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Common fixed point theorems for generalized multivalued contractions on cone metric spaces over a non-normal solid cone
Fixed Point Theory and Applications volume 2014, Article number: 159 (2014)
Abstract
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.
MSC:47H10, 54H25.
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 [1] 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 [3]).
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) , , ;
(P3) .
A cone P is called solid [5] 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.
Definition 1.2 Let P be a cone in a real Banach space E. The cone P is called normal, if there exists a constant such that, for all ,
or, equivalently, if
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 [1])
Let be a cone metric space. We say that a sequence in X is
-
(i)
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;
-
(ii)
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 [1]. Also, in this case from and it need not follow that , as well as from and it need not follow that .
Example 1.8 Let , and the norm be as in Example 1.3. Consider the sequences and . Then and , but
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 [[19], Definition 3.1] introduced a new cone metric . Then he introduced the concept of set-valued contraction of Nadler type [21] and proved a fixed point theorem by assumption that a cone P of E is solid and normal. But, as noted in [3], 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 [22] modified Wardowski’s [19] idea of H-cone metric. They introduced the following notion of H-cone metric.
Definition 1.9 (Arshad and Ahmad [22])
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 .
Example 1.10 Let be a metric space and let be a family of all nonempty, closed, bounded subsets of X. Then the mapping given by the formula
which is called a Hausdorff metric induced by the metric d, is an H-cone metric induced by d.
Arshad and Ahmad [22] extended the theorem of Wardowski [19] to a complete cone metric space without the assumption that a cone P is normal. They proved the following theorem.
Theorem 1.11 (Arshad and Ahmad [22])
Let be a complete cone metric space. Let be a collection of nonempty, closed, and bounded subsets of X and let be an H-cone metric induced by d. If for a map there exists such that, for all ,
then T has a fixed point.
Clearly, Theorem 1.11 is a generalization of the classical theorem of multivalued contractive mappings (Nadler [21]). Recall that some of the initial generalizations of the theorem of Nadler are given in [23] and in [24]. In 1971 Ćirić in [25] introduced the concept of a generalized single-valued contraction, and then in 1972 in [23] he used the following concept of a generalized multivalued contraction.
Definition 1.12 (Ćirić [23])
Let be a metric space and let be a family of nonempty, closed, and bounded subsets of X. A mapping is said to be a generalized multivalued contraction if and only if there exists such that, for all ,
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:
(D1) ;
(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:
(C1) ;
(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.
Consider now the element in the case that, in the one of the inequalities (C1), (C2), (C3) or (C4) which holds, the right hand side is not θ. Let be a fixed element. Since , we have . Thus we have , where . Then from and from the property (H4) of the H-cone metric in Definition 1.9 we find, as , that there exists such that
Consider now the element . Clearly, . Again from (H4) with , as , there exists such that
Continuing this process we can construct a sequence in X such that , and
According to the inequality (2.2) and the inequalities (C1), (C2), (C3), and (C4) with and , we have to consider four cases.
-
(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
Hence
-
(3)
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
Then from (2.2) and by the triangle inequality we have
which implies that
Since , we have , and from (2.5) we obtain
It is easy see that from (2.3), (2.4), and (2.6) we get
Using similar arguments to (2.1) we can prove that
From (2.7) and (2.8) we conclude that
for all . From (2.9) we get
Using mathematical induction it is easy to prove that
By the triangle inequality and (2.10) for any we have
Hence we get
where is the remainder of the convergent series . Since and as , we get
Let with be arbitrary. From (2.12) and (p6) in Lemma 1.7 it follows that we can choose a natural number such that
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 .
Now we shall show that z is a common fixed point of T and S. Since
from the property (H4) of the H-cone metric in Definition 1.9 we see, as , that there exists such that
According to (2.13) and the inequalities (C1), (C2), (C3), and (C4) with and we have to consider four cases.
-
(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
which implies that
-
(4)
If for any and , then we can take and . Thus from (2.13) and by the triangle inequality we get
which implies that
Since for , we have
Thus, from (2.14), (2.15), (2.16), and (2.17) we have
By the triangle inequality and (2.18) 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.
From Theorem 2.4 we can obtain Theorem 3.1 of Arshad and Ahmad [22] and Theorem 3.1 of Wardowski [19].
Now we shall present an example where Theorem 2.4 can be applied, but the theorem of Arshad and Ahmad [22] (Theorem 1.11) and the theorem of Wardowski [19] cannot be applied.
Example 2.5 Let and let , and the norm be as in Example 1.3. Define by
where . Then d is a cone metric on X. Let be a family of all nonempty, closed, bounded subsets of X and let the mapping be defined by
Let be defined by
where ℋ is the usual Hausdorff metric on X induced by the metric .
Now we can show that T satisfies all conditions of Theorem 2.4 with .
-
(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 .
Now we shall show that in this example the theorems of Arshad and Ahmad [22] and Wardowski [19], as well as other theorems known in the literature, cannot be applied. Let and . Then
Clearly, there does not exist such that . Therefore, Theorem 3.1 of Arshad and Ahmad [22] (Theorem 1.11) and Theorem 3.1 of Wardowski [19] cannot be applied in this example.
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 [26], Reich [27], Chatterjea [28] and Ćirić [25] 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.
Corollary 2.7 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 , the mappings T and S satisfy the condition
for each and for each . Then T and S have a common fixed point.
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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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Ðorić, D. Common fixed point theorems for generalized multivalued contractions on cone metric spaces over a non-normal solid cone. Fixed Point Theory Appl 2014, 159 (2014). https://doi.org/10.1186/1687-1812-2014-159
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DOI: https://doi.org/10.1186/1687-1812-2014-159