Abstract
In this paper we extend the Banach contraction for multivalued mappings in a cone b-metric space without the assumption of normality on cones and generalize some attractive results in literature.
MSC:47H10, 54H25.
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1 Introduction
The analysis on existence of linear and nonlinear operators was developed after the Banach contraction theorem [1] presented in 1922. Many generalizations are available with applications in the literature [2–13]. Nadler [14] gave its set-valued form in his classical paper in 1969 on multivalued contractions. A real generalization of Nadler’s theorem was presented by Mizoguchi and Takahashi [15] as follows.
Theorem 1.1 [15]
Let be a complete metric space and let be a multivalued map such that Tx is a closed bounded subset of X for all . If there exists a function such that for all and if
then T has a fixed point in X.
Huang and Zhang [10] introduced a cone metric space with normal cone as a generalization of a metric space. Rezapour and Hamlbarani [16] presented the results of [10] for the case of a cone metric space without normality in cone. Many authors worked on it (see [17]). Cho and Bae [18] presented the result of [15] for multivalued mappings in cone metric spaces with normal cone.
Recently Hussain and Shah [19] introduced the notion of cone b-metric spaces as a generalization of b-metric and cone metric spaces. In [20] the authors presented some fixed point results in cone b-metric spaces without assumption of normality on cone.
In this article we present the generalized form of Cho and Bae [18] for the case of cone b-metric spaces without normality on cone. We also give an example to support our main theorem.
2 Preliminaries
Let be a real Banach space and P be a subset of . By θ we denote the zero element of and by intP the interior of P. The subset P is called a cone if and only if:
-
(i)
P is closed, nonempty, and ;
-
(ii)
, , ;
-
(iii)
.
For a given cone , we define a partial ordering ⪯ with respect to P by if and only if ; will stand for and , while will stand for , where intP denotes the interior of P. The cone P is said to be solid if it has a nonempty interior.
Definition 2.1 [19]
Let X be a nonempty set and be a given real number. A function is said to be a cone b-metric if the following conditions hold:
(C1) for all and if and only if ;
(C2) for all ;
(C3) for all .
The pair is then called a cone b-metric space.
Example 2.1 [20]
Let with , where . Let be defined as
where . Then is a b-metric space. Put , . Letting the map be defined by , we conclude that is a cone b-metric space with the coefficient , but is not a cone metric space.
Example 2.2 [20]
Let , , . Define by
Then is a cone b-metric space with coefficient . But it is not a cone metric space, because
Remark 2.1 [19]
The class of cone b-metric spaces is larger than the class of cone metric spaces since any cone metric space must be a cone metric b-metric space. Therefore, it is obvious that cone b-metric spaces generalize b-metric spaces and cone metric spaces.
Definition 2.2 [19]
Let be a cone b-metric space, , let be a sequence in X. Then
-
(i)
converges to x whenever for every with there is a natural number such that for all . We denote this by ;
-
(ii)
is a Cauchy sequence whenever for every with there is a natural number such that for all ;
-
(iii)
is complete cone b-metric if every Cauchy sequence in X is convergent.
Remark 2.2 [17]
The results concerning fixed points and other results, in case of cone spaces with non-normal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of Lemmas 1-4 in [10] hold. Further, the vector cone metric is not continuous in the general case, i.e., from , it need not follow that .
Let be an ordered Banach space with a positive cone P. The following properties hold [17, 19]:
(PT1) If and , then .
(PT2) If and , then .
(PT3) If and , then .
(PT4) If for each , then .
(PT5) If for each , then .
(PT6) Let be a sequence in . If and (as ), then there exists such that for all , we have .
3 Main result
According to [18], we denote by Λ a family of nonempty closed and bounded subsets of X, and
For , we define
Remark 3.1 Let be a cone b-metric space. If and , then is a b-metric space. Moreover, for , is the Hausdorff distance induced by d.
Now, we start with the main result of this paper.
Theorem 3.1 Let be a complete cone b-metric space with the coefficient and cone P, and let be a multivalued mapping. If there exists a function such that
for any decreasing sequence in P. If for all ,
then T has a fixed point in X.
Proof Let be an arbitrary point in X, then , so . Let and consider
By definition we have
which implies
Since , so we have
We have
So there exists some such that
It gives
By induction we can construct a sequence in X such that
If for some , then T has a fixed point. Assume that , then from (c) the sequence is decreasing in P. Hence from (a) there exists such that
Thus, for any , there exists some such that for all , implies . Now consider, for all ,
where .
Let . Applying (C3) to triples , we obtain
Now and imply that
Now, according to (PT6) and (PT1), we obtain that for a given there exists such that
that is, is Cauchy sequence in . Since is a complete cone b-metric space, so there exists some such that . Take such that for all . Now we will prove . For this let us consider
By definition we have
which implies
Since , so we have
So there exists some such that
It gives
Now consider
which means , since Tu is closed so . □
Corollary 3.1 [18]
Let be a complete cone metric space with a normal cone P, and let be a multivalued mapping. If there exists a function such that
for any decreasing sequence in P. If for all ,
then T has a fixed point in X.
Corollary 3.2 [15]
Let be a complete metric space and let be a multivalued map such that Tx is a closed bounded subset of X for all . If there exists a function such that for all and if
then T has a fixed point in X.
The following is Nadler’s theorem for multivalued mappings in a complete metric space.
Corollary 3.3 [14]
Let be a complete metric space and let be a multivalued map such that Tx is a closed bounded subset of X for all . If there exists such that
then T has a fixed point in X.
Example 3.1 Let and be the set of all real-valued functions on X which also have continuous derivatives on X. Then is a vector space over ℝ under the following operations:
for all , . That is, with the norm and
then P is a non-normal cone. Define as follows:
Then is a cone b-metric space but not a cone metric space. For , set , , so . From the inequality
we have
which implies that
But
is impossible for all . Indeed, taking advantage of the inequality
we have
for all . Thus the triangular inequality in a cone metric space is not satisfied, so is not a cone metric space but is a cone b-metric space.
Let be such that
then we have, for ,
Since
so
Hence, for , we have
All conditions of our main theorems are satisfied, so T has a fixed point.
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Azam, A., Mehmood, N., Ahmad, J. et al. Multivalued fixed point theorems in cone b-metric spaces. J Inequal Appl 2013, 582 (2013). https://doi.org/10.1186/1029-242X-2013-582
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DOI: https://doi.org/10.1186/1029-242X-2013-582