Best Approximations Theorem for a Couple in Cone Banach Space

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.

1. Introduction and Preliminaries

Banach, valued metric space was considered by Rzepecki [1], Lin [2], and lately by Huang and Zhang [3]. Basically, for nonempty set , the definition of metric is replaced by a new metric, namely, by an ordered Banach space : . Such metric spaces are called cone metric spaces (in short CMSs). In 1980, by using this idea Rzepecki [1] generalized the fixed point theorems of Maia type. Seven years later, Lin [2] extends some results of Khan and Imdad [4] by considering this new metric space construction. In 2007, Huang and Zhang [3] 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

(1.1)

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

normal if there is a number such that for all ,

(1.2)

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 towhenever 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.

Let be a CNS and let be the closed unit interval. A continuous mapping is said to be a convex structure on if for all and

(1.3)

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.

Let be a CNS, and and the nonempty convex subsets of . A mapping is said to be almost quasiconvex with respect to if

(1.4)

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]).

Let be partially ordered set and . is said to have mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any ,

(2.1)

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]).

An element is said to be a couple fixed point of the mapping if

(2.2)

Throughout this paper, let be partially ordered set and let be a cone metric on such that is a complete CMS over the normal cone with the normal constant . Further, the product spaces satisfy the following:

(2.3)

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.

Let be a nonempty subset of a CNS . A set-valued map is called KKM map if for every finite subset of

(2.4)

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.

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 , then there exists such that

(2.5)

Proof.

Let by

(2.6)

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.

Let , where and are finite subsets of . Then, there exists

(2.7)

From the first expression in (2.7), one can get that there exist such that and . Set and then , and , . Regarding that is almost quasiconvex with respect to yields

(2.8)

where and

Thus

(2.9)

Taking (2.7) into account, one can get

(2.10)

for all which is a contradiction. Hence is a KKM mapping. It follows that there exists such that for all . Thus,

(2.11)

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.

Due to Theorem 2.10, there exists such that

(2.12)

Since ,

(2.13)

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.

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

(2.14)

for all .

Proof.

If has a coupled fixed point, then we are done. Suppose that has no coupled fixed points. Due to Theorem 2.10, there exists such that

(2.15)

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

Consider the first case: . Suppose . Since is convex, then there exists such that . Thus and

(2.16)

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 :

(i)there exists a such that

(2.17)

(ii)there exists an such that

(2.18)

(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 .