# Fixed Point Theorems in Cone Banach Spaces

- Erdal Karapınar
^{1}Email author

**2009**:609281

**DOI: **10.1155/2009/609281

© Erdal Karapınar. 2009

**Received: **23 October 2009

**Accepted: **15 December 2009

**Published: **15 December 2009

## Abstract

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 [1] 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 [2].

*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 [3] 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 [4] on fixed point theorems were considered for -metric spaces. Without mentioning the papers of Lin and Rzepecki, in 2007, Huang and Zhang [5] 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

for all , has a unique fixed point.

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.

Throughout this paper
stands for real Banach space. Let
always be a closed nonempty subset of
.
is called *cone* if
for all
and nonnegative real numbers
where
and
.

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

The cone is called

*normal* if there is a number
such that for all
:

*regular* if every increasing sequence which is bounded from above is convergent. That is, if
is a sequence such that
for some
, then there is
such that
.

In , the least positive integer , satisfying (1.2), is called the normal constant of .

- (iii)
The cone is regular if every decreasing sequence which is bounded from below is convergent.

Proofs of (i) and (ii) are given in [6] and the last one follows from definition.

Definition 1.2 (see [5]).

Let be a nonempty set. Suppose the mapping satisfies

for all ,

if and only if ,

for all ,

for all ,

then is called cone metric on , and the pair is called a cone metric space (CMS).

Example 1.3.

Let , , and . Define by , where are positive constants. Then is a CMS. Note that the cone is normal with the normal constant

It is quite natural to consider Cone Normed Spaces (CNS).

Definition 1.4 (see [15, 16]).

Let be a vector space over . Suppose the mapping satisfies

for all ,

if and only if ,

for all ,

for all ,

then is called cone norm on , and the pair is called a cone normed space (CNS).

Note that each CNS is CMS. Indeed, .

Definition 1.5.

Let be a CNS, and a sequence in . Then

(i)
*converges to*
whenever for every
with
there is a natural number
, such that
for all
. It is denoted by
or
;

(ii)
is a *Cauchy sequence* whenever for every
with
there is a natural number
, such that
for all
;

(iii) is a complete cone normed space if every Cauchy sequence is convergent.

Complete cone normed spaces will be called cone Banach spaces.

Lemma 1.6.

Let be a CNS, a normal cone with normal constant , and a sequence in . Then,

(i)the sequence converges to if and only if , as ;

(ii)the sequence is Cauchy if and only if as ;

(iii)the sequence converges to and the sequence converges to then .

The proof is direct by applying [5, Lemmas , , and ] to the cone metric space , where , for all .

Let be a CNS over a cone in . Then and , . If then there exists such that implies . For any given and , there exists such that . If are sequences in such that , , and , for all then .

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 [17]).

is called minihedral cone if exists for all , and strongly minihedral if every subset of which is bounded from above has a supremum.

Lemma 1.9 (see [18]).

Every strongly minihedral normal cone is regular.

Example 1.10.

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 1.11.

for all . Then, is said to satisfy the condition .

For , the set of fixed points of is denoted by .

Definition 1.12 (see [14]).

Then is called nonexpansive (resp., quasi-nonexpansive) if it satisfies the condition (1.4) (resp., (1.5)).

## 2. Main Results

From now on, will be a cone Banach space, a normal cone with normal constant and a self-mapping operator defined on a subset of .

Theorem 2.1.

Then, has a unique fixed point.

Proof.

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.

Proposition 2.2.

Every nonexpansive mapping satisfies the condition .

Proposition 2.3.

Let satisfy the condition and , then is a quasi-nonexpansive.

Proof.

Theorem 2.4.

for all , where . Then, has at least one fixed point.

Proof.

Thus, when , one can get , that is, .

Notice that identity map, , satisfies the condition (2.4). Thus, maps that satisfy the condition (2.4) may have fixed points.

From the triangle inequality,

By (2.4),

Thus, letting implies that

Hence we have the following conclusion.

Theorem 2.5.

for all , where . Then has a fixed point.

Theorem 2.6.

for all , where . Then has at least one fixed point.

Proof.

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

Theorem 2.7.

for all . Then, has at least one fixed point.

Proof.

which is equivalent to (2.23) since . Hence, the claim is proved.

as . This last condition is equivalent to saying that as

## Authors’ Affiliations

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