Journal of Fixed Point Theory and Applications

, Volume 7, Issue 2, pp 265–290

Origin and evolution of the Palais–Smale condition in critical point theory


DOI: 10.1007/s11784-010-0019-7

Cite this article as:
Mawhin, J. & Willem, M. J. Fixed Point Theory Appl. (2010) 7: 265. doi:10.1007/s11784-010-0019-7


In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution.

Mathematics Subject Classification (2010)

58E05 58-03 01A60 


Condition (C) Palais–Smale condition critical point theory minimax theorems 

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium