In the paper, there were studied Caristi-like conditions that guaranteed existence of a minimum of a function on a metric space. For functions dependent on a parameter, there were obtained conditions for existence of a minimum for each value of the parameter. These results were applied to derive conditions for fixed point and coincidence point existence for mappings in metric spaces. For mappings dependent on a parameter, there were obtained conditions of coincidence point existence for each value of the parameter. © 2019, House of the Book of Science. All rights reserved.