Ekeland variational principle and its equivalents in $T_1$-quasi-uniform spaces

The present paper is concerned with Ekeland Variational Principle (EkVP) and its equivalents (Caristi-Kirk fixed point theorem, Takahashi minimization principle, Oettli-Th'era equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel, Nonlinear Anal. $\mathbf{\text{62}}$ (2005), 913–924, in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of $F$-quasi-gauge spaces, a non-symmetric version of $F$-gauge spaces introduced by Fang, J. Math. Anal. Appl. $\mathbf{\text{202}}$ (1996), 398–412, is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by Arutyunov and Gel’man, Zh. Vychisl. Mat. Mat. Fiz. $\mathbf{\text{49}}$ (2009), 1167–1174, and Arutyunov, Proc. Steklov Inst. Math. $\mathbf{\text{291}}$ (2015), no.~1, 24–37, in complete metric spaces. Key words: Ekeland Variational Principle, Takahashi minimization principle, equilibrium problems, uniform spaces, quasi-uniform spaces, gauge spaces, quasi-gauge spaces, completeness in quasi-uniform spaces