Remarks on Caristi’s fixed point theorem and Kirk’s problem

Without assumptions on the continuity and the subadditivity of ηη, by means of Caristi’s fixed point theorem, we investigated the existence of fixed points for a Caristi type mapping which partially answered Kirk’s problem and improved Caristi’s fixed point theorem, Jachymski’s fixed point theorem and Khamsi’s fixed point theorem since φφ is not necessarily assumed to be bounded below on XX.