On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces


In this paper we deal with nontrivial solvability in balls of Hammerstein integral equations in Hilbert spaces for nonlinearities of potential type. We use a variational approach based on variants of the mountain pass theorem which are due to Guo-Sun-Qi and Schechter. Our main contribution is a new technique to verify compactness conditions of Palais-Smale type. This technique combines the compactness criterium for countable sets in \(L^p\) with basic properties of the measures of noncompactness and integral inequalities.


Radu Precup
Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania


Compactness; Hammerstein integral equation; Mountain pass theory.

Paper cordinates

R. Precup, On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces, Nonlinear  Anal. 47 (2001), 1233-1244. http://dx.doi.org/10.1016/S0362-546X(01)00261-9


About this paper

Cite this paper as:

Nonlinear Analysis

Publisher Name


Print ISSN

Not available yet.

Online ISSN


Google Scholar Profile

Zbl 1042.47530


[1] A. Ambrosetti, P.H. Rabinowitz,  Dual variational methods in critical point theory and applicaitons,  J. Funct. Anal. 14, 349-381, 1973.
[2] H. Brezis, Analyse fonctionnelle, Mason, Paris, 1983.
[3] D. Guo, V. Lakshmikantham, X. Liu,  Nonlinear integral equations in abstract spaces,  Kluwer Academic Publisher, Dordrecht, 1996.
[4] D. Guo, J. Sun, G. Qi,  Some extensions of the mountain pass lemma,  Differential Integral Equations 1, 351-358, 1988.
[5] O. Kavian, Introduction a le theorie des points critiques,  Springer, Berlin, 1995.
[6] M.A. Krasnoselskii,  Topological methods in the theory of nonlinear integral equations,  Pergamon Press, Oxford, 1964.
[7] J. Mawhin, M. Willem,  Critical point theorem and Hamiltonian system, Springer-Verlag, Berlin, 1989.
[8] V. Moroz, A. Vignoli, P. Zabvreiko,  On the three critical points theorem,  Topol. Methods Nonlinear Anal. 11, 103-113, 1998.
[9] V.B.Moroz, P.P. Zabreiko,  A variant of the mountain pass theorem and its application to Hammerstein integral equations,  Zeit. Anal. Anwendungen 15, 985-997, 1996.
[10] D. O’Regan, R. Precup,  Existence criteria for integral equations in Banasch spaces, J. Inequal. Appl., to appear.
[11] R. Precup, Nonlinear integral equations (In Romanian), University Babes-Bolyai, Cluj, 1993.
[12] R. Precup, Nontrivial solvability of Hammerstein integral equations in Hilbert spaces,  Seminaire de la theorie de la meilleure approximation, convexite et optimisation, Srima, Cluj-Napoca, 255-265, 2000.
[13] M. Schechter,  A bounded mountain pass lemma without the (PS) condition and applciations,  Trans. Amer. Math. Soc., 331, 681-703, 1992.


Related Posts