Applied Nonlinear Analysis

This introductory text offers simple presentations of the fundamentals of nonlinear analysis, with direct proofs and clear applications. Its full treatment ranges from smooth to nonsmooth functions, from convex to nonconvex variational problems, and from economics to mechanics.Both authors are Professors of Mathematics at Paris-Dauphine University.

Background Notes

1

(32)

Set-Valued Maps

1

(4)

Complete Metric Spaces

5

(4)

Banach Spaces

9

(7)

Differentiable Functionals

16

(10)

Support Functions and Barrier Cones of Convex Subsets

26

(7)

Smooth Analysis

33

(70)

Iterative Procedures for Inverting a Map

34

(7)

Milnor's Proof of Brouwer's Fixed Point Theorem

41

(5)

Local Study of the Equation f(x)=0

46

(12)

Birth and Death of Critical Points

58

(12)

Further Degeneracies: Bifurcation

70

(13)

Transversality Theory

83

(8)

Proof of the Transversality Theorem and Applications

91

(12)

Set-Valued Maps

103

(56)

Upper and Lower Semicontinuity of Set-Valued Maps

107

(14)

Maps with Closed Convex Values

121

(9)

Maps with Closed Convex Graphs

130

(16)

Eigenvalues of positive Maps with Closed Convex Graphs

146

(13)

Convex Analysis and Optimization

159

(80)

Tangent and Normal Cones to Convex Subsets

166

(11)

Derivatives and Codifferentials of Set-valued Maps with Convex Graphs

177

(9)

Epiderivatives and Subdifferentials of Convex Functions

186

(14)

Conjugate Functions

200

(14)

The Subdifferential of the Marginal Function and Lagrange Multipliers

214

(6)

Convex Optimization Problems

220

(5)

Regularity of Solutions to Convex Optimization Problems

225

(6)

Lagrangians and Hamiltonians

231

(8)

A General Variational Principle

239

(56)

Walking on Complete Metric Spaces

239

(9)

Fixed Points of Nonexpansive Maps

248

(6)

The ε-Variational Principle

254

(7)

Applications to Convex Optimization

261

(8)

Condition (C) of Palais and Smale

269

(10)

Generic Differentiability

279

(6)

Perturbed Optimization Problems

285

(10)

Solving Inclusions

295

(106)

Main Concepts of Game Theory

299

(13)

Two-Person Zero-Sum Games: The Minimax Theorem

312

(13)

The Ky Fan Inequality

325

(11)

Existence of Zeros of Set-Valued Maps

336

(18)

Walras Equilibria and Price Decentralization

354

(9)

Monotone Maps

363

(16)

Maximal Monotone Maps

379

(17)

Existence and Uniqueness of Solutions to Differential Inclusions

396

(5)

Nonsmooth Analysis

401

(50)

Contingent and Tangent Cones

405

(6)

Contingent Derivatives and Derivatives of a Set-Valued Map

411

(7)

Epicontingent Derivatives and Epiderivatives of Real-Valued Functions

418

(10)

Generalized Second Derivatives of Real-Valued Functions

428

(1)

The Inverse Function Theorem for Set-Valued Maps

429

(10)

Calculus of Contingent and Tangent Cones, Derivatives and Epiderivatives

439

(12)

Hamiltonian Systems

451

(36)

The Least Action Principle

452

(4)

A Dual Action Principle

456

(8)

Nonresonant Problems

464

(5)

Resonant Problems

469

(6)

Transresonant Problems

475

(12)

Comments

487

(8)

Bibliography

495

(22)

Author Index

517

(1)

Subject Index

517

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Applied Nonlinear Analysis

Summary

Table of Contents

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