Variational principles in analysis and existence of minimizers for functions on metric spaces

Functions defined on metric spaces are studied. For these functions, a generalized Caristi-like condition is introduced. It is shown that this condition is sufficient for a bounded below, lower semicontinuous function to attain its minimum. Criteria for a generalized Caristi-like condition to hold are derived. Generalizations of the Ekeland and Bishop-Phelps variational principles are obtained and compared with their prototypes. © 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved.