This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

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Aspects of topology: On dimension theory

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Nonlinear Functional Analysis And Its Applications

Part I Game and Game Theory.- Minimax: Existence and Stability.- Recent Advances in Minimax Theory and Applications.- On Noncooperative Games, Minimax Theorems and Equilibrium Problems.- Nonlinear Games.- Scalar Asymptotic Contractivity and Fixed-Points for Nonexpansive Mappings on Unbounded Sets.- Cooperative Combinatorial Games.- Algorithmic Cooperative Game Theory.- A Survey of Bicooperative Game Theory.- Cost Allocation in Combinatorial Optimization Games.- Time-Dependent Equilibrium Problems.- Differential Games of Multiple Agents and Geometrical Structures.- Convexity in Differential Games.- Game Dynamic Problems for Systems with Fractional Derivatives.- Projected Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational Procedure.- Strategic Audit Policies Without Commitment.- Optimality and Efficiency in Auctions Design: A Survey.- Part II Multiobjective, KKT, Bilevel.- Solution Concepts and an Approximation Kuhn-Tucker Approach for Fuzzy Multi-Objective Linear Bilevel Programming.- Pareto Optimality.- Multiobjective Optimization: A Brief Overview.- Parametric Multiobjective Optimization.- Part III Applications.- The Extended Linear Complementarity Problem and Its Applications in Analysis and Control of Discrete-Event Systems.- Traffic Assignmetn: Equilibrium Models.- Investment Paradoxes in Electricity Networks.- Algorithms for Network Interdiction and Fortification Games.- Game Theoretical Approaches in Wireless Networks.- A Military Application of Viability: Winning Cones, Differential Inclusions and Lanchester Type Models for Combat.- Statics and Dynamics of Global Supply Chain Networks.- Game Theory Models and Their Applications in Inventory Management in Supply Chain.

Pareto optimality, game theory and equilibria

In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5].

On a three critical points theorem

A general variational principle and some of its applications

In this paper, given a reflexive real Banach space X and two sequentially weakly lower semicontinuous functionals Phi, Psi on X with Psi strongly continuous and coercive, we are mainly interested in the existence of infinitely many local minima of the functional capital Phi + r Psi for each sufficiently real r.

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper the following result is proved: Let X be a reflexive real Banach space; I ⊆ R an interval; Φ : X → R a sequentially weakly lower semicontinuous C 1 functional, bounded on each bounded subset of X , whose derivative admits a continuous inverse on X ∗ ; J : X → R a C 1 functional with compact derivative Assume that lim ‖ x ‖ → + ∞ ( Φ ( x ) + λ J ( x ) ) = + ∞ for all λ ∈ I , and that there exists ρ ∈ R such that sup λ ∈ I inf x ∈ X ( Φ ( x ) + λ ( J ( x ) + ρ ) ) inf x ∈ X sup λ ∈ I ( Φ ( x ) + λ ( J ( x ) + ρ ) )  Then, there exist a non-empty open set A ⊆ I and a positive real number r with the following property: for every λ ∈ A and every C 1 functional Ψ : X → R with compact derivative, there exists δ > 0 such that, for each μ ∈ [ 0 , δ ] , the equation Φ ′ ( x ) + λ J ′ ( x ) + μ Ψ ′ ( x ) = 0 has at least three solutions in X whose norms are less than r 

Biagio Ricceri

Papers

A general variational principle and some of its applications

A general variational principle and some of its applications

On a three critical points theorem

Existence of three solutions for a class of elliptic eigenvalue problems

A three critical points theorem revisited