Existence and Multiplicity Results for a Non Linear Hammerstein Integral Equation

  • F. Faraci
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

In this paper we study the solvability of a nonlinear Hammerstein integral equation by using a variational principle of B. Ricceri and methods of critical point theory. In particular we do not require any positivity assumption on the kernel of the equation. Our results can be applied to higher order elliptic boundary value problem with changing sign kernel.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications J. Functional Analysis 14 (1973) 349–381.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. Anello, G. Cordaro, An existence and localization theorem for the solutions of a Dirichlet problem, Ann. Polon. Math., to appear.Google Scholar
  3. [3]
    F. Faraci, Bifurcation theorems for Hammerstein nonlinear integral equations Glasgow Math. J. 44 (2002) 471–481.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997) 589–626.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    H. Hofer, A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. London Math. Soc. (2) 31 (1985), 566–570.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    V. Kozlov, V. Maz’ya, J. Rossmann, Elliptic boundary value problems in domain with point singularities. Mathematical Surveys and Monographs, Vol. 52, A.M.S., 1997.Google Scholar
  7. [7]
    M.A. Krasnoselskii, Topological methods in the theory of non-linear integral equations. Macmillan, New York, 1964.Google Scholar
  8. [8]
    S.G. Krein, Ju.I. Petunin, Scales of Banach spaces (Russian). Uspehi Mat. Nauk (1966) 2 89–168.MathSciNetGoogle Scholar
  9. [9]
    V. Moroz, On the Morse critical groups for indefinite sublinear elliptic problems, Nonlinear Anal. Ser. A: Theory Methods. 52 (2003), 1441–1453.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    V. Moroz, P. Zabreiko, On the Hammerstein equations with natural growth conditions. Z. Anal. Anwendungen 18 (1999), 625–638.MATHMathSciNetGoogle Scholar
  11. [11]
    A. Povolotskii, P. Zabreiko, On the theory of Hammerstein equations (Russian). Ukrain. Mat. Zh. 22 (1970), 150–162.Google Scholar
  12. [12]
    E.H. Spanier, Algebraic topology, McGraw-Hill Book Co., New-York, 1966.MATHGoogle Scholar
  13. [13]
    P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math, 65 A.M.S., R.I., 1986.Google Scholar
  14. [14]
    B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–10.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    P. Zabreiko, On the theory of Integral Operators (Russian), Thesis, Voronej State University, 1968.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • F. Faraci
    • 1
  1. 1.Dept. of MathematicsUniversity of CataniaCataniaItaly

Personalised recommendations