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VARIATIONAL INEQUALITIES, INFINITE-DIMENSIONAL DUALITY, INVERSE PROBLEM AND APPLICATIONS TO OLIGOPOLISTIC MARKET EQUILIBRIUM PROBLEM Annamaria Barbagallo Joint work with Antonino Maugeri Department of Mathematics and Computer Science

  1. VARIATIONAL INEQUALITIES, INFINITE-DIMENSIONAL DUALITY, INVERSE PROBLEM AND APPLICATIONS TO OLIGOPOLISTIC MARKET EQUILIBRIUM PROBLEM Annamaria Barbagallo Joint work with Antonino Maugeri Department of Mathematics and Computer Science University of Catania Conference on Applied Inverse Problems 2009 Wien, July 20 th , 2009 A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 1 / 36

  2. Outlines Outline 1 Introduction Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

  3. Outlines Outline 1 Introduction Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem 2 Existence results Existence theorem A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

  4. Outlines Outline 1 Introduction Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem 2 Existence results Existence theorem 3 Regularity results Mosco’s convergence Regularity results A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

  5. Outlines Outline 1 Introduction Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem 2 Existence results Existence theorem 3 Regularity results Mosco’s convergence Regularity results 4 Duality theorem and inverse problem Duality theory Inverse problem A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

  6. Introduction Some contributions on VI, IDD and OMEP Motivation Problem An oligopolistic market equilibrium problem is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms. A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 3 / 36

  7. Introduction Some contributions on VI, IDD and OMEP Motivation Problem An oligopolistic market equilibrium problem is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms. Motivation The reason for which dynamic oligopolistic market problem and evolutionary variational inequality which expresses dynamic equilibrium condition are studied is that necessary to consider the dynamics of network adjustment process in which a lag response is operating. A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 3 / 36

  8. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  9. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; • Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of variational inequalities; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  10. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; • Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of variational inequalities; • Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational inequalities in infinite-dimensional spaces; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  11. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; • Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of variational inequalities; • Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational inequalities in infinite-dimensional spaces; • Mosco (Adv. Math., 1969): introduction of sets convergence; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  12. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; • Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of variational inequalities; • Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational inequalities in infinite-dimensional spaces; • Mosco (Adv. Math., 1969): introduction of sets convergence; • Dafermos and Nagurney (Regional Science and Urban Economics, 1987): study on oligopolistic and competitive behavior of spatially separated markets in the static case; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  13. Introduction Some contributions on VI, IDD and OMEP Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem • Cournot (Researches into the Mathematical Principles of the Theory of Wealth, 1838): consider a two-firm competitive oligopoly problem in which firms sought to determine their profit-maximizing production quantities; • Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of variational inequalities; • Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational inequalities in infinite-dimensional spaces; • Mosco (Adv. Math., 1969): introduction of sets convergence; • Dafermos and Nagurney (Regional Science and Urban Economics, 1987): study on oligopolistic and competitive behavior of spatially separated markets in the static case; • Daniele-Idone-Giuffr´ e-Maugeri (Math. Ann., 2007): study the infinite-dimensional duality by means of quasi-relative interior. A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

  14. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

  15. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: • m firms P i , i = 1 , 2 , . . . , m ; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

  16. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: • m firms P i , i = 1 , 2 , . . . , m ; • n demand markets Q j , j = 1 , 2 , . . . , n ; A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

  17. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: • m firms P i , i = 1 , 2 , . . . , m ; • n demand markets Q j , j = 1 , 2 , . . . , n ; • are generally spatially separated. A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

  18. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: • m firms P i , i = 1 , 2 , . . . , m ; • n demand markets Q j , j = 1 , 2 , . . . , n ; • are generally spatially separated. Let us assume that: A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

  19. Introduction Dynamic oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem Let us consider: • m firms P i , i = 1 , 2 , . . . , m ; • n demand markets Q j , j = 1 , 2 , . . . , n ; • are generally spatially separated. Let us assume that: • homogeneous commodity, produced by the m firms and consumed at the n markets, is involved during a period of time [0 , T ], T > 0. A. Barbagallo (University of Catania) VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

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