Critical Point Theorems and Ekeland Type Variational Principle with Applications

We introduce the notion of -spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang (2007). We establish some critical point theorems in the setting of -spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. (1983) and the results given by Khanh and Quy (2010) to -spaces and cone metric spaces. As applications of our results, we characterize the completeness of -space (cone metric spaces and quasimetric spaces are special cases of -space) and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.