Common fixed point theorems for generalized multivalued contractions on cone metric spaces over a non-normal solid cone
© Ðori¿; licensee Springer. 2014
Received: 4 April 2014
Accepted: 5 June 2014
Published: 22 July 2014
In this paper we define a new class of multivalued generalized contractions on cone metric spaces. Then, by using a necessary new technique, we prove two common fixed point theorems for a pair of those mappings on complete cone metric spaces over solid, not necessarily normal cone. Our main theorems are generalizations of the theorem of Wardowski (Appl. Math. Lett. 24:275-278, 2011) and many existing theorems in the literature. By using our main theorems, we can obtained some important corollaries which are generalizations of the well-known metric fixed point theorems to setting of cone metric spaces over a solid non-normal cone.
1 Introduction and preliminaries
There exist many generalizations of the concept of metric spaces in the literature. Fixed point theory in abstract (cone) metric, or in K-metric spaces over a Banach space, was developed in the mid-1970s. Huang and Zhang  reintroduced cone metric spaces and defined the convergence via interior points of the cone which determines an order on E. Although they considered and proved several fixed point theorems only in cone metric spaces over a normal cone, their approach enables the investigation of cone metric spaces over a cone which is not necessarily normal. It is well known that many fixed point results in the setting of cone metric spaces can be obtained from the corresponding results in metric spaces (see [2–4]). The results in the setting of cone metric spaces are appropriate only if the underlying cone is not necessarily normal (see ).
Definition 1.1 Let E be a topological vector space and P be a subset of E. The set P is called a cone if
(P1) P is closed, nonempty, and , where θ is the zero vector of E;
(P2) , , ;
A cone P is called solid  if , where intP is the interior of P.
Each cone P of E determines a partial order ⪯ on E by if and only if for each . We write if but , while will denote that . This relation is compatible with the vector structure.
The least positive number K satisfying the above inequality is called the normal constant of P.
The following example shows that there are non-normal cones.
Example 1.3 Let with the norm and consider the cone . For each , put and . Then , and . Since for each there exists such that , we have , but for any . Therefore, the cone P is non-normal.
Let E be a Banach space and θ be the zero vector of E. Let P be a cone in E with and let ⪯ be a partial ordering with respect to P. A mapping is called a cone metric on the nonempty set X if the following axioms are satisfied:
(d1) for all and if and only if ;
(d2) for all ;
(d3) for all .
The pair , where X is a nonempty set and d is a cone metric, is called a cone metric space.
Example 1.5 Let , , and defined by , where is a constant. Then is a cone metric space with the normal cone P, where the normal constant .
Definition 1.6 (Huang and Zhang )
a convergent sequence if, for every c in E with , there is an N such that for all and for some fixed x in X;
a Cauchy sequence if, for every c in E with , there is an N such that for all .
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
In the following lemma we suppose that E is a Banach space, P is a cone in E with , without the assumption of normality of cone P.
Let be a cone metric space. Then the following properties are often used (particularly when dealing with cone metric spaces in which the cone need not be normal).
(p1) If and , then .
(p2) If for each , then .
(p3) If for each , then .
(p4) If and , then .
(p5) If E is a real Banach space with a cone P and if , where and , then .
(p6) If , and , then there exists such that, for all , we have .
From (p6) it follows that the sequence converges to if as and is a Cauchy sequence if as . In the situation with a non-normal cone we have only one part of Lemmas 1 and 4 from . Also, in this case from and it need not follow that , as well as from and it need not follow that .
Therefore, does not converge to 0, although . Thus it follows by (1.1) that P is a non-normal cone.
The study of fixed points of multivalued mappings satisfying certain contractive conditions has many applications and studied by many researchers (see [8–12]). An element is said to be a fixed point of a multivalued map if . Recently many authors proved fixed point theorems for multivalued mappings on complete cone metric spaces assuming that the corresponding cone is regular or normal (see [13–20]). For a cone metric space let be a family of subsets of X. Wardowski [, Definition 3.1] introduced a new cone metric . Then he introduced the concept of set-valued contraction of Nadler type  and proved a fixed point theorem by assumption that a cone P of E is solid and normal. But, as noted in , most of the fixed points results in cone metric spaces over a normal cone can be obtained as a consequences from the corresponding results in metric spaces. Very recently Arshad and Ahmad  modified Wardowski’s  idea of H-cone metric. They introduced the following notion of H-cone metric.
Definition 1.9 (Arshad and Ahmad )
Let be a cone metric space and let be a family of all nonempty, closed, bounded subsets of X. A map is called an H-cone metric on induced by d if the following conditions hold:
(H1) for all and if and only if ;
(H2) for all ;
(H3) for all ;
(H4) If , with , then for each there exists such that .
which is called a Hausdorff metric induced by the metric d, is an H-cone metric induced by d.
Theorem 1.11 (Arshad and Ahmad )
then T has a fixed point.
Clearly, Theorem 1.11 is a generalization of the classical theorem of multivalued contractive mappings (Nadler ). Recall that some of the initial generalizations of the theorem of Nadler are given in  and in . In 1971 Ćirić in  introduced the concept of a generalized single-valued contraction, and then in 1972 in  he used the following concept of a generalized multivalued contraction.
Definition 1.12 (Ćirić )
where for is the Hausdorff metric (1.2) induced by metric d.
In the present paper we will introduce the concept of a generalized multivalued contraction on cone metric spaces and then, using a new technique of proof, we prove two common fixed point theorems for a pair of those multivalued mappings on cone metric spaces over solid non-normal cones. As a consequence, we also obtain some important corollaries which are generalizations of the well-known metric fixed point theorems.
2 Main results
Inspired by Definition 1.12 of Ćirić we shall introduce the notion of the cone generalized multivalued contraction.
Definition 2.1 Let E be a Banach space and let be a cone metric space over E. Let be a family of nonempty, closed, and bounded subsets of X and let there exists an H-cone metric induced by d. A mapping is said to be a cone generalized multivalued contraction if and only if there exists such that, for all , a mapping T satisfies one of the following contractive conditions:
(D2) for each fixed ;
(D3) for each fixed ;
(D4) for each fixed and each fixed .
It is easy to show that the generalized multivalued contraction defined in Definition 1.12 is an example of the cone generalized multivalued contraction defined in Definition 2.1.
Example 2.2 Let and be the usual metric space ordered by a usual ordering ≤. Let be a family of all nonempty, closed, bounded subsets of X and be a Hausdorff metric. Suppose that a mapping is a generalized multivalued contraction defined in Definition 1.12. If we set , , and for , we define if and only if , then is a cone metric space over cone P and is a cone generalized multivalued contraction.
Now we prove our main theorem.
Theorem 2.3 Let E be a Banach space, let P be a solid not necessarily normal cone of E and let be a cone metric space over E. Let be a family of nonempty, closed, and bounded subsets of X and let there exists an H-cone metric induced by d. Suppose that are two cone multivalued mappings and suppose that there is such that, for all , at least one of the following conditions holds:
(C2) for each fixed ;
(C3) for each fixed ;
(C4) for each fixed , .
Then T and S have a common fixed point.
Proof Let and be arbitrary. Consider the element . If each right hand side of (C1), (C2), (C3), and (C4) with and is θ in E, then and hence from the property (d1) of the metric d it follows . This and imply . Further, for each fixed implies . Hence . Therefore, in this case is a common fixed point of S and T and proof is done.
- (1)If , then from (2.2) we have(2.3)
- (2)If for any , then we can take . So, we obtain and from (2.2) we get
If for any , then we may take and we obtain . Then from (2.2) we again have (2.3).
- (4)If for any and , then we may take , . So we obtain
for all . Thus, by (2.11), for all . Therefore, by (ii) in Definition 1.6, we conclude that is Cauchy sequence. Since X is complete, there exists such that .
- (1)If , then from (2.13) we have(2.14)
- (2)If for any fixed , then we can take . Thus from (2.13) we get(2.15)
- (3)If for any fixed , then we can take . Thus from (2.13) and by the triangle inequality we get
- (4)If for any and , then we can take and . Thus from (2.13) and by the triangle inequality we get
Since converges to z and since and by (2.10) as , the right hand side of the inequality (2.19) converges to θ as . Therefore, from (p6) in Lemma 1.7 and (2.19) we can choose a natural number such that for all , where with is arbitrary. By (i) in Definition 1.6 we conclude that converges to z. Since and Sz is closed, we get .
Analogously, we can get . So, we proved that z is a common fixed point of T and S. □
If we take in Theorem 2.3, then we obtain the following fixed point theorem in complete non-normal cone metric spaces.
Theorem 2.4 Let be a complete cone metric space over a solid non-normal cone and let be a family of nonempty, closed, and bounded subsets of X. Suppose that there exists an H-cone metric induced by d and suppose that is a cone generalized multivalued contraction. Then T has a fixed point.
where ℋ is the usual Hausdorff metric on X induced by the metric .
- (1)If and , then we have
Thus T satisfies the contractive condition (D1) in Definition 1.12 with .
- (2)If and , then we have
for all . Therefore, in this case T satisfies the contractive condition (D2) in Definition 1.12 with .
From (1) and (2) we see that the mapping T satisfies all of the conditions of Theorem 2.4 and has a fixed point .
In a cone P of an ordered Hausdorff topological vector space , from it does not need to follow that nor . Thus in addition to the conditions (C1)-(C4) of Theorem 2.3 we can consider the condition
(C5) for each fixed and .
The following theorem is a generalization of Theorem 2.3.
Theorem 2.6 Let be a complete cone metric space over a solid non-normal cone, let be a family of non-empty, closed, and bounded subsets of X and let there exists an H-cone metric induced by d. Suppose that are two cone multivalued mappings and suppose that there is such that, for all , at least one of the conditions (C1)-(C5) holds. Then T and S have a common fixed point.
We shall omit the proof of this theorem since it is similar to the proof of Theorem 2.3.
By using Theorem 2.3 and Theorem 2.6 we can obtain corollaries which are generalizations of the well-known metric fixed point theorems of Kannan , Reich , Chatterjea  and Ćirić  to non-normal cone metric spaces. For example, the following corollary is a cone multivalued version of Kannan’s fixed point theorem, and it easily follows from Theorem 2.6.
for each and for each . Then T and S have a common fixed point.
The author is thankful to the Ministry of Education, Science and Technological Development of the Republic of Serbia. The author thanks the editor and the referees for their valuable comments and suggestions that helped to improve the text.
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