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Journal of Applied Mathematics
Volume 2011, Article ID 985797, 22 pages
http://dx.doi.org/10.1155/2011/985797
Research Article

The Existence of Cone Critical Point and Common Fixed Point with Applications

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 6 May 2011; Accepted 15 August 2011

Academic Editor: Ya Ping Fang

Copyright © 2011 Wei-Shih Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus our research on the equivalence between scalar versions and vectorial versions of some results of fixed point and others.