Skip to main content
SpringerLink
Log in

Recent Advances in Minimax Theory and Applications

  • Chapter
Pareto Optimality, Game Theory And Equilibria

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 17))

In this chapter, we give an overview of various applications of a recent minimax theorem. Among them, there are some multiplicity theorems for nonlinear equations as well as a general well-posedness result for functionals with locally Lipschitzian derivative.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bai, Z.B., Ge, W.G.: Existence of Three Positive Solutions for Some Second-Order Boundary Value Problems. Comput. Math. Appl., 48, 699–707 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonanno, G.: Existence of Three Solutions for a Two Point Boundary Value Problem. Appl. Math. Lett., 13, 53–57 (2000)

    Article  MathSciNet  Google Scholar 

  3. Bonanno, G.: A Minimax Inequality and Its Applications to Ordinary Differential Equations. J. Math. Anal. Appl., 270, 210–229 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonanno, G.: Some Remarks on a Three Critical Points Theorem. Nonlinear Anal., 54, 651–665 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonanno, G., Livrea, R.: Multiplicity Theorems for the Dirichlet Problem Involving the p-Laplacian. Nonlinear Anal., 54, 1–7 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cammaroto, F., Chinnì, A., Di Bella, B.: Multiple Solutions for a Two Point Boundary Value Problem. J. Math. Anal. Appl., 323, 530–534 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cammaroto, F., Chinnì, A., Di Bella, B.: Multiple Solutions for a Quasilinear Elliptic Variational System on Strip-Like Domains. Proc. Edinb. Math. Soc., 50, 597–603 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Cammaroto, F., Chinnì, A., Di Bella, B.: Multiplicity Results for Nonlinear Schrödinger Equation. Glasg. Math. J., 49, 423–429 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cordaro, G.: On a Minimax Problem of Ricceri. J. Inequal. Appl., 6, 261–285 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cordaro, G.: Further Results Related to a Minimax Problem of Ricceri. J. Inequal. Appl., 523–533 (2005)

    Google Scholar 

  11. Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, 1543, Springer-Verlag (1993)

    Google Scholar 

  12. Efimov, N.V., Stechkin, S.B.: Approximate Compactness and Chebyshev Sets. Soviet Math. Dokl., 2, 1226–1228 (1961)

    Google Scholar 

  13. Fan, K.: Fixed-Point and Minimax Theorems in Locally Convex Topological Linear Spaces. Proc. Nat. Acad. Sci. U.S.A., 38, 121–126 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  14. Faraci, F., Iannizzotto, A.: An Extension of a Multiplicity Theorem by Ricceri with an Application to a Class of Quasilinear Equations. Studia Math., 172, 275–287 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Faraci, F., Iannizzotto, A., Lisei, H., Varga, C.: A Multiplicty Result for Hemivariational Inequalities. J. Math. Anal. Appl., 330, 683–698 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Geraghty, M.A., Lin, B.-L.: Topological Minimax Theorems. Proc. Amer. Math. Soc., 91, 377–380 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Horvath, C.: Quelques Théorèmes en Théorie des Mini-Max. C. R. Acad. Sci. Paris, Série I, 310, 269–272 (1990)

    MATH  Google Scholar 

  18. König, H.: A General Minimax Theorem Based on Connectedness. Arch. Math. (Basel), 59, 55–64 (1992)

    MATH  MathSciNet  Google Scholar 

  19. Kristály, A.: Multiplicity Results for an Eigenvalue Problem for Hemivariational Inequalities in Strip-Like Domains. Set-Valued Anal., 13, 85–103 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kristály, A.: Existence of Two Nontrivial Solutions for a Class of Quasilinear Elliptic Variational Systems on Strip-Like Domains. Proc. Edinb. Math. Soc., 48, 465–477 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kristály, A.: Multiple Solutions for a Sublinear Schrödinger Equation. NoDEA Nonlinear Differential Equations Appl., 14, 291–301 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kristály, A., Varga, C.: On a Class of Nonlinear Eigenvalue Problems in R N. Math. Nachr., 278, 1756–1765 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Livrea, R.: Existence of Three Solutions for a Quasilinear Two Point Boundary Value Problem. Arch. Math. (Basel), 79, 288–298 (2002)

    MATH  MathSciNet  Google Scholar 

  24. Marano, S.A., Motreanu, D.: On a Three Critical Points Theorem for Non-Differentiable Functions and Applications to Nonlinear Boundary Value Problems. Nonlinear Anal., 48, 37–52 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Naselli, O.: On a Class of Functions with Equal Infima Over a Domain and Its Boundary. J. Optim. Theory Appl., 91, 81–90 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Naselli, O.: On the Solution Set of an Equation of the Type f(t,Φ(u)(t)) = 0. Set-Valued Anal., 4, 399–405 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nikaidô, H.: On von Neumann’s Minimax Theorem. Pacific J. Math., 4, 65–72 (1954)

    MATH  MathSciNet  Google Scholar 

  28. Pucci, P., Serrin, J.: A Mountain Pass Theorem. J. Differential Equations, 60, 142–149 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ricceri, B.: Some Topological Mini-Max Theorems via an Alternative Principle for Multifunctions. Arch. Math. (Basel), 60, 367–377 (1993)

    MATH  MathSciNet  Google Scholar 

  30. Ricceri, B.: A Variational Property of Integral Functionals on L p-Spaces of Vector-Valued Functions. C. R. Acad. Sci. Paris, Série I, 318, 337–342 (1994)

    MATH  MathSciNet  Google Scholar 

  31. Ricceri, B.: On the Integrable Selections of Certain Multifunctions. Set-Valued Anal., 4, 91–99 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ricceri, B.: More on a Variational Property of Integral Functionals. J. Optim. Theory Appl., 94, 757–763 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ricceri, B.: On a Topological Minimax Theorem and Its Applications. In: Ricceri, B., Simons, S. (eds) Minimax Theory and Applications, Kluwer Academic Publishers, 191–216 (1998)

    Google Scholar 

  34. Ricceri, B.: On a Three Critical Points Theorem. Arch. Math. (Basel), 75, 220–226 (2000)

    MATH  MathSciNet  Google Scholar 

  35. Ricceri, B.: Further Considerations on a Variational Property of Integral Functionals. J. Optim. Theory Appl., 106, 677–681 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ricceri, B.: Sublevel Sets and Global Minima of Coercive Functionals and Local Minima of Their Perturbations. J. Nonlinear Convex Anal., 5, 157–168. (2004)

    MATH  MathSciNet  Google Scholar 

  37. Ricceri, B.: A General Multiplicity Theorem for Certain Nonlinear Equations in Hilbert Spaces. Proc. Amer. Math. Soc., 133, 3255–3261 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ricceri, B.: Minimax Theorems for Limits of Parametrized Functions Having at Most One Local Minimum Lying in a Certain Set. Topology Appl., 153, 3308–3312 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ricceri, B.: Uniqueness Properties of Functionals with Lipschitzian Derivative. Port. Math. (N.S.), 63, 393–400 (2006)

    MATH  MathSciNet  Google Scholar 

  40. Ricceri, B.: On the Existence and Uniqueness of Minima and Maxima on Spheres of the Integral Functional of the Calculus of Variations. J. Math. Anal. Appl., 324, 1282–1287 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)

    Google Scholar 

  42. Saint Raymond, J.: Connexité des Sous-Niveaux des Fonctionnelles Intégrales. Rend. Circ. Mat. Palermo, 44, 162–168 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. Shilgba, L.K.: Multiplicity of Periodic Solutions for a Boundary Eigenvalue Problem. Dyn. Syst., 20, 223–232 (2005)

    MATH  MathSciNet  Google Scholar 

  44. Simons, S.: Minimax Theorems and Their Proofs. In: Du, D.-Z., Pardalos, P.M. (eds) Minimax and Applications. Kluwer Academic Publishers, 1–23 (1995)

    Google Scholar 

  45. Simons, S.: Minimax and Monotonicity. Lecture Notes in Mathematics, 1693, Springer-Verlag (1998)

    Google Scholar 

  46. Sion, M.: On General Minimax Theorems. Pacific J. Math., 8, 171–176 (1958)

    MATH  MathSciNet  Google Scholar 

  47. Tsar’kov, I.G.: Nonunique Solvability of Certain Differential Equations and Their Connection with Geometric Approximation Theory. Math. Notes, 75, 259–271 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  48. Tuy, H.: On a General Minimax Theorem. Soviet Math. Dokl., 15, 1689–1693 (1974)

    MATH  Google Scholar 

  49. Von Neumann, J.: Zur Theorie der Gesellschaftspiele. Math. Ann., 100, 295–320 (1928)

    Article  MATH  MathSciNet  Google Scholar 

  50. Wu, W.T.: A Remark on the Fundamental Theorem in the Theory of Games. Sci. Rec. (New Ser.), 3, 229–233 (1959)

    Google Scholar 

  51. Wu, X.: Saddle Point Characterization and Multiplicity of Periodic Solutions of Non-Autonomous Second-Order Systems. Nonlinear Anal., 58, 899–907 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  52. Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific (2002)

    Google Scholar 

  53. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Vol. II/B, Springer-Verlag (1985)

    Google Scholar 

  54. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Vol. III, Springer-Verlag (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Catania, Catania, Italy

    Biagio Ricceri

Authors
  1. Biagio Ricceri
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos , Altannar Chinchuluun  & Panos M. Pardalos ,  & 

  2. Department of Production Engineering and Management, Technical University of Crete, 73100, Greece

    Athanasios Migdalas

  3. Department of Production Engineering and Management, Aristotle University of Thessaloniki, Greece

    Leonidas Pitsoulis

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Ricceri, B. (2008). Recent Advances in Minimax Theory and Applications. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds) Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77247-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation