Some remarks on variational principles
The authors analyze three variational principles published between 1976 and 1991 by Ekeland, Borwein and Preiss, and DeVille. The main result of the article is an extension of the variational principle of Borwein and Preiss to the case of complete metric space. The method of proof is based on the construction of a Cauchy sequence which converges to a point meeting the requirements of the theorem. Apparently due to this circumstance, the theorem is called a constructive variational principle. The above mentioned sequence is constructed using a complete family of bump-functions. There is a certain freedom in selection of such a family. Selecting this family in a special way the authors obtain as an implication of their result ``almost'' Ekeland's variational principle, where a strict minimum is replaced by a non-strict minimum. They also extend DeVille's principle to complete metric spaces. In the concluding part of the article it is demonstrated that existence theorems obtained from iterative procedures can also be obtained using variational principles.
scientific article; zbMATH DE number 1238019