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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 $\left(X,d\right)$ be a complete CMS, $P$ a strongly minihedral normal cone, and $\varphi :X\to P\subset E$ ($E$ is a real Banach space) a lower-semicontinuous function. Then each selfmap $T:X\to X$ satisfying

$d\left(x,Tx\right)\le \varphi \left(x\right)-\varphi \left(Tx\right),\phantom{\rule{1.em}{0ex}}\forall x\in X,$

has a fixed point in $X$.

Theorem 2.5. Let $\varphi :X\to E$ be a lower-semicontinuous function on a complete CMS, where $P$ is a strongly minihedral normal cone. If $\varphi$ is bounded below, then there exists $y\in X$ such that

$\varphi \left(y\right)<\varphi \left(x\right)+d\left(x,y\right),\phantom{\rule{1.em}{0ex}}\forall x\in X,\phantom{\rule{4pt}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}x\ne y·$

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
##### Keywords:
cone; fixed points
##### References:
 [1] [2] [3] [4] [5]