L.-G. Huang and X. Zhang [J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)] introduced the notion of cone metric spaces (CMS) by replacing the real numbers by an ordered Banach space. In the present paper the authors extend some of the results in J. Caristi [Trans. Am. Math. Soc. 215, 241–251 (1976; Zbl 0305.47029)] and I. Ekeland [C. R. Acad. Sci., Paris, Sér. A 275, 1057–1059 (1972; Zbl 0249.49004)] to CMS and quasicone metric spaces. They prove, amongst others, the following theorems:
Theorem 2.4. Let be a complete CMS, a strongly minihedral normal cone, and ( is a real Banach space) a lower-semicontinuous function. Then each selfmap satisfying
has a fixed point in .
Theorem 2.5. Let be a lower-semicontinuous function on a complete CMS, where is a strongly minihedral normal cone. If is bounded below, then there exists such that