Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem
© T. Abdeljawad and E. Karapinar. 2009
Received: 4 July 2009
Accepted: 3 December 2009
Published: 9 December 2009
Cone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.
1. Introduction and Preliminaries
In 2007, Huang and Zhang  introduced the notion of cone metric spaces (CMSs) by replacing real numbers with an ordering Banach space. The authors there gave an example of a function which is contraction in the category of cone metric spaces but not contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that series of articles about cone metric spaces started to appear. Some of those articles dealt with the extension of certain fixed point theorems to cone metric spaces (see, e.g., [2–5]), and some other with the structure of the spaces themselves (see, e.g., [3, 6]). Very recently, some authors have used regular cones to extend some fixed point theorems. For example, in  a result about Meir-Keeler type contraction mappings has been extended to regular cone metric spaces. In other works, some results about fixed points of multifunctions on cone metric spaces with normal cones have been obtained as well . For the use of lower semicontinuous functions in obtaining fixed point theorems in cone metric spaces we refer to .
In this manuscript, we use cone-valued lower semicontinuous functions to extend some of the results in Caristi  and Ekeland  to CMS and quasicone metric space (QCMS). The cones under consideration are assumed to be strongly minihedral and normal and hence regular. In particular the cone in the real line is strongly minihedral and normal; hence the results mentioned in the above references are recovered.
For a given cone , one can define a partial ordering (denoted by or ) with respect to by if and only if . The notation indicates that and while will show , where denotes the interior of . From now on, it is assumed that
In , the least positive integer , satisfying (1.1), is called the normal constant of . Note that, in [1, 2], normal constant is stated a positive real number, ( ). However, later on and in [2, Lemma ] it was proved that there is no normal cone with constant .
Every regular cone is normal.
The proof of (i) and (ii) were given in  and the last one just follows from definition.
Example 1.2 (see ).
Lemma 1.6 (see ).
Definition 1.8 (see ).
It is easy to see that every strongly minihedral normal cone is regular.
Let with the supremum norm and Then is a cone with normal constant which is not regular. This is clear, since the sequence is monotonicly decreasing, but not uniformly convergent to . Thus, is not strongly minihedral. It is easy to see that the cone mentioned in Example 1.4 is strongly minihedral.
Definition 1.10 (see ).
Remark 1.11 (see ).
2. Main Results
The following Lemma will be used to prove the next results.
Since is an increasing sequence, , for and for all . Then closeness of implies that for all . To see that is the greatest lower bound of , assume that some satisfies for all . From and the closeness of we get or which shows that
The following theorem is an extension of the result of Caristi ([10, Theorem ]).
The following theorem is a generalization of the result in .
The following theorem is a generalization of [13, Theorem ].
This work is partially supported by the Scientific and Technical Research Council of Turkey.
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