## 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 $$\varphi: X\to P\subset E$$ ($$E$$ is a real Banach space) a lower-semicontinuous function. Then each selfmap $$T: X\to X$$ satisfying
$d(x, Tx)\leq\varphi(x)-\varphi(Tx),\quad\forall x\in X,$
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 $\varphi(y)< \varphi(x)+ d(x,y),\quad\forall x\in X,\;\text{where }x\neq y.$

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems

### Keywords:

cone; fixed points

### Citations:

Zbl 1118.54022; Zbl 0305.47029; Zbl 0249.49004
Full Text:

### References:

 [1] Huang, L-G; Zhang, X, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 332, 1468-1476, (2007) · Zbl 1118.54022 [2] Rezapour, Sh; Hamlbarani, R, Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings”, Journal of Mathematical Analysis and Applications, 345, 719-724, (2008) · Zbl 1145.54045 [3] Turkoglu D, Abuloha M: Cone metric spaces and fixed point theorems in diametrically contractive mappings.Acta Mathematica Sinica, English Series, submitted · Zbl 1203.54049 [4] Turkoglu, D; Abuloha, M; Abdeljawad, T, KKM mappings in cone metric spaces and some fixed point theorems, Nonlinear Analysis: Theory, Methods and Applications, 72, 348-353, (2010) · Zbl 1197.54076 [5] Şahin, İ; Telci, M, Fixed points of contractive mappings on complete cone metric spaces, Hacettepe Journal of Mathematics and Statistics, 345, 719-724, (2008) · Zbl 1190.47058 [6] Rezapour Sh, Derafshpour M, Hamlbarani R: A Review on Topological Properties of Cone Metric Spaces.Analysis, Topology and Applications 2008 (ATA2008), the 30th of May to the 4th of June, 2008, Technical Faculty, Cacak, University of Kragujevac Vrnjacka Banja, Serbia [7] Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces.Expositiones Mathematicae. In press · Zbl 1193.47058 [8] Rezapour, Sh; Haghi, RH, Fixed point of multifunctions on cone metric spaces, Numerical Functional Analysis and Optimization, 30, 825-832, (2009) · Zbl 1171.54033 [9] Klim, D; Wardowski, D, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Analysis: Theory, Methods and Applications, 71, 5170-5175, (2009) · Zbl 1203.54042 [10] Caristi, J, Fixed point theorems for mappings satisfying inwardness conditions, Transactions of the American Mathematical Society, 215, 241-251, (1976) · Zbl 0305.47029 [11] Ekeland, I, Sur LES problèmes variationnels, Comptes Rendus de l’Académie des Sciences, 275, a1057-a1059, (1972) · Zbl 0249.49004 [12] Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450. · Zbl 1257.47059 [13] Ćirić, LB, On a common fixed point theorem of a Greguš type, Publications de l’ Institut Mathématique, 49, 174-178, (1991) · Zbl 0753.54023
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