Review

# Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric spaces

Seong-Hoon Cho1*, Jong-Sook Bae2 and Kwang-Soo Na2

Author Affiliations

1 Department of Mathematics, Hanseo University, Seosan, Chungnam, 356-706, South Korea

2 Department of Mathematics, Moyngji University, Yongin, 449-728, South Korea

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Fixed Point Theory and Applications 2012, 2012:133  doi:10.1186/1687-1812-2012-133

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/133

 Received: 30 April 2012 Accepted: 2 August 2012 Published: 16 August 2012

© 2012 Cho et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we establish a fixed-point theorem for multivalued contractive mappings in complete cone metric spaces. We generalize Caristi’s fixed-point theorem to the case of multivalued mappings in complete cone metric spaces. We give examples to support our main results. Our results are extensions of the results obtained by Feng and Liu (J. Math. Anal. Appl. 317:103-112, 2006) to the case of cone metric spaces.

MSC: 47H10, 54H25.

##### Keywords:
fixed point; multivalued map; cone metric space

### 1 Introduction

Banach’s contraction principle plays an important role in several branches of mathematics. Because of its importance for mathematical theory, it has been extended in many directions (see [10,11,14,19,21,37,46]); especially, the authors [36,37,39] generalized Banach’s principle to the case of multivalued mappings. Feng and Liu gave a generalization of Nadler’s fixed-point theorem. They proved the following theorem in [21].

Theorem 1.1Letbe a complete metric space and letbe a multivalued map such thatTxis a closed subset ofXfor all. Let, where.

If there exists a constantsuch that for any, there existssatisfying

thenThas a fixed point inX, i.e., there existssuch thatprovidedand the function, is lower semicontinuous.

Recently, in [22], the authors used the notion of a cone metric space to generalize the Banach contraction principle to the case of cone metric spaces. Since then, many authors [1-3,7,9,13,15,18,22-28,32-34,41,43,44,48] obtained fixed-point theorems in cone metric spaces. The cone Banach space was first used in [4,6]. Since then, the authors [29,30] obtained fixed-point results in cone Banach spaces. The authors [8] proved a Caristi-type fixed-point theorem for single valued maps in cone metric spaces. The author [5] studied the structure of cone metric spaces.

Especially, the authors [16,31,35,42,45,47] proved fixed point theorems for multivalued maps in cone metric spaces.

In this paper, we give a generalization of Theorem 1.1 to the case of cone metric spaces and we establish a Caristi-type fixed-point theorem for multivalued maps in cone metric spaces.

Consistent with Huang and Zhang [22], the following definitions will be needed in the sequel.

Let E be a topological vector space. A subset P of E is a cone if the following conditions are satisfied:

(1) P is nonempty, closed, and ,

(2) , whenever and (),

(3) .

Given a cone , we define a partial ordering ⪯ with respect to P by if and only if . We write to indicate that but .

For , stand for , where is the interior of P.

If E is a normed space, a cone P is called normal whenever there exists a number such that for all , implies .

A cone P is minihedral[20] if exists for all . A cone P is strongly minihedral[20] if every upper bounded nonempty subset A of E, supA exists in E. Equivalently, a cone P is strongly minihedral if every lower bounded nonempty subset A of E, infA exists in E (see also [1,8]).

If E is a normed space, a strongly minihedral cone P is continuous whenever, for any bounded chain , and .

From now on, we assume that E is a normed space, is a solid cone (that is, ), and ⪯ is a partial ordering with respect to P.

Let X be a nonempty set. A mapping is called cone metric[22] on X if the following conditions are satisfied:

(1) for all and if and only if ,

(2) for all ,

(3) for all .

Let be a cone metric space, and let be a sequence. Then

is convergent[22] to a point (denoted by or ) if for any , there exists N such that for all , .

is Cauchy[22] if for any , there exists N such that for all , . A cone metric space is called complete[22] if every Cauchy sequence is convergent.

Remark 1.1 (1) If , then . The converse is true if E is a normed space and P is a normal cone.

(2) If , then is a Cauchy sequence in X. If E is a normed space and P is a normal cone, then is a Cauchy sequence in X if and only if .

We denote by (resp. , , ) the set of nonempty (resp. bounded, closed, closed and bounded) subsets of a cone metric space or a metric space.

The following definitions are found in [16].

Let for , and for and .

For , we denote

Lemma 1.1 ([16])

Letbe a cone metric space, and letbe a cone.

(1) Let. If, then.

(2) Letand. If, then.

(3) Letand letand. If, then.

Remark 1.2 Let be a cone metric space. If and , then is a metric space. Moreover, for , is the Hausdorff distance induced by d.

Remark 1.3 Let be a cone metric space. Then for .

Lemma 1.2 ([16,40])

Ifwith, then for eachthere existsNsuch thatfor all.

### 2 Fixed-point theorems for multivalued contractive mappings

Let be a cone metric space, and let .

A function defined by is called sequentially lower semicontinuous if for any , there exists such that for all , whenever for any sequence and .

Let be a multivalued mapping. We define a function as .

For a , let .

Theorem 2.1Letbe a complete cone metric space and letbe a multivalued map. If there exists a constantsuch that for anythere existssatisfying

(2.1)

thenThas a fixed point inXprovidedandhis sequentially lower semicontinuous.

Proof Let . Then there exists such that . For , there exists such that .

Continuing this process, we can find a sequence such that

and

(2.2)

for all  .

We now show that is a Cauchy sequence in X.

Since , .

From (2.2), we have . Thus, . Hence,

for all  , where .

So we have

For , we have

By Lemma 1.2, is a Cauchy sequence in X. It follows from the completeness of X that there exists such that .

We now show that .

Suppose that .

Since Tz is closed, there exists such that implies .

But since h is sequentially lower semicontinuous, there exists N such that and .

Thus, there exists such that . Hence, , which is a contradiction. □

Remark 2.1 By Remark 1.1, Theorem 2.1 generalizes Theorem 1.1 ([12], Theorem 3.1]).

Corollary 2.2Letbe a complete cone metric space and letbe a multivalued map. If there exists a constantsuch that for any,

thenThas a fixed point inXprovidedhis sequentially lower semicontinuous.

By Lemma 1.1(3), we have the following result, which is Nadler’s fixed-point theorem in the cone metric space.

Corollary 2.3Letbe a complete cone metric space, and letbe a multivalued map. If there exists a constant, such that

for all, , thenThas a fixed point inXprovidedhis sequentially lower semicontinuous.

By Remark 1.1, we have the following corollaries.

Corollary 2.4 ([21])

Letbe a complete metric space and letbe a multivalued map. If there exists a constantsuch that

for all, , thenThas a fixed point inXprovidedhis sequentially lower semicontinuous.

Corollary 2.5Letbe a complete metric space and letbe a multivalued map. If there exists a constantsuch that

for all, , thenThas a fixed point inXprovidedhis sequentially lower semicontinuous.

The following example illustrates our main theorem.

Example 2.1 Let , and . Define by , where . Then d is a complete cone metric on X. Consider a mapping defined by

where is defined by .

Obviously, is sequentially lower semicontinuous.

For any , we can prove . To see this, we compute for

Since , we have , and hence we obtain .

Put . Then we have and for

Thus, we have , and .

Therefore, all conditions of Theorem 2.1 are satisfied and T has a fixed point .

### 3 Fixed-point theorems for multivalued Caristi type mappings

Let be a cone metric space with a preordering ⊑.

A sequence of points in X is called ⊑-decreasing if for all . The set is ⊑-complete if every decreasing Cauchy sequence in converges in it.

A function is called lower semicontinuous from above if, for every sequence conversing to some point and satisfying for all , we have .

Lemma 3.1Letbe a cone metric space, and letbe a multivalued mapping. Suppose thatis a function andis a nondecreasing, continuous, and subadditive function such thatif and only if.

We define a relationonXas follows:

(3.1)

Thenis a partial order onX.

Proof The proof follows by using the cone metric axioms, properties (2) and (3) for the cone, and (3.1). □

Lemma 3.2 ([17])

Letbe a strongly minihedral and continuous cone, and letbe a preordered set. Suppose that a mappingsatisfies the following conditions:

(1) andimply;

(2) for every ⊑-decreasing sequence, there existssuch thatfor all;

(3) ψis bounded from below.

Then, for each, has a minimal element in, where.

Theorem 3.1Letbe a cone metric space such thatPis strongly minihedral and continuous, and letbe a multivalued mapping andbe a mapping bounded from below. Suppose that, for each, is-complete, whereis a partial ordering onXdefined as (3.1).

If for any, there existssatisfying

thenThas a fixed point inX.

Proof We define a partial ordering on X as (3.1).

If and , then and , and so . Hence, .

Let be a -decreasing sequence in X. Then for all , and is bounded from below, because ϕ is bounded from below. Hence, is bounded. Since P is strongly minihedral, exists in E. Also, since P is continuous, . Hence, and for all .

For , since , . Hence . Thus, . Since η is continuous, . So .

Hence, is a -decreasing Cauchy sequence in . Since is -complete and for all , there exists such that . Thus, for all .

By Lemma 3.2, has a minimal element in . By assumption, there exists such that . Hence, . Since is minimal element in , . Thus, . □

Corollary 3.2Letbe a cone metric space such thatPis strongly minihedral and continuous, and letbe a multivalued mapping andbe a mapping bounded from below. Suppose that, for each, is-complete, whereis a partial ordering onXdefined as (3.1).

If for anyand for any,

then there existssuch that.

Theorem 3.3Letbe a complete cone metric space such thatPis strongly minihedral and continuous. Suppose thatis a multivalued mapping andis lower semicontinuous from above and bounded from below.

If for any, there existssatisfying

thenThas a fixed point inX.

Proof We define a partial ordering on X as (3.1). It suffices to show that, for each , is -complete.

Let be a fixed, and let be a -decreasing Cauchy sequence in . Then it is a -decreasing Cauchy sequence in X. Hence, for all . Since X is complete, there exists such that . Since ϕ is lower semicontinuous from above, . Thus, for all . Since for , we obtain

Hence,

Letting in above inequality, we have because η and d are continuous. Hence, .

Thus, we have , and so . Hence, , and hence is -complete. From Theorem 3.3, T has a fixed point in X. □

Corollary 3.4Letbe a complete cone metric space such thatPis strongly minihedral and continuous. Suppose thatis a multivalued mapping andis lower semicontinuous from above and bounded from below.

If for anyand for any,

then there existssuch that.

We now give an example to support Theorem 3.3.

Example 3.1 Let , and let and . We define by , where . Then is a complete cone metric space, and P is strongly minihedral and continuous.

Let for all .

We define a multivalued mapping by

and we define a mapping by

Then ϕ is lower semicontinuous from above and bounded from below.

For any , put . Then we have , and so .

Thus, all conditions of Theorem 3.3 are satisfied and T has a fixed point .

Remark 3.1 Theorem 3.3 (resp. Corollary 3.4) is a generalization of Theorem 4.2 (resp. Corollary 4.3) in [21], and also results in [38,50] to the case of cone metric spaces.

If in Theorem 3.3 (resp. Corollary 3.4), then we have generalizations of the results in [12,36,49] to the case of cone metric spaces.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the referees for careful reading and giving valuable comments. This research (S.H. Cho) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (No. 2011-0012118).

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