Abstract
In this paper, we establish new fixed point results for multi-valued operator on a cone metric space with respect to a solid cone by using the idea of the generalized Hausdorff distance due to Cho and Bae (Fixed Point Theory Appl. 2011:87, 2011). We also furnish some interesting examples which support our main theorems and give many results as corollaries of our result. As applications of our results, we obtain fixed point results for the multi-valued contraction operators in cone metric spaces endowed with graph.
MSC:47H10, 54H25.
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1 Introduction
In 2007, Huang and Zhang [1] re-introduced the concept of cone metric space which is replacing the set of real numbers (as the co-domain of a metric) by an ordered Banach space. They described the convergence sequence in this space via interior points of the cone and also introduced the concept of completeness. By using these concepts, they proved some fixed point theorem for contractive type mappings which is a generalization of Banach’s contraction principle. Afterward, many authors studied and extended fixed point theorems in cone metric spaces (see [2–4] and references therein).
In 2011, Janković et al. [5] showed that most of the fixed point results for mappings satisfying a contractive type condition in cone metric spaces with normal cone can be reduced to the corresponding results from metric space theory. They also give some examples and showed that fixed point theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces whenever the cone is non-normal. In fact, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is non-normal with non-empty interior (such cones are called solid) because the results concerning fixed points and other results in this case cannot be proved by reducing to metric spaces. So fixed point results in this trend are still of interest and importance in some ways.
On the other hand, a study of fixed point for multi-valued (set-valued) operators was originally initiated by von Neumann [6] in the study of game theory. In 1969, the development of geometric fixed point theory for a multi-valued operator via the concept of Hausdorff distance was initiated with the work of Nadler [7], usually referred to as Nadler’s contraction principle. Many authors in [8–11] proved fixed point theorems for multi-valued operators in cone metric spaces which are generalizations of classical Nadler’s contraction principle.
Recently, Cho and Bae [12] first introduced the concept of the generalized Hausdorff distance operator in cone metric spaces and initially studied fixed point results for a multi-valued operator via such a concept in cone metric spaces. Most recently, Kutbi et al. [13] established fixed point results for multi-valued operator under the generalized contractive condition via the generalized Hausdorff distance operator of Cho and Bae [12].
In this paper, we establish new fixed point results for multi-valued operator by using the idea of generalized Hausdorff distance in the context of cone metric spaces with respect to a solid cone. Our results unify, generalize, and complement results of Kutbi et al. [13] and many results from the literature. Also, we furnish some interesting examples which support our main theorems. As an application, we analyze the fixed points for the multi-valued contraction operators in cone metric spaces endowed with a graph.
2 Preliminaries
Throughout this paper, we denote by ℕ, and ℝ the sets of positive integers, non-negative real numbers and real numbers, respectively.
Now, we recall some definitions and lemmas in cone metric spaces and the notion of the generalized Hausdorff distance operator of Cho and Bae [12], which will be required in the sequel.
Let be a real Banach space and θ be a zero element in . A subset P of is called a cone if the following conditions are satisfied:
(C1) P is non-empty closed and ;
(C2) for all and ;
(C3) .
For a given cone , we define a partial ordering ⪯ with respect to P by if and only if and stands for and , while stands for , where denotes the interior of P.
The cone P is said to be normal if there exists a real number such that for all ,
The least positive number K satisfying the above statement is called the normal constant of P. In 2008, Rezapour and Hamlbarani [14] showed that there are no normal cones with normal constant .
Definition 2.1 ([1])
Let X be a non-empty set and P be a cone in real Banach space . If a function satisfies the following conditions:
(CM1) for all and if and only if ;
(CM2) for all ;
(CM3) for all ,
then d is called a cone metric on X and is called a cone metric space.
Definition 2.2 ([1])
Let be a cone metric space with cone P in real Banach space , be a sequence in X and .
-
1.
If for every with , there is such that for all , then is said to converge to x. This limit is denoted by or as .
-
2.
If for every with , there is such that for all , then is called a Cauchy sequence in X.
-
3.
If every Cauchy sequence in X is convergent in X, then is called a complete cone metric space.
Remark 2.1 The cone metric is not continuous in the general case, i.e., from , as it need not follow that as . However, if is a cone metric space with a normal cone P, then the cone metric d is continuous (see Lemma 5 in [1]).
Definition 2.3 ([1])
Let be a cone metric space. A subset is called closed if for any sequence in A convergent to x, we have .
Lemma 2.1 ([5])
Let be a cone metric space (particularly when dealing with cone metric spaces in which the cone need not be normal). Then the following properties hold:
(PT1) If and , then .
(PT2) If and , then .
(PT3) If and , then .
(PT4) If for each , then .
(PT5) If , for each , then .
(PT6) If and is a sequence in such that for all and as , then there exists such that for all , we have .
For a cone metric space , denote
and
For we denote the generalized Hausdorff distance induced by d (see [12]) by the following notation:
The following remarks are found in [12].
Remark 2.2 Let be a cone metric space. The functional defined by
for all , is the Hausdorff distance induced by d.
Remark 2.3 Let be a cone metric space. Then for all .
Lemma 2.2 ([12])
Let be a cone metric space with cone P in real Banach space .
(L1) Let . If , then .
(L2) Let and . If , then .
(L3) Let and . If , then for all and for all .
(L4) Let and let , then .
In 2012, Samet et al. [15] introduced the idea of an α-admissible mapping and proved a fixed point theorem for a single valued mapping by using this concept. They showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and also applied the main results to ordinary differential equations. Afterward, Asl et al. [16] introduced the concept of -admissible operator, which is multi-valued version of the α-admissible mapping provided in [15].
Definition 2.4 ([16])
Let X be a non-empty set, , where is a collection of non-empty subset of X and be a mapping. We say that F is -admissible if the following condition holds:
where .
Recently, Mohammadi et al. [17] extended the concept of -admissible operator to α-admissible operator as follows.
Definition 2.5 ([17])
Let X be a non-empty set, , where is a collection of non-empty subset of X and be a mapping. We say that F is α-admissible whenever for each and with , we have for all .
Remark 2.4 If F is -admissible, then F is also α-admissible.
3 Fixed point results for multi-valued operators
In the following, we always suppose that Ψ be a family of functions such that
-
(i)
and for ,
-
(ii)
for all ,
-
(iii)
for every ,
-
(iv)
ψ is a strictly increasing function, i.e., whenever .
Remark 3.1 From (i), we obtain for all .
Definition 3.1 Let be a cone metric space with cone P in real Banach space , and be two given mappings. The multi-valued operator is said to be an α-ψ-contraction if
for all ,
Theorem 3.1 Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space , be a given mapping, and be an α-ψ-contraction multi-valued operator. Suppose that the following conditions hold:
-
(a)
F is α-admissible operator;
-
(b)
there exist and such that ;
-
(c)
if is a sequence in X such that for all and as , then for all .
Then there exists a point such that , that is, F has a fixed point in X.
Proof We start from and in (b). Since F is an α-ψ-contraction multi-valued operator, we have
Using Lemma 2.2(L3), we get
By definition of , we can take such that
By Lemma 2.2(L4), we have
This implies that
Now, we have
It follows from F being an α-ψ-contraction multi-valued operator that
From Lemma 2.2(L3), we obtain
By definition of , we can choose such that
From Lemma 2.2(L4), we obtain
and hence
Since , , and F is α-admissible operator, we have . Therefore,
Inductively, we can construct a sequence in X such that
and
for all . If there exists such that , then is a fixed point of F. This finishes the proof. Therefore, we may assume that for all . From (3.3), we obtain
for all . Since , we have as .
Fix such that . Using the property of ψ, we get . By (PT6), there is such that and for . Therefore, and for . For fixed , we have
This shows that is a Cauchy sequence. By the completeness of X, we get for some . Using (3.4) and (c), we obtain for all . Now, we have
for all . By Lemma 2.2(L3), we have
for all . Therefore, we can choose such that
for all . Using Lemma 2.2(L4), we get
and so
for all . Therefore, we have
for all . Since as , for a given with , there is such that
for all . It follows that
for all . Hence, we have . Since Fu is closed and for all , we obtain . This implies that u is a fixed point of F. This completes the proof. □
Now we give an example to support Theorem 3.1.
Example 3.2 Let be a real Banach space with norm defined by
for each , where which is defined by for all . Define cone P in by
This cone is solid but non-normal (see [18]).
Let and define cone metric for each as follows:
for all . Define a mapping and a multi-valued operator by
and
Now we show that F is an α-ψ-contraction multi-valued operator with define by for all . For , we obtain
which implies that . Otherwise, it is easy to see that . Therefore, F is an α-ψ-contraction multi-valued operator.
Next, we show that our results in this paper can be used for this case. It is easy to see that F is an α-admissible operator and there exist and such that . Also, we see that condition (c) in Theorem 3.1 holds. Therefore, all the conditions of Theorem 3.1 are satisfied and so F has a fixed point.
Corollary 3.3 ([13])
Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space , be a given mapping, and be an α-ψ-contraction multi-valued operator. Suppose that the following conditions hold:
-
(a)
F is -admissible operator;
-
(b)
there exist and such that ;
-
(c)
if is a sequence in X such that for all and as , then for all .
Then there exists a point such that , that is, F has a fixed point in X.
Proof We can prove this result by using Theorem 3.1 and Remark 2.4. □
Corollary 3.4 Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space , be a given mapping and be multi-valued operator such that
for all , where . Suppose that the following conditions hold:
-
(a)
F is α-admissible operator;
-
(b)
there exist and such that ;
-
(c)
if is a sequence in X such that for all and as , then for all .
Then there exists a point such that , that is, F has a fixed point in X.
Proof By taking for all in Theorem 3.1, we get this result. □
Corollary 3.5 Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space , and be multi-valued operator such that
for all . Then there exists a point such that , that is, F has a fixed point in X.
Proof By taking for all in Theorem 3.1, we get this result. □
Corollary 3.6 Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space and be multi-valued operator such that
for all , where . Then there exists a point such that , that is, F has a fixed point in X.
Proof By taking for all and for all in Theorem 3.1, we get this result. □
By Remark 2.2, we get the following results.
Corollary 3.7 Let be a complete metric space, be a given mapping and be multi-valued operator such that
for all , where . Suppose that the following conditions hold:
-
(a)
F is α-admissible operator;
-
(b)
there exist and such that ;
-
(c)
if is a sequence in X such that for all and as , then for all .
Then there exists a point such that , that is, F has a fixed point in X.
Corollary 3.8 ([7])
Let be a complete metric space and be multi-valued operator such that
for all , where . Then there exists a point such that , that is, F has a fixed point in X.
Next, we give a lemma which is useful to prove the second main result. Moreover, this lemma is also a tool for analyzing fixed point results in a directed graph in Section 4.
Lemma 3.9 Let be a cone metric space with cone P in real Banach space , be a given mapping. If α has transitive property (i.e., if and , then ), then the following conditions are equivalent:
-
(c)
if is a sequence in X such that for all and as , then for all ;
(c′) if is a sequence in X such that for all and as , then there is a subsequence with for all .
Proof It easy to see that (c) ⟹ (c′). Now assume (c′) holds and is as in (c). By transitivity, if . From (c′), there is a subsequence such that for . Since , we have . By transitivity, we have . □
Using Lemma 3.9, we have the following result.
Corollary 3.10 Let be a complete cone metric space with solid (normal or non-normal) cone P in a real Banach space , be a mapping satisfying transitive property, and be an α-ψ-contraction multi-valued operator. Suppose that the following conditions hold:
-
(a)
F is α-admissible operator;
-
(b)
there exist and such that ;
(c′) if is a sequence in X such that for all and as , then there is a subsequence with for all .
Then there exists a point such that , that is, F has a fixed point in X.
Corollary 3.11 Let be a complete cone metric space with solid (normal or non-normal) cone P in real Banach space , be a mapping satisfying transitive property, and be an α-ψ-contraction multi-valued operator. Suppose that the following conditions hold:
-
(a)
F is -admissible operator;
-
(b)
there exist and such that ;
(c′) if is a sequence in X such that for all and as , then there is a subsequence with for all .
Then there exists a point such that , that is, F has a fixed point in X.
4 Fixed point analysis with graph
Throughout this section let be a cone metric space. A set is called a diagonal of the Cartesian product and is denoted by Δ. Consider a directed graph G such that the set of its vertices coincides with X and the set of its edges contains all loops, i.e., . We assume G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N () is a sequence of vertices such that , and for .
A graph G is connected if there is a path between any two vertices. Also, G is weakly connected if is connected, where is the undirected graph obtained from G by ignoring the direction of the edges.
Recently, some results have appeared providing sufficient conditions for a single valued mapping and a multi-valued operator on some space which is endowed with a graph to be a Picard operator. The first result in this direction was given by Jachymski [19].
In this section, we give the existence of fixed point theorems for multi-valued operator in a cone metric space endowed with a graph under the generalized Hausdorff distance. Before presenting our results, we will introduce new definitions in a cone metric space endowed with a graph.
Definition 4.1 Let be a cone metric space (with cone P in real Banach space ) endowed with a graph G and be multi-valued operator. We say that F weakly preserves edges of G if for each and with it is implied that for all .
Definition 4.2 Let be a cone metric space (with cone P in real Banach space ) endowed with a graph G and . A multi-valued operator is said to be a ψ-G-contraction if
for all with . If ψ defined by for all , where , then F is said to be a generalized Banach G-contraction.
Remark 4.1 Note that a Banach G-contraction in the sense of being multi-valued, i.e.,
for all for which , where , is a ψ-G-contraction by using Remark 2.2.
Example 4.1 Any mapping defined by , where , is a ψ-G-contraction for any graph G with and all .
Example 4.2 Any mapping is trivially a ψ-G-contraction, where .
Definition 4.3 Let be a cone metric space (with cone P in real Banach space ) endowed with a graph G. We say that X has the G-regular property if given and sequence in X such that as and for all , then for all .
Definition 4.4 Let be a cone metric space (with cone P in real Banach space ) endowed with a graph G. We say that is quasi-ordered (or transitive) if and imply for all .
Here, we give two fixed point results for multi-valued operator in a cone metric space endowed with a graph.
Theorem 4.3 Let be a complete cone metric space (with cone P in real Banach space ) endowed with a graph G and be a ψ-G-contraction. Suppose that the following conditions hold:
-
(A)
F weakly preserves edges of G;
-
(B)
there exist and such that ;
-
(C)
X has the G-regular property.
Then there exists a point such that , that is, F has a fixed point in X.
Proof Consider the mapping defined by
Since F is a ψ-G-contraction, we have, for all ,
This implies that F satisfies (3.1). By construction of α and hypothesis (A), we see that F is an α-admissible operator. From condition (B) and the definition of α, we get . Using the G-regular property of X, we find that condition (c) in Theorem 3.1 holds. Now all the hypotheses of Theorem 3.1 are satisfied and so the existence of the fixed point of F follows from Theorem 3.1. □
Corollary 4.4 Let be a complete cone metric space (with cone P in real Banach space ) endowed with a graph G and be a generalized Banach G-contraction. Suppose that the following conditions hold:
-
(A)
F weakly preserves edges of G;
-
(B)
there exist and such that ;
-
(C)
X has the G-regular property.
Then there exists a point such that , that is, F has a fixed point in X.
By Remark 2.2, we get the multi-valued version of Jachymski’s result in [19].
Corollary 4.5 Let be a complete cone metric space (with cone P in real Banach space ) endowed with a graph G and be a multi-valued operator such that
for all with , where . Suppose that the following conditions hold:
-
(A)
F weakly preserves edges of G;
-
(B)
there exist and such that ;
-
(C)
X has the G-regular property.
Then there exists a point such that , that is, F has a fixed point in X.
Next, we give the fixed point result for a multi-valued operator in a connected graph.
Theorem 4.6 Let be a complete cone metric space (with cone P in real Banach space ) endowed with a connected graph G and be a ψ-G-contraction. Suppose that the following conditions hold:
-
(A)
F weakly preserves edges of G;
-
(B)
there exist and such that ;
(C′) if and a sequence in X are given such that as and for all , then there is a subsequence with for all .
Then there exists a point such that , that is, F has a fixed point in X.
Proof Since G is a connected graph, is a quasi-ordered. Consider the mapping defined as in the proof of Theorem 4.3. It follows from being quasi-ordered that α has the transitive property. By Lemma 3.9, we note that condition (C′) is in equivalence to condition (C) in Theorem 4.3. Thus we can prove this theorem similarly to the proof of Theorem 4.3 and consequently F has a fixed point in X. □
Corollary 4.7 Let be a complete cone metric space (with cone P in real Banach space ) endowed with a connected graph G and be a generalized Banach G-contraction. Suppose that the following conditions hold:
-
(A)
F weakly preserves edges of G;
-
(B)
there exist and such that ;
(C′) if and a sequence in X are given such that as and for all , then there is a subsequence with for all .
Then there exists a point such that , that is, F has a fixed point in X.
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Acknowledgements
The authors thank the editor and the referees for their valuable and insightful comments and suggestions which improved greatly the quality of this paper. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
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Kutbi, M.A., Sintunavarat, W. Fixed point analysis for multi-valued operators with graph approach by the generalized Hausdorff distance. Fixed Point Theory Appl 2014, 142 (2014). https://doi.org/10.1186/1687-1812-2014-142
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DOI: https://doi.org/10.1186/1687-1812-2014-142