## Abstract

In this note some well-known existence and multiplicity results of nontrivial solutions for scalar Hammerstein equations [1], [3] are extended to equations in Hilbert spaces. The tools are a mountain pass theorem on closed convex substes of a Hilbert space due to Guo-Sun-Qi [1] and a new technique of checking the Palais-Smale compactness condition which was first presented in [4]. The results compplement those established in [4].

## Authors

**Radu Precup**

Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

Hammerstein integral equation; compactness; critical point theory.

## Paper coordinates

R. Precup, *Nontrivial solvability of Hammerstein integral equations in Hilbert spaces*, Seminaire de la Theorie de la Mielleure Approximation Convexite et Optimisation, Cluj-Napoca, 26 octombre – 29 octobre, 2000, pp. 255-265.

## About this paper

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Seminaire de la Theorie de la Meilleure Approximation, Convexite et Optimisation

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[1] D. Guo, J. Sun, G. Qi, *Some extensions of the mountain pass lemma*, Differential Integral Equaitons 1 (1988), 351-358.

[2] M.A. Krasnoselskii, *Topological Methods in the Theory of Nonlinear Integral equations,* Pergamon Press, Oxford, 1964.

[3] R. Precup, *Nonlinear Integral Equations* (Romanian), Babes-Bolyai Univ. Cluj, 1993.

[4] R. Precup, *On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces*, to appear.

[5] P.H. Rabinowits, *Minimax Methods in Critical Point Theory with Applications to Differential Equations*, Amer., Math. Soc., Providence, 1986.