Best Approximations Theorem for a Couple in Cone Banach Space
© Erdal Karapınar and Duran Türkoğlu. 2010
Received: 23 March 2010
Accepted: 8 June 2010
Published: 29 June 2010
The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham, (2006). In this manuscript, some result of Mitrović, (2010) extended to the class of cone Banach spaces.
1. Introduction and Preliminaries
for all , has a unique fixed point. Recently, many results on fixed point theorems have been extended to cone metric spaces (see, e.g., [3, 5–11]). In , the authors extends to cone metric spaces over regular cones. In this manuscript, some results of some result of Mitrović in  are extended to the class of cone metric spaces.
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 . It can be easily shown that and where . Throughout this manuscript .
In , the least positive integer satisfying (1.2) is called the normal constant of . Note that, in [3, 5], normal constant is considered a positive real number, ( ), although it is proved that there is no normal cone for in (see e.g., Lemma , ).
Lemma 1.1 (see e.g., ).
One has the following.
(i)Every regular cone is normal.
Definition 1.2 (see ).
It is quite natural to consider Cone Normed Spaces (CNSs).
Complete cone-normed spaces will be called cone Banach spaces.
The proof is direct by applying Lemmas 1, 4, and 5 in  to the cone metric space , where , for all .
2. Couple Fixed Theorems on Cone Metric Spaces
Let be a CMS and . Then the mapping such that forms a cone metric on . A sequence is said to be a double sequence of . A sequence is convergent to if, for every , there exists a natural number such that for all .
Lemma 2.3 (see ).
Definition 2.6 (see ).
Remark 2.7 (see ).
Let be a CNS over strongly minidhedral cone , and let be a nonempty convex compact subset of . If is continuous mapping and is continuous almost quasiconvex mapping with respect to such that , then and have a coupled coincidence point.
Let be a CNS over strongly minidhedral cone , and let be a nonempty convex compact subset of . Suppose that is continuous mapping. Then has a coupled fixed point if one of the following conditions is satisfied for all such that :
It is clear that (iii) (ii) (i). To finalize proof, it is sufficient to show that is satisfied. Suppose that holds but has no coupled fixed point. Take Theorem 2.13 into account; then there exist such that (2.14) holds which contradicts .
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