Abstract
The purpose of this paper is to present some strict fixed point theorems for multivalued operators satisfying a Reich-type condition on a metric space endowed with a graph. The well-posedness of the fixed point problem is also studied.
MSC:47H10, 54H25.
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1 Preliminaries
A new approach in the theory of fixed points was recently given by Jachymski [1] and Gwóźdź-Lukawska and Jachymski [2] by using the context of metric spaces endowed with a graph. Other recent results for single-valued and multivalued operators in such metric spaces are given by Nicolae, O’Regan and Petruşel in [3] and by Beg, Butt and Radojevic in [4].
Let be a metric space and let Δ be the diagonal of . Let G be a directed graph such that the set of its vertices coincides with X and , where is the set of the edges of the graph. Assume also that G has no parallel edges and, thus, one can identify G with the pair .
If x and y are vertices of G, then a path in G from x to y of length is a finite sequence of vertices such that , and for . Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if is connected, where denotes the undirected graph obtained from G by ignoring the direction of edges.
Denote by the graph obtained from G by reversing the direction of edges. Thus,
Since it is more convenient to treat as a directed graph for which the set of its edges is symmetric, under this convention, we have that
If G is such that is symmetric, then for , the symbol denotes the equivalence class of the relation ℜ defined on by the rule:
Let us consider the following families of subsets of a metric space :
The gap functional between the sets A and B in the metric space is given by
In particular, if then .
The Pompeiu-Hausdorff functional is defined by
The diameter generalized functional generated by d is given by
In particular, we denote by the diameter of the set A.
Let be a metric space. If is a multivalued operator, then is called a fixed point for T if and only if . The set is called the fixed point set of T, while is called the strict fixed point set of T. denotes the graph of T.
Definition 1.1 Let be a mapping. Then φ is called a strong comparison function if the following assertions hold:
-
(i)
φ is increasing;
-
(ii)
as for all ;
-
(iii)
for all .
Definition 1.2 Let be a complete metric space, let G be a directed graph, and let be a multivalued operator. By definition, T is called a -G-contraction if there exists , a strong comparison function, such that
In this paper, we present some fixed point and strict fixed point theorems for multivalued operators satisfying a contractive condition of Reich type involving the functional δ (see [5, 6]). The equality between and and the well-posedness of the fixed point problem are also studied.
Our results also generalize and extend some fixed point theorems in partially ordered complete metric spaces given in Harjani and Sadarangani [7], Nicolae et al. [3], Nieto and Rodríguez-López [8] and [9], Nieto et al. [10], O’Regan and Petruşel [11], Petruşel and Rus [12], and Ran and Reurings [13].
2 Fixed point and strict fixed point theorems
We begin this section by presenting a strict fixed point theorem for a Reich type contraction with respect to the functional δ.
Theorem 2.1 Let be a complete metric space and let G be a directed graph such that the triple satisfies the following property:
Let be a multivalued operator. Suppose that the following assertions hold:
-
(i)
There exists with and such that
for all .
-
(ii)
For each , the set
is nonempty.
Then we have:
-
(a)
;
-
(b)
If we additionally suppose that
then .
Proof (a) Let . Since , there exists and such that and
By (i) we have that
Hence,
For , since , we get again that there exists such that and . Then
On the other hand, by (i), we have that
Using (2.2) we obtain
For , we have , and so there exists such that and .
Then
By these procedures, we obtain a sequence with the following properties:
-
(1)
for each ;
-
(2)
for each ;
-
(3)
for each .
From (2) we obtain that the sequence is Cauchy. Since the metric space X is complete, we get that the sequence is convergent, i.e., as . By the property (P), there exists a subsequence of such that for each .
We have
But as and as . Hence, , which implies that . Thus .
We shall prove now that .
Because , we need to show that .
Let . Because , we have that . Using (ii) with , we obtain
So, . Because , we get that . Hence, we have
Suppose that . This implies that . Thus from (2.6) we obtain that , which contradicts the hypothesis .
Thus , i.e., and .
Hence, .
(b) Suppose that there exist with . We have that
-
;
-
;
-
.
Using (i) we obtain
Thus, , which implies that , which is a contradiction.
Hence, . □
Next we present some examples and counterexamples of multivalued operators which satisfy the hypothesis in Theorem 2.1.
Example 2.1 Let and be given by
Let .
Notice that all the hypotheses in Theorem 2.1 are satisfied (the condition (i) is verified for , and so .
The following remarks show that it is not possible to have elements in .
Remark 2.1 If we suppose that there exists , then, since , we get (using the condition (i) in the above theorem with ) that , which is a contradiction with .
Remark 2.2 If, in the previous theorem, instead of the property (P), we suppose that T has a closed graph, then we obtain again the conclusion .
Remark 2.3 If, in the above remark, we additionally suppose that
then .
The next result presents a strict fixed point theorem where the operator T satisfies a -G-contractive condition on .
Theorem 2.2 Let be a complete metric space and let G be a directed graph such that the triple satisfies the following property:
Let be a multivalued operator. Suppose that the following assertions hold:
-
(i)
T is a -G-contraction.
-
(ii)
For each , the set
is nonempty.
Then we have:
-
(a)
;
-
(b)
If, in addition, the following implication holds:
then .
Proof (a) Let . Then, since is nonempty, there exist and such that and
By (i) we have that
For , by the same approach as before, there exists such that and .
We have
On the other hand, by (i) we have that
By the same procedure, for there exists such that and . Thus
We have
By these procedures, we obtain a sequence with the following properties:
-
(1)
for each ;
-
(2)
for each ;
-
(3)
for each .
By (2), using the properties of φ, we obtain that the sequence is Cauchy. Since the metric space is complete, we have that the sequence is convergent, i.e., as . By the property (P), we get that there exists a subsequence of such that for each .
We shall prove now that . We have
Since as and φ is continuous in 0 with , we get that .
Hence, .
We shall prove now that .
Because , we need to show that .
Let . Because , we have that . Using (i) with , we obtain
Hence, and the proof of this conclusion is complete.
(b) Suppose that there exist with . We have that
-
;
-
;
-
.
Using (i) we obtain
This is a contradiction. Hence, . □
In the next result, the operator T satisfies another contractive condition with respect to δ on .
Theorem 2.3 Let be a complete metric space, let G be a directed graph, and let be a multivalued operator. Suppose that defined is a lower semicontinuous mapping. Suppose that the following assertions hold:
-
(i)
There exist , with and , such that
-
(ii)
For each , the set
is nonempty.
Then .
Proof Let . Then, since is nonempty, there exist and such that
and . Since , we get that .
By (i), taking and , we have that
Hence,
For (since ), there exists such that and . But and so .
Then
By (i), taking and , we have that
By these procedures, we obtain a sequence with the following properties:
-
(1)
for each ;
-
(2)
for each ;
-
(3)
for each .
From (2) we obtain that the sequence is Cauchy. Since the metric space X is complete, we have that the sequence is convergent, i.e., as . Now, by the lower semicontinuity of the function f, we have
Thus , which means that . Thus .
Let . Then and hence .
Using (i) with , we obtain
So, . If we suppose that , then . Thus, , which contradicts the hypothesis.
Thus and so . The proof is now complete. □
Remark 2.4 Example 2.1 satisfies the conditions from Theorem 2.3 for and .
3 Well-posedness of the fixed point problem
In this section we present some well-posedness results for the fixed point problem. We consider both the well-posedness and the well-posedness in the generalized sense for a multivalued operator T.
We begin by recalling the definition of these notions from [14] and [15].
Definition 3.1 Let be a metric space and let be a multivalued operator. By definition, the fixed point problem is well posed for T with respect to H if:
-
(i)
;
-
(ii)
If is a sequence in X such that as , then as .
Definition 3.2 Let be a metric space and let be a multivalued operator. By definition, the fixed point problem is well posed in the generalized sense for T with respect to H if:
-
(i)
;
-
(ii)
If is a sequence in X such that as , then there exists a subsequence of such that as .
In our first result we will establish the well-posedness of the fixed point problem for the operator T, where T is a Reich-type δ-contraction.
Theorem 3.1 Let be a complete metric space and let G be a directed graph such that the triple satisfies the property (P).
Let be a multivalued operator. Suppose that
-
(i)
conditions (i) and (ii) in Theorem 2.1 hold;
-
(ii)
if , then ;
-
(iii)
for any sequence , with as , we have .
In these conditions the fixed point problem is well posed for T with respect to H.
Proof From (i) and (ii) we obtain that . Let be a sequence which satisfies (iii). It is obvious that ,
Thus
Hence, as . □
Remark 3.1 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.
The next result deals with the well-posedness of the fixed point problem in the generalized sense.
Theorem 3.2 Let be a complete metric space and let G be a directed graph such that the triple satisfies the property (P).
Let be a multivalued operator. Suppose that
-
(i)
conditions (i) and (ii) in Theorem 2.1 hold;
-
(ii)
for any sequence , with as , there exists a subsequence such that and .
In these conditions the fixed point problem is well posed in the generalized sense for T with respect to H.
Proof From (i) we have that . Let be a sequence which satisfies (ii). Then there exists a subsequence such that .
We have ,
Thus
Hence, . □
Remark 3.2 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.
Next we consider the case where the operator T satisfies a φ-contraction condition.
Theorem 3.3 Let be a complete metric space and let G be a directed graph such that the triple satisfies the property (P).
Let be a multivalued operator. Suppose that
-
(i)
conditions (i) and (ii) in Theorem 2.2 hold;
-
(ii)
the following implication holds: implies ;
-
(iii)
the function , given by , has the following property: if as , then as ;
-
(iv)
for any sequence with as , we have for all .
In these conditions the fixed point problem is well posed for T with respect to H.
Proof From (i) and (ii) we obtain that . Let , be a sequence which satisfies (iv). It is obvious that ,
Thus
Using condition (iii), we get that as . Hence, . □
Remark 3.3 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.
The next result gives a well-posedness (in the generalized sense) criterion for the fixed point problem.
Theorem 3.4 Let be a complete metric space and let G be a directed graph such that the triple satisfies the property (P).
Let be a multivalued operator. Suppose that
-
(i)
the conditions (i) and (ii) in Theorem 2.2 hold;
-
(ii)
the function , given , has the following property: for any sequence , there exists a subsequence such that if as , then as ;
-
(iii)
for any sequence , with as , there exists a subsequence such that and .
In these conditions the fixed point problem is well posed in the generalized sense for T with respect to H.
Proof From (i) we have that . Let , be a sequence which satisfies (iii). Then there exists a subsequence such that .
We have ,
Thus
Using condition (ii), we get that as . Hence, . □
Remark 3.4 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.
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Acknowledgements
The third author is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.
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Chifu, C., Petruşel, G. & Bota, MF. Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph. Fixed Point Theory Appl 2013, 203 (2013). https://doi.org/10.1186/1687-1812-2013-203
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DOI: https://doi.org/10.1186/1687-1812-2013-203