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Handbook of applied analysis. (English) Zbl 1189.49003

Advances in Mechanics and Mathematics 19. New York, NY: Springer (ISBN 978-0-387-78906-4/hbk; 978-0-387-78907-1/ebook). xvii, 783 p. (2009).
The aim of the handbook is to present a survey of the basic modern aspects of nonlinear analysis. The treatment of all subjects is rigorous and mathematically precise; every chapter ends with a survey of the literature.
In Chapter 1 the Gâteaux and Fréchet derivatives for smooth functions on Banach spaces are presented and their calculus is developed in full detail; in particular, two basic theorems, namely the implicit function theorem and the inverse function theorem, are provided. In the direction of nonsmooth functions, convex functions provide the link that connects smooth and nonsmooth calculus. First the notion of subdifferential is introduced for convex functions, and a duality theory and a theory of subdifferentiation is developed. Subsequently, a generalization of the notion of subdifferential for locally Lipschitz functions (Clarke’s theory) is presented. Subsequently, the authors introduce and study related geometrical concepts (such as tangent and normal cones) for various kinds of sets. Finally, they investigate a kind of variational convergence of functions, known as \(\Gamma\)-convergence, which is suitable in the stability (sensitivity) analysis of variational problems.
Chapter 2 is devoted to the study of variational problems and of the mathematical tools needed to deal with them. First, in order to investigate the so-called direct method of the calculus of variations, a detailed study of the notion of lower semicontinuity of functions on locally convex spaces is presented. Infinite-dimensional optimization problems with constraints are analyzed by developing the method of Lagrange multipliers. This leads to mimimax theorems, saddle points and the theory of KKM-multimaps, which are useful in producing coincidence theorems for families of multifunctions and also lead to existence theorems for variational inequalities. Section 2.4 deals with some modern aspects of the direct method, which involve the so-called variational principles, central among them being the so-called “Ekeland variational principle”. The last two sections concern the calculus of variations, where problems with integral cost functional and fixed endpoints are considered, and the optimal control, with systems monitored by ordinary differential equations. An existence theory is developed in the presence of convexity; otherwise, four relaxation methods are shown. A necessary condition for optimality in the form of a maximum principle (Pontryagin’s maximum principle) is developed.
In Chapter 3 certain classes of nonlinear operators that arise naturally in applications are studied. Section 3.1 concerns compact operators that were the first attempt to deal with infinite-dimensional operator equations. The basics of both the linear (compact self–adjoint operators on a Hilbert space) and nonlinear theories are presented. In order to broaden this class, monotone operators from a Banach space \(X\) to its dual \(X^*\), and accretive operators from \(X\) in itself are considered. Monotone operators exhibit remarkable surjectivity properties which play a central role in the existence theory of nonlinear boundary value problems, while accretive operators are closely related to the generation theory of linear and nonlinear semigroups. In particular, maximal monotone operators are studied in detail. In Section 3.3 the basic Brouwer degree (finite-dimensional) and the Leray-Schauder degree theory (extension to the infinite-dimensional setting) are presented. Having these degree maps, one is led smoothly to the fixed point theory: metric fixed points and topological fixed points are studied and the interplay between order and fixed point theory is investigated.
In Chapter 4 the abstract results that are necessary in order to implement the variational method in the study of nonlinear boundary value problems are developed. First of all the critical point theory for smooth functionals is considered; the approach followed by the authors to characterize the critical values of the functional is based on deformation arguments. The classical results such as the mountain pass theorem, the saddle point theorem, and the generalized mountain pass theorem are derived. In Section 4.2 the Ljusternik-Schnirelman theory for smooth functionals defined on a Banach space is presented. Topological indices are introduced as substitutes of the notion of dimension. This way one has all the necessary tools to develop the spectral properties of the Laplacian and of the \(p\)–Laplacian (under Dirichlet, Neumann and periodic boundary conditions). Then, using the Lagrange multipliers method, one can deal with abstract eigenvalue problems. Finally some basic notions and results from bifurcation theory are presented.
Chapter 5 uses the tools developed in Chapters 3 and 4 in order to study nonlinear boundary value problems (involving ordinary differential equations and elliptic partial differential equations). First the authors illustrate the variational method based on the minimax principles of critical point theory and then they present the method of upper and lower solutions and the degree–theoretic method. Subsequently they illustrate the degree-theoretical approach in the study of nonlinear boundary value problems, producing constant sign and nodal (sign changing) solutions. Then some maximum and comparison principles involving Laplacian and \(p\)-Laplacian differential operators are proved. Finally periodic Hamiltonian systems are studied, and two existence theorems are proved: one for a prescribed minimal period, and the other for a prescribed energy level.
The aim of Chapter 6 is to introduce a survey of some of the basic aspects of the theory of multivalued analysis. The notion of continuity and the measurability properties of multifunctions are presented. Then, the fundamental problem of the existence of measurable selectors for multifunctions is investigated (Michael’s theorem and Kuratowski-Ryll Nardzewski, and the Yankov-von Neumann-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. Via the notion of decomposability (an effective substitute of convexity) that plays a central role in this direction, the authors prove fixed point theorems for multifunctions and also study Carathéodory multifunctions. Finally they introduce and study various notions of convergence of sets that arise naturally in applications.
Chapter 7 focuses on two particular topics dealing with mathematical economics that hold a central position in economic theory: the theory of competitive markets (in particular their equilibrium theory) and the theory of growth. First of all, a static model of an exchange economy is discussed. Assuming that perfect competition prevails, which is modelled by a continuum (nonatomic measure space) of agents, a “core Walras equivalence theorem” is proved and the existence of Walras allocations is established. Then the attention is turned to growth theory (dynamic models). Infinite horizon, discrete-time, multisector growth models are considered and the existence of optimal programs for both discounted and undiscounted models are established. For the latter, the notion of “weak maximality” is introduced and it is proved that the model admits maximal program. Then asymptotic properties of optimal programs are determined via weak and strong turnpike theorems. Growth under uncertainty is investigated for both nonstationary, discounted and stationary, undiscounted models; existence and characterization results that are the analogous to the ones proved for the deterministic case are shown. Continuous-time discounted models are then considered, and finally the authors characterize choice behavior consistent with the “expected utility hypothesis”.
In Chapter 8 the authors deal with different models in game theory, which provide a substantial amount of generalization of some of the notions considered in the previous chapter. Starting with noncooperative \(n\)-players games, for which the notion of “Nash equilibrium” is introduced, the existence of such equilibria is proved. Then cooperative \(n\)-players games are considered, in case of both side-payment and no-side-payment: theorems are provided showing the nonemptiness of the core. Then random games are investigated with a continuum of players and an infinite-dimensional strategy space. For such games, the existence of “Cournot–Nash equilibria” is proved. Subsequently, using the formalism of dynamic programming, stochastic, 2-player, zero-sum games are considered. Finally, using approximate subdifferentials for convex function, approximate Nash equilibria for noncooperative games with noncompact strategy sets are produced.
Chapter 9 studies how information can be incorporated as a variable in various decision models (in particular in ones with asymmetric information structure). First the mathematical framework is presented, which will allow the analytical treatment of the notion of information. To this purpose, two comparable metric topologies are defined and studied in detail. Then the ex-post view is examined, in modelling systems with uncertainty; the continuity of this model with respect to the information variable is proved. An alternative approach, the ex-ante view, is examined. Subsequently, a third mode of convergence of information is introduced, and it is shown that it is suitable in the analysis of prediction sequences. Also games with incomplete information and games with a general state space and an unbounded cost function are studied.
The final chapter (Chapter 10) is devoted to the evolution equations and the mathematical tools associated with them. These tools are developed in the first section and a central role among them is played by the notion of “evolution triple”; some basic function spaces associated to these evolution triples are examined. Moreover, semilinear evolution equations are investigated using the semigroup method. Subsequently, two classes of nonlinear evolution equations are considered, one involving time-invariant subdifferential operators, including variational inequalities, the other one involving operators of monotone type and formulated in the framework of evolution triples. The first class is treated using nonlinear semigroup theory while the second one requires Galerkin approximations.
Reviewer: Rita Pini (Milano)

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
34G20 Nonlinear differential equations in abstract spaces
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
91A10 Noncooperative games
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