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Preface to the Dover Edition | |

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Preface (1982) | |

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Summary of Results: A Guideline for the Reader | |

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Contents of Other Possible Courses | |

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Notations | |

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Optimization and Convex Analysis | |

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Minimization Problems and Convexity | |

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Strategy sets and loss functions | |

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Optimization problem | |

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Allocation of available commodities | |

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Resource and service operators | |

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Extension of loss functions | |

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Sections and epigraphs | |

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Decomposition principle | |

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Product of a loss function by a linear operator | |

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Example: Inf-convolution of functions | |

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Decomposition principle | |

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Another decomposition principle | |

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Mixed strategies and convexity | |

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Motivation: extension of strategy sets and loss functions | |

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Mixed strategies and linearized loss functions | |

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Interpretation of mixed strategies | |

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Case of finite strategy sets | |

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Representation by infinite sequences of pure strategies | |

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Linearized extension of maps and the barycentric operator | |

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Interpretation of convex functions in terms of risk aversion | |

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Elementary properties of convex subsets and functions | |

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Indicators, support functions and gauges | |

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Indicators and support functions | |

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Reformulation of the Hahn-Banach theorem | |

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The bipolar theorem | |

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Recession cones and barrier cones | |

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Interpretation: production sets and profit functions | |

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Gauges | |

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Existence, Uniqueness and Stability of Optimal Solutions | |

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Existence and uniqueness of an optimal solution | |

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Structure of the optimal set | |

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Existence of an optimal solution | |

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Continuity versus compactness | |

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Lower semi-continuity of convex functions in infinite dimensional spaces | |

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Fundamental property of lower semi-continuous and compact functions | |

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Uniqueness of an optimal solution | |

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Non-satiation property | |

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Minimization of quadratic functionals on convex sets | |

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Hilbert spaces | |

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Existence and uniqueness of the minimal solution | |

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Characterization of the minimal solution | |

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Projectors of best approximation | |

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The duality map from an Hilbert space onto its dual | |

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Minimization of quadratic functionals on subspaces | |

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The fundamental formula | |

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Orthogonal right inverse | |

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Orthogonal left inverse | |

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Another decomposition property | |

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Interpretation | |

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Perturbation by linear forms: conjugate functions | |

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Conjugate functions | |

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Characterization of lower semi-continuous convex functions | |

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Examples of conjugate functions | |

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Elementary properties of conjugate functions | |

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Interpretation: cost and profit functions | |

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Stability properties: an introduction to correspondences | |

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Upper semi-continuous correspondences | |

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Lower semi-continuous correspondences | |

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Closed correspondences | |

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Construction of upper semi-continuous correspondences | |

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Compactness and Continuity Properties | |

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Lower semi-compact functions | |

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Coercive and semi-coercive functions | |

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Functions such that f* is continuous at 0 | |

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Lower semi-compactness of linear forms | |

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Constraint qualification hypothesis | |

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Case of infinite dimensional spaces | |

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Extension to compact subsets of mixed strategies | |

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Proper maps and preimages of compact subsets | |

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Proper maps | |

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Compactness of some strategy sets | |

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Examples where the map L* + 1 is proper | |

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Continuous convex functions | |

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A characterization of lower semi-continuous convex functions | |

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A characterization of continuous convex functions | |

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Examples of continuous convex functions | |

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Continuity of gL and Lf | |

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Continuous convex functions (continuation) | |

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Strong continuity of lower semi-continuous convex functions | |

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Estimates of lower semi-continuous convex functions | |

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Characterization of continuous convex functions | |

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Continuity of support functions | |

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Maximum of a convex function: extremal points | |

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Differentiability and Subdifferentiability: Characterization of Optimal Solutions | |

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Subdifferentiability | |

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Definitions | |

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Examples of subdifferentials | |

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Subdifferentiability of continuous convex functions | |

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Upper semi-continuity of the subdifferential | |

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Characterization of subdifferentiable convex functions | |

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Differentiability and variational inequalities | |

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Definitions | |

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Differentiability and subdifferentiability | |

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Legendre transform | |

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Interpretation: marginal profit | |

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Variational inequalities | |

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Differentiability from the right | |

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Definition and main inequalities | |

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Derivatives from the right and the support function of the subdifferential | |

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Derivative of a pointwise supremum | |

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Local [epsilon]-subdifferentiability and perturbed minimization problems | |

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Approximate optimal solutions in Banach spaces | |

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The approximate variational principle | |

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Local [epsilon]-subdifferentiability | |

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Perturbation of minimization problems | |

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Proof of Ekeland-Lebourg's theorem | |

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Introduction to Duality Theory | |

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Dual problem and Lagrange multipliers | |

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Lagrangian | |

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Lagrange multipliers and dual problem | |

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Marginal interpretation of Lagrange multipliers | |

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Example | |

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Case of linear constraints: extremality relations | |

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Generalized minimization problem | |

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Extremality relations | |

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The fundamental formula | |

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Minimization problem under linear constraints | |

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Minimization of a quadratic functional under linear constraints | |

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Minimization problem under linear equality constraints | |

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Duality and the decomposition principle | |

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The decentralization principle | |

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Conjugate function of gL | |

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Conjugate function of f[subscript 1]+f[subscript 2] | |

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Minimization of the projection of a function | |

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Minimization on the diagonal of a product | |

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Existence of Lagrange multipliers in the case of a finite number of constraints | |

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The Fenchel existence theorem | |

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Stability properties | |

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Applications to subdifferentiability | |

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Case of nonlinear constraints: The Uzawa existence theorem | |

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Game Theory and the Walras Model of Allocation of Resources | |

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Two-Person Games: An Introduction | |

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Some solution concepts | |

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Description of the game | |

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Shadow minimum | |

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Conservative solutions and values | |

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Non-cooperative equilibrium | |

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Pareto minimum | |

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Core of a two-person game | |

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Selection of strategy of the core | |

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Examples: some finite games | |

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Example | |

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Coordination game | |

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Prisoner's dilemma | |

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Game of chicken | |

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The battle of the sexes | |

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Example: Analysis of duopoly | |

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The model of a duopoly | |

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The set of Pareto minima | |

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Conservative solutions | |

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Non-cooperative equilibria | |

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Stackelberg equilibria | |

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Stackelberg disequilibrium | |

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Example: Edgeworth economic game | |

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The set of feasible allocations | |

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The biloss operator | |

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The Edgeworth box | |

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Pareto minima | |

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Core | |

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Walras equilibria | |

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Two-person zero-sum games | |

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Duality gap and value | |

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Saddle point | |

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Perturbation by linear functions | |

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Case of finite strategy sets: Matrix games | |

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Two-Person Zero-Sum Games: Existence Theorems | |

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The fundamental existence theorems | |

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Existence of conservative solutions | |

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Decision rules | |

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Finite topology on convex subsets | |

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Existence of an optimal decision rule | |

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The Ky-Fan inequality | |

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The Lasry theorem | |

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The minisup theorem | |

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The Nikaido theorem | |

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Existence of saddle points | |

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Another existence theorem for saddle points | |

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Extension of games without and with exchange of informations | |

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Definition of extensions of games | |

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Mixed extensions | |

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Extensions without exchange of information | |

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Sequential extensions | |

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Extensions with exchange of information | |

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Iterated games | |

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Iterated extensions | |

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The Moulin theorem | |

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Proof of playability of iterated extensions | |

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A system of functional equations | |

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A lemma on successive approximations | |

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Proof of existence of saddle decision rules | |

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The Fundamental Economic Model: Walras Equilibria | |

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Description of the model | |

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The subset of available commodities | |

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Appropriation of the economy | |

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Demand correspondences | |

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Walras equilibrium | |

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Examples of subsets of available commodities and of appropriations | |

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Example: Quadratic demand functions | |

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Existence of a Walras equilibrium | |

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Existence of a Walras pre-equilibrium | |

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Surjectivity of correspondences: the Debreu-Gale-Nikaido theorem | |

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Demand correspondences defined by loss functions | |

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Statement of the existence theorem | |

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Upper semi-continuity of the demand correspondence | |

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Compactification of an economy | |

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Proof of the existence of a Walras equilibrium | |

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Economies with producers | |

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Description of the model | |

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Statement of the existence theorem | |

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Compactification | |

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Proof of the existence of a Walras equilibrium | |

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Non-Cooperative n-Person Games | |

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Existence of a non-cooperative equilibrium | |

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Games described in strategic form | |

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Conservative values and multistrategies | |

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Non-cooperative equilibria | |

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The Nash theorem | |

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Stability | |

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Associated variational inequalities | |

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Case of quadratic loss functions; application to Walras-Cournot equilibria | |

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Non-cooperative games with quadratic loss functions | |

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Existence of solutions of variational inequalities | |

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Examples | |

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Multistrategy sets defined by linear constraints | |

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Walras-Cournot equilibria | |

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Constrained non-cooperative games and fixed point theorems | |

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Selection of a fixed point | |

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Equilibria of constrained non-cooperative games | |

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Fixed-point theorems | |

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Non-cooperative Walras equilibria | |

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Description of the model | |

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Existence of a non-cooperative Walras equilibrium: the Arrow-Debreu theorem | |

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Non-cooperative Walras equilibria of economies with producers | |

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Main Solution Concepts of Cooperative Games | |

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Behavior of the whole set of players: Pareto strategies | |

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Pareto strategies | |

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Rates of transfer | |

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Pareto multipliers | |

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Pareto allocations | |

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Selection of Pareto strategies and imputations | |

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Normalized games | |

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Pareto strategies obtained by using selection functions | |

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Closest strategy to the shadow minimum | |

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The best compromise | |

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Existence of Pareto strategies | |

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Interpretation: threat functionals | |

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Imputations: the Nash bargaining solution | |

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Behavior of coalitions of players: the core | |

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Coalitions | |

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Cooperative game described in strategic form and its core | |

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The multiloss operator F[superscript A]# of the coalition A | |

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Examples of multistrategy sets X(A) | |

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Economic games and core of an economy | |

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Cooperative game described in characteristic form and its core | |

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Behavior of fuzzy coalitions: the fuzzy core | |

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Fuzzy coalitions | |

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Extension of a family of coalitions | |

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Debreu-Scarf coalitions | |

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Fuzzy coalitions on a continuum of players | |

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Fuzzy games described in characteristic form | |

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Characterization of the core of a (fuzzy) game | |

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Fuzzy economic games and fuzzy core of an economy | |

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Fuzzy games described in strategic form and fuzzy core | |

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Selection of elements of the core: cooperative equilibrium and nucleolus | |

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Canonical cooperative equilibrium | |

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Least-core | |

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Nucleolus | |

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Games With Side-Payments | |

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Core of a fuzzy game with side-payments | |

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Core of a game with side-payments | |

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Linear games | |

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Non-emptiness of the core of fuzzy games with side-payments | |

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Core of fuzzy market games | |

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Core of a game with side-payments | |

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Convex cover of a game | |

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Non-emptiness of the core of a balanced game | |

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Balanced family of multistrategy sets | |

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Balanced characteristic functions and convex loss functions | |

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Further properties of convex functions and balances | |

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Values of fuzzy games | |

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The diagonal property | |

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Sequence of fuzzy values | |

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Existence and uniqueness of a sequence of fuzzy values | |

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Relations between core and fuzzy value | |

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Best approximation property of fuzzy values | |

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Generalized solution to locally Lipschitz games | |

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Shapley value and nucleolus of games with side-payments | |

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The Shapley value | |

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Existence and uniqueness of a Shapley value | |

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Simple games | |

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Nucleolus of games with side-payments | |

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Games Without Side-Payments | |

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Equivalence between the fuzzy core and the set of equilibria | |

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Representation of a game | |

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Equilibrium of a representation | |

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Cover associated with a representation | |

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Fuzzy core of a representation | |

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The equivalence theorem | |

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Non-emptiness of the fuzzy core of a balanced game | |

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Statement of theorems of non-emptiness of the fuzzy core | |

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Upper semi-continuity of the associated side-payment games | |

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Existence of approximate cooperative equilibria | |

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Proof of the non-emptiness of the core | |

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Equivalence between the fuzzy core of an economy and the set of Walras allocations | |

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Representation of economic games | |

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Fuzzy core and Walras allocations | |

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The equivalence theorem | |

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Non-Linear Analysis and Optimal Control Theory | |

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Minimax Type Inequalities, Monotone Correspondences and [gamma]-Convex Functions | |

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Relaxation of compactness assumptions | |

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Existence of a conservative solution | |

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Proof of existence of a conservative solution | |

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Existence of optimal decision rules and minisup under weaker compactness assumptions | |

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Relaxation of continuity assumptions: variational inequalities for monotone correspondences | |

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Variational inequalities | |

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Existence of a solution to variational inequalities for completely upper semi-continuous correspondences | |

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Pseudo-monotone functions: the Brezis-Nirenberg-Stampacchia theorem | |

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Existence of a solution to variational inequalities for pseudo-monotone maps | |

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Pseudo-monotonicity of monotone maps | |

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Monotone and cyclically monotone correspondences | |

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Maximal monotone correspondences | |

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Relaxation of convexity assumptions | |

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Definition of [gamma]-convex functions | |

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The fundamental characteristic property of families of [gamma]-convex functions | |

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The minisup theorem for [gamma subscript x]-convex-[gamma subscript y]-concave functions | |

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Existence of optimal decision rules for functions [gamma subscript y]-concave with respect to y | |

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Example: Image of a cone of convex functions by [pi]* | |

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Relations between convexity and [gamma]-convexity | |

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Example: [beta]-convex set functions | |

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Example: Convex functions of atomless vector measures | |

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Introduction to Calculus of Variations and Optimal Control | |

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Duality in infinite dimensional spaces | |

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Lagrangian of a minimization problem under linear constraints | |

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Extremality relations | |

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Existence of a Lagrange multiplier under the Slater condition | |

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Relaxation of the Slater condition | |

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Generalized Lagrangian of a minimization problem | |

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Characterization of a Lagrangian by perturbations of the minimization problem | |

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Duality in the case of non-convex integral criterion and contraints | |

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Modulus of non-convexity of a function | |

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Estimate of the duality gap | |

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The Shapley-Folkman theorem | |

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Sharp estimate of the duality gap | |

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Applications | |

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Extremality relations | |

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The Aumann-Perles duality theorem | |

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The approximation procedure | |

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Duality in calculus of variations | |

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The Green formula | |

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Abstract problem of calculus of variations | |

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The Hamiltonian system | |

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Lagrangian of a problem of calculus of variations | |

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Existence of a Lagrange multiplier | |

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Example: the Dirichlet variational problem | |

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The maximum principle for optimal control problems | |

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Optimal control and impulsive control problems | |

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The Hamilton-Jacobi-Bellman equation of a control problem | |

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Construction of the closed loop control | |

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The principle of optimality | |

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The quadratic case: Riccati equations | |

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The Bensoussan-Lions variational inequalities of a stopping time problem | |

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Construction of the optimal stopping time | |

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The Bensoussan-Lions quasi-variational inequalities of an impulsive control problem | |

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Construction of the optimal impulsive control | |

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Fixed Point Theorems, Quasi-Variational Inequalities and Correspondences | |

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Fixed point and surjectivity theorems for correspondences | |

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The Browder-Ky-Fan existence theorem for critical points | |

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Properties of inward and outward correspondences | |

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Critical points of homotopic correspondences | |

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Other existence theorems for critical points | |

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Quasi-variational inequalities | |

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Selection of fixed point by pseudo-monotone functions | |

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Fixed point theorem for increasing maps | |

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Quasi-variational inequalities for increasing correspondences | |

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Other properties and examples of upper and lower semi-continuous correspondences | |

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Lower semi-continuity of preimages of linear operators | |

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Lower semi-continuity of correspondences defined by constraints | |

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Continuous selection theorem | |

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Weak Hausdorff topology on the family of closed subsets of topological vector spaces | |

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Relations between hemi-continuity and semi-continuity | |

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Summary of Linear Functional Analysis | |

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Hahn-Banach theorems | |

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Paired spaces | |

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Topologies of uniform convergence | |

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Topologies associated with a duality pairing | |

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The Banach-Steinhauss theorem | |

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The Knaster-Kuratowski-Mazurkiewicz Lemma | |

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Barycentric subdivision of simplexes | |

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Sequence of barycentric subdivisions | |

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The Sperner lemma | |

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The Knaster-Kuratowski-Mazurkiewicz lemma | |

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The Brouwer theorem | |

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Lyapunov's Theorem on the Range of A Vector Valued Measure | |

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Comments | |

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References | |

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Subject Index | |