Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem

Abstract

Cone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.

1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced the notion of cone metric spaces (CMSs) by replacing real numbers with an ordering Banach space. The authors there gave an example of a function which is contraction in the category of cone metric spaces but not contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that series of articles about cone metric spaces started to appear. Some of those articles dealt with the extension of certain fixed point theorems to cone metric spaces (see, e.g., [2–5]), and some other with the structure of the spaces themselves (see, e.g., [3, 6]). Very recently, some authors have used regular cones to extend some fixed point theorems. For example, in [7] a result about Meir-Keeler type contraction mappings has been extended to regular cone metric spaces. In other works, some results about fixed points of multifunctions on cone metric spaces with normal cones have been obtained as well [8]. For the use of lower semicontinuous functions in obtaining fixed point theorems in cone metric spaces we refer to [9].

In this manuscript, we use cone-valued lower semicontinuous functions to extend some of the results in Caristi [10] and Ekeland [11] to CMS and quasicone metric space (QCMS). The cones under consideration are assumed to be strongly minihedral and normal and hence regular. In particular the cone in the real line is strongly minihedral and normal; hence the results mentioned in the above references are recovered.

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

,

for all and non-negative 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 . From now on, it is assumed that

The cone is called

()normal if there is a number such that for all ,

(1.1)

()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 (1.1), is called the normal constant of . Note that, in [1, 2], normal constant is stated a positive real number, (). However, later on and in [2, Lemma ] it was proved that there is no normal cone with constant .

Lemma 1.1.

• (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.

The proof of (i) and (ii) were given in [2] and the last one just follows from definition.

Example 1.2 (see [2]).

Let with the norm , and consider the cone

For each , put and Then, , and Since , is not normal constant of and hence is a nonnormal cone.

Definition 1.3.

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

() for all ,

() if and only if ,

(), for all .

Then is said to be a quasicone metric on , and the pair is called a quasicone metric space (QCMS). Additionally, if also satisfies

() for all ,

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

Example 1.4.

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

Definition 1.5.

Let be a CMS, , and let be a sequence in . Then

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

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

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

Lemma 1.6 (see [1]).

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

(i)the sequence converges to if and only if d (,) 0 or equivalently ;

(ii)the sequence is Cauchy if and only if (or equivalently );

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

Lemma 1.7 (see [1, 2]).

Let be a CMS over a cone in . Thenone has the following.

(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 1.8 (see [12]).

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

It is easy to see that every strongly minihedral normal cone is regular.

Example 1.9.

Let with the supremum norm and Then is a cone with normal constant which is not regular. This is clear, since the sequence is monotonicly decreasing, but not uniformly convergent to . Thus, is not strongly minihedral. It is easy to see that the cone mentioned in Example 1.4 is strongly minihedral.

Definition 1.10 (see [1]).

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 1.11 (see [6]).

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

2. Main Results

Let be a CMS, , and a function on . Then, the function is called a lower semicontinuous (l.s.c) on whenever

(2.1)

Also, let be an arbitrary selfmapping on such that

(2.2)

Then, is called a Caristi map on .

The following Lemma will be used to prove the next results.

Lemma 2.1.

If is a decreasing sequence (via the partial ordering obtained by the closed cone ) such that , then

Proof.

Since is an increasing sequence, , for and for all . Then closeness of implies that for all . To see that is the greatest lower bound of , assume that some satisfies for all . From and the closeness of we get or which shows that

Proposition 2.2.

Let be a compact CMS, a strongly minihedral cone, and a lower semicontinuous function. Then, attains a minimum on .

Proof.

Let which exists by strong minihedrality. For each , there is an such that , where . Since is compact, then has a convergent subsequence. Let be this sequence and let .

From the definition of lower semicontinuity and Lemma 2.1 it follows that

(2.3)

But then, by the definition of , for all . This completes the proof.

Theorem 2.3.

Let be a CMS, a compact subset of , a strongly minihedral normal cone, and a lower semicontinuous function. Then, each selfmap satisfying (2.2) has a fixed point in .

Proof.

By Proposition 2.2, attains its minimum at some point of , say . Since is the minimum point of , we have . By (2.2),

(2.4)

Thus, and so

The following theorem is an extension of the result of Caristi ([10, Theorem ]).

Theorem 2.4.

Let be a complete CMS, a strongly minihedral normal cone, and a lower semicontinuous function. Then, each selmap satisfying (2.2) has a fixed point in .

Proof.

Let have the normal constant . Let and for all . Since , and so .

For , set and construct a sequence in the following way: let be such that , where . Thus, one can observe that

(i),

(ii)

for all . Note that, (i) implies that the sequence is a decreasing sequence in and is regular cone. So, the sequence is convergent. Thus, for each , there exists such that for all , . For , the triangular inequality implies that

(2.5)

Hence, . By Lemma 1.6, yields that the sequence is a Cauchy in . Completeness of implies that the sequence is convergent to some point in , say

By (2.5), and so

(2.6)

for all . By regarding (2.6), Lemma 1.6, and lower semicontinuity of the function , one can obtain that

(2.7)

for all . Thus,

(2.8)

for all . Hence, and it is trivial that for all . Note that (ii) implies that

(2.9)

Thus, for all . On the other hand, by lower semicontinuity of and (2.9), one can obtain that

(2.10)

Therefore, .

Since for each and , the following inequalities are obtained:

(2.11)

Hence, for all . This implies that for all .

By (2.9), is obtained. As is observed by (2.2) and that , then

(2.12)

is achieved. Hence, . Finally, by (2.2) we have .

The following theorem is a generalization of the result in [11].

Theorem 2.5.

Let be a function on a complete CMS, where is a strongly minihedral normal cone. If is bounded below, then there exits such that

(2.13)

Proof.

It is enough to show that the point , obtained in Theorem 2.4, satisfies the statement of the theorem. Following the same notation in the proof of Theorem 2.4, it is needed to show that for . Assume the contrary that for some , we have . Then, implies . By triangular inequality,

(2.14)

which implies that and thus for all . Taking the limit when tends to infinity, one can obtain , which is in contradiction with . Thus, for any , implies , that is,

(2.15)

Let be defined by .

Theorem 2.6.

Let be a sequentially complete QCMS and let be a strongly minihedral normal cone. Assume that for each , the function defined above is continuous on and is a family of mappings . If there exists a l.s.c function such that

(2.16)

then for each there is a common fixed point of such that

(2.17)

Proof.

Let be strongly minihedral normal cone with normal constant . First note that strong minihedrality of guarantees that exists. Let and for all . Note that , so and also .

For , set and construct a sequence as in the proof of Theorem 2.4: such that , . Thus, one can observe that for each ,

(i),

(ii).

Similar to the proof of Theorem 2.4, (ii) implies that

(2.18)

Also, by using the same method in the proof of Theorem 2.4, it can be shown that is a Cauchy sequence and converges to some and .

We shall show that for all . Assume the contrary that there is such that . Then (2.16) with implies that Thus, by definition of , there is such that . Since ,

(2.19)

which implies that . Hence which is in a contradiction with Thus, for all . Since , we have

(2.20)

is obtained.

The following theorem is a generalization of [13, Theorem ].

Theorem 2.7.

Let be a set, as in Theorem 2.6, a surjective mapping, and a family of arbitrary mappings . If there exists a function such that

(2.21)

and each , then and have a common coincidence point, that is, for some , for all .

Proof.

Let be arbitrary and as in Theorem 2.6. Since is surjective, for each there is some such that . Let be a fixed mapping. Define by a mapping of into itself such that , where , that is, . Let be a family of all mappings . Then, (2.21) yields that

(2.22)

Thus, by Theorem 2.6, for all . Hence for all , where is such that .