Takahashi’s minimization theorem and some related results in quasi-metric spaces

In this paper, we establish Takahashi’s minimization theorem in the setting of quasi-metric spaces and provide its equivalence with Ekeland’s variational principle given in Cobzaş (Topol Appl 158:1073–1084, 2011). We present an equilibrium version of Ekeland’s variational principle and extended Takahashi’s minimization theorem in the setting of quasi-metric spaces but without using the triangle inequality of the involved bifunction. We establish an equivalent chain of theorems containing Takahashi’s minimization theorem, Ekeland’s variational principle, the equilibrium version of Ekeland’s variational principle and Caristi–Kirk’s fixed point theorem for set-valued maps in the setting of quasi-metric spaces. As applications, we give an error bound for the solution set of the equilibrium problems and provide sufficient conditions for the existence of weak sharp solutions of equilibrium problems.