Fixed Point Theorems in Cone Banach Spaces
© Erdal Karapınar. 2009
Received: 23 October 2009
Accepted: 15 December 2009
Published: 15 December 2009
In this manuscript, a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset of a cone Banach space with the norm , if there exist , , and satisfies the conditions and for all , then has at least one Fixed point.
1. Introduction and Preliminaries
In 1980, Rzepecki  introduced a generalized metric on a set in a way that , where is Banach space and is a normal cone in with partial order . In that paper, the author generalized the fixed point theorems of Maia type .
Let be a nonempty set endowed in two metrics , and a mapping of into itself. Suppose that for all , and is complete space with respect to , and is continuous with respect to , and is contraction with respect to , that is, for all , where . Then has a unique fixed point in .
Seven years later, Lin  considered the notion of -metric spaces by replacing real numbers with cone in the metric function, that is, . In that manuscript, some results of Khan and Imdad  on fixed point theorems were considered for -metric spaces. Without mentioning the papers of Lin and Rzepecki, in 2007, Huang and Zhang  announced the notion of cone metric spaces (CMS) by replacing real numbers with an ordering Banach space. In that paper, they also discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces: any mapping of a complete cone metric space into itself that satisfies, for some , the inequality
Recently, many results on fixed point theorems have been extended to cone metric spaces (see, e.g., [5–9]). Notice also that in ordered abstract spaces, existence of some fixed point theorems is presented and applied the resolution of matrix equations (see, e.g., [10–12]).
In this manuscript, some of known results (see, e.g., [13, 14]) are extended to cone Banach spaces which were defined and used in [15, 16] where the existence of fixed points for self-mappings on cone Banach spaces is investigated.
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
Proofs of (i) and (ii) are given in  and the last one follows from definition.
Definition 1.2 (see ).
It is quite natural to consider Cone Normed Spaces (CNS).
Complete cone normed spaces will be called cone Banach spaces.
The proof is direct by applying [5, Lemmas , , and ] to the cone metric space , where , for all .
The proofs of the first two parts followed from the definition of . The third part is obtained by the second part. Namely, if is given then find such that implies . Then find such that and hence . Since is closed, the proof of fourth part is achieved.
Definition 1.8 (see ).
Lemma 1.9 (see ).
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 monotonically decreasing, but not uniformly convergent to . This cone, by Lemma 1.9, is not strongly minihedral. However, it is easy to see that the cone mentioned in Example 1.3 is strongly minihedral.
Definition 1.12 (see ).
2. Main Results
By [5, Theorem ], has a unique fixed point which is equivalent to saying that has a unique fixed point.
The following statement is consequence of Definition 1.11.
From the triangle inequality,
Hence we have the following conclusion.
and thus, . Since , the sequence is a Cauchy sequence that converges to some . Since also converges to as in the proof of Theorem 2.4, the inequality (2.16) (under the assumption and ) by the help of Lemma 1.6(iii) yields that which is equivalent to saying that
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