- Research Article
- Open Access

# Best Approximations Theorem for a Couple in Cone Banach Space

- Erdal Karapınar
^{1}Email author and - Duran Türkoğlu
^{2}

**2010**:784578

https://doi.org/10.1155/2010/784578

© Erdal Karapınar and Duran Türkoğlu. 2010

**Received:**23 March 2010**Accepted:**8 June 2010**Published:**29 June 2010

## Abstract

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.

## Keywords

- Fixed Point Theorem
- Normal Constant
- Finite Subset
- Unique Fixed Point
- Couple Fixed Point

## 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 [3], the authors extends to cone metric spaces over regular cones. In this manuscript, some results of some result of Mitrović in [12] are extended to the class of cone metric spaces.

Throughout this paper
stands for real Banach space. Let
always be closed subset of
.
is called *cone* if the following conditions are satisfied:

,

for all and nonnegative real numbers ,

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 . It can be easily shown that and where . Throughout this manuscript .

The cone is called

*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 . 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 , [5]).

Lemma 1.1 (see e.g., [13]).

One has the following.

(i)Every regular cone is normal.

(ii)For each , there is a normal cone with normal constant .

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

Definition 1.2 (see [14]).

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

Example 1.3.

Let with the supremum norm and Since the sequence is monotonically decreasing, but not uniformly convergent to , thus, is not strongly minihedral.

Definition 1.4.

Let be nonempty set. Suppose that the mapping satisfies the following:

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 1.5.

Let and 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 (CNSs).

Definition 1.6 (see e.g., [9, 15, 16]).

Let be a vector space over . Suppose that the mapping satisfies the following:

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 1.7.

Let be a CNS, and a sequence in . Then one has the following.

(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 1.8.

Let be a CNS, let be a normal cone with normal constant , and let be a sequence in . Then, one has the following:

(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 and then .

The proof is direct by applying Lemmas 1, 4, and 5 in [3] to the cone metric space , where , for all .

Lemma 1.9 (see, e.g., [6, 7]).

Let be a CNS over a cone in . Then (1) and . (2) If then there exists such that implies . (3) For any given and there exists such that . (4) If are sequences in such that , and for all then .

Definition 1.10.

holds for all . A CNS together with a convex structure is said to be convex CNS. A subset is convex, if holds for all and .

Definition 1.11.

*almost quasiconvex with respect*to if

where for all and .

## 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 2.1.

Let and . Then, if and only if and .

Proof.

Suppose . Thus, for any , there exist such that for all . Hence, and for all , that is, and .

Conversely, assume and . Thus, for any , there exist such that for all , and also for all . Hence, for all , where .

Definition 2.2.

Let be a CMS. A function is said to be sequentially continuous if implies that . Analogously, a function is sequentially continuous if implies that .

Lemma 2.3 (see [6]).

Let be a CNS. Then is continuous if and only if is sequentially continuous.

Definition 2.4 (see [10, 17, 18]).

Note that this definition reduces the notion of mixed monotone function on where represents usual total order in .

Definition 2.5 (see [10, 17, 18]).

Definition 2.6 (see [3]).

Let be a CMS and . is said to be sequentially compact if for any sequence in there is a subsequence of such that is convergent in .

Remark 2.7 (see [19]).

Every cone metric space is a topological space which is denoted by . Moreover, a subset is sequentially compact if and only if is compact.

Definition 2.8.

where denotes the convex hull.

Lemma 2.9.

Let be a topological vector space, let be a nonempty subset of and let be called KKM map with closed values. If is compact for at least one then .

Theorem 2.10.

Proof.

for each . Since , then . Regarding that the mappings and are continuous, is closed for each . Since is compact, then is compact for each . Thus, is a KKM map.

where and

Theorem 2.11.

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.

Proof.

then .

Thus, and .

If we take as an identity, , in Theorem 2.11, then we get the following result.

Theorem 2.12.

Let be a CNS over strongly minidhedral cone , and let be a nonempty convex compact subset of . If is continuous mapping, then has a coupled fixed point.

Theorem 2.13.

for all .

Proof.

Take which implies (2.14). It is sufficient to show that . The inequality (2.14) implies that either or .

This is a contradiction. Analogously one can get the contradiction from the case . Thus, .

Theorem 2.14.

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 :

(iii)

Proof.

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 .

## Authors’ Affiliations

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