-x

sup ( (f' (x ,y) ,h) + E: ilhl!} ;;,: 0 yEM(x) x E: E:

E:

provided by (2,4), that is, V h E E-x . E:

( 2. 5)

By definition,

(2.1) reads as inf sup f(x,y) xEE yEF

infG(x) :.G(xE:) xEE

=

f(xE:,yE:) s;

Let

{E:n]

Ex F,

inf sup f(x,y) xEE yEF

(2. 6)

E: j O. Then, the sequence n possesses a convergent subsequence (xE:.'YE:.l -» (x,y)

n

provided by (T.P.S.).

(f' (x,y) ,h) ;;,: 0 X

(x,y)

(2. 5).

J

E

min max f(x,y) xEE yEF

yEF

by (2.6); and

J

Thus we have

= max f(x,y)

f(x,y)

by

+ E:.

be a positive sequence with

l (XE: , y E: ) } n

sup f(xE:,y) yEF

V h E E-x

is the solution of our theorem.

Now we turn to the proof of Theorem A.

It remains to prove that

(G.P.S.) of the functions y I----> -f(x,y)

(4) and (5) of theorem 2.1.

implies the assumptions Lemma 2.2. g E c 1 (E,JR 1 )

Let

E

be a closed convex set of a Banach space, and let

be a function bounded below and satisfying (G.P.S.).

Then the set M

=

rx

E E

I g {x)

is nonempty and compact. If

OM

V x EE

is a neighborhood of

M,

inf g (x)] xEE with boundary

oOM,

g(x) > inf g(x). inf xEE xEEfloOM Proof.

(2. 7)

The first assertion follows directly from the Ekeland's Varia-

tional Principle.

In fact,

VE:> 0 3 x

g(x) 8

and

then

s;

E:

EE

inf g(x) + E: xEE

such that

A Local Minimax Theorem Without Compactness

i.e.

217

I

Inf (g' (x ) ,____b__) hEE-x e I hll

:?.

-e.

e

[x

We obtain a convergent subsequence Therefore

e.

} -a> x,

provided by (G.P.S.).

l

x EM.

Again, by (G.P.S.), the set

M

is compact.

Denote (2. 8)

which is positive because

M

is compact.

We shall prove the second assertion by contradiction. were

[xn} c

En

oOM

If there

such that g(xn) -a> inf g(x), xEE

then we would have

z

n

EE

such that

g(zn) s g(x ) < inf g(x) + e, n xEE

for all

e > g (xn) - inf g (x) xEE

:?.

0,

and which imply

Again, by (G.P.S.), -,, x.

Therefore

[ z

n

}

possesses a convergent subsequence g (x)

i.e.,

xEM,

but

. nJ

inf g (x) , xEM

dist(x,oOM) s llx-z s

z

½+

-11

nJ

+ llz

.-x -II nJ nJ

llx-znjll _,,

½,

which contradicts with (2.8). Remark 2.1.

In lemma 2.2, the same conclusion holds true, if the con-

dition (G.P.S.) is weakened as follows: any sequence (G.P .S.) *

[x] n

CE,

along which

inf (g' (x ) , II hhll) hEE-x n n a convergent subsequence.

and

g(xn) -a> inf g(x) xEE possesses

It is interesting to note that (G.P.S.)* is also a necessary condition for the conclusion.

In fact, if

[x] n

c E

is a sequence such that

218

Shi and Chang g(xn} -;;,, inf g(x) xEE

h

and

inf (g' (xn), -llhll) -;;,, i:l hEE-x n

Then we have

;;e

0.

dist(xn,M) -;;,, 0 provided by (2. 7). we have

[x~]

x~ E M

II xn -x~II
0

such that (2. 9)

y 0 E M(x 0 ) arbitrarily, and connecting y 0 by a segment, the segment must intersect with the set oO, Choosing

3 An E (0,1)

with i.e.,

such that

=

zn

(1-An)y 0 + AnYn E oO,

n = 1,2, • · ·.

On one hand, by lemma 2.2, we have sup f(x 0 ,zn) n

sup f(x 0 ,y)

~

oO

< sup f(x 0 ,y) yEF

(2

.10)

On the other hand, f(x ,z) n n

2:

min[f(x ,y ),f(x ,y 0 )J = f(xn,y 0 ), n n n

provided by the quasi-convexity of the function For large

y

I---?>

-f(x,y).

n,

We obtain, lim f(x 0 ,zn)

n-tQ;I

2:

lim f(xn,Yol

n-tro

which contradicts with (2.10). Proof of Theorem A. We mentioned before that we only need to verify the assumptions (4) and (5) of theorem 2.1. Actually, assumption (4) was verified in lemma 2.4. According to lemma 2.4, we know that

V h E E-x,

V positive sequence

choose arbitrarily point

y

E M(x),

Theorem A

yn E M(xn).

x t--> M(x)

is u.s.c.

Now,

{tnJ

with

Then

{ynJ possesses an accumulation

provided by lemma 2.3.

tn I 0,

xn

x+tnh,

we

Assumption (5) is verified.

follows directly from theorem 2.1.

220 3.

Shi and Chang Extensions and Remarks. It is our purpose of this section to weaken the smoothness condi-

tion of the function Theorem A.

f

and the convexity condition of the set

E

of

In addition, we shall discuss how important the convexity

of the function

x I--> f(x,y)

plays a role in von Neumann-Sion-Ky Fan

Theorem. (I) Locally Lip. function. Carefully analyzing the proof of Theorem A, the the function

f

c 1 -condition of

was used in the following points:

(1)

A formulation of the P.S. condition.

(2)

The inequality (2.3), where a mean value property and the

. . C1 -continuity are use d .

(3)

A locally Lipschitzian property in (2.9).

These inspire us to generalize our results to loc. Lip. functions. Let us recall the Clarke directional derivative for Loe. lip. functions [6]. (x 0 ,y 0 ) EE

For each

x F,

(h,l)

~

E (E-x 0 )

x (F-y 0 ),

we define

= lim f[f(x'+th,y' )-f(x' ,y' )]

f~(x 0 ,y 0 ;h)

tlO

(x' ,y') ~ (xo,Yo) fO(x 0 ,y 0 ;1} = lim t[f(x' ,y'+tl)-f(x 1 ,y' )] y tlO (x'

,y') _,,, (x 0 ,y 0 )

to be the Clarke partial directional derivatives. f

is said to be regular in

at

x0,

if

= lim f[f(x 0 +th,y 0 )-f(x 0 ,y 0 )]

f~x(x 0 ,y 0 ;h) exists for each

x

uo

h E E-x 0 ,

and

f~x(x 0 ,y 0 ;h) Employing these notions, we use

0 -f y (x , y·l) '

(or

0

fx(x,y;h))

(-f' (x,y) ,1) (or (f' (x,y) ,h) resp.) in the definition of y X (f (x,y)). ((T.P.S.) resp.) of the function y 1--> -f(x,y) In the deduction of (2.3), we carry out as follows lim tl [f(x +t h,y )-f(x ,y )] G+' (x , h) ~ e:n n e:n e:

n...,"'

~

n

fox(xe:,yh;h). -

to replace (G.P.S.)

221

A Local Minimax Theorem Without Compactness In order to obtain the quasi-concavity of the function

0

fx(x 8 ,y;h),

we shall assume that the function

f(x,y)

y

i----,,.

is regular in

x. In summary, Theorem A holds true for loc. Lip. functions is regular in E

(II)

f, which

x.

without convexity.

The convexity of

E

was used only in defining the directional de-

rivative of the function

f.

Let us recall the notions of the contin-

gent cone and the Clarke tangential cone of a nonempty subset Banach space

X

[2].

V x O EE,

E

of a

let such that

x 0 +t h

EE]

[h EX IV xn-,.. x 0 ,xn EE, V tn •+O, 3 hn ---,,>h,

such

n n

and

that

x

n

+ t h E EJ• n n

They are called the contingent cone and the Clarke tangential cone of E

at

respectively. By definition,

CE(x 0 )

is convex, and both

say that

E

TE(x 0 )

and

CE(x 0 ) c TE(x 0 ). If CE(x 0 ) = TE(x 0 ), is regular at x 0 in the Clarke sense. It is easily seen

are closed cones with that

U .!. (E-x 0 )

V x 0 E E,

A.>0 II.

and

CE (x 0 ) = TE (x 0 ) = the tangent space of Let g

at

at

E

1 g : E ------:;, .IR , and let x 0 E E. along a direction h E TE(x 0 )

c;i:_'+E (xO; h)

=

if

E

x 0 , if

is convex,

is

E

at

The contingent derivative of is defined to be

½[g (x 0 +th ')-g (x 0 )].

lim

h' -+h

x 0 +th' EE t!O Similarly, the Clarke directional derivative for a loc. Lip. function And

g

g

x O,

lim

h1 ➔h

x O+th' EE t ♦ O

exists with

0

(x O;h)

is defined along directions

is said to be regular at g'+E(x O ;h) =

g

if

E

xO

h E CE(x O).

is regular at

t[g(x O+th')-g(x O)]

at x O,

and

Shi and Chang

222

Employing these notions, we use f~x(xn,yn;ll~IT) -~a~ 0

CE(x)

to replace

E-x

and

inf (f~ (x ,y ) , hEE-x X n n

to replace

n

h

llhlf ~a~ 0

in the definition of (T.P.S.), Theorem A holds true

also. In summary, Theorem A holds true for any closed set convex sets x EE

F,

on which

in direction

f(x,y)

E

and closed

is loc. Lip. and regular for each

x.

(III) Further discussion. We shall emphasize the main difficult point of Theorem A being the lack of convexity

of the function

x

f-->

f(x,y).

Provided by the Lop-Sided Minimax Theorem via weak topology we have the following Theorem 3.1.

Suppose that

reflexive Banach spaces. (1)

'

E

and

F

are two closed convex sets of

We assume that f(x,y)

is l.s.c. and quasi-convex,

3 Yo E F such that the function below and coercive, (2)

(3)

'

-f(x,y)

below and coercive. Then there exists a saddle point f(x,y)

~

f(x,y)

~

X 1-->

f(x,yo)

is bounded

is 1. s .c. and quasi-convex, y

I--,>

(x,y) EE x F

f(x,y)

'

!Ix II

as

=

[x EE

I g(x) =

For

b E :JR1 ,

is nonempty and compact. gb and c

=

C

< +°',

=

{x E E

sup{b E

We shall prove that

°',

x E E.

then

c

=

]R

1

inf g(x)} xEE define the level set

I g (x) gb

+°',

s: b},

is bounded}.

so that

g

is coercive.

is a bounded set, but

g

g

-n c-2 g -n c BR, an open ball with radius c-2 n

This implies

g(x) > c-2-n

V x E E\BR

In fact,

-n

c+2 Rn;;,: n,

is not. i.e.,

n

-n with llx n -n is unbounded, there is xn E g c+2 c+2 Applying Ekeland's Variational Principle to the function g

Since Rn+2.

-.-

According to lemma 2.2, the set M

if not,

+°',

g

the metric space

E\BR, n

we obtain

x~ E E\BR, n

II> on

such that

c-2-n < g(x') s:g(x) s: c+2-n, n n

( 3. 2) ( 3. 3)

and V

X

,f

( 3. 4)

x E E\BR. n

The inequalities (3.2) and (3.4) imply , ( , 9'..+E xn,

Nh )

;;,:

2

1-n

,

Therefore there is a convergent subsequence of (G.P.S. )'.

However, from (3.3),

It is impossible. Corollary 3.1. x ~ f(x,y 0 )

(x' J n

provided by

The contradiction proves the theorem.

Theorem 3.1 holds, if the coerciveness of the functions and

y I--> -f(x 0 ,y)

is replaced by (G.P.S.)'.

We compare Theorem A with Corollary 3.1.

The conditions (1) and

(2) of Corollary 3.1 are weaker than the conditions (1) and (2) of Theorem A.

Since now the convexity of the function

y

~

not assumed, we need a more complicated condition (T.P.S.).

-f(x 0 ,y)

is

224

Shi and Chang

(IV) An improvement of Theorem A. Employing the contingent derivative of a function, we have generalized (G.P.S.) to (G.P.S.)'. A can be weakened by

Now, we mention that (T.P.S.) in Theorem

(G.P.S.)' of the function

G(x)

=

One can verify it directly, the proof is even shorter.

sup f(x,y). yEF

We point out here that (G.P.S.)' plus assumption (1) of Theorem A imply (T.P.S.).

In fact, if

( (x ,y l} CE X F, n n

along which

is bounded and inf G'+E (xn, h) /[[ h [[ ~ a. "" 0, hEE-xn then we conclude that there is a convergent subsequence via (G.P.S.)'.

According to lemma 2.4 and lemma 2.3,

accumulate point, provided by

ynj E M(xnj).

x . ~ x* {yn;J

has an

We arrive at (T.P.S.).

4. Applications In this section, we present three applications of Theorem A: (1)

A variational inequality,

(2)

An elliptic system BVP,

(3)

Infinite dimensional nonlinear programming.

We shall give some new existence theorems, in which the solutions are saddle points rather than loc. minima of functionals. (I) A variational inequality. in

Let M, N be two bounded closed convex sets containing the origin JRm and JRn respectively. Let O CJRP be a bounded open set.

Assume that 1

2

m

1

m

u = (u (x),u (x), •.. ,u (x)) E H0 (0,JR ), 1 2 m 1 n (v (x),v (x), ... ,v (x)) E H0 (0,JR );

v and denote

Suppose that

E

{u E

F

[v E

H; H~

g E C1 (M

(0,JRm)

u(x) E M a.e.},

(O,JRn)

v(x) E N

X

N,JR 1 ) ,

Vu EM, Theorem 4.1.

and that

v ~> g(u,v)

There exists a pair of solutions

ing variational inequalities:

a.e.}.

is concave. (u 0 ,v 0 )

( 4 .1)

of the follow-

A Local Minimax Theorem Without Compactness

V qi E Ht(O,JRm)

Proof.

225

~ [vu 0 vqi + g~ (u 0 ,v 0 )qi]dx ;;e 0,

(4.2)

J0

(4 . 3 J

with

[ v v Ov l/J - g ~ (u O, v O) l/J Jdx qi E E-u 0 ,

o,

;;e

V ljJ E Ht(O,JRn)

and

with

ljJ E F-v 0 .

Define a functional I (u,v) =

J [½l'vu 12 -

½/'vv / 2 + g (u,v) ]dx

0

on the closed convex set

Ex F.

We shall verify that

fies all assumptions of Theorem A. function

v t---> -I(u,v)

V u 0 EE;

[vk} c F

satis-

Actually, only the (G.P.S.) of the

and (T.P.S.) of

we consider the function

Assume that

I(u,v)

I

are needed to prove

v I--> I(u 0 ,v).

is a sequence, along which (4. 4)

is bounded, and ~ ['vvk'vlji- g~(u 0 ,vk)lji]dx

[vk]

From (4.4),

O (Ill/JIil,

1 n H0 (0,JR),

is bounded in

V ljJ E F-vk.

(4. 5)

so that

1 n (HO (0,JR ) ) ,

v*

~

;;e

a. e.,

for a subsequence.

Thus

v* E F.

t0

Now for each

E F-v*, we shall prove (4. 6)

In fact, let

l/Jk.

V

J

f

0

* + lj!O-vk.,

'vv*'vl)! 0 dx

~ 'vvkj vl)! 0 dx + 0 (1)

J 'vvk 0

;;e

and

then

J

j

J

'vljik dx+ 'vvk 'v' (vk -v*)dx+ 0 (1) j O j j

O(1) + Jvvk _vl)ik. dx, 0

J

(4. 7)

J

~ g~(u 0 ,vk/l)! 0 dx+ 0(1)

= provided by Lebesgue Theorem. obtain (4.6).

f

(,

g' (u 0 ,vk )ljik dx+ 0(1) V

j

j

Combining (4. 5) ,

(4. 8)

(4. 7) with ( 4. 8) , we

226

Shi and Chang Putting

in (4.5), and

~ = v*-vk

in (4.6), and adding

vk-v*

~O

these two i~equalities, we obtain

Thus

1 n (HO (0,JR ) ) .

-» v* This verifies

(G.P.S.) of

[ukJ c E,

Next, we assume

J[l

-I(u 0 ,v). {vk] c F

such that

E½lvukl 2 -½l'vvkl 2 +g(uk,vk)]dx

is bounded,

( 4. 9)

(4.10) and (4.11)

From (4.10), take

v 0 -vk

~

for any fixed

v 0 E F,

we obtain

1 n is bounded in H0 (0,JR). SubM1 , so that {vk] 1 m stituting this into (4.9), we see that [ uk} is bounded in H0 (0,JR ) •

for some constant

Then we have subsequences Uk.

J

uk.

-~

1 m HO (0,JR ) ,

u*

vk.

__,.,

V*

--,>

u*

J

u*

a.e.

vk.

J

1 n HO (0,JR ) , a.e.

J

Similarly, we prove

f[l and

['vu*vcp + g' (u* ,v*)cp]dx u

O

2:

V cp E E-u*

(4 .12)

u* EE. Combining (4.10) with (4.12), we have lluk.-u*ll 2 ,;; O(lluk.-u*II). J

Similarly, we have O(llvk. -v*II J

verified.

>.

Thus

( 4. 6) with * uk. -» u I J

* uo = u

I(u 0 ,v 0 ) and

--,>

vk.

J

Applying Theorem A, we obtain

=

J I

and then u*

.

sup I(u 0 ,v)

J

The (T.P.S.) condition is

(u 0 ,v 0 ) EE x F

vEF

llvk. -v*II 2 ,;;

such that (4.13)

(4 .14)

A Local Minimax Theorem Without Compactness

227

Now (4.14) is just (4.2), and (4.13) implies (4.3). Remark 4.1.

The same conclusion holds true, if

g

depends on

x E 0

with some dominant conditions, say, lg (x,u,v) U

g 0 E L1 (0),

Remark 4.2. and

Ig' I

g 0 (x)

I, lg'V (x,u,v) I :.

lg' (x,u,v) where

I :.

V (u,v) E M x N

g 1 (x)

V (u,v) E M x N

g 1 E L 2 (O).

and

The same conclusion holds if is bounded on

M, N

are closed convex sets

M x N.

II. An elliptic system.

Employing the same notations as before we are looking for a weak solution of the following elliptic system: {

Liu = g~ (u,v)

(4 .15)

Liv = -g~ (u,v),

i.e.,

J0 [vuvcp + g'u (u,v)cp]dx

J0 ['ii'v\i'ljJ- g'V (u,v)ljJ]dx Here we assume

g E c 1 (JR.m x JR.n,JR.l)

(a)

V u E JR.m,

(bl

3 constants

and

c 1 > O,

lg(u,O) 3 constant

I :.

I ~

0

1 n V ljJ E HO (0,JR. ) .

3 constants

By Young's inequality,

cl (l+lul 13 J

m

)

1

0

~

I

~

c 2 +g(u,v).

(c) implies a constant

~

C 3 ( l+

Iv I l +y

+

c3

such that

g,_(l+y)

Iu Iy C.

The system (4.16) has a weak solution

x Ho (0, R

n

.17)

13 < 1,

such that

Hereafter, we denote various constants by 1

(4

c 1 (l+lula+lvlal,

~ c 1 (l+lula+lvlYl, 1 c 2 > 0 and e E (0, 2 ) such that

I g ( u, v) I

Ho (G., R

.16)

such that

p-

' 1 gv ' ( u,v)v Sgu(u,v) · u+ 2

Theorem 4.2.

(4

satisfying

is concave, and a E (0 ,p+22)

y E [0,1)

jg'u (u,v) (d)

V cp E H~ (0,JR.m)

v I--> g (u,v)

lg~ (u,v)

(cl

0

) •

(u 0 ,v 0 J E

228

Shi and Chang

Proof.

We introduce the functional I (u,v)

Obviously,

v 1----7" I(u,v)

is quasi-concave, via (a), and is

bounded above via (4.17). It remains to verify the P.S. condition and (T.P.S.).

[vk]

assume that

c H~(O,JRn)

Firstly,

along which ( 4 .18)

and (4.19) is bounded, for a fixed

1 m u 0 E H0 (0,JR ) .

According to (4.17) and (4.19), fvk] is bounded in H~(O,JRn), and then by (4.18) and (c), there is a subsequence v ~ v* in 1 n kj H0 (0,JR ). Therefore v 1----.,> I(u 0 ,v) satisfies P.S. condition V u 0 E

1 m HO (0,JR ) .

Next, we turn to verify (T.P.S.). 1 n [vk ] c H0 (0,JR), along which

f

n

and

Assume that

c½l'vukl 2 -½ l

gi

i

) t- 0

(x)

=

0

and

j

to which

for all for these

and then

for infi-

J

= 0.

h. (x)

f (x N) +

)

provided by the compactness of Nk V j, i f h. (x ) = 0 Similarly,

So we shall restrict ourselves to those gi (x

Nk

O,

$

i

the compactness of nitely many

- x,

) - ~ g~ (x),

g~ (x

many

[xN]

f(xN)

Thus

k, i

and

Sim-

j.

Ng. (x N) + N

L..

i

gi (x )>O This implies g. (x

Nk

i

)

+

L

is bounded.

h. (x

N J h. (x k)=O J

Therefore g. (xl

lim gi (x k ..."'

h. (x) J

lim h. (x

i

and

Thus we proved

k....,.

Nk

)

0

)

0.

x EK.

Now we t urn t o prove t ha t quence.

J

Nk

(,N,µN) ~

possesses convergen t su b se-

Combining (4.25) with (4.26), we have N

f' (x k) +

L

iEI (x)

N

N

N

N

,.._ k.g'. (x k) + µ k.h' (x k) i

i

e.

According to the K.T. condition (assumption (1)), (g' (x

(4. 29)

Nk

) ,h. (x J

Nk

)]

232 i

Shi and Chang

E I (x),

j=l, ... ,q

are linearly independent for

Nk Ai

determinants do not vanish. (4. 28).

for

i

Therefore we can solve Nk is bounded, all Ai and

Since

E I (x).

ity, denote by Nk have x - ~ x.

large; their Gram Nk and µ by Nk µ are bounded

We obtain a convergent subsequence, no loss of generalN N (A k,µ k) --.,,. (f,µ). By the way, again via (4.28), we

Finally, we shall verify that fact,

k

(x,f,µ)

solves our problem.

In

from (4.24), we have f (xl + f • g (xl +

v·h (xl

f(x)+

sup

[A·g(x)+µ·h(x)]

(A,µ)EJR~xlRq f (xl

:. f(x) +

sup

[A·g(x) +µ•h(x)]

(A,µ) EJR~ XlRq ¥ x EX.

This implies f (x) :. f (x) ,

¥

E K,

X

and f•g(x) = O,

and then

figi(x) = 0,

i=l,2, ••• ,p.

The conclusion (2) follows from a limiting process of

(4.25).

The

proof is complete. As an example, we consider the following problem. Suppose that ···•~p'

OE lRn

n 1 , ... ,nq

is a bounded open domain, and that

are Caratheodory functions defined on

q:,,~ 1 ,

Ox JR 1 ,

sat-

isfying the following growth condition: Jg(x,t) where n

I :.

a. E (1 n+2) 'n-2 '

is a constant, and

C

= 1,2. Assume that

J

1

jS

q:,(x,t)dt,

f·

Q

J

t

n .: 3,

and

JSO

l

(x,t)dt,

n- (x,t)dt, J

i = 1, .•. , p; j = 1, •.. , q;

and

i=l, ... ,p; j=l, ..• ,q, 1 ElR. Let

f (u) =

0

r ~A. (X,$) =

if

(~.(x,t),n.(x,t)},

ear independent for each fixed

i (x, s)

C(l+JtJa.)

J ½Ivu I 2 + 0

J0 'i' l. ( x , u) dx ,

gi (u) h. (u) · J

!f ( x , u) dx ,

=

1 u E HO (0).

J

0

A . ( x , u) dx , J

a. .: 1

if

is lin-

233

A Local Minimax Theorem Without Compactness If

where with

x JRq

-a< A1 ,

c 1 is a constant, and 0-Dirichlet data in 0,

the first eigenvalue of

then there exists

-t:.

(u,X°,µ) E H~(O) X JR~

such that

(1)

f(u) = min[f(u)

Iu

t H~(O), gi(u)

j=l,2, ... ,q}, ( 2)

-t:.u(x)+qJ(x,u(x))+

( 3)

X°.f l [l

hj (x,u(x) )dx

(5)

J

Yi(x,u(x))dx s. 0.

Remark 4.3.

If

o,

i= 1,2, ... ,p;

r

0,

l

J0

ever, in case

0, hj(u)

tX-.\jJ.(x,u(x))+ µJ.T]J.(x,u(x)) i=l l l i=l Y. (x,u(x))dx = O,

(4)

0

s.

dim X < +w, dim X

+w,

O,

then Theorem 4.3 is obviously true.

How-

it seems worth proving. REFERENCES

1.

Aubin, J.P., Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam-New York-Oxford, Rev. ed. 1982.

2.

Aubin, J.P. and Ekeland, I., Applied Nonlinear Analysis, WileyInterscience, New York, 1984.

3.

Barbu, V. and Precupanu, Th., Convexity and Optimization in Banach Spaces, Sijthoff & Noordhoff, Bucharest, 1978.

4.

Brezis, H., Nirenberg, L. and Stampacchia, G., A remark on Ky Fan's minimax principle, Boll. Un. Math. Ital. 5(1973), 293-300.

5.

Chang, Kung-ching and Eells, J., Unstable minimal surface coboundaries, preprint, Univ. of Warwick, 1985. (To appear in Acta Math. Sinica.)

6.

Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.

7.

Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc. (n.s.) 1(1979), 443-474.

8.

Fan, Ky, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39(1953), 42-47.

9.

Fan, Ky, A minimax inequality and applications, in "Inequalities III" (Shisha, 0., ed.), Academic Press, New York, 1972, 151-156.

10.

Li, Shujie, A multiple critical point theorem and its application in nonlinear partial differential equations (in Chinese), Acta Mathematica Scientia, 4(1984), 135-340.

11.

Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (n.s.), 4 (1981), 267-302.

12.

Shi, Shuzhong, Ekeland's variational principle and the mountain pass lemma, Cahiers de CEREMADE, Univ. de Paris-Dauphine, n°8425. (To appear in Acta Math. Sinica.) Sion, M., On general minimax theorems, Pacific J. Math. 8(1958), 171-176.

13.

Theorems of Convex Sets Related to Fixed-Point Theorems MAU-HSIANG SHIH Department of Mathematics, Chung Yuan University, ChungLi, Taiwan, Republic of China KOK-KEONG TAN Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada

l.

A Conman Generalization of Theorems of Knaster-Kuratowski-Mazurkiewicz, Shapley and Ky Fan.

The a in of this rarer is to prove the foll owing (Jenera l coverin9 theorem of convex sets in locally convex topological vector spaces. THEOREM l. Let X be a paracompact convex subset of a real locally convex Hausdorff topological vector space E , xO a non-empty compact convex convex ~ubs_et of X and K a non-empty compact subset of X . Let {Ai: c I} be a locally finite family of closed subsets of X such that X = u A. , and let id

1

{C.: , I} .ti_~ a family (indexed al so by I) of non-empty subsets of E Let 1 s: X ➔ P(E) be an upper hemi-continuous set-valued map with each s(x) a weakly compact_convex set . If for each x e (Kn ax) u (X\K) , the closed convex hull of ,,{Ci+ s(x): X \} meets the tangent cones e

X

+

IJ

,\>O

X

+

lJ

:\>O

:\(X-x)

if

X

' Kn

:\(X O-x)

if

X

E

ax

'

X\I(

then there exist a non-~~pty finite subset I O of and a roint x c X such that X is in the closed convex hull of u{Ci + s(x): i E Io} as well as in A

In Theoren l, P(E) stands for the collection of all non-emnty subsets of E denote the boundary operation and the closure the synbol s a and operation, respectively. Theorem l remains valid if the tan(Jent cones are replaced by X + u :\(X-x) if X (Kn ax , and X + u :\(Xo-xl if XE X\K, 11

11

11 - 11

:\ O

x E f(x)

•

Proof. We first observe that if x is an interior point of X , then x + u A(X-xJ = E ; thus condition (a) is equivalent to the following condition: A>O (a') For each XE K, f(x) meets X + ·-;;--~(X-x) A>O Suppose that the assertion of the theorem is false. Let x E X ; then x 4: f (x) , so by the Hahn-Banach theorem, there exists 1;x E E' and tx E ~ such that sup{ : v E f(x}} < tx < . As f is upper he~i-continuous and hood Ux of x satisfying

1;x

is continuous, there is an open neighbor-

(2) sup < tx < for all u E Ux yEf(U) As {Ux: x EX} is an open covering of the paracompact space X , there is a continuous partition of unity {¢x: x EX} subordinate to this open coverin~. Thus (i) for each x EX , ¢x is a non-negative real-valued continuous function on X such that its support supp ¢x c Ux , (ii) the fa~ily {supp ¢x: x EX} is a locally finite covering of X and (iii) z ¢x(y) = l for all y EX Define XEX p: X ➔ E' by setting

p(y) := -

Z ¢2 (y)1; 2 for all y E X . ZEX Let y, X ; note that whenever ¢2 (y) t O , we have y E supp ¢2 we must have

thus

c

U2

so by (2),

Covering Theorems Related to Fixed-Point Theorems

< t 2 < As

¢2 (y)

r

O for at least one

z

241

for all

v

f(y) .

c

(3)

X , it follows from (3) that

c

= - l:

zcX

¢z(y)

< - z ¢ (y) ZEX

for all

z

X

v

c

f(y) .

Thus we have shown that for each ye X , < Now we define w: Xx X ➔ JR w(x,y)

for all

v

(4)

f(y)

c

by

for all

(x,y)

Xx X .

E

We observe that (I) For each fixed x e X , w(x,y) is a lower semi-continuous function of y be a net in X for which on X • Indeed, let t e JR and let y ➔ y E X and w(x,y ) ~ t for all Because {supp ¢2 : z EX} is a a Ny of y such that tly n SUJ1Jl ¢ 2 f 0 locally finite, there is an open neiqhborhood • for at most finitely many z c X say, {z , X: N n supp ¢2 r 0} {zl ,z2•· .. ,zn}. y Choose an a 0 c I such that ya e NY for all a? a 0 , ~,e have t

~

w(x,ya)

¢z(ya) ¢z_(ya) < r,z_, x-ya> 1

1

(if

a? ao)

n

➔ -

z ¢z. (y)

i=l

1

1

- z ¢z(y) zeX w(x,y) Hence w(x,y) is a lower semi-continuous function of y on X . (II) For each fixed y EX, w(x,y) is an affine function and hence a quasiconcave function of x on X (III) Clearly w(x,x) 0 for all x e X • (IV) The set C := {y e X: w(x,y) ;i O for all x e x0} is compact. Indeed, by (I), it is sufficient to show that Cc K Suppose ye X\K ; then by hypothesis (b), there exists u E f(y) n [y + u t.(X 0-y)] . As u e f(y) , it ).>0

follows from (4) that < . By continuity of there is an open neighborhood Nu of u such that < O u A>O there exist x E x0 and r > 0 for which u' = y + r(x-y) . Therefore

As

u Ey +

~(x,y)

- z ¢ (y) ZEX

Z

l z

r ZEX

lr

Z

¢ (y) > O

Hence C c K . This shows that y E X\C Thus all hypotheses in Theorem 3 are satisfied, it follows that there exists y EX such that ~(x,y); 0 for all x EX; that is, ; for all Note that

x EX.

y E K ; thus by (a'), there exists w E f(y)

(6)

u >-(X-y)] He ;\>O shall verify that ~ Indeed, let E > 0 be given; because p(y) is continuous at w , there is an open neighborhood U of w such that

J - I < As w E

yt

u

J->O

>-(X-y) , we can find

< +

E

and

u0

n

for all

E

u0 E Wn [y +

u

J->0

U

E

W

>-(X-y)] , it follows that

for some

y + r(u - y)

[y +

u E X and

r >0

By

(6), we have ; , and hence = r ~ o .

Therefore

< + f

+

E

E.

E > 0 is arbitrary, we must have ; which contradicts (4) w E f(y) . This completes the proof. □ We remark that our proof of Theorem 4 follows the method given in Ky Fan [8]. THEOREM 5. Let X be a _2_aracoll1J}act convex subset of a real locally convex lj_~l!_sdorff topolog_f_~~Lv_~cto_r:__~JJ..~~~ E , x0 ~_n_o_r~::.empty compact convex subset of X , and K a non-elllJ}t,r conp ct subset of X • Let f: X ➔ P(E) be up~er hemicontinuous with each f(x) a closed convex subset of E ~l!.~~-t~~t (a) For each x EK n ax , f(x) ----·--meets x + Au-(X-x) - - - ---

As

as

(b)

l:_o__r:__~~ch

X E X\K ' f(x)

Then there exists

x EX

meets

such that

X

+ ~--:\(Xo-xf J-0

x

c

Kn

ax ,

if

f(x)

Similarly, for each

243

meets

x

x, X\K , if

u 11(X 0-x) , then g(x) meets x + u A(X 0-x) 11O a fixed point and therefore f has a fixed point. x

u

;.0 11>0 Applying Theorem 4, there exists a point x c X such that x, f(i) . If we take I 0 := I(x), then the proof of Theorem l is complete. REFERENCES l.

G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58(1977), l-10.

2.

J.-P. Aubin, Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, Revised Edition, 1982.

3.

J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.

4.

H. Brezis, L. Nirenberg and G. Stampacchia, A remark on Ky Fan's minimax principle, Boll. Un. Mat. Ital. 6(1972), 293-300.

5.

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. fJat. Acad. Sci. USA 38(1952), 121-126.

6.

K. Fan, A miniriax inequality and applications, in "Inequalities III", Proceedings Third Symposium on Inequalities (O. Shisha, Ed.), pp. 103-113, Academic Press, New York, 1972.

244

Shih and Tan

7.

K. Fan, A further generalization of Shapley's generalization of the KnasterKuratowski-Mazurkiewicz theorem, in "Game Theory and Related Topics", (0. Moeschlin and D. Pallaschke, Eds.), pp. 275-279, North-Holland, Amsterdam, 1981.

8.

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266(1984), 519-537.

9.

I.L. Glicksberg, A further generalization of the Kakutani fixed point theore~, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3(1952), 170-174.

10.

B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe. Fund. Math. 14(1929), 132-137.

'n. L.S. Shapley, On balanced games without side payments, in "Mathematical

Programming", (T.C. Hu and S.M. Robinson, Eds.), pp. 261-290, Academic Press, New York, 1973.

Selections and Covering Theorems of Simplexes MAU-HSIANG SHIH Department of Mathematics, Chung Yuan University, ChungLi, Taiwan, Republic of China KOK-KEONG TAN Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada

§l.

KKMS Theorem

Throughout this paper, 0 will denote a simplex in a euclidean snace. The family of all faces of o (of a 11 dimensions) is denoted by F For each T c F , let c(T) denote the barycenter of 1 . A set V of faces of 0 is said to be balanced if the convex hull of the set {c(1): 1 c V} contains the barycenter c(o) of o The collection of all subsets of o is denoted by 2° . To formulate the results, we need the following DEFINITION. 1

E

A set-valued map

B : F-----+ 2° is called a Shanley-map if for each

f ,

The fol lowing remarkable generalization, which plays an important role in game theory, of the classical Knaster-Kuratowski-Mazurkiewicz Theorem [5] is due to Shapley [6]. KKMS Theorem. If A : F-----+ 2° is a Shaoley-map with each A(p) a closed subset of o , then there exists a balanced set V of faces of o such that n TEV

A(T) 'f 0 .

In case A(p) 'f 0 only for 0-dimensional faces of o , KKMS theorem becomes the classical Knaster-Kuratowski-Mazurkiewicz theorem [SJ. Shapley's Proof of the KKMS theorem is based on a generalization [6] of the Snerner combinatorial lemma [7]. A oroof usin9 a Ky Fan's coincidence theorem [1] has been recently given by Ichiishi [3]. In the present paper, we first orove a selection theorem for Shapley-maps, and 245

DOI: 10.1201/9781003420040-19

Shih and Tan

246

then apply it together with the KKMS theorem to give two new covering theorems of simplexes with facial structures. As a consequence, we obtain a general and more direct method for dealing with a recent basic covering theorem of Ky Fan [2, Theorem 2]. §2.

A Selection Theorem.

Let A,B : F->- 2° be set-valued mans. If A(p) c B(p) for each p E F , then A is called a selection for B . If A is both a Shapley-man and a selection for B , then A is called a Shapley-selection for B . The closure of a set U in a is denoted by TI. We now establish THEOREM l. If B : F-.. 2° is a Shapley-map with each B(p) an open subset of a , then B admits a Shapley-selection A : F->- 2° such that each A(p) is a a closed subset of o Since B is a Shapley-map , o = u B(p) .

Proof.

pd

Hy := n{B(pj : y

E

For each

y

E

o, let

B(p)}

Then Hy is an open set in a containing y , and therefore there exists an ooen neighbourhood Uy of y in a such that For each ,

E

F , we have

u{B{p) : pc,}= u{UY: y As B is a Shapley-map, the compactness of , set BT of u{B{p) : pc,} such that We now let K is a finite set .

then

K := u{B, : ,

E

E

B{p)

for some

pc,} .

ensures that there exists a finite

F} ;

Define A: F-.. 2° A(p) := ulUY :y

E

by K and

Uy c B(p)}

for each p E F . Then for each p E F , A(p) is a closed subset of B(p) A(p) lt remains to verity that A is a Shapley-map. Let arbitrarily fixed. For each z E, , there exist p E F with pc, such that z E Uy; it follows that y E K and Uy c TIy c Hy c B(p) z E A(p) for some pc, . Thus C

A(p). u ropEF lhis concludes the proof of our theorem. L

o such that

, E F be and y E B(p) , and hence

C

D

Shapley Selections and Covering Theorems of Simplexes

§3.

247

Some Covering Theorems of Simplexes KKMS and Theorem 1 imply the following

THEOREM 2. lf B : F-+ 2CT is a Shapley-map with each B(p) an ooen subset of CT , then there exists a balanced set V of faces of CT such that 8(1:) f

n

i:cV

0

In what follows we shall show that Theorem 2 also imolies the KKMS theorem. Indeed, ·1et A : F-+ 2CT be a Shapley-map with each A(p) a closed subset of CT For each k = 1,2, ... , we define B(k) : F-,. 2CT by B(k) (p) : = {x

CT : dist(x,A(p))

E

1/k}


(e,,,.,)=

t Mjk(-J--s(1,,.,1!e))·

. k-1

J, -

T/k

T/J

3. Hitherto, suppose Aj,1. = 1, 2 are two subalgebras of the algebra A satisfying (11) [a1, a2] = 0 for ai E Aj, 1· = 1, 2,

where [a 1,a2] is the commutator

aja2 - a2a 1 .

For¢ E Zl(A), define

for 1· = 1, 2, where/ = 2 for 1· = 1 and/= 1 for 1· = 2.

Lemma 2. The functionals ¢j, 1· = 1, 2 are tri-linear,

1>i(l, TJ; ~) = -¢i(TJ, c; ~), (b¢i)(fo, 6, 6; ~) = ¢(~; fo66),

¢1(fo, c1; [TJo, T/1]) = ¢2(TJ0, T/1; [co, c1]),

(13) (14) (15)

1>(c1T/1, c2TJ2) = 1>1(c1, 6; ,,.,1,,.,2) + 1>2(TJ1, T/2, 66)) = ¢1(c1, 6; 8(TJ1TJ2)) + ¢2(r11, T/2; 8(6c2)).

(16) (17)

and

Proof. It is obvious that ¢i are tri-linear. To prove (13), notice that

1>i (e, ,,.,; ~)

+ 1>i (,,.,, e; ~)

1

=

2(¢(~€,,,.,) + ¢(,,.,~, c) + ¢( e, ~,,.,) + ¢(,,.,, e~))

=

2(¢(~, e,,.,J -

1

1>(~, TJc)) =

1

21>(~, [e, ,,.,]),

(18)

by (1) and (4). As functions of€ and T/, ¢i(C.TJ;~) + (~, 1e, ,,.,])

= o,

(19)

Xia

302

for ~ E AJ,, c, r, E AJ. From ¢(~6, 6fo) ¢(~fo, 66)

and

+ c/>(fo, ~66) + c1>(c2, fo~6) = o, + ¢(6, 6~fo) + ¢(6, ~foci)= o

(14) follows. To prove (15), observe that and Thus, if [Ci, r:,3]

= 0 for

i,j = 1, 2, then

(20) and similarly

(21)

From (20) and (21) it follows (15) immediately. By a simple calculation, it is easy to see (16) and (17). Theorem 2. Let the algebra A over C be generated by .4 1 and .4 2 satisfying (11). Then¢ E Zl(A), if and only if¢(-, ·) is a bilinear functional on A X A satisfying (17), where ¢ 3 ,j = 1,2 are tri-linear functionals satisfying (13), (14) and

(15).

Proof. By Lemma 2, we only have to prove the "if" part of the theorem. Suppose ¢(·, ·) is a bilinear functional satisfying (13), (14), (15) and (17). It is obvious that ¢(·, •) is skew-symmetric, by (13) and (17). From (15), there is a bilinear functional 1/J(·, ·) on .4 1 x .4 2 such that

To prove (4), we have to prove ¢1(fo, c1c2; S(r,0[111112])) + ¢1(6, c2fo; S(171[r,2r:,o])) + ¢1((2, foci; S(r:,2[110111])) + ¢2(110, r11112; S(fo[ci6])) + ¢2(111, 112110; S(c1 [c2fo])) + ¢2(112, r/or/1; S(c2[foci])) = 0.

(23)

Trace Formula for Almost Lie Algebra of Operators

303

By means of (14) and (22), we ha.ve

and

¢( fo, Cl c2; 110111112) + ¢i( Cl' C2 fo; 111112110) + ¢( 6' fo C1; 112110111) = ¢(110111112, foE1c2) + '1j;([E1, E2fol, [111112, 110]) + '1j;([c2, foE1l, [112, 110111]), ¢1(fo, E16; 111112110) + ..-

174 viability

3,5,8

Range condition 194-195,200 Riesz-Dunford integral 15 Selection 102-103 Shapley- 245-246

balanced 37,245,247-249 contractible 99-105 D-convex 158,162,164 finite-concave 65,67 finite-convex 65,70-71 reachable 51-52,54 weakly compact 77-78,80-82

Side-payment

23,31-32 25,32

C

Set

Space ab- 42 bornological 42 Frechet 38,40 Lorentz sequence 170 Montel 38,40 Spectral theorem for unitary operator 16 Strategy correlated 108,115,117,

120-122

mixed 108-115 pure 109,111-114,119-122 randomized 108,116-119,

121,123

repeated-game- 119-121 Strongly summable sequence Sum ll. 00 -direct 168 11. 1 -direct

168,170

Supremization problem

259,265

Surrogate duality

264

298

253-254,

255-256,259,

Theorem cone-compression 176,185 cone-expansion 176-185 Debranges 78-79 Fan 6-7,15,56,61,63-64,

69,74,78,80,99,108, 112-113 fixed-point 103 Heinz 15

Knaster-Kuratowski-Mazurkiewicz

99-101,236,245,248 235,245,247,250 Krein 82 lop sided minimax 213,215, 222 minimax 61,64-68,211,213 separation 17,56 Simons 82 von Neumann 15,61,64,66,68 Total expected multiloss 151-152 Trace formula 299 Transition probability 151,159, 164 KKMS

Ulitmately bounded filter

42

37,40,

311

Index

Ultrafilter 38,42 Upper demi-continuity Upper hemi-continuity Utility 107,109,121 expected 115,117 Variational inequality 224

237-239 235-239

131,214,

Viability domains 5-9 largest 6 Viable solutions 1-5,8-11 heavy 3,8, 11 Volterra equation 83 Weak convergence 84,87-93, 271,273,275,286