An Order on Subsets of Cone Metric Spaces and Fixed Points of Set-Valued Contractions

Abstract

In this paper at first we introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.

1. Introduction and Preliminary

Cone metric spaces were introduced by Huang and Zhang [1]. They replaced the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractions [1]. The study of fixed point theorems in such spaces followed by some other mathematicians, see [2–8]. Recently Wardowski [9] was introduced the concept of set-valued contractions in cone metric spaces and established some end point and fixed point theorems for such contractions. In this paper at first we will introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.

Let be a real Banach space. A nonempty convex closed subset is called a cone in if it satisfies.

(i) is closed, nonempty, and ,

(ii) and imply that

(iii) and imply that

The space can be partially ordered by the cone ; that is, if and only if . Also we write if , where denotes the interior of .

A cone is called normal if there exists a constant such that implies .

In the following we always suppose that is a real Banach space, is a cone in and is the partial ordering with respect to .

Definition 1.1 (see [1]).

Let be a nonempty set. Assume that the mapping satisfies

(i) for all and iff

(ii) for all

(iii) for all .

Then is called a cone metric on , and is called a cone metric space.

In the following we have some necessary definitions.

(1)Let be a cone metric space. A set is called closed if for any sequence convergent to , we have

(2)A set is called sequentially compact if for any sequence , there exists a subsequence of is convergent to an element of

(3)Denote a collection of all nonempty subsets of , a collection of all nonempty closed subsets of and a collection of all nonempty sequentially compact subsets of

(4)An element is said to be an endpoint of a set-valued map if We denote a set of all endpoints of by

(5)An element is said to be a fixed point of a set-valued map if Denote

(6)A map is called lower semi-continuous, if for any sequence in and such that as , we have

(7)A map is called have lower semi-continuous property, and denoted by lsc property if for any sequence in and such that as , then there exists that for all

(8) called minihedral cone if exists for all , and strongly minihedral if every subset of which is bounded from above has a supremum [10]. Let a cone metric space, cone is strongly minihedral and hence, every subset of has infimum, so for , we define

Example 1.2.

Let with . The cone is normal, minihedral and strongly minihedral with .

Example 1.3.

Let be a compact set, and . The cone is normal and minihedral but is not strongly minihedral and .

Example 1.4.

Let be a finite measure space, countably generated, , and . The cone is normal, minihedral and strongly minihedral with .

For more details about above examples, see [11].

Example 1.5.

Let with norm and that is not normal cone by [12] and not minihedral by [10].

Example 1.6.

Let and . This is strongly minihedral but not minihedral by [10].

Throughout, we will suppose that is strongly minihedral cone in with nonempty interior and be a partial ordering with respect to

2. Main Results

Let be a cone metric space and . For Let

(2.1)

At first we prove the closedness of without the assumption of normality.

Lemma 2.1.

Let be a complete cone metric space and . If the function for is lower semi-continuous, then is closed.

Proof.

Let and We show that Since

(2.2)

so which implies for some . Let with then, there exists such that for , Now, for we have,

(2.3)

So is a Cauchy sequence in complete metric space, hence there exist such that . Since is closed, thus Now by uniqueness of limit we conclude that

Definition 2.2.

Let and are subsets of , we write if and only if there exist such that for all , Also for , we write if and only if and similarly if and only if

Note that , for every scaler and subsets of .

The following lemma is easily proved.

Lemma 2.3.

Let , , and .

(1)If and then

(2)

(3)If then

(4)If then

(5)

(6)If then

The order "" is not antisymmetric, thus this order is not partially order.

Example 2.4.

Let and . Put and so but

Theorem 2.5.

Let be a complete cone metric space, , a set-valued map and the function defined by , with lsc property. If there exist real numbers and with such that for all there exists :

(2.4)

then

Proof.

Let , then there exists such that

(2.5)

Let , there exist such that and Continuing this process, we can iteratively choose a sequence in such that , and So, for we have,

(2.6)

Therefore, for every , Let and be given. Choose such that where Also, choose a such that for all Then for all Thus for all Namely, is Cauchy sequence in complete cone metric space, therefore for some Now we show that

Let hence there exists such that for all Now as so for all there exists such that for all

According to lsc property of , for all there exists such that for all

(2.7)

So for all Namely, thus for some and by the closedness of we have

We notice that implies that for all there exists such that for all but the inverse is not true.

Example 2.6.

Let with norm and that is not normal cone by [12]. Consider and so and , (see [10]) Define cone metric with , for . Since namely, but . Indeed in but in Even for and in particular but .

Example 2.7.

Let with norm and that is not normal cone. Define cone metric with , for and set-valued mapping by . In this space every Cauchy sequence converges to zero. The function have lsc property. Also we have and . Now for and for all take . Therefore, it satisfies in all of the hypothesis of Theorem 2.5. So has a fixed point For sample take and

Theorem 2.8.

Let be a complete cone metric space, , a set-valued map, and a function defined by , with lsc property. The following conditions hold:

(i)if there exist real numbers and with such that for all there exists :

(2.8)

then

(ii)if there exist real numbers and with such that for all and :

(2.9)

then

Proof.

• (i) It is obvious that It is enough to show that for all However for some , it implies for some and this is a contradiction.

• (ii) By (i), there exists such that Then for and we have . Therefore, This implies that

Corollary 2.9.

Let be a complete cone metric space, , a set-valued map, and the function defined by , for with lsc property. If there exist real numbers and with such that for all there exists with

(2.10)

then

To have Theorems??3.1 and ??3.2 in [9], as the corollaries of our theorems we need the following lemma and remarks.

Lemma 2.10.

Let be a cone metric space, a normal cone with constant one and , a set-valued map, then

(2.11)

Proof.

Put and we show that

Let then and so which implies

For the inverse, let for all . Then for all

Since , for every that there exists such that so for all Thus

Remark 2.11.

By Proposition ?1.7.59, page 117 in [11], if is an ordered Banach space with positive cone , then is a normal cone if and only if there exists an equivalent norm on which is monotone. So by renorming the we can suppose is a normal cone with constant one.

Remark 2.12.

Let be a cone metric space, a normal cone with constant one, , a set-valued map, the function defined by , with lsc property, and with . Then is lower semi-continuous.

Now the Theorems ?3.1 and ?3.2 in [9] is stated as the following corollaries without the assumption of normality, and by Lemma ?2.10 and Remarks ?2.11, ?2.12 we have the same theorems.

Corollary 2.13 (see [9, Theorem ?3.1]).

Let be a complete cone metric space, , a set-valued map and the function defined by , with lsc property. If there exist real numbers , such that for all there exists one has and then

Corollary 2.14 (see [9, Theorem ?3.2]).

Let be a complete cone metric space, , a set-valued map and the function defined by , with lsc property. The following hold:

(i)if there exist real numbers , such that for all there exists one has and then

(ii)if there exist real numbers , such that for all and every one has and then

Definition 2.15.

For , where is a set-valued map we define

(2.12)

Note that for

The following theorem is a reform of Theorem 2.5.

Theorem 2.16.

Let be a complete cone metric space, , a set-valued map, and the function defined by , with lsc property. If there exists such that

(2.13)

for all Then

Proof.

For every , then there exist and such that , for all . Let , there exist and such that since . Thus The remaining is same as the proof of Theorem 2.5.