An Order on Subsets of Cone Metric Spaces and Fixed Points of Set-Valued Contractions
© M. Asadi et al. 2009
Received: 16 April 2009
Accepted: 22 September 2009
Published: 11 October 2009
In this paper at first we introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.
1. Introduction and Preliminary
Cone metric spaces were introduced by Huang and Zhang . They replaced the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractions . The study of fixed point theorems in such spaces followed by some other mathematicians, see [2–8]. Recently Wardowski  was introduced the concept of set-valued contractions in cone metric spaces and established some end point and fixed point theorems for such contractions. In this paper at first we will introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.
Definition 1.1 (see ).
In the following we have some necessary definitions.
(8) called minihedral cone if exists for all , and strongly minihedral if every subset of which is bounded from above has a supremum . Let a cone metric space, cone is strongly minihedral and hence, every subset of has infimum, so for , we define
For more details about above examples, see .
Let and . This is strongly minihedral but not minihedral by .
2. Main Results
The following lemma is easily proved.
Therefore, for every , Let and be given. Choose such that where Also, choose a such that for all Then for all Thus for all Namely, is Cauchy sequence in complete cone metric space, therefore for some Now we show that
Let with norm and that is not normal cone. Define cone metric with , for and set-valued mapping by . In this space every Cauchy sequence converges to zero. The function have lsc property. Also we have and . Now for and for all take . Therefore, it satisfies in all of the hypothesis of Theorem 2.5. So has a fixed point For sample take and
To have Theorems??3.1 and ??3.2 in , as the corollaries of our theorems we need the following lemma and remarks.
By Proposition ?1.7.59, page 117 in , if is an ordered Banach space with positive cone , then is a normal cone if and only if there exists an equivalent norm on which is monotone. So by renorming the we can suppose is a normal cone with constant one.
Now the Theorems ?3.1 and ?3.2 in  is stated as the following corollaries without the assumption of normality, and by Lemma ?2.10 and Remarks ?2.11, ?2.12 we have the same theorems.
Corollary 2.13 (see [9, Theorem ?3.1]).
Corollary 2.14 (see [9, Theorem ?3.2]).
The following theorem is a reform of Theorem 2.5.
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