For other titles published in this series, go towww.springer.com/series/5613

HANDBOOK OF APPLIED ANALYSIS

Advances in Mechanics and Mathematics VOLUME 19 Series Editors David Y. Gao (Virginia Polytechnic Institute and State University) Ray W. Ogden (University of Glasgow) Advisory Board Ivar Ekeland (University of British Columbia, Vancouver) Tim Healey (Cornell University, USA) Kumbakonam Rajagopal (Texas A&M University, USA) Tudor Ratiu (École Polytechnique Fédérale, Lausanne)

David J. Steigmann (University of California, Berkeley) Aims and Scope Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, thermal, quantum effects, biomechanics, nanomechanics, multiscale modeling, dynamical systems, optimization and control, and computational methods. Driven by increasingly elaborate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. The increasingly large number of specialist journals has generated a complementarity gap between the partners, and this gap continues to widen. Advances in Mechanics and Mathematics is a series dedicated to the publication of the latest developments in the interaction between mechanics and mathematics and intends to bridge the gap by providing interdisciplinary publications in the form of monographs, graduate texts, edited volumes, and a special annual book consisting of invited survey articles.

By

123

HANDBOOK OF APPLIED ANALYSIS

NIKOLAOS S. PAPAGEORGIOU National Technical University, Athens, Greece

SOPHIA TH. KYRITSI-YIALLOUROU Hellenic Naval Academy, Piraeu s, Greece

Series Editors: David Y. Gao Ray W. Ogden Department of Mathematics Department of Mathematics Virginia Polytechnic Institute University of Glasgow

Glasgow, Scotland, UK rwo@maths.gla.ac.uk

N.S. Papageorgiou S. Th. Kyritsi-Yiallourou National Technical University Hellenic Naval Academy Department of Mathematics Military Institute of University Education 157 80 Athens Leoforos Chatzikyriakou Zografou Campus 185 39 Piraeus Greece Greece npapg@math.ntua.gr skyrits@math.ntua.gr

ISBN 978-0-387-78906-4 e-ISBN 978-0-387-78907-1 DOI 10.1007/b120946 Springer Dordrecht Heidelberg London New York

Library of Congress Control Number:

Mathematics Subject Classification (2000): 34xx, 35xx, 46xx, 47xx, 49xx, 90xx, 91xx

© Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

2009927202

Blacksburg, VA 24061 gao@vt.edu

and State University

To the memory of my fatherS. Th. K.

The mathematical sciences particularly exhibit order, symmetry andlimitation; and these are the greatest forms of the beautiful.

AristotleMetaphysica, 3M, 1078b

Series Preface

As any human activity needs goals, mathematical research needs problems.—David Hilbert

Mechanics is the paradise of mathematical sciences.—Leonardo da Vinci

Mechanics and mathematics have been complementary partners since New-ton’s time, and the history of science shows much evidence of the beneficialinfluence of these disciplines on each other. Driven by increasingly elabo-rate modern technological applications, the symbiotic relationship betweenmathematics and mechanics is continually growing. However, the increas-ingly large number of specialist journals has generated a duality gap betweenthe partners, and this gap is growing wider.

Advances in Mechanics and Mathematics (AMMA) is intended to bridgethe gap by providing multidisciplinary publications that fall into the two fol-lowing complementary categories:

1. An annual book dedicated to the latest developments in mechanicsand mathematics;

2. Monographs, advanced textbooks, handbooks, edited volumes, andselected conference proceedings.

The AMMA annual book publishes invited and contributed comprehensiveresearch and survey articles within the broad area of modern mechanics andapplied mathematics. The discipline of mechanics, for this series, includes rel-evant physical and biological phenomena such as: electromagnetic, thermal,

x Series Preface

and quantum effects, biomechanics, nanomechanics, multiscale modeling, dy-namical systems, optimization and control, and computation methods. Es-pecially encouraged are articles on mathematical and computational modelsand methods based on mechanics and their interactions with other fields. Allcontributions will be reviewed so as to guarantee the highest possible scien-tific standards. Each chapter will reflect the most recent achievements in thearea. The coverage should be conceptual, concentrating on the methodologi-cal thinking that will allow the nonspecialist reader to understand it. Discus-sion of possible future research directions in the area is welcome. Thus, theannual volumes will provide a continuous documentation of the most recentdevelopments in these active and important interdisciplinary fields. Chap-ters published in this series could form bases from which possible AMMAmonographs or advanced textbooks could be developed.

Volumes published in the second category contain review/research con-tributions covering various aspects of the topic. Together these will providean overview of the state-of-the-art in the respective field, extending from anintroduction to the subject right up to the frontiers of contemporary research.Certain multidisciplinary topics, such as duality, complementarity, and sym-metry in mechanics, mathematics, and physics are of particular interest.

The Advances in Mechanics and Mathematics series is directed to all sci-entists and mathematicians, including advanced students (at the doctoral andpostdoctoral levels) at universities and in industry who are interested in me-chanics and applied mathematics.

David Y. GaoRay W. Ogden

Preface

The aim of this book is to present the basic modern aspects of nonlinearanalysis and then to illustrate their use in different applied problems.

Nonlinear analysis was born from the need to deal with nonlinear equa-tions which arise in various problems of science, engineering, and economicsand which are often notoriously difficult to solve. On a theoretical level, non-linear analysis is a remarkable mixture of various areas of mathematics such astopology, measure theory, functional analysis, nonsmooth analysis, and multi-valued analysis. On an applied level, nonlinear analysis provides the necessarytools to formulate and study realistic and accurate models describing variousphenomena in different areas of physical sciences, engineering and economics.For this reason, the theoretically inclined nonmathematician (physicist, engi-neer, or economist), needs to have a working knowledge of at least some ofthe basic aspects of nonlinear analysis. This knowlegde can help him buildgood models for the phenomena he studies, study them in detail and extractfrom them important information which is crucial to the design process. Asa consequence, nonlinear analysis has acquired an interdisciplinary characterand is a prerequisite for many nonmathematicians, who wish to investigatetheir problems in detail with the greatest possible generality. This leads to acontinuously increasing need for books that survey this large area of mathe-matical analysis and present its applications.

There should be no misunderstanding. The subject is vast, it touchesmany different areas of mathematics and its applications cover several otherfields in science and engineering. In this volume, we make an effort to presentthe basic theoretical aspects and the main applications of nonlinear analysis.Of course the treatment is not exhaustive; such a project would require severalvolumes. Nevertheless, we believe that we touch the main parts of the theoryand of the applications. Mathematicians and nonmathematicians alike canfind in this volume material that covers their interests and can be useful intheir research and/or teaching.

Chapter 1 begins with the calculus of smooth and nonsmooth functions.We present the Gateaux and Frechet derivatives and develop their calculus in

xii Preface

full detail. In the direction of nonsmooth functions, first we deal with convexfunctions for which we develop a duality theory and a theory of subdifferen-tiation. Subsequently, we generalize to locally Lipschitz functions (Clarke’stheory). We also introduce and study related geometrical concepts (such astangent and normal cones) for various kinds of sets. Finally we investigate akind of variational convergence of functions, known as Γ-convergence, whichis suitable in the stability (sensitivity) analysis of variational problems.

In Chapter 2, we use the tools of the previous chapter in order to studyextremal and optimal control problems. We begin with a detailed study ofthe notion of lower semicontinuity of functions. Next we examine constrainedminimization problems and develop the method of Lagrange multipliers. Thisleads to mimimax theorems, saddle points and the theory of KKM-multimaps.Section 2.4 deals with some modern aspects of the direct method, which in-volve the so-called variational principles, central among them being the so-called “Ekeland variational principle”. The last two sections deal with thecalculus of variations and optimal control. In optimal control, we focus onexistence theorems, relaxation and the necessary conditions for optimality(Pontryagin’s maximum principle).

Chapter 3 deals with some important families of nonlinear maps and ex-amines their uses in fixed point theory. We start with compact and Fredholmoperators which are the natural generalizations of finite rank maps. Subse-quently we pass to operators of monotone type and to accretive operators.Monotone operators exhibit remarkable surjectivity properties which play acentral role in the existence theory of nonlinear boundary value problems.Accretive operators are closely related to the generation theory of linearand nonlinear semigroups. We investigate this connection. Then we intro-duce the Brouwer degree (finite-dimensional) and the Leray–Schauder andBrowder–Skrypnik degrees (the latter for operators of monotone type) whichare infinite-dimensional. Having these degree maps, we can move to the fixedpoint theory. We deal with metric fixed points, topological fixed points andinvestigate the interplay between order and fixed point theory.

Chapter 4 presents the main aspects of critical point theory which is the ba-sic tool in the so-called “variational method” in the study of nonlinear bound-ary value problems. We start with minimax theorems describing the criticalvalues of a C1-functional. Then we present the Ljusternik–Schnirelmann the-ory for multiple critical points of nonlinear homogeneous maps. This way wehave all the necessary tools to develop the spectral properties of the Laplacianand of the p-Laplacian (under Dirichlet, Neumann and periodic boundary con-ditions). Then using the Lagrange multipliers method we deal with abstracteigenvalue problems. Finally we present some basic notions and results frombifurcation theory.

Chapter 5 uses the tools developed in Chapters 3 and 4 in order to studynonlinear boundary value problems (involving ordinary differential equationsand elliptic partial differential equations). First we illustrate the variationalmethod based on the minimax principles of critical point theory and then

Preface xiii

we present the method of upper and lower solutions and the degree-theoreticmethod. Subsequently we consider nonlinear eigenvalue problems, for whichwe produce constant sign and nodal (sign changing) solutions. Then we provemaximum and comparison principles involving the Laplacian and p-Laplaciandifferential operators. Finally we deal with periodic Hamiltonian systems.We consider the problem of prescribed minimal period and the problem of aprescribed energy level. For both we prove existence theorems.

In Chapter 6, we deal with the properties of maps which have as valuessets (multifunctions or set-valued maps). We introduce and study their conti-nuity (Section 6.1) and measurability (Section 6.2) properties. Then for suchmultifunctions (continuous or measurable), we investigate whether they ad-mit continuous or measurable selectors (Michael’s theorem and Kuratowski–Ryll Nardzewski, and the Yankov–von Neuman–Aumann selection theorems).This leads to the study of the sets of integrable selectors of a multifunction,which in turn permits a detailed set-valued integration. The notion of de-composability (an effective substitute of convexity) plays a central role in thisdirection. Then we prove fixed point theorems for multifunctions and alsostudy Caratheodory multifunctions. Finally we introduce and study variousnotions of convergence of sets that arise naturally in applications.

In Chapter 7, we consider applications to problems of mathematical eco-nomics. We consider both static and dynamic models. We start with the staticmodel of an exchange economy. Assuming that perfect competition prevails,which is modelled by a continuum (nonatomic measure space) of agents. Weprove a “core Walras equivalence theorem” and we also establish the exis-tence of Walras allocations. We then turn our attention to growth models(dynamic models). First we deal with an infinite horizon, discrete-time, mul-tisector growth model and we establish the existence of optimal programs forboth discounted and undiscounted models. For the latter, we use the notionof “weak maximality”. Then we determine the asymptotic properties of op-timal programs via weak and strong turnpike theorems. We then examineuncertain growth models and optimal programs for both nonstationary dis-counted and stationary undiscounted models. We also characterize them usinga price system. Continuous-time discounted models are then considered, andfinally we characterize choice behavior consistent with the “Expected UtilityHypothesis”.

Chapter 8 deals with deterministic and stochastic games, which provide asubstantial amount of generalization of some of the notions considered in theprevious chapter. We start with noncooperative n-players games, for whichwe introduce the notion of “Nash equilibrium”. We show the existence ofsuch equilibria. Then we consider cooperative n-players games, for which wedefine the notion of “core” and show its nonemptiness. We continue with ran-dom games with a continuum of players and an infinite-dimensional strategyspace. For such games, we prove the existence of “Cournot–Nash equilibria”.We also study corresponding Bayesian games. Subsequently, using the for-malism of dynamic programming, we consider stochastic, 2-player, zero-sum

xiv Preface

games. Finally, using approximate subdifferentials for convex function, weproduce approximate Nash equilibria for noncooperative games with noncom-pact strategy sets.

Chapter 9 studies how information can be incorporated as a variable invarious decision models (in particular in ones with asymmetric informationstructure). First we present the mathematical framework, which will allowthe analytical treatment of the notion of information. For this purpose, wedefine two comparable metric topologies, which we study in detail. Then weexamine the ex-post view and the ex-ante view, in the modelling of systemswith uncertainty. In both cases we establish the continuity of the model inthe information variable. Subsequently, we introduce a third mode of conver-gence of information and study prediction sequences. We also study gameswith incomplete information and games with a general state space and anunbounded cost function.

The final chapter (Chapter 10) deals with evolution equations and themathematical tools associated with them. These tools are developed in thefirst section and central among them is the notion of “evolution triple”. Weconsider semilinear evolutions, which we study using the semigroup method.We then move on to nonlinear evolutions. We consider evolutions drivenby subdifferential operators (this class of problems incorporates variationalinequalities) and problems with operators of monotone type, defined withinthe framework of an evolution triple. The first class is treated using nonlinearsemigroup theory, while the second requires Galerkin approximations. Weconclude with an analogous study of second-order evolutions.

The treatment of all subjects is rigorous and every chapter ends with anextensive survey of the literature.

We hope that both mathematicians and nonmathematicians alike, will findsome interesting and useful for their needs in this volume.

Acknowledgments: We would like to thank Professor D. Y. Gao for recom-mending this book for publication in the Advances in Mechanics and Math-ematics (AMMA) series. Many thanks are also due to Elizabeth Loew andJessica Belanger for their patience and professional assistance.

Athens, September 2008 Nikolaos S. PapageorgiouSophia Th. Kyritsi–Yiallourou

Contents

Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Smooth and Nonsmooth Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Gateaux and Frechet Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Locally Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4 Variational Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5 Γ-Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2 Extremal Problems and Optimal Control . . . . . . . . . . . . . . . . . . 632.1 Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.2 Constrained Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . 722.3 Saddle Points and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.4 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.5 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.6 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3 Nonlinear Operators and Fixed Points . . . . . . . . . . . . . . . . . . . . . 1473.1 Compact and Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . 1483.2 Monotone and Accretive Operators . . . . . . . . . . . . . . . . . . . . . . . . 1623.3 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1963.4 Metric Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2243.5 Topological Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2373.6 Order and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2453.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

xvi Contents

4 Critical Point Theory and Variational Methods . . . . . . . . . . . . 2674.1 Critical Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2684.2 Ljusternik–Schnirelmann Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2894.3 Spectrum of the Laplacian and of the p-Laplacian . . . . . . . . . . . 2974.4 Abstract Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3344.5 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3404.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

5 Boundary Value Problems–Hamiltonian Systems . . . . . . . . . . 3515.1 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.2 Method of Upper–Lower Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3805.3 Degree-Theoretic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4045.4 Nonlinear Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.5 Maximum and Comparison Principles . . . . . . . . . . . . . . . . . . . . . . 4325.6 Periodic Solutions for Hamiltonian Systems . . . . . . . . . . . . . . . . . 4405.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

6 Multivalued Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4556.1 Continuity of Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4566.2 Measurability of Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4706.3 Continuous and Measurable Selectors . . . . . . . . . . . . . . . . . . . . . . 4756.4 Decomposable Sets and Set-Valued Integration . . . . . . . . . . . . . . 4876.5 Fixed Points and Caratheodory Multifunctions . . . . . . . . . . . . . . 5046.6 Convergence of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5146.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

7 Economic Equilibrium and Optimal Economic Planning . . . 5297.1 Perfectly Competitive Economies: Core and Walras Equilibria 5307.2 Infinite Horizon Multisector Growth Models . . . . . . . . . . . . . . . . 5427.3 Turnpike Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5587.4 Stochastic Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5677.5 Continuous-Time Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . 5897.6 Expected Utility Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5997.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

8 Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6098.1 Noncooperative Games–Nash Equilibrium . . . . . . . . . . . . . . . . . . 6108.2 Cooperative Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6178.3 Cournot–Nash Equilibria for Random Games . . . . . . . . . . . . . . . 6248.4 Bayesian Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6288.5 Stochastic Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6328.6 Approximate Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6448.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Contents xvii

9 Uncertainty, Information, Decision Making . . . . . . . . . . . . . . . . 6519.1 Mathematical Space of Information . . . . . . . . . . . . . . . . . . . . . . . . 6529.2 The ex-Post View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6629.3 The ex-ante View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6689.4 Convergence of σ-Fields and Prediction Sequences . . . . . . . . . . . 6739.5 Games with Incomplete Information . . . . . . . . . . . . . . . . . . . . . . . 6809.6 Markov Decision Chains with Unbounded Costs . . . . . . . . . . . . . 6849.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

10 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69110.1 Lebesgue–Bochner Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69210.2 Semilinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70110.3 Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71810.4 Second-Order Nonlinear Evolution Equations . . . . . . . . . . . . . . . 74010.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783