1. Introduction
Without doubt, the two most important results in fixed point theory are the Banach contraction principle and the Tarski fixed point theorem. These results have been the subject of many generalizations and extensions, either by extending the contractive condition or changing the structure of the space. In [
1], Echenique gave a proof of Tarski fixed point theorem by using graphs combining fixed point techniques and graph theory. Then, Jachymski [
2] developed another structure in the fixed point theory of metric spaces by replacing the structure of orders by the structure of graphs on metric spaces. He obtained sufficient conditions for self-mappings to be a Picard operator on a metric space endowed with a graph. Fixed point theory and graph theory provide an intersection between the theories of fixed point results, which give the conditions under which (single or multivalued) mappings have solutions, and graph theory, which uses mathematical structures to illustrate the relationship between ordered pairs of elements in terms of their vertices and directed edges. Note that metric fixed point and graph theory have common application environments. In the multivalued case, the authors in [
3] proved a fixed point theorem for Mizoguchi–Takahashi-type contractions on a metric space endowed with a graph. For further results in this direction, we refer to [
4,
5,
6,
7,
8,
9,
10,
11]. Recently, in [
12], the authors introduced a new concept of contractions called
F-Khan contractions and proved a related fixed point theorem.
In this paper, we present new fixed point results for variant multivalued mappings via a graph structure. To do this, we introduce new types of contractions called rational-type multivalued G-contractions and F-Khan-type multivalued contractions. We ensure that related mappings (involving a graph structure) have a fixed point. At the end, we consider an example to show the validity of our obtained results and to ensure that the known results in the literature are not applicable.
2. Preliminaries
Let
be a metric space.
denotes the family of all nonempty subsets of
X;
denotes the family of all nonempty, closed, and bounded subsets of
X; and
denotes the family of all nonempty compact subsets of
X. The Pompeiu–Hausdorff metric
is defined as
where
and
Using the Pompeiu–Hausdorff metric
Nadler [
13] established that every multivalued contraction mapping on a complete metric space has a fixed point. Inspired by his result, various fixed point results concerning multivalued contraction mappings appeared in the last several decades [
14,
15,
16,
17,
18,
19,
20,
21,
22].
On the other hand, Jachymski [
2] used a graph structure instead of partial orders to prove some important fixed point results. Additionally, Bojor [
23] established some fixed point results in a metric space endowed with a graph. We can also find some fixed point results on a metric space with a graph structure in [
4,
5,
7,
9,
24].
Now, we recall some definitions and lemmas.
Definition 1. Ref. [2] Let X be a non-empty set and Δ
denote the diagonal of the cartesian product . A directed graph or digraph G is characterized by a nonempty set of its vertices and the set of its directed edges. A digraph is reflexive if any vertex admits a loop. For a digraph , If whenever , then the digraph G is called an oriented graph.
A digraph G is transitive whenever and for any .
A path of G is a sequence with for each
G is connected if there is a path between each two vertices, and it is weakly connected if is connected, where corresponds to the undirected graph obtained from G by ignoring the direction of edges.
Let be the graph obtained from G by reversing the direction of edges. Thus, We call a subgraph of G if and and for any edge , .
We refer to [
25] for more details on graph theory.
Definition 2. Ref. [4] Let be a metric space endowed with a graph G such that and let be a multivalued mapping. T has the weakly graph-preserving (WGP) property, whenever for each and with implies for all . Definition 3. Ref. [21] Let be a function verifying the following conditions: For all such that
For any positive real sequence , There is so that
for all with
We mean by if F verifies the conditions –, and by if the conditions – hold. Clearly, .
Lemma 1. Ref. [4] Let be a metric space and be an upper semi-continuous mapping such that is closed for all . If , and , then A function (where ) is said to be a comparison function if it satisfies the following conditions:
- (i)
is monotonically increasing;
- (ii)
for all .
Meanwhile, if satisfies (i) and the following condition:
- (iii)
is convergent for each ,
then is named to be a -comparison function.
Lemma 2. Ref. [26] Let be a metric space. Let , and Then, for every there exists such that 3. Main Results
3.1. On Rational-Type Multivalued G-Contractions
Theorem 1. Let G be a directed graph on a complete metric space and be a multivalued mapping. Suppose that f is upper semi-continuous and a WGP mapping. Suppose that:
f is a rational multivalued G-contraction of type I, that is, there is a strictly increasing -comparison function φ such thatfor all , where is nonempty.
Then, f admits a fixed point.
Proof. Set
There exists
such that
. So, we can use the condition (i) for
and
. Then, we have
Given
an arbitrary constant. Therefore, from Lemma 2, there exists
such that
Recall that
, so one writes
Since
is strictly increasing, we have
Set
. Then,
. Since
,
and
, using the WGP property, one writes
. Then,
Therefore, from Lemma 2 there exists
such that
Since
is strictly increasing, we have
Get
. Then,
. Continuing, we construct the sequence
in
X so that
,
and
To prove that
is a Cauchy sequence, take
with
Consider
Since is a -comparison function, the series on the right-hand side converges, and so as . That is, is a Cauchy sequence in , which is complete, so is convergent to some , that is, Since f is upper semi-continuous, using Lemma 1, we find . That is, f has a fixed point in X. □
Consider the following property:
Theorem 2. Let G be a directed graph on a complete metric space and be a multivalued mapping such that the following conditions are verified:
f is a rational multivalued G-contraction of type II, that is, there is a strictly increasing -comparison function φ such thatfor all , where (defined in Theorem 1) is nonempty;
The (P)-property is satisfied;
f is a WGP mapping.
Then, f has a fixed point.
Proof. Take
There is
such that
. So, we can use the condition (i) for
and
. Then, we have
where
is a constant. Therefore, from Lemma 2, there exists
such that
Since
is strictly increasing, we have
Take
. We have
. In view of
,
,
, and using the WGP property, one writes
. Then,
Therefore, from Lemma 2, there exists
such that
Since
is strictly increasing, we have
Set
. Then,
. Proceeding again, we construct the sequence
in
X such that
,
and
We will show that
is a Cauchy sequence. Let
with
Consider
Since
is a
-comparison function, the series on the right-hand side converges, and so we get
as
. That is,
is a Cauchy sequence in
, which is complete, hence
is convergent to some
, that is,
By the
-property, there is a subsequence
of
such that
for each
Now, assume that
. Since
,
, there is
such that for
and there is
so that for
If we take
then by (
2) and (
3), we have
Taking , we have which is a contradiction. Thus, and since is closed, we deduce . That is, f admits a fixed point. □
If we assume that in the previous theorems, we do not need the strictly increasing property of This corresponds to the following result.
Theorem 3. Let be a complete metric space, G be a directed graph on X and be a multivalued mapping. Assume that f is upper semi-continuous and a WGP mapping. Suppose that:
There is a -comparison function φ such thatfor all ; is nonempty.
Then, f has a fixed point.
Proof. Set
There is
such that
. So, we can use the condition (i) for
and
. Then, we have
Since
is compact, there exists
such that
So,
Since
,
, and
, using the WGP property, we get
. Then,
Since
is compact, again there exists
such that
. Therefore, we have
Continuing this process, we construct the sequence
in
X such that
,
, and
The rest of the proof can be finished as in Theorem 2. □
3.2. On F-Khan Contractions
Let
G be a directed graph on a metric space
X and
T be a mapping from
X to
. Define
and
Definition 4. Let be a metric space and be a mapping. We say that T is a multivalued F-Khan contraction if there are and such that for all , we havefor with . With the assumptions of upper semi-continuity and having the weak graph preservation of the mapping Theorems 4 and 5 hold.
Theorem 4. Let be a complete metric space endowed with a directed graph G and be a multivalued F-Khan contraction. If the set is nonempty, then T admits a fixed point.
Proof. If
T has no fixed point,
for all
. Let
Then,
for some
. Then, we get
Thus,
. Using condition (
4) for
and
we have
Due to the compactness of
, there is
such that
. So, we get
Since
,
, and
, by the WGP property, one writes
. In view of
, we get that
. Then,
Proceeding in a similar way, we can get
Again, the compactness of
implies that there is
such that
. We have
Similarly, we construct the sequence
in
such that
,
and
Denote
, then
and from (
8),
is a decreasing real sequence, so there is
such that
We have
The right-hand side of (
9) goes to
when
. Hence,
Using
, one writes
Due to
, there is
such that
By Inequality (
9), we have
which is verified for all
. Letting
in (
10), we find that
From (
11), there is
such that
for all
. Thus,
for all
. Now, we claim that the sequence
is Cauchy. For this, take
with
Hence,
Since , the series converges, so as , that is, is a Cauchy sequence in the complete metric space . Then, is convergent to some . Using the upper semi-continuity of T and Lemma 1, we get , which is a contradiction with our assumption. Then, T has a fixed point. □
Theorem 5. Let be a complete metric space endowed with a directed graph G and be a multivalued F-Khan contraction (with ). If is nonempty, then T has a fixed point.
Proof. If there is no fixed point of
T, then
for each
. Let
Then
for some
. Then, we get
Thus,
. In view of
and using the condition (
4) for
and
we have
and by (
5) we have
Thus, there is
so that
The rest of the proof follows as in the proof of Theorem 4. □
Theorem 6. Let be a complete metric space endowed with a directed graph G such that: Let be a multivalued F-Khan contraction (resp. be a multivalued F-Khan contraction with ). Suppose that T is a WGP mapping and is nonempty. If F is continuous, then T admits a fixed point.
Proof. Assume that
T has no fixed point. In a similar way as in the proof of Theorem 4 (resp. Theorem 5), we construct the sequence
such that
for some
. By the property (
13), there is a subsequence
of
such that
for each
Since
and
, then there is no natural number
such that
for each
. Therefore, for all
thus
for all
. From (
4) and (F1), we have
for all
. Letting
and due to the continuity of
we obtain that
which is a contradiction, so
T admits a fixed point. □
Corollary 1. Let be a complete metric space endowed with a directed graph G and be a mapping. Suppose that there are and such thatfor with If T is upper semi-continuous and a WGP mapping and the set is nonempty, then T has a fixed point. Corollary 2. Let be a complete metric space endowed with a directed graph G and be a mapping. Suppose that there are and such thatfor with . Assume that T is upper semi-continuous and a WGP mapping and the set is nonempty. Then, T admits a fixed point. Example 1. Let and . Then is a complete metric space. Define byand a graph on X by and Then, T is upper semi-continuous and a WGP mapping. Now, we show that T is a multivalued F-Khan contraction with and . Then, for any with we consider two cases:
If and we have If , we have
Thus, all assumptions of Theorem 4 (or Theorem 5) hold. Therefore, T has a fixed point. Moreover, if the graph on X is not considered, the contractive condition is not satisfied. In fact, taking and we have and so for all and , we get 4. Concluding Remarks
As it is known, combining some branches is an activity in different fields of science, especially in mathematics. One of them is the combination of fixed point theory and graph theory. In this paper, we present new fixed point results for multivalued mappings. To do this, we introduce new types of contractions called rational-type multivalued G-contractions and F-Khan-type multivalued contractions. We ensure that related mappings have a fixed point. At the end, we consider an example that shows the importance of graph on the contractive condition. Thus, we show here that the defined contraction does not even satisfy the condition of contraction in a metric space without graph, but it provides the condition of a contraction in a metric space with graph and has a fixed point, which shows us the importance of graph structure.
Author Contributions
Formal analysis, Ö.A., H.A. and M.D.l.S.; Investigation, Ö.A., H.A. and M.D.l.S.; Methodology, Ö.A. All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported in part by the Basque Government under Grant IT1207-19.
Institutional Review Board Statement
The authors are grateful to the Spanish Government and the European Commission for Grant IT1207-19.
Data Availability Statement
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests concerning the publication of this article.
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