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Quasicone metric spaces and generalizations of Caristi Kirk’s theorem. (English) Zbl 1197.54051

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 (X,d) be a complete CMS, P a strongly minihedral normal cone, and φ:XPE (E is a real Banach space) a lower-semicontinuous function. Then each selfmap T:XX satisfying


has a fixed point in X.

Theorem 2.5. Let φ:XE be a lower-semicontinuous function on a complete CMS, where P is a strongly minihedral normal cone. If φ is bounded below, then there exists yX such that


54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
cone; fixed points